--- /srv/rebuilderd/tmp/rebuilderdFldIyd/inputs/macaulay2-common_1.24.11+ds-5_all.deb +++ /srv/rebuilderd/tmp/rebuilderdFldIyd/out/macaulay2-common_1.24.11+ds-5_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2025-02-09 22:54:37.000000 debian-binary │ --rw-r--r-- 0 0 0 504688 2025-02-09 22:54:37.000000 control.tar.xz │ --rw-r--r-- 0 0 0 29430680 2025-02-09 22:54:37.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 504940 2025-02-09 22:54:37.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 29427920 2025-02-09 22:54:37.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.24.11+ds-5 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 287461 │ │ │ +Installed-Size: 287438 │ │ │ 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) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -2795,49 +2795,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5118 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AInfinity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 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--rw-r--r-- 0 root (0) root (0) 4554 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_random__Sub__Simplicial__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_random__Abstract__Simplicial__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 4872 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_random__Sub__Simplicial__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6182 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_reduced__Simplicial__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5157 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_simplicial__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15216 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13092 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8539 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AbstractToricVarieties/dump/ │ │ │ @@ -2878,25 +2878,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11537 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7717 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4457 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 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22:54:37.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9211 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8759 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11597 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ @@ -3060,15 +3060,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BGG/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3464 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_beilinson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_bgg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 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./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 283502 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -3308,15 +3308,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3407 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/_twist__Inv__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3375 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/_twist__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 72605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 60186 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 30193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/dump/ │ │ │ --rw-r--r-- 0 root (0) root (0) 177228 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump │ │ │ +-rw-r--r-- 0 root (0) root (0) 177232 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___B_sq__Constants.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Bertini.out │ │ │ -rw-r--r-- 0 root (0) root (0) 659 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Bertini_spinput_spconfiguration.out │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Is__Projective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Number__To__B_sq__String.out │ │ │ @@ -3354,22 +3354,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3232 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Main__Data__Directory.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5562 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Number__To__B_sq__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3492 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Path__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4346 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7129 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Incidence__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8305 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Main__Data__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5862 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Parameter__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8211 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Solutions__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12090 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_make__B_sq__Input__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_make__B_sq__Section.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11234 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Bertini/html/_make__B_sq__Slice.html │ │ │ @@ -3793,15 +3793,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism_lp..._cm__Length__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3452 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3451 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7178 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ @@ -3868,19 +3868,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4375 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1503 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3411 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2014 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4378 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ --rw-r--r-- 0 root (0) root (0) 760 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 665 2025-02-09 22:54:37.000000 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2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/___Strat.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/___Vanishing__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_alphabet.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_ambient__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 377 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_bitflip__Decode.out │ │ │ --rw-r--r-- 0 root (0) root (0) 12327 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cartesian__Code.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 12279 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cartesian__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 197 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│ │ │ --rw-r--r-- 0 root (0) root (0) 9438 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_order__Code.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 9444 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_order__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 935 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_parity__Check__To__Generator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2718 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_quasi__Cyclic__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_rand__L__D__P__C.out │ │ │ -rw-r--r-- 0 root (0) root (0) 219 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_rand__No__Repeats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2521 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_random__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 466 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_reduced__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 254 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_reed__Muller__Code.out │ │ │ @@ -4096,21 +4096,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4937 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Generators.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22200 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Linear__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11772 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Linear__Code_sp_eq_eq_sp__Linear__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6952 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Parity__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6170 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Parity__Check__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6405 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Parity__Check__Rows.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5448 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Polynomial__Set.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8180 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Sets.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8142 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Sets.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Strat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5490 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Vanishing__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5132 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_alphabet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5981 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_ambient__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6487 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_bitflip__Decode.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24564 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cartesian__Code.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24516 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cartesian__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5799 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_choose__Strat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5485 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_codewords.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12844 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cyclic__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8615 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cyclic__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6194 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_dim_lp__Linear__Code_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5702 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_dual__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5575 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_enumerate__Vectors.html │ │ │ @@ -4126,15 +4126,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6070 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_hyp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5831 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_information__Rate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5682 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_length_lp__Linear__Code_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 39797 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_linear__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8785 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_locally__Recoverable__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6666 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_messages.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6906 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_minimum__Weight.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22439 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_order__Code.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22445 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_order__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6744 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_parity__Check__To__Generator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_quasi__Cyclic__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6689 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_rand__L__D__P__C.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5531 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_rand__No__Repeats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8579 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_random__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5432 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_reduced__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6140 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_reed__Muller__Code.html │ │ │ @@ -4257,15 +4257,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231124 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3103 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4608 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4126 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2721 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 980 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module__Data.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2092 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1739 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Lift.html │ │ │ @@ -5171,136 +5171,136 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 14011 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CotangentSchubert/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10920 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CotangentSchubert/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CotangentSchubert/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 238666 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2313 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22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_get__Linear__Diophantine__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 501 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_get__Prime__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_get__Prime__Divisors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_ideal__Power.out │ │ │ @@ -5667,16 +5667,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 441 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_is__Zero__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_map__To__Projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_non__Cartier__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 780 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_positive__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_ramification__Divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4356 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexify.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1095 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexive__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4354 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1096 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexive__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_ring_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_to__Q__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 458 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_to__R__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_to__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_torsion__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 364 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_trim_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_zero__Divisor.out │ │ │ @@ -5706,15 +5706,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6088 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_ceiling_lp__R__Weil__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5044 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_clean__Support.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5700 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_clear__Cache.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6167 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6288 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7766 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_coefficients_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18847 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_divisor.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_dualize.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11557 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_dualize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12606 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_embed__As__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7288 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_find__Element__Of__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6937 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_gbs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8027 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_get__Linear__Diophantine__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6838 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_get__Prime__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5441 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_get__Prime__Divisors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6072 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_ideal__Power.html │ │ │ @@ -5739,16 +5739,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5611 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_is__Zero__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7851 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_map__To__Projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8042 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_non__Cartier__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6265 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_positive__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6751 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8698 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9757 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_ramification__Divisor.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19080 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_reflexify.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9039 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_reflexive__Power.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19078 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_reflexify.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9040 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_reflexive__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4888 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_ring_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5845 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_to__Q__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6614 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_to__R__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6866 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_to__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6584 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_torsion__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5901 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_trim_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4680 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_zero__Divisor.html │ │ │ @@ -5866,15 +5866,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_isolated__Vertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 544 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_line__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_neighbors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 903 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_num__Connected__Components.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_num__Connected__Graph__Components.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_num__Triangles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 693 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Graph.out │ │ │ --rw-r--r-- 0 root (0) root (0) 846 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 560 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 808 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Uniform__Hyper__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 269 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_ring_lp__Hyper__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_simplicial__Complex__To__Hyper__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_smallest__Cycle__Size.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_spanning__Tree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 847 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_vertex__Cover__Number.out │ │ │ -rw-r--r-- 0 root (0) root (0) 735 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_vertex__Covers.out │ │ │ @@ -5946,15 +5946,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6536 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_isolated__Vertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6003 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_line__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7122 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_neighbors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9832 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_num__Connected__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8707 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_num__Connected__Graph__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6396 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_num__Triangles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6573 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Graph.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9488 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9172 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6149 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph_lp..._cm__Branch__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5824 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph_lp..._cm__Time__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7209 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Uniform__Hyper__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5641 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_ring_lp__Hyper__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6513 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_simplicial__Complex__To__Hyper__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7793 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_smallest__Cycle__Size.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6061 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_spanning__Tree.html │ │ │ @@ -5976,21 +5976,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8422 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4932 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3096 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13740 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 891 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 893 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 889 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9030 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9079 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 6789 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6791 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7548 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15750 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15149 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6461 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5392 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3602 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/ │ │ │ @@ -6181,35 +6181,35 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EngineTests/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10834 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 367 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_multiple__Cover.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2167 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 48 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4985 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4993 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_multiple__Cover.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10964 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10968 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6367 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5540 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3551 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52732 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Equivariant__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Priority__Queue.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Monomial__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring_lp__Ring_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 222 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_delete__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1274 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 348 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_exponent__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 449 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_inc__Orbit.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2025-02-09 22:54:37.000000 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5860 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_exponent__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6955 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_inc__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5664 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_insert_lp__Priority__Queue_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4996 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_length_lp__Priority__Queue_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6075 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_merge__P__Q.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4972 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_min_lp__Priority__Queue_rp.html │ │ │ @@ -6423,70 +6423,70 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9220 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FGLM/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4551 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FGLM/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3008 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FGLM/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 142955 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 29583 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 29587 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ 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./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Random__Submatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Largest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Smallest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_get__Submatrix__Of__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1785 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Rank__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ -rw-r--r-- 0 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22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Det__Strategy.html │ │ │ --rw-r--r-- 0 root (0) root (0) 48696 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 48700 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5140 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Max__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4122 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Min__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4154 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Modulus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6499 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Point__Options.html │ │ │ --rw-r--r-- 0 root (0) root (0) 26935 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14553 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 26932 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14552 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9083 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Good__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5457 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Random__Submatrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6073 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Largest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5989 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./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25936 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1849 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1848 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 334 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_affine__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_co1__Fitting.out │ │ │ -rw-r--r-- 0 root (0) root (0) 848 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gauss__Col.out │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gotzmann__Test.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_next__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 197 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_quot__Scheme.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 10278 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5938 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_affine__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7195 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_co1__Fitting.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7113 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gauss__Col.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7927 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gotzmann__Test.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7042 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_next__Degree.html │ │ │ -rw-r--r-- 0 root (0) root 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./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Foreign__Object.out │ │ │ -rw-r--r-- 0 root (0) root (0) 226 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out │ │ │ --rw-r--r-- 0 root (0) root (0) 525 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 524 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 92 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Void__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 321 2025-02-09 22:54:37.000000 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--rw-r--r-- 0 root (0) root (0) 6118 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6117 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5256 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Void__Type.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6853 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6454 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6286 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5427 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11686 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/_fast_sp__Fourier_sptransform_spexample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6330 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Array__Type.html │ │ │ @@ -6708,16 +6708,16 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 105269 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25101 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23557 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25099 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23555 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12804 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12106 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9354 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18981 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20424 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7695 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -6789,15 +6789,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7420 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6622 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5424 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5303 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27535 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -7844,42 +7844,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75720 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Graphs/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 59550 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Graphs/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 32717 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Graphs/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56327 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 33117 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 33164 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 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./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13044 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 15158 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6625 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 675 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_smaller__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 960 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_standard__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1344 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_tail__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5775 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___All__Standard.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4680 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___Minimalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9272 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Constraints.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8150 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 41520 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10010 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Stratum.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5657 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_linear__Part.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23060 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23349 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7047 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_smaller__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7401 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_standard__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8226 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_tail__Monomials.html │ │ │ --rw-r--r-- 0 root (0) root (0) 49360 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 49407 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11472 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6158 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18005 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 705 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -8024,15 +8024,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50640 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 920 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/___Appell__F1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2967 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/___Canonical_sp__Series_sp__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 630 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Expts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1007 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Expts__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4892 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4893 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5004 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_diff__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_distraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 768 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_euler__Operators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_gkz.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_is__Torus__Fixed.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4657 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_put__Weyl__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2252 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out │ │ │ @@ -8042,15 +8042,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6444 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Appell__F1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13276 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Canonical_sp__Series_sp__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3089 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Wto__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3523 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_create__Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6608 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7394 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10799 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10800 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11975 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_diff__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8119 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_distraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7349 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_euler__Operators.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8628 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_gkz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7242 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_indicial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6102 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_is__Torus__Fixed.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3680 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_nilsson__Support.html │ │ │ @@ -8202,16 +8202,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1444 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__P__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_idealizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Keep_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 28084 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 370 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Variable_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2362 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3674 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2371 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3672 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2595 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1221 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_is__Normal_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2065 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_make__S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 496 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ring__From__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1935 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_test__Huneke__Question.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/ │ │ │ @@ -8237,16 +8237,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4732 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer_lp..._cm__Index_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10501 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5388 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6918 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Keep_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8320 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 62512 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12568 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Variable_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9830 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14131 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9839 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14129 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14275 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12467 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5862 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_is__Normal_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9425 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10403 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7546 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_ring__From__Fractions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8797 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_test__Huneke__Question.html │ │ │ @@ -8271,17 +8271,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 677 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_generators_lp__Finite__Group__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 555 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_generators_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 636 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_group__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1302 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hilbert__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hilbert__Series_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hironaka__Decomposition.out │ │ │ --rw-r--r-- 0 root (0) root (0) 9241 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 9240 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariant__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1300 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1302 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 513 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Coefficient__Ring_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1158 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1326 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Diagonal__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Finite__Group__Action_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2647 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Finite__Group__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 773 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Linearly__Reductive__Action_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 798 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Linearly__Reductive__Action_rp.out │ │ │ @@ -8327,18 +8327,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6293 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Finite__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6224 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 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./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Coefficient__Ring_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8676 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12839 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Diagonal__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9913 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Finite__Group__Action_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11079 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Finite__Group__Action_rp.html │ │ │ @@ -8457,41 +8457,41 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7130 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/JSON/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6337 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/JSON/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3318 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/JSON/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 78217 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2737 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2738 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9513 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2465 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___J__J.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5997 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5996 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1511 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets__Radical.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1210 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Affine__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1209 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2492 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1463 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Quotient__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1393 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_lifting__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_lifting__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_principal__Component.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 675 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 70 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 8541 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8542 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8900 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17138 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9876 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5001 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___J__J.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3928 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Saturate.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15225 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15224 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4869 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6089 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Base.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4301 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Info.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4651 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4429 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Max__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6932 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Projection.html │ │ │ @@ -8515,59 +8515,59 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 102072 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1940 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2827 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5879 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1003 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2263 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3425 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1001 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 268 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2261 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3427 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1004 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 269 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_product__Of__Projective__Spaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1031 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1638 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution__Twists.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2220 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2219 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 129 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_scheme__In__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3076 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4628 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4195 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7601 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9237 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6471 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13084 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11799 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9474 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10205 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6616 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6590 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9472 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10207 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6619 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6591 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17353 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7148 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9146 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14516 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4922 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8161 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7918 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5901 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7827 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6949 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7056 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8055 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8112 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8111 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5703 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7256 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10598 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_schreyer__Name.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6161 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_small__Diagonal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 32580 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22924 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9790 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/toc.html │ │ │ @@ -8691,27 +8691,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LAYOUT.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33508 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2858 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2411 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1437 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1436 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 476 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_gcd__L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_is__L__L__L.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 28 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3787 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___B__K__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3733 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Engine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3847 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Top__Level.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3650 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Givens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3210 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Hermite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10658 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9716 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 30774 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 30773 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3695 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___N__T__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3496 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__F__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3516 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3698 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__R__R.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3565 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__X__D.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4195 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Threshold.html │ │ │ @@ -8727,15 +8727,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 70556 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/___Working_spwith_splattice_sppolytopes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_adjoint__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_ambient__Halfspaces.out │ │ │ --rw-r--r-- 0 root (0) root (0) 596 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 593 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_cayley.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_codegree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 281 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_degree__Of__Jet__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_epsilon__Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 373 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gaussk__Fiber.out │ │ │ @@ -8752,15 +8752,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_toric__Div.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_torus__Embedding.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4549 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/___Working_spwith_splattice_sppolytopes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5468 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_adjoint__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5506 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_ambient__Halfspaces.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7428 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7425 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9293 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_cayley.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4749 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_codegree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6623 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_degree__Of__Jet__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6268 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_epsilon__Bounds.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6425 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6459 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6744 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gaussk__Fiber.html │ │ │ @@ -9175,15 +9175,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5830 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Fast__Nonminimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1543 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___File_sp_lt_lt_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 414 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 460 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Graded__Module__Map_sp_vb_sp__Graded__Module__Map.out │ │ │ @@ -9243,15 +9243,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 248 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 767 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_st_st_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1114 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 377 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 484 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_vb_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_vb_vb_sp__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1015 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1014 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2609 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 313 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_pl_pl_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_sl_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 299 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_st_st_sp__Ring.out │ │ │ @@ -9296,15 +9296,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 587 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 753 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_sl_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 322 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1854 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D.out │ │ │ --rw-r--r-- 0 root (0) root (0) 349 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 348 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5127 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schreyer_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1024 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schubert_lp__Z__Z_cm__Z__Z_cm__Visible__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Self__Initializing__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 950 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp-_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ @@ -9397,15 +9397,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__sh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 334 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__sh_qu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 368 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__sh_sp__Basic__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 387 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__sl.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__sl_sl.out │ │ │ -rw-r--r-- 0 root (0) root (0) 276 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__sl_sl_sl.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1059 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/__st.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7213 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7212 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1028 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_about.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_abs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1025 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_accumulate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_acos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_acosh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_acot.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_acoth.out │ │ │ @@ -9439,15 +9439,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assigning_spvalues.out │ │ │ --rw-r--r-- 0 root (0) root (0) 235 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -9463,15 +9463,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between_lp__Thing_cm__Visible__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 713 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 531 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache__Value.out │ │ │ --rw-r--r-- 0 root (0) root (0) 591 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 592 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1783 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_chain__Complex_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_chain__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_chain__Complex_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ @@ -9496,24 +9496,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Rank__Profile.out │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1523 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1545 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 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4287 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1283 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spresolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1282 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spresolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spsyzygies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1767 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenating_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 392 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conditional_spexecution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cone_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 148 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate_lp__Partition_rp.out │ │ │ @@ -9525,15 +9525,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 740 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 113 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cosh.out │ │ │ 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root (0) 483 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_span_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_spand_spwriting_spfiles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csc.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csch.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Column__Number.out │ │ │ @@ -9593,15 +9593,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1642 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21210 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21211 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_else.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3570 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ @@ -9744,15 +9744,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_install__Assignment__Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances_lp__Type_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances_lp__Type_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inverse__Erf.out │ │ │ @@ -9888,19 +9888,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 125 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max_lp__Graded__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1636 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1638 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 970 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6279 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2730 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods_spfor_spnormal_spforms_spand_spremainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ @@ -9988,31 +9988,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1091 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Package_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_override.out │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pairs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1735 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8672 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1737 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8673 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 535 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pdim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare__N.out │ │ │ -rw-r--r-- 0 root (0) root (0) 580 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4045 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ @@ -10035,27 +10035,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 78 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_product_lp__Z__Z_cm__Function_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1542 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1543 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1426 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_promote.out │ │ │ -rw-r--r-- 0 root (0) root (0) 236 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_protect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 838 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pseudo__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pseudocode.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3606 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2486 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Type_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1087 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Z__Z_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1151 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Z__Z_cm__Ring_rp.out │ │ │ @@ -10114,15 +10114,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1627 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_run__Program.out │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_same.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2025-02-09 22:54:37.000000 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22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_schreyer__Order_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 99 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sec.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sech.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1142 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_select__In__Subring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_select__Keys.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_select__Pairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1331 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_select__Variables_lp__List_cm__Polynomial__Ring_rp.out │ │ │ @@ -10146,15 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./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Basis__Element__Limit.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10200 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Basis__Element__Limit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3362 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Bayer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4053 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Before__Print.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5368 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Bessel__J.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5368 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Bessel__Y.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5378 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Beta.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16504 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Betti__Tally.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3646 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Binary.html │ │ │ @@ -10391,16 +10391,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3690 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Chain__Complex_sp_st_st_sp__Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5372 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Chain__Complex_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6957 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Chain__Complex_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Chain__Complex_sp_us_sp__Z__Z_sp_eq_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4947 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Change__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3832 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Check__Documentation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3679 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Closest__Fit.html │ │ │ --rw-r--r-- 0 root (0) root (0) 4491 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Codimension__Limit.html │ │ │ --rw-r--r-- 0 root (0) root (0) 4700 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Coefficient__Ring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 4477 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Codimension__Limit.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 4690 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3949 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Cofactor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11951 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html │ │ │ -rw-r--r-- 0 root (0) root (0) 30972 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Compiled__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4000 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Compiled__Function__Body.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4776 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Compiled__Function__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3684 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3849 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Complete__Intersection.html │ │ │ @@ -10416,18 +10416,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7980 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html │ │ │ -rw-r--r-- 0 root (0) root 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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1182 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multirational__Map_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2768 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_parametrize_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 490 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 498 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 497 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 849 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 322 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 381 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5788 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8037 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8034 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9525 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7352 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12743 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7949 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9170 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9073 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8296 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6559 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6287 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7531 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5730 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7246 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6251 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6076 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6818 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9200 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Graph.html │ │ │ @@ -14043,15 +14043,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 178746 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2350 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2349 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 915 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_differential__Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_eliminating__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 270 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_evaluate_lp__Diff__Op_cm__Abstract__Point_rp.out │ │ │ @@ -14084,15 +14084,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8128 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8547 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6987 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4762 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7020 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7460 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7388 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7387 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3723 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Tolerance_sp_lp__Noetherian__Operators_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6442 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_amult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5183 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_colon.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8074 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_coordinate__Change__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3816 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6698 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6272 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -14146,15 +14146,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7452 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5484 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4158 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 608945 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3510 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3508 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1905 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1922 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1977 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -14218,15 +14218,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 937 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1560 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1693 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1963 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 700 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ @@ -14256,15 +14256,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 12193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12191 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11640 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 28057 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8658 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8968 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12263 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8320 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Z__Z.html │ │ │ @@ -14333,15 +14333,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10043 2025-02-09 22:54:37.000000 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│ -rw-r--r-- 0 root (0) root (0) 307 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_projective__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points__Mat.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 14 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4229 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/___All__Random.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4374 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/___Verify__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9752 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9753 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6807 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6158 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Make__Ring__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7954 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6778 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6757 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__Mat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5904 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_expected__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5049 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/html/_min__Max__Resolution.html │ │ │ @@ -16386,15 +16386,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_filtration.out │ │ │ -rw-r--r-- 0 root (0) root (0) 525 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flag__Chains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flag__Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 210 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flagf__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 244 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flagh__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_gap__Convert__Poset.out │ │ │ --rw-r--r-- 0 root (0) root (0) 591 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 590 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_h__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_hasse__Diagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 96 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_height_lp__Poset_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 258 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_hibi__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 907 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_hibi__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_incomparability__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 269 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_index__Labeling.out │ │ │ @@ -16515,15 +16515,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5936 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7063 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_filtration.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6744 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_flag__Chains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6947 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_flag__Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6655 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_flagf__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6658 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_flagh__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7494 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_gap__Convert__Poset.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7919 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7918 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6185 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_h__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5977 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_hasse__Diagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5145 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_height_lp__Poset_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6608 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_hibi__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9522 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_hibi__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5854 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_incomparability__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6667 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Posets/html/_index__Labeling.html │ │ │ @@ -16642,15 +16642,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2223 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1389 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1390 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17065 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4283 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ @@ -16658,15 +16658,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8019 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9798 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8175 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11507 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20676 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5348 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9615 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9616 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7109 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9948 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_strategies_spfor_spcomputing_spprimary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7287 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_top__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20337 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13327 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6093 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Probability/ │ │ │ @@ -16953,15 +16953,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 13630 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3258 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1713 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 28361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21988 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2741 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4056 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2380 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1645 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2176 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -16995,15 +16995,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8290 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 9103 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 6474 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15710 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19483 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8732 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7974 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 51851 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 45478 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10221 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3812 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11982 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7789 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 13447 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11236 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8513 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -17140,27 +17140,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2621 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1465 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4961 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1096 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1072 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3641 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3987 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3843 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9567 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6887 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6833 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6587 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13744 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7713 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7689 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10830 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9658 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5151 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1330 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ @@ -17217,23 +17217,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 894 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 247 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 249 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 421 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8869 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8758 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1615 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Toric__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_reg__Seq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_square__Free.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4894 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Alexander__Probability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9887 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Finding_sp__Extreme_sp__Examples.html │ │ │ @@ -17242,23 +17242,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5499 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7140 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7501 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6408 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7570 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Elements__From__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7433 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5983 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5985 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6844 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7519 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Pure__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7191 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7295 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal__Chain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9608 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7966 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Sparse__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7126 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17620 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17509 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8134 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Toric__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5498 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_reg__Seq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5785 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5526 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free_lp__Z__Z_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25799 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17897 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8907 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ @@ -17453,15 +17453,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 117445 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1111 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/___Rational__Mapping.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1086 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/___Rational__Mapping_sp_st_sp__Rational__Mapping.out │ │ │ -rw-r--r-- 0 root (0) root (0) 608 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_base__Locus__Of__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_ideal__Of__Image__Of__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4960 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4959 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Birational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1041 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Birational__Onto__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1413 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Regular__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 588 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Same__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 613 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_jacobian__Dual__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 749 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_map__Onto__Image.out │ │ │ @@ -17480,15 +17480,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8014 2025-02-09 22:54:37.000000 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22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19020 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12076 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Birational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11771 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Birational__Onto__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14409 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Embedding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6230 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Regular__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7630 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Same__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9207 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7138 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 122874 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5792 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 122875 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5793 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1268 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_veronese.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 712 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 41 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5812 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5062 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Proj.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5017 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Assume__Ordinary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4798 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Duality.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8812 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11914 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4536 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6173 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8466 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10356 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13908 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24838 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10359 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13907 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24836 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7807 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20680 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11671 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8958 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11670 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./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8012 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12190 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9636 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9634 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14053 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 130212 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16088 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 130213 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16089 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/_veronese.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19678 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21347 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8335 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Resultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37595 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 418 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6287 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 416 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6321 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4982 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Keep__Statistics__Command.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5254 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Pre__Run__Script.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4839 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Child.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4905 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7314 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23002 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7312 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23036 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5489 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2__Return__Answer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6095 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2_lp..._cm__Keep__Files_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7295 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_suggestions_spfor_spusing_sp__Run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8971 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7557 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SCMAlgebras/ │ │ │ @@ -18292,28 +18292,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 26032 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44792 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2826 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11853 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complexes.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3436 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3437 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5471 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_are__Pseudo__Inverses.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2031 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6927 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7251 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_laplacians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_numeric__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9599 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9183 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_pseudo__Inverse.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3432 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Laplacian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3446 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Projection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11394 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12445 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12446 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14495 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3307 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9606 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14034 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13796 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3647 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8632 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_numeric__Rank.html │ │ │ @@ -18340,27 +18340,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52986 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1858 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1856 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8444 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6028 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6783 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10737 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Basis__Element__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12751 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10232 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Pair__Limit_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25840 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25838 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14657 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11378 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12109 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13607 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11857 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5138 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -18793,37 +18793,37 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 48932 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_chow__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 690 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_contained__In__Singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 895 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_intersection__Product.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1525 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1524 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Multi__Hom.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Product__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 932 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 876 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre.out │ │ │ --rw-r--r-- 0 root (0) root (0) 721 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 720 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7409 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_chow__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8087 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_contained__In__Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9553 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_intersection__Product.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9207 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9206 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6270 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Multi__Hom.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6244 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Product__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7343 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9004 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8810 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8581 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8441 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8440 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17183 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13920 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7233 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44343 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/example-output/ │ │ │ @@ -18902,25 +18902,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42166 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_ar__Xiv.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1096 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_package__Template.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_simple__Doc__Frob.out │ │ │ -rw-r--r-- 0 root (0) root (0) 261 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_stacks__Project.out │ │ │ --rw-r--r-- 0 root (0) root (0) 135 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 136 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_wikipedia.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6187 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_ar__Xiv.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20190 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_doc.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6720 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_doc__Example.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5916 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_package__Template.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10506 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_simple__Doc__Frob.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5686 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_stacks__Project.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5298 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5299 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/_wikipedia.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11013 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7409 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4291 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimplicialComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 390036 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -19345,18 +19345,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 646 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 597 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Ring__Element_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Discriminant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 930 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Resultant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Resultant_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 485 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 484 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 814 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 931 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1363 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 929 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dim_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_entries_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 394 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 471 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Resultant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1329 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_flattening.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Laurent__Polynomials.out │ │ │ @@ -19370,15 +19370,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_random__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2126 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_rank_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1772 2025-02-09 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│ │ │ -rw-r--r-- 0 root (0) root (0) 911 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 724 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 35 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 10027 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6153 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp-_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6036 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp_eq_eq_sp__Multidimensional__Matrix.html │ │ │ @@ -19388,18 +19388,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5975 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Ring__Element_sp_st_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5143 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5671 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6200 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4969 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/SparseResultants/html/_ring_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5357 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7655 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sort__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19922 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 65066 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 65058 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6790 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22049 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21490 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12465 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178022 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump │ │ │ @@ -19448,15 +19448,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 210 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_cycle__Decomposition_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 252 2025-02-09 22:54:37.000000 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4965 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_size_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6181 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5339 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7243 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Module__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7116 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Polynomial_lp__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8530 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Polynomials_lp__Partition_cm__Polynomial__Ring_rp.html │ │ │ @@ -19559,40 +19559,40 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 203652 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 914 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___G__Mtables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1325 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_ambient__Fivefold.out │ │ │ --rw-r--r-- 0 root (0) root (0) 955 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 954 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1005 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1191 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1173 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1172 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1555 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 298 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 299 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 653 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 652 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1972 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6387 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 530 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__External__String_lp__Hodge__Special__Fourfold_rp.out │ │ │ @@ -19608,47 +19608,47 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9733 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Hodge__Special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5972 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Intersection__Of__Three__Quadrics__In__P7.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4384 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Singular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9292 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9852 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8467 2025-02-09 22:54:37.000000 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22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10280 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6505 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11896 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11897 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11226 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7752 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7531 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7123 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7471 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8807 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5892 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ @@ -20503,25 +20503,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1328 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1329 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ --rw-r--r-- 0 root (0) root (0) 488 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1824 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 489 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1820 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1493 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1513 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1514 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2752 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 697 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4011 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4077 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4850 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -20554,25 +20554,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8819 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6888 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9079 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4988 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10181 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11795 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4837 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16359 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16360 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8210 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7828 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14966 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7829 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14962 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9851 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9220 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15663 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15664 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7524 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6079 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14360 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17076 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38028 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35805 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14203 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 134990 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -20764,32 +20764,32 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38211 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26568 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24627 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1429 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8263 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 572 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1592 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7172 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 535 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1147 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1030 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2666 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1155 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2076 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 712 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6460 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6606 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6769 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6349 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6791 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6701 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12946 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6826 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12356 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7787 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22196 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 21105 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7093 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4834 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66622 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Topcom/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25169 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Topcom/example-output/___An_spexample_spuse_spof_sp__Top__Com.out │ │ │ @@ -20868,25 +20868,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20320 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 944 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Volumes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_dual__Deg__Codim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1537 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1536 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1032 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_polar__Degrees.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4189 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Force__Amat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4006 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3998 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Text__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8467 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Volumes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7178 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_dual__Deg__Codim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8559 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8558 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8495 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_polar__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7242 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4946 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricTopology/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23591 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump │ │ │ @@ -21847,15 +21847,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11419 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 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│ +-rw-r--r-- 0 root (0) root (0) 20950 2025-02-09 22:54:37.000000 ./usr/share/info/Bertini.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31682 2025-02-09 22:54:37.000000 ./usr/share/info/BettiCharacters.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3077 2025-02-09 22:54:37.000000 ./usr/share/info/BinomialEdgeIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10618 2025-02-09 22:54:37.000000 ./usr/share/info/Binomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19492 2025-02-09 22:54:37.000000 ./usr/share/info/BoijSoederberg.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 16173 2025-02-09 22:54:37.000000 ./usr/share/info/Book3264Examples.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1561 2025-02-09 22:54:37.000000 ./usr/share/info/BooleanGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 833 2025-02-09 22:54:37.000000 ./usr/share/info/Browse.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 6248 2025-02-09 22:54:37.000000 ./usr/share/info/Bruns.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 19460 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./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 9259 2025-02-09 22:54:37.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 13221 2025-02-09 22:54:37.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6682 2025-02-09 22:54:37.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10548 2025-02-09 22:54:37.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10208 2025-02-09 22:54:37.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5696 2025-02-09 22:54:37.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8207 2025-02-09 22:54:37.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4443 2025-02-09 22:54:37.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8523 2025-02-09 22:54:37.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 30035 2025-02-09 22:54:37.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 18956 2025-02-09 22:54:37.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 50801 2025-02-09 22:54:37.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 17996 2025-02-09 22:54:37.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 18002 2025-02-09 22:54:37.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12909 2025-02-09 22:54:37.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10007 2025-02-09 22:54:37.000000 ./usr/share/info/Visualize.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8884 2025-02-09 22:54:37.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 31762 2025-02-09 22:54:37.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 5857 2025-02-09 22:54:37.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31748 2025-02-09 22:54:37.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 5862 2025-02-09 22:54:37.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8457 2025-02-09 22:54:37.000000 ./usr/share/info/XML.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 48526 2025-02-09 22:54:37.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11341 2025-02-09 22:44:40.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ ├── ./usr/share/doc/Macaulay2/A1BrouwerDegrees/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ bWFrZURpYWdvbmFsRm9ybQ== │ │ │ #:len=2204 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIEdyb3RoZW5kaWVjay1XaXR0IGNs │ │ │ YXNzIG9mIGEgZGlhZ29uYWwgZm9ybSIsICJsaW5lbnVtIiA9PiAyNSwgSW5wdXRzID0+IHtTUEFO │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aGFzTWluaW1hbE11bHQoUmluZyxJbmZpbml0ZU51bWJlcik= │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQ4Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaGFzTWluaW1hbE11bHQsUmluZyxJbmZpbml0ZU51 │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 1.63212s elapsed │ │ │ + -- 2.71557s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ - -- 1.93782s elapsed │ │ │ + -- 3.38896s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -86,26 +86,26 @@ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 1.63212s elapsed
│ │ │ + -- 2.71557s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o3 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o3 : Complex
│ │ │ │ │ │ │ │ │ 
i4 : elapsedTime burkeResolution(M, 7, Check => true)
│ │ │ - -- 1.93782s elapsed
│ │ │ + -- 3.38896s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o4 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -24,24 +24,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 1.63212s elapsed │ │ │ │ + -- 2.71557s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o3 : Complex │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ │ - -- 1.93782s elapsed │ │ │ │ + -- 3.38896s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=92 │ │ │ aW5kdWNlZFJlZHVjZWRTaW1wbGljaWFsQ2hhaW5Db21wbGV4TWFwKEFic3RyYWN0U2ltcGxpY2lh │ │ │ bENvbXBsZXgsQWJzdHJhY3RTaW1wbGljaWFsQ29tcGxleCk= │ │ │ #:len=502 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzMyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/___Calculations_spwith_sprandom_spsimplicial_spcomplexes.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 6456613336951100320 │ │ │ │ │ │ i1 : setRandomSeed(currentTime()); │ │ │ │ │ │ i2 : K = randomAbstractSimplicialComplex(4) │ │ │ │ │ │ o2 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{2}, {3}} │ │ │ - 1 => {{2, 3}} │ │ │ + 0 => {{3}, {4}} │ │ │ + 1 => {{3, 4}} │ │ │ │ │ │ o2 : AbstractSimplicialComplex │ │ │ │ │ │ i3 : prune HH simplicialChainComplex K │ │ │ │ │ │ 1 │ │ │ o3 = ZZ │ │ │ @@ -19,17 +19,18 @@ │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : setRandomSeed(currentTime()); │ │ │ │ │ │ i5 : L = randomAbstractSimplicialComplex(6,3) │ │ │ │ │ │ -o5 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{3}, {6}} │ │ │ - 1 => {{3, 6}} │ │ │ +o5 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{2}, {3}, {6}} │ │ │ + 1 => {{2, 3}, {2, 6}, {3, 6}} │ │ │ + 2 => {{2, 3, 6}} │ │ │ │ │ │ o5 : AbstractSimplicialComplex │ │ │ │ │ │ i6 : prune HH simplicialChainComplex L │ │ │ │ │ │ 1 │ │ │ o6 = ZZ │ │ │ @@ -38,50 +39,50 @@ │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ i7 : setRandomSeed(currentTime()); │ │ │ │ │ │ i8 : M = randomAbstractSimplicialComplex(6,3,2) │ │ │ │ │ │ -o8 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ +o8 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ 0 => {{1}, {2}, {3}, {4}, {5}, {6}} │ │ │ - 1 => {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {3, 6}, {5, 6}} │ │ │ - 2 => {{1, 2, 5}, {2, 3, 4}, {3, 5, 6}} │ │ │ + 1 => {{1, 3}, {1, 4}, {1, 5}, {2, 5}, {2, 6}, {3, 4}, {4, 5}, {5, 6}} │ │ │ + 2 => {{1, 3, 4}, {1, 4, 5}, {2, 5, 6}} │ │ │ │ │ │ o8 : AbstractSimplicialComplex │ │ │ │ │ │ i9 : prune HH simplicialChainComplex M │ │ │ │ │ │ - 1 1 │ │ │ -o9 = ZZ <-- ZZ │ │ │ - │ │ │ - 0 1 │ │ │ + 1 │ │ │ +o9 = ZZ │ │ │ + │ │ │ + 0 │ │ │ │ │ │ o9 : Complex │ │ │ │ │ │ i10 : setRandomSeed(currentTime()); │ │ │ │ │ │ i11 : K = randomAbstractSimplicialComplex(4) │ │ │ │ │ │ o11 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{2}, {3}} │ │ │ - 1 => {{2, 3}} │ │ │ + 0 => {{3}, {4}} │ │ │ + 1 => {{3, 4}} │ │ │ │ │ │ o11 : AbstractSimplicialComplex │ │ │ │ │ │ i12 : J = randomSubSimplicialComplex(K) │ │ │ │ │ │ o12 = AbstractSimplicialComplex{-1 => {{}}} │ │ │ - 0 => {{2}} │ │ │ + 0 => {{4}} │ │ │ │ │ │ o12 : AbstractSimplicialComplex │ │ │ │ │ │ i13 : inducedSimplicialChainComplexMap(K,J) │ │ │ │ │ │ 2 1 │ │ │ o13 = 0 : ZZ <--------- ZZ : 0 │ │ │ - | 1 | │ │ │ | 0 | │ │ │ + | 1 | │ │ │ │ │ │ o13 : ComplexMap │ │ │ │ │ │ i14 : │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/_random__Abstract__Simplicial__Complex.out │ │ │ @@ -1,36 +1,34 @@ │ │ │ -- -*- M2-comint -*- hash: 9222441599761629245 │ │ │ │ │ │ i1 : setRandomSeed(currentTime()); │ │ │ │ │ │ i2 : K = randomAbstractSimplicialComplex(4) │ │ │ │ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{1}, {2}, {4}} │ │ │ - 1 => {{1, 2}, {1, 4}, {2, 4}} │ │ │ - 2 => {{1, 2, 4}} │ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{1}, {3}} │ │ │ + 1 => {{1, 3}} │ │ │ │ │ │ o2 : AbstractSimplicialComplex │ │ │ │ │ │ i3 : setRandomSeed(currentTime()); │ │ │ │ │ │ i4 : L = randomAbstractSimplicialComplex(6,3) │ │ │ │ │ │ -o4 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{1}, {4}, {6}} │ │ │ - 1 => {{1, 4}, {1, 6}, {4, 6}} │ │ │ - 2 => {{1, 4, 6}} │ │ │ +o4 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{1}, {3}} │ │ │ + 1 => {{1, 3}} │ │ │ │ │ │ o4 : AbstractSimplicialComplex │ │ │ │ │ │ i5 : setRandomSeed(currentTime()); │ │ │ │ │ │ i6 : M = randomAbstractSimplicialComplex(6,3,2) │ │ │ │ │ │ -o6 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{1}, {2}, {3}, {4}, {5}, {6}} │ │ │ - 1 => {{1, 3}, {1, 5}, {2, 4}, {2, 5}, {3, 5}, {4, 5}, {4, 6}, {5, 6}} │ │ │ - 2 => {{1, 3, 5}, {2, 4, 5}, {4, 5, 6}} │ │ │ +o6 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{1}, {2}, {3}, {4}, {5}} │ │ │ + 1 => {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {3, 5}} │ │ │ + 2 => {{1, 3, 4}, {1, 3, 5}, {2, 3, 5}} │ │ │ │ │ │ o6 : AbstractSimplicialComplex │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/_random__Sub__Simplicial__Complex.out │ │ │ @@ -1,18 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 13473104809235542297 │ │ │ │ │ │ i1 : setRandomSeed(currentTime()); │ │ │ │ │ │ i2 : K = randomAbstractSimplicialComplex(4) │ │ │ │ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}}} │ │ │ - 0 => {{1}} │ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{2}, {3}, {4}} │ │ │ + 1 => {{2, 3}, {2, 4}, {3, 4}} │ │ │ + 2 => {{2, 3, 4}} │ │ │ │ │ │ o2 : AbstractSimplicialComplex │ │ │ │ │ │ i3 : J = randomSubSimplicialComplex(K) │ │ │ │ │ │ -o3 = AbstractSimplicialComplex{-1 => {{}}} │ │ │ +o3 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{2}, {3}, {4}} │ │ │ + 1 => {{2, 3}, {2, 4}, {3, 4}} │ │ │ + 2 => {{2, 3, 4}} │ │ │ │ │ │ o3 : AbstractSimplicialComplex │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/___Calculations_spwith_sprandom_spsimplicial_spcomplexes.html │ │ │ @@ -53,16 +53,16 @@ │ │ │ │ │ │ 
i1 : setRandomSeed(currentTime());
│ │ │ │ │ │ │ │ │ 
i2 : K = randomAbstractSimplicialComplex(4)
│ │ │  
│ │ │  o2 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ -                               0 => {{2}, {3}}
│ │ │ -                               1 => {{2, 3}}
│ │ │ +                               0 => {{3}, {4}}
│ │ │ +                               1 => {{3, 4}}
│ │ │  
│ │ │  o2 : AbstractSimplicialComplex
│ │ │ │ │ │ │ │ │ 
i3 : prune HH simplicialChainComplex K
│ │ │  
│ │ │         1
│ │ │ @@ -79,17 +79,18 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i4 : setRandomSeed(currentTime());

│ │ │  		          
              
i5 : L = randomAbstractSimplicialComplex(6,3)
│ │ │  
│ │ │ -o5 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ -                               0 => {{3}, {6}}
│ │ │ -                               1 => {{3, 6}}
│ │ │ +o5 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ +                               0 => {{2}, {3}, {6}}
│ │ │ +                               1 => {{2, 3}, {2, 6}, {3, 6}}
│ │ │ +                               2 => {{2, 3, 6}}
│ │ │  
│ │ │  o5 : AbstractSimplicialComplex

│ │ │  		          
              
i6 : prune HH simplicialChainComplex L
│ │ │  
│ │ │         1
│ │ │ @@ -106,28 +107,28 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i7 : setRandomSeed(currentTime());

│ │ │  		          
              
i8 : M = randomAbstractSimplicialComplex(6,3,2)
│ │ │  
│ │ │ -o8 = AbstractSimplicialComplex{-1 => {{}}                                                                   }
│ │ │ +o8 = AbstractSimplicialComplex{-1 => {{}}                                                           }
│ │ │                                 0 => {{1}, {2}, {3}, {4}, {5}, {6}}
│ │ │ -                               1 => {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {3, 6}, {5, 6}}
│ │ │ -                               2 => {{1, 2, 5}, {2, 3, 4}, {3, 5, 6}}
│ │ │ +                               1 => {{1, 3}, {1, 4}, {1, 5}, {2, 5}, {2, 6}, {3, 4}, {4, 5}, {5, 6}}
│ │ │ +                               2 => {{1, 3, 4}, {1, 4, 5}, {2, 5, 6}}
│ │ │  
│ │ │  o8 : AbstractSimplicialComplex

│ │ │  		          
              
i9 : prune HH simplicialChainComplex M
│ │ │  
│ │ │ -       1       1
│ │ │ -o9 = ZZ  <-- ZZ
│ │ │ -              
│ │ │ -     0       1
│ │ │ +       1
│ │ │ +o9 = ZZ
│ │ │ +      
│ │ │ +     0
│ │ │  
│ │ │  o9 : Complex

│ │ │  		          

│ │ │          

│ │ │            
Creates a random sub-simplicial complex of a given simplicial complex.

│ │ │          

│ │ │ @@ -135,34 +136,34 @@
│ │ │            
              
i10 : setRandomSeed(currentTime());

│ │ │  		          
              
i11 : K = randomAbstractSimplicialComplex(4)
│ │ │  
│ │ │  o11 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ -                                0 => {{2}, {3}}
│ │ │ -                                1 => {{2, 3}}
│ │ │ +                                0 => {{3}, {4}}
│ │ │ +                                1 => {{3, 4}}
│ │ │  
│ │ │  o11 : AbstractSimplicialComplex

│ │ │  		          
              
i12 : J = randomSubSimplicialComplex(K)
│ │ │  
│ │ │  o12 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ -                                0 => {{2}}
│ │ │ +                                0 => {{4}}
│ │ │  
│ │ │  o12 : AbstractSimplicialComplex

│ │ │  		          
              
i13 : inducedSimplicialChainComplexMap(K,J)
│ │ │  
│ │ │              2              1
│ │ │  o13 = 0 : ZZ  <--------- ZZ  : 0
│ │ │ -                 | 1 |
│ │ │                   | 0 |
│ │ │ +                 | 1 |
│ │ │  
│ │ │  o13 : ComplexMap

│ │ │  		          

│ │ │        
│ │ │      
│ │ │    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,33 +10,34 @@
│ │ │ │  be performed on random simplicial complexes.
│ │ │ │  Create a random abstract simplicial complex with vertices supported on a subset
│ │ │ │  of [n] = {1,...,n}.
│ │ │ │  i1 : setRandomSeed(currentTime());
│ │ │ │  i2 : K = randomAbstractSimplicialComplex(4)
│ │ │ │  
│ │ │ │  o2 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ │ -                               0 => {{2}, {3}}
│ │ │ │ -                               1 => {{2, 3}}
│ │ │ │ +                               0 => {{3}, {4}}
│ │ │ │ +                               1 => {{3, 4}}
│ │ │ │  
│ │ │ │  o2 : AbstractSimplicialComplex
│ │ │ │  i3 : prune HH simplicialChainComplex K
│ │ │ │  
│ │ │ │         1
│ │ │ │  o3 = ZZ
│ │ │ │  
│ │ │ │       0
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  Create a random simplicial complex on [n] with dimension at most equal to r.
│ │ │ │  i4 : setRandomSeed(currentTime());
│ │ │ │  i5 : L = randomAbstractSimplicialComplex(6,3)
│ │ │ │  
│ │ │ │ -o5 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ │ -                               0 => {{3}, {6}}
│ │ │ │ -                               1 => {{3, 6}}
│ │ │ │ +o5 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ │ +                               0 => {{2}, {3}, {6}}
│ │ │ │ +                               1 => {{2, 3}, {2, 6}, {3, 6}}
│ │ │ │ +                               2 => {{2, 3, 6}}
│ │ │ │  
│ │ │ │  o5 : AbstractSimplicialComplex
│ │ │ │  i6 : prune HH simplicialChainComplex L
│ │ │ │  
│ │ │ │         1
│ │ │ │  o6 = ZZ
│ │ │ │  
│ │ │ │ @@ -48,43 +49,43 @@
│ │ │ │  binomial(binomial(n,d+1),m) possibilities.
│ │ │ │  i7 : setRandomSeed(currentTime());
│ │ │ │  i8 : M = randomAbstractSimplicialComplex(6,3,2)
│ │ │ │  
│ │ │ │  o8 = AbstractSimplicialComplex{-1 => {{}}
│ │ │ │  }
│ │ │ │                                 0 => {{1}, {2}, {3}, {4}, {5}, {6}}
│ │ │ │ -                               1 => {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5},
│ │ │ │ -{3, 4}, {3, 5}, {3, 6}, {5, 6}}
│ │ │ │ -                               2 => {{1, 2, 5}, {2, 3, 4}, {3, 5, 6}}
│ │ │ │ +                               1 => {{1, 3}, {1, 4}, {1, 5}, {2, 5}, {2, 6},
│ │ │ │ +{3, 4}, {4, 5}, {5, 6}}
│ │ │ │ +                               2 => {{1, 3, 4}, {1, 4, 5}, {2, 5, 6}}
│ │ │ │  
│ │ │ │  o8 : AbstractSimplicialComplex
│ │ │ │  i9 : prune HH simplicialChainComplex M
│ │ │ │  
│ │ │ │ -       1       1
│ │ │ │ -o9 = ZZ  <-- ZZ
│ │ │ │ +       1
│ │ │ │ +o9 = ZZ
│ │ │ │  
│ │ │ │ -     0       1
│ │ │ │ +     0
│ │ │ │  
│ │ │ │  o9 : Complex
│ │ │ │  Creates a random sub-simplicial complex of a given simplicial complex.
│ │ │ │  i10 : setRandomSeed(currentTime());
│ │ │ │  i11 : K = randomAbstractSimplicialComplex(4)
│ │ │ │  
│ │ │ │  o11 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ │ -                                0 => {{2}, {3}}
│ │ │ │ -                                1 => {{2, 3}}
│ │ │ │ +                                0 => {{3}, {4}}
│ │ │ │ +                                1 => {{3, 4}}
│ │ │ │  
│ │ │ │  o11 : AbstractSimplicialComplex
│ │ │ │  i12 : J = randomSubSimplicialComplex(K)
│ │ │ │  
│ │ │ │  o12 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ │ -                                0 => {{2}}
│ │ │ │ +                                0 => {{4}}
│ │ │ │  
│ │ │ │  o12 : AbstractSimplicialComplex
│ │ │ │  i13 : inducedSimplicialChainComplexMap(K,J)
│ │ │ │  
│ │ │ │              2              1
│ │ │ │  o13 = 0 : ZZ  <--------- ZZ  : 0
│ │ │ │ -                 | 1 |
│ │ │ │                   | 0 |
│ │ │ │ +                 | 1 |
│ │ │ │  
│ │ │ │  o13 : ComplexMap
│ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_random__Abstract__Simplicial__Complex.html
│ │ │ @@ -50,54 +50,52 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i1 : setRandomSeed(currentTime());

│ │ │  		          
              
i2 : K = randomAbstractSimplicialComplex(4)
│ │ │  
│ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ -                               0 => {{1}, {2}, {4}}
│ │ │ -                               1 => {{1, 2}, {1, 4}, {2, 4}}
│ │ │ -                               2 => {{1, 2, 4}}
│ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ +                               0 => {{1}, {3}}
│ │ │ +                               1 => {{1, 3}}
│ │ │  
│ │ │  o2 : AbstractSimplicialComplex

│ │ │  		          

│ │ │          

│ │ │            
Create a random simplicial complex on [n] with dimension at most equal to r.

│ │ │          

│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i3 : setRandomSeed(currentTime());

│ │ │  		          
              
i4 : L = randomAbstractSimplicialComplex(6,3)
│ │ │  
│ │ │ -o4 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ -                               0 => {{1}, {4}, {6}}
│ │ │ -                               1 => {{1, 4}, {1, 6}, {4, 6}}
│ │ │ -                               2 => {{1, 4, 6}}
│ │ │ +o4 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ +                               0 => {{1}, {3}}
│ │ │ +                               1 => {{1, 3}}
│ │ │  
│ │ │  o4 : AbstractSimplicialComplex

│ │ │  		          

│ │ │          

│ │ │            
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.

│ │ │          

│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i5 : setRandomSeed(currentTime());

│ │ │  		          
              
i6 : M = randomAbstractSimplicialComplex(6,3,2)
│ │ │  
│ │ │ -o6 = AbstractSimplicialComplex{-1 => {{}}                                                           }
│ │ │ -                               0 => {{1}, {2}, {3}, {4}, {5}, {6}}
│ │ │ -                               1 => {{1, 3}, {1, 5}, {2, 4}, {2, 5}, {3, 5}, {4, 5}, {4, 6}, {5, 6}}
│ │ │ -                               2 => {{1, 3, 5}, {2, 4, 5}, {4, 5, 6}}
│ │ │ +o6 = AbstractSimplicialComplex{-1 => {{}}                                                   }
│ │ │ +                               0 => {{1}, {2}, {3}, {4}, {5}}
│ │ │ +                               1 => {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {3, 5}}
│ │ │ +                               2 => {{1, 3, 4}, {1, 3, 5}, {2, 3, 5}}
│ │ │  
│ │ │  o6 : AbstractSimplicialComplex

│ │ │  		          

│ │ │        
│ │ │        

│ │ │          
See also

│ │ │ ├── html2text {}
│ │ │ │ @@ -6,42 +6,40 @@
│ │ │ │  ************ rraannddoommAAbbssttrraaccttSSiimmpplliicciiaallCCoommpplleexx ---- CCrreeaattee aa rraannddoomm ssiimmpplliicciiaall sseett ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Create a random abstract simplicial complex with vertices supported on a subset
│ │ │ │  of [n] = {1,...,n}.
│ │ │ │  i1 : setRandomSeed(currentTime());
│ │ │ │  i2 : K = randomAbstractSimplicialComplex(4)
│ │ │ │  
│ │ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ │ -                               0 => {{1}, {2}, {4}}
│ │ │ │ -                               1 => {{1, 2}, {1, 4}, {2, 4}}
│ │ │ │ -                               2 => {{1, 2, 4}}
│ │ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ │ +                               0 => {{1}, {3}}
│ │ │ │ +                               1 => {{1, 3}}
│ │ │ │  
│ │ │ │  o2 : AbstractSimplicialComplex
│ │ │ │  Create a random simplicial complex on [n] with dimension at most equal to r.
│ │ │ │  i3 : setRandomSeed(currentTime());
│ │ │ │  i4 : L = randomAbstractSimplicialComplex(6,3)
│ │ │ │  
│ │ │ │ -o4 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ │ -                               0 => {{1}, {4}, {6}}
│ │ │ │ -                               1 => {{1, 4}, {1, 6}, {4, 6}}
│ │ │ │ -                               2 => {{1, 4, 6}}
│ │ │ │ +o4 = AbstractSimplicialComplex{-1 => {{}}     }
│ │ │ │ +                               0 => {{1}, {3}}
│ │ │ │ +                               1 => {{1, 3}}
│ │ │ │  
│ │ │ │  o4 : AbstractSimplicialComplex
│ │ │ │  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.
│ │ │ │  i5 : setRandomSeed(currentTime());
│ │ │ │  i6 : M = randomAbstractSimplicialComplex(6,3,2)
│ │ │ │  
│ │ │ │  o6 = AbstractSimplicialComplex{-1 => {{}}
│ │ │ │  }
│ │ │ │ -                               0 => {{1}, {2}, {3}, {4}, {5}, {6}}
│ │ │ │ -                               1 => {{1, 3}, {1, 5}, {2, 4}, {2, 5}, {3, 5},
│ │ │ │ -{4, 5}, {4, 6}, {5, 6}}
│ │ │ │ -                               2 => {{1, 3, 5}, {2, 4, 5}, {4, 5, 6}}
│ │ │ │ +                               0 => {{1}, {2}, {3}, {4}, {5}}
│ │ │ │ +                               1 => {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 5},
│ │ │ │ +{3, 4}, {3, 5}}
│ │ │ │ +                               2 => {{1, 3, 4}, {1, 3, 5}, {2, 3, 5}}
│ │ │ │  
│ │ │ │  o6 : AbstractSimplicialComplex
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m -- get a random object
│ │ │ │      * randomSquareFreeMonomialIdeal (missing documentation)
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommAAbbssttrraaccttSSiimmpplliicciiaallCCoommpplleexx:: **********
│ │ │ │      * randomAbstractSimplicialComplex(ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_random__Sub__Simplicial__Complex.html
│ │ │ @@ -50,23 +50,28 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i1 : setRandomSeed(currentTime());

│ │ │  		          
              
i2 : K = randomAbstractSimplicialComplex(4)
│ │ │  
│ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ -                               0 => {{1}}
│ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ +                               0 => {{2}, {3}, {4}}
│ │ │ +                               1 => {{2, 3}, {2, 4}, {3, 4}}
│ │ │ +                               2 => {{2, 3, 4}}
│ │ │  
│ │ │  o2 : AbstractSimplicialComplex

│ │ │  		          
              
i3 : J = randomSubSimplicialComplex(K)
│ │ │  
│ │ │ -o3 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ +o3 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ +                               0 => {{2}, {3}, {4}}
│ │ │ +                               1 => {{2, 3}, {2, 4}, {3, 4}}
│ │ │ +                               2 => {{2, 3, 4}}
│ │ │  
│ │ │  o3 : AbstractSimplicialComplex

│ │ │  		          

│ │ │        

│ │ │        

│ │ │          
Ways to use randomSubSimplicialComplex:

│ │ │ ├── html2text {}
│ │ │ │ @@ -6,20 +6,25 @@
│ │ │ │  ************ rraannddoommSSuubbSSiimmpplliicciiaallCCoommpplleexx ---- CCrreeaattee aa rraannddoomm ssuubb--ssiimmpplliicciiaall ccoommpplleexx
│ │ │ │  ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Creates a random sub-simplicial complex of a given simplicial complex.
│ │ │ │  i1 : setRandomSeed(currentTime());
│ │ │ │  i2 : K = randomAbstractSimplicialComplex(4)
│ │ │ │  
│ │ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ │ -                               0 => {{1}}
│ │ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ │ +                               0 => {{2}, {3}, {4}}
│ │ │ │ +                               1 => {{2, 3}, {2, 4}, {3, 4}}
│ │ │ │ +                               2 => {{2, 3, 4}}
│ │ │ │  
│ │ │ │  o2 : AbstractSimplicialComplex
│ │ │ │  i3 : J = randomSubSimplicialComplex(K)
│ │ │ │  
│ │ │ │ -o3 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ │ +o3 = AbstractSimplicialComplex{-1 => {{}}                   }
│ │ │ │ +                               0 => {{2}, {3}, {4}}
│ │ │ │ +                               1 => {{2, 3}, {2, 4}, {3, 4}}
│ │ │ │ +                               2 => {{2, 3, 4}}
│ │ │ │  
│ │ │ │  o3 : AbstractSimplicialComplex
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommSSuubbSSiimmpplliicciiaallCCoommpplleexx:: **********
│ │ │ │      * randomSubSimplicialComplex(AbstractSimplicialComplex)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_S_u_b_S_i_m_p_l_i_c_i_a_l_C_o_m_p_l_e_x is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/AbstractToricVarieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  QWJzdHJhY3RUb3JpY1ZhcmlldGllcw==
│ │ │  #:len=379
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibGlua3MgYWJzdHJhY3Qgc2ltcGxpY2lh
│ │ │  bCAobm9ybWFsKSB0b3JpYyB2YXJpZXRpZXMgdG8gU2NodWJlcnQyIiwgRGVzY3JpcHRpb24gPT4g
│ │ ├── ./usr/share/doc/Macaulay2/AdjointIdeal/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  dHJhY2VNYXRyaXgoSWRlYWwsTWF0cml4KQ==
│ │ │  #:len=268
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE2MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VNYXRyaXgsSWRlYWwsTWF0cml4KSwidHJh
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  YWRqdW5jdGlvblByb2Nlc3MoSWRlYWwsWlop
│ │ │  #:len=310
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjc3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGp1bmN0aW9uUHJvY2VzcyxJZGVhbCxaWiksImFk
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │  o8 : BettiTally
│ │ │  
│ │ │  i9 : c=codim I
│ │ │  
│ │ │  o9 = 4
│ │ │  
│ │ │  i10 : elapsedTime fI=res I
│ │ │ - -- .079856s elapsed
│ │ │ + -- .0884815s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │                                                    
│ │ │        0       1        2        3        4       5
│ │ │  
│ │ │  o10 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out
│ │ │ @@ -87,30 +87,30 @@
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : phi=map(P2,Pn,H);
│ │ │  
│ │ │  o14 : RingMap P2 <-- Pn
│ │ │  
│ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .699484s elapsed
│ │ │ + -- .913919s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │  o15 : BettiTally
│ │ │  
│ │ │  i16 : I'== I
│ │ │  
│ │ │  o16 = true
│ │ │  
│ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 7.27713s elapsed
│ │ │ + -- 11.6095s elapsed
│ │ │  
│ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │                         0: 1 2
│ │ │                            0 1
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out
│ │ │ @@ -79,40 +79,40 @@
│ │ │            1: . .
│ │ │            2: . .
│ │ │            3: . 8
│ │ │  
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : elapsedTime sub(I,H)
│ │ │ - -- .0547829s elapsed
│ │ │ + -- .0700664s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │  
│ │ │  i15 : phi=map(P2,Pn,H);
│ │ │  
│ │ │  o15 : RingMap P2 <-- Pn
│ │ │  
│ │ │  i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .192756s elapsed
│ │ │ + -- .104621s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │  
│ │ │  i17 : I'== I
│ │ │  
│ │ │  o17 = true
│ │ │  
│ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 2.20464s elapsed
│ │ │ + -- 3.40021s elapsed
│ │ │  
│ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │                          0: 1  .
│ │ │                          1: .  .
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html
│ │ │ @@ -130,15 +130,15 @@
│ │ │            
│ │ │                
i9 : c=codim I
│ │ │  
│ │ │  o9 = 4

│ │ │            
│ │ │            
│ │ │                
i10 : elapsedTime fI=res I
│ │ │ - -- .079856s elapsed
│ │ │ + -- .0884815s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │                                                    
│ │ │        0       1        2        3        4       5
│ │ │  
│ │ │  o10 : ChainComplex

│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │           2: . 12
│ │ │ │  
│ │ │ │  o8 : BettiTally
│ │ │ │  i9 : c=codim I
│ │ │ │  
│ │ │ │  o9 = 4
│ │ │ │  i10 : elapsedTime fI=res I
│ │ │ │ - -- .079856s elapsed
│ │ │ │ + -- .0884815s elapsed
│ │ │ │  
│ │ │ │          1       14       33       28       8
│ │ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │ │  
│ │ │ │        0       1        2        3        4       5
│ │ │ │  
│ │ │ │  o10 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html
│ │ │ @@ -192,15 +192,15 @@
│ │ │            
│ │ │                
i14 : phi=map(P2,Pn,H);
│ │ │  
│ │ │  o14 : RingMap P2 <-- Pn

│ │ │            
│ │ │            
│ │ │                
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .699484s elapsed
│ │ │ + -- .913919s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -209,15 +209,15 @@
│ │ │            
│ │ │                
i16 : I'== I
│ │ │  
│ │ │  o16 = true

│ │ │            
│ │ │            
│ │ │                
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 7.27713s elapsed

│ │ │ + -- 11.6095s elapsed

│ │ │            
│ │ │            
│ │ │                
i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │                         0: 1 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -111,28 +111,28 @@
│ │ │ │            6: . 7
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o14 : RingMap P2 <-- Pn
│ │ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .699484s elapsed
│ │ │ │ + -- .913919s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 7.27713s elapsed
│ │ │ │ + -- 11.6095s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -167,28 +167,28 @@
│ │ │            2: . .
│ │ │            3: . 8
│ │ │  
│ │ │  o13 : BettiTally

│ │ │            
│ │ │            
│ │ │                
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0547829s elapsed
│ │ │ + -- .0700664s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2

│ │ │            
│ │ │            
│ │ │                
i15 : phi=map(P2,Pn,H);
│ │ │  
│ │ │  o15 : RingMap P2 <-- Pn

│ │ │            
│ │ │            
│ │ │                
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .192756s elapsed
│ │ │ + -- .104621s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally

│ │ │ @@ -196,15 +196,15 @@
│ │ │            
│ │ │                
i17 : I'== I
│ │ │  
│ │ │  o17 = true

│ │ │            
│ │ │            
│ │ │                
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 2.20464s elapsed

│ │ │ + -- 3.40021s elapsed
│ │ │ │ │ │ │ │ │ 
i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │                          0: 1  .
│ │ │ ├── html2text {}
│ │ │ │ @@ -83,36 +83,36 @@
│ │ │ │            0: 1 .
│ │ │ │            1: . .
│ │ │ │            2: . .
│ │ │ │            3: . 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : elapsedTime sub(I,H)
│ │ │ │ - -- .0547829s elapsed
│ │ │ │ + -- .0700664s elapsed
│ │ │ │  
│ │ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o14 : Ideal of P2
│ │ │ │  i15 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o15 : RingMap P2 <-- Pn
│ │ │ │  i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .192756s elapsed
│ │ │ │ + -- .104621s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 2.20464s elapsed
│ │ │ │ + -- 3.40021s elapsed
│ │ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                             0  1
│ │ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │ │                          0: 1  .
│ │ │ │                          1: .  .
│ │ │ │                          2: .  .
│ │ ├── ./usr/share/doc/Macaulay2/AlgebraicSplines/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c3RhbmxleVJlaXNuZXIoTGlzdCxMaXN0KQ==
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3RhbmxleVJlaXNuZXIsTGlzdCxMaXN0KSwic3Rh
│ │ ├── ./usr/share/doc/Macaulay2/AnalyzeSheafOnP1/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  YW5hbHl6ZShNb2R1bGUp
│ │ │  #:len=251
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbmFseXplLE1vZHVsZSksImFuYWx5emUoTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/AssociativeAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2VxdWVuY2VUb1ZhcmlhYmxlU3ltYm9scw==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InNlcXVlbmNl
│ │ ├── ./usr/share/doc/Macaulay2/BGG/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  dGF0ZVJlc29sdXRpb24=
│ │ │  #:len=2024
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluaXRlIHBpZWNlIG9mIHRoZSBUYXRl
│ │ │  IHJlc29sdXRpb24iLCAibGluZW51bSIgPT4gNzY5LCBJbnB1dHMgPT4ge1NQQU57VFR7Im0ifSwi
│ │ ├── ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
│ │ │ @@ -114,26 +114,26 @@
│ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │  
│ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.691952s (cpu); 0.476372s (thread); 0s (gc)
│ │ │ + -- used 0.849482s (cpu); 0.525012s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │  o14 : BettiTally
│ │ │  
│ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.796378s (cpu); 0.57534s (thread); 0s (gc)
│ │ │ + -- used 0.706791s (cpu); 0.545274s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -228,15 +228,15 @@
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.691952s (cpu); 0.476372s (thread); 0s (gc)
│ │ │ + -- used 0.849482s (cpu); 0.525012s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -245,15 +245,15 @@
│ │ │          
│ │ │          

│ │ │            
With the form pureResolution(p,q,D) we can directly create the situation of pureResolution(M,D) 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.

│ │ │          

│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,26 +192,26 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = ChainComplex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- .855753s elapsed
│ │ │ │ + -- 1.25961s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 37.8601s elapsed
│ │ │ │ + -- 58.1123s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 17.2337s elapsed
│ │ │ │ + -- 23.0969s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over R
│ │ │ │  
│ │ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/BinomialEdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=3
│ │ │  YmVp
│ │ │  #:len=336
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQmlub21pYWwgZWRnZSBpZGVhbCIsIERl
│ │ │  c2NyaXB0aW9uID0+ICgiYmVpIGlzIGEgc3lub255bSBmb3IgIixUT3tuZXcgRG9jdW1lbnRUYWcg
│ │ ├── ./usr/share/doc/Macaulay2/Binomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Ymlub21pYWxJc1ByaW1l
│ │ │  #:len=1312
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGVzdCBmb3IgcHJpbWVuZXNzIG9mIGEg
│ │ │  Ymlub21pYWwgaWRlYWwiLCAibGluZW51bSIgPT4gMTU4MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJ
│ │ ├── ./usr/share/doc/Macaulay2/BoijSoederberg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bWF0cml4KEJldHRpVGFsbHksWlop
│ │ │  #:len=291
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkyNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF0cml4LEJldHRpVGFsbHksWlopLCJtYXRyaXgo
│ │ ├── ./usr/share/doc/Macaulay2/Book3264Examples/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  SW50ZXJzZWN0aW9uIFRoZW9yeSBTZWN0aW9uIDUuMg==
│ │ │  #:len=1578
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQmFzaWNzIG9mIHZlY3RvciBidW5kbGVz
│ │ │  IGFuZCBDaGVybiBjbGFzc2VzIiwgRGVzY3JpcHRpb24gPT4gKERJVntQQVJBe1RFWHsiSW4gU2No
│ │ ├── ./usr/share/doc/Macaulay2/BooleanGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Z2JCb29sZWFuKElkZWFsKQ==
│ │ │  #:len=1781
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZSBHcm9lYm5lciBCYXNpcyBm
│ │ │  b3IgSWRlYWxzIGluIEJvb2xlYW4gUG9seW5vbWlhbCBRdW90aWVudCBSaW5nIiwgImxpbmVudW0i
│ │ ├── ./usr/share/doc/Macaulay2/Browse/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  QnJvd3Nl
│ │ │  #:len=397
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBtZXRob2QgZm9yIGJyb3dzaW5nIGFu
│ │ │  ZCBleGFtaW5pbmcgTWFjYXVsYXkyIGRhdGEgc3RydWN0dXJlcyIsIERlc2NyaXB0aW9uID0+ICgi
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNTeXp5Z3koTW9kdWxlLFpaKQ==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc1N5enlneSxNb2R1bGUsWlopLCJpc1N5enlneShN
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out
│ │ │ @@ -230,15 +230,15 @@
│ │ │            0: 1 . . . .
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │  i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.140644s (cpu); 0.139198s (thread); 0s (gc)
│ │ │ + -- used 0.308002s (cpu); 0.310916s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -336,15 +336,15 @@
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,27 +27,27 @@
│ │ │ │  i3 : P2 = convexHull matrix {{2,-2,0},{1,1,0}};
│ │ │ │  i4 : P3 = convexHull matrix {{-2,-2,0},{1,-1,0}};
│ │ │ │  i5 : P4 = convexHull matrix {{-2,2,0},{-1,-1,0}};
│ │ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │ │  i7 : C = cellComplex(R,F);
│ │ │ │  i8 : facePoset C
│ │ │ │  
│ │ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 |
│ │ │ │ -                      | 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 |
│ │ │ │ -                      | 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 |
│ │ │ │ -                      | 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |
│ │ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 |
│ │ │ │ +                      | 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 |
│ │ │ │ +                      | 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 |
│ │ │ │ +                      | 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │  
│ │ │ │  o8 : Poset
│ │ │ │  The labels on the vertices can be controlled via the optional parameter Labels
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Polyhedron_rp.html
│ │ │ @@ -128,15 +128,15 @@
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.796378s (cpu); 0.57534s (thread); 0s (gc)
│ │ │ + -- used 0.706791s (cpu); 0.545274s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -162,30 +162,30 @@
│ │ │ │        | -30a-29b -29a-24b -47a-39b 38a+2b   |
│ │ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o13 : Matrix A  <-- A
│ │ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ │ - -- used 0.691952s (cpu); 0.476372s (thread); 0s (gc)
│ │ │ │ + -- used 0.849482s (cpu); 0.525012s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o14 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ │ │  
│ │ │ │  o14 : BettiTally
│ │ │ │  With the form pureResolution(p,q,D) we can directly create the situation of
│ │ │ │  pureResolution(M,D) 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.
│ │ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ │ - -- used 0.796378s (cpu); 0.57534s (thread); 0s (gc)
│ │ │ │ + -- used 0.706791s (cpu); 0.545274s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BIBasis/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  YmlCYXNpcyguLi4sdG9Hcm9lYm5lcj0+Li4uKQ==
│ │ │  #:len=240
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODMsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JpQmFzaXMsdG9Hcm9lYm5lcl0sImJpQmFzaXMoLi4u
│ │ ├── ./usr/share/doc/Macaulay2/BeginningMacaulay2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  QmVnaW5uaW5nTWFjYXVsYXky
│ │ │  #:len=29455
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTWF0aGVtYXRpY2lhbnMnIEludHJvZHVj
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│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  QmVuY2htYXJr
│ │ │  #:len=347
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│ │ │  bWFya3MiLCBEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMgcGFja2FnZSBwcm92
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sun Feb  9 23:59:51 UTC 2025
│ │ │ --- Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 (2025-02-07) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Sat Jul 19 03:45:16 UTC 2025
│ │ │ +-- Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.35-1 (2025-07-03) x86_64 GNU/Linux
│ │ │ +-- AMD EPYC-Milan Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │  -- Macaulay2 1.24.11, compiled with gcc 14.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .113587 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .130482 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -73,19 +73,19 @@
│ │ │          
Description

│ │ │          

│ │ │            
The tests available are: 
"deg2generic" -- gb of a generic ideal of codimension 2 and degree 2
"gb4by4comm" -- gb of the ideal of generic commuting 4 by 4 matrices over ZZ/101
"gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
"gbB148" -- gb of Bayesian graph ideal #148
"res39" -- res of a generic 3 by 9 matrix over ZZ/101
"resG25" -- res of the coordinate ring of Grassmannian(2,5)
"yang-gb1" -- an example of Yang-Hui He arising in string theory
"yang-subring" -- an example of Yang-Hui He

│ │ │          

│ │ │          
│ │ │            
│ │ │  
│ │ │          
              
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sun Feb  9 23:59:51 UTC 2025
│ │ │ --- Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 (2025-02-07) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Sat Jul 19 03:45:16 UTC 2025
│ │ │ +-- Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.35-1 (2025-07-03) x86_64 GNU/Linux
│ │ │ +-- AMD EPYC-Milan Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │  -- Macaulay2 1.24.11, compiled with gcc 14.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .113587 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .130482 seconds
│ │ │  		          

│ │ │        
│ │ │        

│ │ │          
For the programmer

│ │ │          
The object runBenchmarks is a command.

│ │ │        

│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  "gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
│ │ │ │  "gbB148" -- gb of Bayesian graph ideal #148
│ │ │ │  "res39" -- res of a generic 3 by 9 matrix over ZZ/101
│ │ │ │  "resG25" -- res of the coordinate ring of Grassmannian(2,5)
│ │ │ │  "yang-gb1" -- an example of Yang-Hui He arising in string theory
│ │ │ │  "yang-subring" -- an example of Yang-Hui He
│ │ │ │  i1 : runBenchmarks "res39"
│ │ │ │ --- beginning computation Sun Feb  9 23:59:51 UTC 2025
│ │ │ │ --- Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 (2025-
│ │ │ │ -02-07) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │ +-- beginning computation Sat Jul 19 03:45:16 UTC 2025
│ │ │ │ +-- Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.35-1 (2025-07-03) x86_64 GNU/Linux
│ │ │ │ +-- AMD EPYC-Milan Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │  -- Macaulay2 1.24.11, compiled with gcc 14.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .113587 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .130482 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ ├── ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
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│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RGxvY2FsaXplQWxsKElkZWFsLFJpbmdFbGVtZW50KQ==
│ │ │  #:len=298
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│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
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│ │ │  #:len=29
│ │ │  aW1wb3J0SW5jaWRlbmNlTWF0cml4KFN0cmluZyk=
│ │ │  #:len=281
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIyMiwgc3ltYm9sIERvY3VtZW50VGFn
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│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints",               
│ │ │                
  • 
│ │ │  RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number              
    • 
│ │ │                
  • 
│ │ │  RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number              
    • 
│ │ │                
  • 
│ │ │ -TopDirectory => ..., default value "/tmp/M2-71481-0/0", Option to change directory for file storage.              
    • 
│ │ │ +TopDirectory => ..., default value "/tmp/M2-124034-0/0", Option to change directory for file storage.              
│ │ │                
  • 
│ │ │  Verbose => ..., default value false, Option to silence additional output              
    • 
│ │ │              
│ │ │            
│ │ │            
  • 
│ │ │  Outputs:            
      
│ │ │                
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-71481-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-124034-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │                
    • 
│ │ │  RandomComplex (missing documentation)
│ │ │   => ..., default value {},               
      • 
│ │ │                
    • 
│ │ │  RandomReal (missing documentation)
│ │ │   => ..., default value {},               
      • 
│ │ │                
    • 
│ │ │ -TopDirectory => ..., default value "/tmp/M2-71481-0/0", Option to change directory for file storage.              
      • 
│ │ │ +TopDirectory => ..., default value "/tmp/M2-124034-0/0", Option to change directory for file storage.              
│ │ │                
    • 
│ │ │  Verbose => ..., default value false, Option to silence additional output              
      • 
│ │ │              
    
│ │ │            
    • 
│ │ │            
  • 
│ │ │  Outputs:            
      
│ │ │                
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │              value {},
│ │ │ │            o HomVariableGroup (missing documentation) => ..., default value {},
│ │ │ │            o M2Precision (missing documentation) => ..., default value 53,
│ │ │ │            o OutputStyle (missing documentation) => ..., default value
│ │ │ │              "OutPoints",
│ │ │ │            o RandomComplex (missing documentation) => ..., default value {},
│ │ │ │            o RandomReal (missing documentation) => ..., default value {},
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-71481-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-124034-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │  OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints",               
      • 
│ │ │                
    • 
│ │ │  RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number              
      • 
│ │ │                
    • 
│ │ │  RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number              
      • 
│ │ │                
    • 
│ │ │ -TopDirectory => ..., default value "/tmp/M2-71481-0/0", Option to change directory for file storage.              
      • 
│ │ │ +TopDirectory => ..., default value "/tmp/M2-124034-0/0", Option to change directory for file storage.              
│ │ │                
    • 
│ │ │  UseRegeneration (missing documentation)
│ │ │   => ..., default value -1,               
      • 
│ │ │                
    • 
│ │ │  Verbose => ..., default value false, Option to silence additional output              
      • 
│ │ │              
    
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-71481-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-124034-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o UseRegeneration (missing documentation) => ..., default value -1,
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list of points that are contained in the variety of F
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aW52ZXJzZVJpbmdBY3RvcnMoLi4uLFN1Yj0+Li4uKQ==
│ │ │  #:len=264
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjg0OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW52ZXJzZVJpbmdBY3RvcnMsU3ViXSwiaW52ZXJz
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = ChainComplex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .444232s elapsed
│ │ │ + -- 1.07415s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = ChainComplex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- .512547s elapsed
│ │ │ + -- .874536s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,27 +187,27 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = ChainComplex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- .855753s elapsed
│ │ │ + -- 1.25961s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 37.8601s elapsed
│ │ │ + -- 58.1123s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -297,15 +297,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 17.2337s elapsed
│ │ │ + -- 23.0969s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o8 = ChainComplex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
    
│ │ │      		          
                  
    i9 : elapsedTime c = character A
│ │ │ - -- .444232s elapsed
│ │ │ + -- 1.07415s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = ChainComplex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .444232s elapsed
│ │ │ │ + -- 1.07415s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over R
│ │ │ │  
│ │ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o6 = ChainComplex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │      		          
                  
    i7 : elapsedTime c=character A
│ │ │ - -- .512547s elapsed
│ │ │ + -- .874536s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = ChainComplex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- .512547s elapsed
│ │ │ │ + -- .874536s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over R
│ │ │ │  
│ │ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -264,28 +264,28 @@
│ │ │  
│ │ │  o19 = ChainComplex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │      		          
                  
    i20 : elapsedTime a1 = character A1
│ │ │ - -- .855753s elapsed
│ │ │ + -- 1.25961s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o20 : Character
    
│ │ │      		          
                  
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 37.8601s elapsed
│ │ │ + -- 58.1123s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -397,15 +397,15 @@
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │      		          
                  
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 17.2337s elapsed
│ │ │ + -- 23.0969s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
    
│ │ │      		          
              
                  
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.140644s (cpu); 0.139198s (thread); 0s (gc)
│ │ │ + -- used 0.308002s (cpu); 0.310916s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
    
│ │ │      		          
                  
    i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -231,15 +231,15 @@
│ │ │ │  o22 = total: 1 5 8 5 1
│ │ │ │            0: 1 . . . .
│ │ │ │            1: . 4 2 . .
│ │ │ │            2: . 1 6 5 1
│ │ │ │  
│ │ │ │  o22 : BettiTally
│ │ │ │  i23 : time j=bruns F.dd_3;
│ │ │ │ - -- used 0.140644s (cpu); 0.139198s (thread); 0s (gc)
│ │ │ │ + -- used 0.308002s (cpu); 0.310916s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of S
│ │ │ │  i24 : betti res j
│ │ │ │  
│ │ │ │               0 1 2 3 4
│ │ │ │  o24 = total: 1 3 6 5 1
│ │ │ │            0: 1 . . . .
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c3ViY29tcGxleChDZWxsQ29tcGxleCxMaXN0KQ==
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTUwNCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ViY29tcGxleCxDZWxsQ29tcGxleCxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.out
│ │ │ @@ -12,27 +12,27 @@
│ │ │  
│ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │  
│ │ │  i7 : C = cellComplex(R,F);
│ │ │  
│ │ │  i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 |
│ │ │ -                      | 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 |
│ │ │ -                      | 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 |
│ │ │ -                      | 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 |
│ │ │ +                      | 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 |
│ │ │ +                      | 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 |
│ │ │ +                      | 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Polyhedron_rp.out
│ │ │ @@ -32,12 +32,12 @@
│ │ │  
│ │ │  i8 : H = hashTable {v_0 => x*y, v_1 => y*z, v_2 => x*w, v_3 => y*z};
│ │ │  
│ │ │  i9 : labeledC = cellComplex(S, P, Labels => H);
│ │ │  
│ │ │  i10 : for i to dim labeledC list cells(i,labeledC)/cellLabel
│ │ │  
│ │ │ -o10 = {{x*w, x*y, y*z, y*z}, {y*z, x*y*z, x*y*z*w, x*y*w}, {x*y*z*w}}
│ │ │ +o10 = {{x*w, x*y, y*z, y*z}, {x*y*w, y*z, x*y*z, x*y*z*w}, {x*y*z*w}}
│ │ │  
│ │ │  o10 : List
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.out
│ │ │ @@ -22,17 +22,17 @@
│ │ │  
│ │ │  o7 = C
│ │ │  
│ │ │  o7 : CellComplex
│ │ │  
│ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │ -                      3 2   2 3     4   5   5   4
│ │ │ -o8 = HashTable{0 => {x y , x y , x*y , x , x , x y}                                       }
│ │ │ -                      5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3
│ │ │ -               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y }
│ │ │ +                      5   4    3 2   2 3     4   5
│ │ │ +o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ +                      5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5
│ │ │ +               1 => {x y , x y , x y , x y , x y , x y, x y , x y , x y , x y , x y , x y}
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_face__Poset_lp__Cell__Complex_rp.out
│ │ │ @@ -20,18 +20,18 @@
│ │ │  
│ │ │  i10 : f = newCell({e12,e23,e34,e41});
│ │ │  
│ │ │  i11 : C = cellComplex(R,{f});
│ │ │  
│ │ │  i12 : facePoset C
│ │ │  
│ │ │ -o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ -                       | 0 1 0 0 1 0 1 0 1 |
│ │ │ -                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ -                       | 0 0 0 1 0 1 0 1 1 |
│ │ │ +o12 = Relation Matrix: | 1 0 0 0 1 0 1 0 1 |
│ │ │ +                       | 0 1 0 0 0 0 1 1 1 |
│ │ │ +                       | 0 0 1 0 0 1 0 1 1 |
│ │ │ +                       | 0 0 0 1 1 1 0 0 1 |
│ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o12 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.html
│ │ │ @@ -101,27 +101,27 @@
│ │ │  
    		          
                  
    i7 : C = cellComplex(R,F);
    
│ │ │      		          
                  
    i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 |
│ │ │ -                      | 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 |
│ │ │ -                      | 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 |
│ │ │ -                      | 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 |
│ │ │ +                      | 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 |
│ │ │ +                      | 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 |
│ │ │ +                      | 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
    
│ │ │      		          
              
                  
    i9 : labeledC = cellComplex(S, P, Labels => H);
    
│ │ │      		          
                  
    i10 : for i to dim labeledC list cells(i,labeledC)/cellLabel
│ │ │  
│ │ │ -o10 = {{x*w, x*y, y*z, y*z}, {y*z, x*y*z, x*y*z*w, x*y*w}, {x*y*z*w}}
│ │ │ +o10 = {{x*w, x*y, y*z, y*z}, {x*y*w, y*z, x*y*z, x*y*z*w}, {x*y*z*w}}
│ │ │  
│ │ │  o10 : List
    
│ │ │      		          
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o7 : Matrix QQ  <-- QQ
│ │ │ │  i8 : H = hashTable {v_0 => x*y, v_1 => y*z, v_2 => x*w, v_3 => y*z};
│ │ │ │  i9 : labeledC = cellComplex(S, P, Labels => H);
│ │ │ │  i10 : for i to dim labeledC list cells(i,labeledC)/cellLabel
│ │ │ │  
│ │ │ │ -o10 = {{x*w, x*y, y*z, y*z}, {y*z, x*y*z, x*y*z*w, x*y*w}, {x*y*z*w}}
│ │ │ │ +o10 = {{x*w, x*y, y*z, y*z}, {x*y*w, y*z, x*y*z, x*y*z*w}, {x*y*z*w}}
│ │ │ │  
│ │ │ │  o10 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x_(_R_i_n_g_,_P_o_l_y_h_e_d_r_a_l_C_o_m_p_l_e_x_) -- creates cell complex from given
│ │ │ │        polyhedral complex
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x_(_R_i_n_g_,_P_o_l_y_h_e_d_r_o_n_) -- creates cell complex from given
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -116,18 +116,18 @@
│ │ │  o7 = C
│ │ │  
│ │ │  o7 : CellComplex
    │ │ │ │ │ │ │ │ │ 
    i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │ -                      3 2   2 3     4   5   5   4
│ │ │ -o8 = HashTable{0 => {x y , x y , x*y , x , x , x y}                                       }
│ │ │ -                      5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3
│ │ │ -               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y }
│ │ │ +                      5   4    3 2   2 3     4   5
│ │ │ +o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ +                      5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5
│ │ │ +               1 => {x y , x y , x y , x y , x y , x y, x y , x y , x y , x y , x y , x y}
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -39,21 +39,21 @@ │ │ │ │ i7 : C = cellComplex(S,Delta,Labels=>H) │ │ │ │ │ │ │ │ o7 = C │ │ │ │ │ │ │ │ o7 : CellComplex │ │ │ │ i8 : applyValues(cells C, l -> apply(l,cellLabel)) │ │ │ │ │ │ │ │ - 3 2 2 3 4 5 5 4 │ │ │ │ -o8 = HashTable{0 => {x y , x y , x*y , x , x , x y} │ │ │ │ + 5 4 3 2 2 3 4 5 │ │ │ │ +o8 = HashTable{0 => {x , x y, x y , x y , x*y , x } │ │ │ │ } │ │ │ │ - 5 4 5 5 2 5 3 5 4 4 2 4 4 5 3 3 5 2 │ │ │ │ -2 4 5 3 │ │ │ │ - 1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , │ │ │ │ -x y , x y } │ │ │ │ + 5 2 5 3 5 4 4 2 4 4 5 3 3 5 2 2 4 5 3 │ │ │ │ +5 4 5 │ │ │ │ + 1 => {x y , x y , x y , x y , x y , x y, x y , x y , x y , x y , │ │ │ │ +x y , x y} │ │ │ │ 5 2 5 4 5 3 5 4 5 2 5 4 5 3 5 4 │ │ │ │ 2 => {x y , x y , x y , x y , x y , x y , x y , x y } │ │ │ │ │ │ │ │ o8 : HashTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_e_l_l_C_o_m_p_l_e_x -- create a cell complex │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_face__Poset_lp__Cell__Complex_rp.html │ │ │ @@ -105,18 +105,18 @@ │ │ │ │ │ │ │ │ │ 
    i11 : C = cellComplex(R,{f});
    │ │ │ │ │ │ │ │ │ 
    i12 : facePoset C
│ │ │  
│ │ │ -o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ -                       | 0 1 0 0 1 0 1 0 1 |
│ │ │ -                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ -                       | 0 0 0 1 0 1 0 1 1 |
│ │ │ +o12 = Relation Matrix: | 1 0 0 0 1 0 1 0 1 |
│ │ │ +                       | 0 1 0 0 0 0 1 1 1 |
│ │ │ +                       | 0 0 1 0 0 1 0 1 1 |
│ │ │ +                       | 0 0 0 1 1 1 0 0 1 |
│ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o12 : Poset
    │ │ │ ├── html2text {} │ │ │ │ @@ -26,18 +26,18 @@ │ │ │ │ i7 : e23 = newCell({v2,v3}); │ │ │ │ i8 : e34 = newCell({v3,v4}); │ │ │ │ i9 : e41 = newCell({v4,v1}); │ │ │ │ i10 : f = newCell({e12,e23,e34,e41}); │ │ │ │ i11 : C = cellComplex(R,{f}); │ │ │ │ i12 : facePoset C │ │ │ │ │ │ │ │ -o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 | │ │ │ │ - | 0 1 0 0 1 0 1 0 1 | │ │ │ │ - | 0 0 1 0 1 0 0 1 1 | │ │ │ │ - | 0 0 0 1 0 1 0 1 1 | │ │ │ │ +o12 = Relation Matrix: | 1 0 0 0 1 0 1 0 1 | │ │ │ │ + | 0 1 0 0 0 0 1 1 1 | │ │ │ │ + | 0 0 1 0 0 1 0 1 1 | │ │ │ │ + | 0 0 0 1 1 1 0 0 1 | │ │ │ │ | 0 0 0 0 1 0 0 0 1 | │ │ │ │ | 0 0 0 0 0 1 0 0 1 | │ │ │ │ | 0 0 0 0 0 0 1 0 1 | │ │ │ │ | 0 0 0 0 0 0 0 1 1 | │ │ │ │ | 0 0 0 0 0 0 0 0 1 | │ │ │ │ │ │ │ │ o12 : Poset │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ SG9tKENoYWluQ29tcGxleCxDaGFpbkNvbXBsZXgp │ │ │ #:len=1256 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlIHRoZSBob21vbW9ycGhpc20g │ │ │ Y29tcGxleCBvZiBhIHBhaXIgb2YgY2hhaW4gY29tcGxleGVzLiIsICJsaW5lbnVtIiA9PiA4MDcs │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize.out │ │ │ @@ -63,15 +63,15 @@ │ │ │ o11 : ChainComplex │ │ │ │ │ │ i12 : isMinimalChainComplex E │ │ │ │ │ │ o12 = false │ │ │ │ │ │ i13 : time m = minimize (E[1]); │ │ │ - -- used 0.281351s (cpu); 0.204197s (thread); 0s (gc) │ │ │ + -- used 0.330028s (cpu); 0.275004s (thread); 0s (gc) │ │ │ │ │ │ i14 : isQuasiIsomorphism m │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : E[1] == source m │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ @@ -27,18 +27,18 @@ │ │ │ i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5); │ │ │ │ │ │ i6 : mods = for i from 0 to max C list pushForward(f, C_i); │ │ │ │ │ │ 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)); │ │ │ │ │ │ i8 : time m = resolutionOfChainComplex C; │ │ │ - -- used 0.096258s (cpu); 0.0958462s (thread); 0s (gc) │ │ │ + -- used 0.124058s (cpu); 0.123077s (thread); 0s (gc) │ │ │ │ │ │ i9 : time n = cartanEilenbergResolution C; │ │ │ - -- used 0.202116s (cpu); 0.149978s (thread); 0s (gc) │ │ │ + -- used 0.304619s (cpu); 0.254382s (thread); 0s (gc) │ │ │ │ │ │ i10 : betti source m │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ o10 = total: 1 19 80 181 312 484 447 156 │ │ │ 0: 1 3 3 1 . . . . │ │ │ 1: . . 1 3 3 . . . │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize.html │ │ │ @@ -155,15 +155,15 @@ │ │ │ │ │ │
    │ │ │

    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ o11 : ChainComplex │ │ │ │ i12 : isMinimalChainComplex E │ │ │ │ │ │ │ │ o12 = false │ │ │ │ Now we minimize the result. The free summand we added to the end maps to zero, │ │ │ │ and thus is part of the minimization. │ │ │ │ i13 : time m = minimize (E[1]); │ │ │ │ - -- used 0.281351s (cpu); 0.204197s (thread); 0s (gc) │ │ │ │ + -- used 0.330028s (cpu); 0.275004s (thread); 0s (gc) │ │ │ │ i14 : isQuasiIsomorphism m │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : E[1] == source m │ │ │ │ │ │ │ │ o15 = true │ │ │ │ i16 : E' = target m │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ @@ -116,19 +116,19 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -274,15 +274,15 @@ │ │ │ o21 = 9h h + 9h h + 9h h + 3h + 7h h + 3h + 3h + 2h │ │ │ 1 2 1 2 1 2 1 1 2 2 1 2 │ │ │ │ │ │ o21 : A │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -161,15 +161,15 @@ │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o14 = ideal (x x - x x x , x x ) │ │ │ │ 0 3 1 2 4 2 5 │ │ │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ i15 : time csmK=CSM(A,K) │ │ │ │ - -- used 1.51962s (cpu); 0.529515s (thread); 0s (gc) │ │ │ │ + -- used 4.16843s (cpu); 0.857075s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o15 = 7h h + 5h h + 4h h + h + 3h h + h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o15 : A │ │ │ │ i16 : csmKHash= CSM(A,K,Output=>HashForm) │ │ │ │ @@ -200,15 +200,15 @@ │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o21 = 9h h + 9h h + 9h h + 3h + 7h h + 3h + 3h + 2h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 1 2 │ │ │ │ │ │ │ │ o21 : A │ │ │ │ i22 : time CSM(A,K,m) │ │ │ │ - -- used 0.268601s (cpu); 0.100341s (thread); 0s (gc) │ │ │ │ + -- used 0.306531s (cpu); 0.123968s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o22 = 7h h + 5h h + 4h h + h + 3h h + h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o22 : A │ │ │ │ In the case where the ambient space is a toric variety which is not a product │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ @@ -60,29 +60,29 @@ │ │ │ │ │ │ o2 = U │ │ │ │ │ │ o2 : NormalToricVariety │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -182,21 +182,21 @@ │ │ │ 
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.281351s (cpu); 0.204197s (thread); 0s (gc)
    │ │ │ + -- used 0.330028s (cpu); 0.275004s (thread); 0s (gc) │ │ │ 
    i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
    │ │ │ 
    i6 : mods = for i from 0 to max C list pushForward(f, C_i);
    │ │ │ 
    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));
    │ │ │ 
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.096258s (cpu); 0.0958462s (thread); 0s (gc)
    │ │ │ + -- used 0.124058s (cpu); 0.123077s (thread); 0s (gc) │ │ │ 
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.202116s (cpu); 0.149978s (thread); 0s (gc)
    │ │ │ + -- used 0.304619s (cpu); 0.254382s (thread); 0s (gc) │ │ │ 
    i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,17 +50,17 @@
│ │ │ │  
│ │ │ │  o4 : RingMap R <-- S
│ │ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │ │  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));
│ │ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ │ - -- used 0.096258s (cpu); 0.0958462s (thread); 0s (gc)
│ │ │ │ + -- used 0.124058s (cpu); 0.123077s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.202116s (cpu); 0.149978s (thread); 0s (gc)
│ │ │ │ + -- used 0.304619s (cpu); 0.254382s (thread); 0s (gc)
│ │ │ │  i10 : betti source m
│ │ │ │  
│ │ │ │               0  1  2   3   4   5   6   7
│ │ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ │            1: .  .  1   3   3   .   .   .
│ │ │ │            2: .  1  3   3   2   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexOperations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  cmV2ZXJzZUZhY3RvcnMoQ2hhaW5Db21wbGV4LENoYWluQ29tcGxleCk=
│ │ │  #:len=335
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  TXVsdGlQcm9qQ29vcmRSaW5n
│ │ │  #:len=2206
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQSBxdWljayB3YXkgdG8gYnVpbGQgdGhl
│ │ │  IGNvb3JkaW5hdGUgcmluZyBvZiBhIHByb2R1Y3Qgb2YgcHJvamVjdGl2ZSBzcGFjZXMiLCAibGlu
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
│ │ │ @@ -83,15 +83,15 @@
│ │ │                2              2
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time csmK=CSM(A,K)
│ │ │ - -- used 1.51962s (cpu); 0.529515s (thread); 0s (gc)
│ │ │ + -- used 4.16843s (cpu); 0.857075s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
│ │ │  
│ │ │ @@ -124,15 +124,15 @@
│ │ │          2 2     2         2     2             2
│ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │  
│ │ │  o21 : A
│ │ │  
│ │ │  i22 : time CSM(A,K,m)
│ │ │ - -- used 0.268601s (cpu); 0.100341s (thread); 0s (gc)
│ │ │ + -- used 0.306531s (cpu); 0.123968s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out
│ │ │ @@ -9,28 +9,28 @@
│ │ │  i2 : U = toricProjectiveSpace 7
│ │ │  
│ │ │  o2 = U
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │  
│ │ │  i3 : time CSM U
│ │ │ - -- used 0.261703s (cpu); 0.119713s (thread); 0s (gc)
│ │ │ + -- used 0.556236s (cpu); 0.138143s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │  o3 : -----------------------------------------------------------------------------------------------
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
│ │ │  
│ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.544499s (cpu); 0.265961s (thread); 0s (gc)
│ │ │ + -- used 0.670811s (cpu); 0.408204s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │  
│ │ │  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));
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 1.27964s (cpu); 0.355065s (thread); 0s (gc)
│ │ │ + -- used 2.08303s (cpu); 0.674085s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o5 : ------
│ │ │          6
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.15421s (cpu); 1.73354s (thread); 0s (gc)
│ │ │ + -- used 3.72797s (cpu); 3.18582s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -53,29 +53,29 @@
│ │ │  i8 : S=QQ[s_0..s_3];
│ │ │  
│ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │  
│ │ │  o9 : Ideal of S
│ │ │  
│ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.560715s (cpu); 0.263507s (thread); 0s (gc)
│ │ │ + -- used 0.546113s (cpu); 0.360224s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.319881s (cpu); 0.163292s (thread); 0s (gc)
│ │ │ + -- used 0.466255s (cpu); 0.0945574s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out
│ │ │ @@ -21,20 +21,20 @@
│ │ │               2                                                        2
│ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.134571s (cpu); 0.0605471s (thread); 0s (gc)
│ │ │ + -- used 0.34766s (cpu); 0.151011s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.316118s (cpu); 0.167256s (thread); 0s (gc)
│ │ │ + -- used 0.43878s (cpu); 0.1816s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │  
│ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │  
│ │ │ @@ -62,20 +62,20 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x )
│ │ │          0 3
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.30978s (cpu); 0.0932417s (thread); 0s (gc)
│ │ │ + -- used 0.547485s (cpu); 0.13849s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.203774s (cpu); 0.0900468s (thread); 0s (gc)
│ │ │ + -- used 0.274604s (cpu); 0.115878s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 4.50668s (cpu); 1.29022s (thread); 0s (gc)
│ │ │ + -- used 17.1081s (cpu); 2.4191s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 4.15312s (cpu); 1.21255s (thread); 0s (gc)
│ │ │ + -- used 17.3167s (cpu); 2.23042s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out
│ │ │ @@ -3,43 +3,43 @@
│ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 1.38977s (cpu); 0.505458s (thread); 0s (gc)
│ │ │ + -- used 2.96638s (cpu); 0.791167s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.178227s (cpu); 0.0505374s (thread); 0s (gc)
│ │ │ + -- used 0.248913s (cpu); 0.0411849s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0448355s (cpu); 0.0317107s (thread); 0s (gc)
│ │ │ + -- used 0.406346s (cpu); 0.0316756s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 2.95931s (cpu); 0.979177s (thread); 0s (gc)
│ │ │ + -- used 9.82452s (cpu); 1.63685s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          7
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.989463s (cpu); 0.347815s (thread); 0s (gc)
│ │ │ + -- used 2.64769s (cpu); 0.446117s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -221,15 +221,15 @@
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
    │ │ │ 
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 1.51962s (cpu); 0.529515s (thread); 0s (gc)
│ │ │ + -- used 4.16843s (cpu); 0.857075s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
    │ │ │ 
    i22 : time CSM(A,K,m)
│ │ │ - -- used 0.268601s (cpu); 0.100341s (thread); 0s (gc)
│ │ │ + -- used 0.306531s (cpu); 0.123968s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
    │ │ │ 
    i3 : time CSM U
│ │ │ - -- used 0.261703s (cpu); 0.119713s (thread); 0s (gc)
│ │ │ + -- used 0.556236s (cpu); 0.138143s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │  o3 : -----------------------------------------------------------------------------------------------
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    │ │ │ 
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.544499s (cpu); 0.265961s (thread); 0s (gc)
│ │ │ + -- used 0.670811s (cpu); 0.408204s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,30 +16,30 @@
│ │ │ │  o1 : Package
│ │ │ │  i2 : U = toricProjectiveSpace 7
│ │ │ │  
│ │ │ │  o2 = U
│ │ │ │  
│ │ │ │  o2 : NormalToricVariety
│ │ │ │  i3 : time CSM U
│ │ │ │ - -- used 0.261703s (cpu); 0.119713s (thread); 0s (gc)
│ │ │ │ + -- used 0.556236s (cpu); 0.138143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ │ │  o3 : --------------------------------------------------------------------------
│ │ │ │  ---------------------
│ │ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x ,
│ │ │ │  - x  + x , - x  + x )
│ │ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5
│ │ │ │  0    6     0    7
│ │ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ │ - -- used 0.544499s (cpu); 0.265961s (thread); 0s (gc)
│ │ │ │ + -- used 0.670811s (cpu); 0.408204s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -76,30 +76,30 @@
│ │ │            
    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));
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │ 
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 1.27964s (cpu); 0.355065s (thread); 0s (gc)
│ │ │ + -- used 2.08303s (cpu); 0.674085s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o5 : ------
│ │ │          6
│ │ │         h
│ │ │          1
    │ │ │ 
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.15421s (cpu); 1.73354s (thread); 0s (gc)
│ │ │ + -- used 3.72797s (cpu); 3.18582s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -116,30 +116,30 @@
│ │ │            
    i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │  
│ │ │  o9 : Ideal of S
    │ │ │ 
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.560715s (cpu); 0.263507s (thread); 0s (gc)
│ │ │ + -- used 0.546113s (cpu); 0.360224s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
    │ │ │ 
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.319881s (cpu); 0.163292s (thread); 0s (gc)
│ │ │ + -- used 0.466255s (cpu); 0.0945574s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,28 +32,28 @@
│ │ │ │  using the regenerative cascade implemented in Bertini. This is done by choosing
│ │ │ │  the option bertini, provided Bertini is _i_n_s_t_a_l_l_e_d_ _a_n_d_ _c_o_n_f_i_g_u_r_e_d.
│ │ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │ │  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));
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 1.27964s (cpu); 0.355065s (thread); 0s (gc)
│ │ │ │ + -- used 2.08303s (cpu); 0.674085s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │ │         1      1      1      1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o5 : ------
│ │ │ │          6
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ │ - -- used 2.15421s (cpu); 1.73354s (thread); 0s (gc)
│ │ │ │ + -- used 3.72797s (cpu); 3.18582s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          6
│ │ │ │ @@ -62,28 +62,28 @@
│ │ │ │  
│ │ │ │  o7 = 2
│ │ │ │  i8 : S=QQ[s_0..s_3];
│ │ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │ │  
│ │ │ │  o9 : Ideal of S
│ │ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.560715s (cpu); 0.263507s (thread); 0s (gc)
│ │ │ │ + -- used 0.546113s (cpu); 0.360224s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.319881s (cpu); 0.163292s (thread); 0s (gc)
│ │ │ │ + -- used 0.466255s (cpu); 0.0945574s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -130,21 +130,21 @@
│ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ 
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.134571s (cpu); 0.0605471s (thread); 0s (gc)
│ │ │ + -- used 0.34766s (cpu); 0.151011s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    │ │ │ 
    i5 : time Euler I
│ │ │ - -- used 0.316118s (cpu); 0.167256s (thread); 0s (gc)
│ │ │ + -- used 0.43878s (cpu); 0.1816s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    │ │ │ 
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    │ │ │
    │ │ │
    │ │ │

    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.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.30978s (cpu); 0.0932417s (thread); 0s (gc)
│ │ │ + -- used 0.547485s (cpu); 0.13849s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    │ │ │ 
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.203774s (cpu); 0.0900468s (thread); 0s (gc)
│ │ │ + -- used 0.274604s (cpu); 0.115878s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
    │ │ │
    │ │ │
    │ │ │

    Now consider an example in \PP^2 \times \PP^2.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -75,19 +75,19 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 │ │ │ │ - 14254x - 11226x x + 2653x x + 12365x x - 10226x x - 12696x ) │ │ │ │ 3 0 4 1 4 2 4 3 4 4 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : time Euler(I,InputIsSmooth=>true) │ │ │ │ - -- used 0.134571s (cpu); 0.0605471s (thread); 0s (gc) │ │ │ │ + -- used 0.34766s (cpu); 0.151011s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : time Euler I │ │ │ │ - -- used 0.316118s (cpu); 0.167256s (thread); 0s (gc) │ │ │ │ + -- used 0.43878s (cpu); 0.1816s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 4 │ │ │ │ i6 : EulerIHash=Euler(I,Output=>HashForm); │ │ │ │ i7 : A=ring EulerIHash#"CSM" │ │ │ │ │ │ │ │ o7 = A │ │ │ │ │ │ │ │ @@ -115,19 +115,19 @@ │ │ │ │ o9 : Ideal of R │ │ │ │ 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. │ │ │ │ i10 : time Euler(J,Method=>DirectCompleteInt) │ │ │ │ - -- used 0.30978s (cpu); 0.0932417s (thread); 0s (gc) │ │ │ │ + -- used 0.547485s (cpu); 0.13849s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = 2 │ │ │ │ i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1}) │ │ │ │ - -- used 0.203774s (cpu); 0.0900468s (thread); 0s (gc) │ │ │ │ + -- used 0.274604s (cpu); 0.115878s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = 2 │ │ │ │ Now consider an example in \PP^2 \times \PP^2. │ │ │ │ i12 : R=MultiProjCoordRing({2,2}) │ │ │ │ │ │ │ │ o12 = R │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ @@ -58,30 +58,30 @@ │ │ │ │ │ │ 
    i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 4.50668s (cpu); 1.29022s (thread); 0s (gc)
│ │ │ + -- used 17.1081s (cpu); 2.4191s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 4.15312s (cpu); 1.21255s (thread); 0s (gc)
│ │ │ + -- used 17.3167s (cpu); 2.23042s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,28 +16,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ │ - -- used 4.50668s (cpu); 1.29022s (thread); 0s (gc)
│ │ │ │ + -- used 17.1081s (cpu); 2.4191s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ │ │  o3 : ----------
│ │ │ │          3   3
│ │ │ │        (h , h )
│ │ │ │          1   2
│ │ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ │ - -- used 4.15312s (cpu); 1.21255s (thread); 0s (gc)
│ │ │ │ + -- used 17.3167s (cpu); 2.23042s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -54,30 +54,30 @@
│ │ │            
│ │ │                
    i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.38977s (cpu); 0.505458s (thread); 0s (gc)
│ │ │ + -- used 2.96638s (cpu); 0.791167s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.178227s (cpu); 0.0505374s (thread); 0s (gc)
│ │ │ + -- used 0.248913s (cpu); 0.0411849s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -89,15 +89,15 @@
│ │ │          
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,49 +23,49 @@
│ │ │ │  o3 = EvaluationCode{cache => CacheTable{}
│ │ │ │  }
│ │ │ │                                                                 9
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │ -                                             Code => image | 1   a+1 |
│ │ │ │ +                                             Code => image | 1   0   |
│ │ │ │ +                                                           | 0   0   |
│ │ │ │ +                                                           | 0   0   |
│ │ │ │                                                             | a+1 0   |
│ │ │ │                                                             | a+1 0   |
│ │ │ │                                                             | a   a   |
│ │ │ │ -                                                           | 1   0   |
│ │ │ │ +                                                           | 1   a+1 |
│ │ │ │                                                             | a   a   |
│ │ │ │ -                                                           | 0   0   |
│ │ │ │ -                                                           | 0   0   |
│ │ │ │                                                             | 1   1   |
│ │ │ │ -                                             GeneratorMatrix => | 1   a+1 a+1 a
│ │ │ │ -1 a 0 0 1 |
│ │ │ │ -                                                                | a+1 0   0   a
│ │ │ │ -0 a 0 0 1 |
│ │ │ │ -                                             Generators => {{1, a + 1, a + 1,
│ │ │ │ -a, 1, a, 0, 0, 1}, {a + 1, 0, 0, a, 0, a, 0, 0, 1}}
│ │ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 a
│ │ │ │ -0 0 0 a+1 |
│ │ │ │ -                                                                  | 0 1 0 0 a+1
│ │ │ │ -0 0 0 0   |
│ │ │ │ -                                                                  | 0 0 1 0 a+1
│ │ │ │ -0 0 0 0   |
│ │ │ │ -                                                                  | 0 0 0 1 0
│ │ │ │ -0 0 0 a   |
│ │ │ │ -                                                                  | 0 0 0 0 0
│ │ │ │ -1 0 0 a   |
│ │ │ │ -                                                                  | 0 0 0 0 0
│ │ │ │ -0 1 0 0   |
│ │ │ │ -                                                                  | 0 0 0 0 0
│ │ │ │ -0 0 1 0   |
│ │ │ │ +                                             GeneratorMatrix => | 1 0 0 a+1 a+1
│ │ │ │ +a 1   a 1 |
│ │ │ │ +                                                                | 0 0 0 0   0
│ │ │ │ +a a+1 a 1 |
│ │ │ │ +                                             Generators => {{1, 0, 0, a + 1, a
│ │ │ │ ++ 1, a, 1, a, 1}, {0, 0, 0, 0, 0, a, a + 1, a, 1}}
│ │ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0
│ │ │ │ +a+1 0 a |
│ │ │ │ +                                                                  | 0 1 0 0 0 0
│ │ │ │ +0   0 0 |
│ │ │ │ +                                                                  | 0 0 1 0 0 0
│ │ │ │ +0   0 0 |
│ │ │ │ +                                                                  | 0 0 0 1 0 0
│ │ │ │ +a   0 1 |
│ │ │ │ +                                                                  | 0 0 0 0 1 0
│ │ │ │ +a   0 1 |
│ │ │ │ +                                                                  | 0 0 0 0 0 1
│ │ │ │ +0   0 a |
│ │ │ │ +                                                                  | 0 0 0 0 0 0
│ │ │ │ +0   1 a |
│ │ │ │                                               ParityCheckRows => {{1, 0, 0, 0,
│ │ │ │ -a, 0, 0, 0, a + 1}, {0, 1, 0, 0, a + 1, 0, 0, 0, 0}, {0, 0, 1, 0, a + 1, 0, 0,
│ │ │ │ -0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 0, 1, 0, 0, a}, {0, 0, 0, 0,
│ │ │ │ -0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ │ -                    Points => {{a, a}, {0, a}, {a, 0}, {a, 1}, {0, 0}, {1, a},
│ │ │ │ -{1, 0}, {0, 1}, {1, 1}}
│ │ │ │ +0, 0, a + 1, 0, a}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0},
│ │ │ │ +{0, 0, 0, 1, 0, 0, a, 0, 1}, {0, 0, 0, 0, 1, 0, a, 0, 1}, {0, 0, 0, 0, 0, 1, 0,
│ │ │ │ +0, a}, {0, 0, 0, 0, 0, 0, 0, 1, a}}
│ │ │ │ +                    Points => {{0, 0}, {0, 1}, {1, 0}, {0, a}, {a, 0}, {a, 1},
│ │ │ │ +{a, a}, {1, a}, {1, 1}}
│ │ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │ │                                                3           2         3
│ │ │ │  2
│ │ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a +
│ │ │ │  1)y  + a*y)
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_cartesian__Code.html
│ │ │ @@ -191,74 +191,74 @@
│ │ │  
│ │ │              
│ │ │  
│ │ │              
│ │ │  
│ │ │              
│ │ │  
│ │ │            
                  
    i5 : time Chern I
│ │ │ - -- used 0.0448355s (cpu); 0.0317107s (thread); 0s (gc)
│ │ │ + -- used 0.406346s (cpu); 0.0316756s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,42 +9,42 @@
│ │ │ │  input ideal is known to define a smooth subscheme setting this option to true
│ │ │ │  will speed up computations (it is set to false by default).
│ │ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 1.38977s (cpu); 0.505458s (thread); 0s (gc)
│ │ │ │ + -- used 2.96638s (cpu); 0.791167s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.178227s (cpu); 0.0505374s (thread); 0s (gc)
│ │ │ │ + -- used 0.248913s (cpu); 0.0411849s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o4 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o4 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  Note that one could, equivalently, use the command _C_h_e_r_n instead in this case.
│ │ │ │  i5 : time Chern I
│ │ │ │ - -- used 0.0448355s (cpu); 0.0317107s (thread); 0s (gc)
│ │ │ │ + -- used 0.406346s (cpu); 0.0316756s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -58,30 +58,30 @@
│ │ │            
                  
    i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │      		          
                  
    i3 : time CSM I
│ │ │ - -- used 2.95931s (cpu); 0.979177s (thread); 0s (gc)
│ │ │ + -- used 9.82452s (cpu); 1.63685s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          7
│ │ │         h
│ │ │          1
    
│ │ │      		          
                  
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.989463s (cpu); 0.347815s (thread); 0s (gc)
│ │ │ + -- used 2.64769s (cpu); 0.446117s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,28 +18,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 2.95931s (cpu); 0.979177s (thread); 0s (gc)
│ │ │ │ + -- used 9.82452s (cpu); 1.63685s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o3 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          7
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.989463s (cpu); 0.347815s (thread); 0s (gc)
│ │ │ │ + -- used 2.64769s (cpu); 0.446117s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  UmluZ01hcCBDaG9yZGFsTmV0
│ │ │  #:len=1424
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgcmluZyBtYXAgdG8gYSBjaG9y
│ │ │  ZGFsIG5ldHdvcmsiLCAibGluZW51bSIgPT4gODg5LCBJbnB1dHMgPT4ge1NQQU57VFR7ImYifSwi
│ │ ├── ./usr/share/doc/Macaulay2/Classic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bW9ub21pYWxJZGVhbChTdHJpbmcp
│ │ │  #:len=1149
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibWFrZSBhIG1vbm9taWFsIGlkZWFsIHVz
│ │ │  aW5nIGNsYXNzaWMgTWFjYXVsYXkgc3ludGF4IiwgImxpbmVudW0iID0+IDE0NCwgSW5wdXRzID0+
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  VmFuaXNoaW5nSWRlYWw=
│ │ │  #:len=1237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmFuaXNoaW5nIGlkZWFsIG9mIGFuIGV2
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│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/___Sets.out
│ │ │ @@ -2,40 +2,40 @@
│ │ │  
│ │ │  i1 : F=GF(4);
│ │ │  
│ │ │  i2 : R=F[x,y];
│ │ │  
│ │ │  i3 : C=cartesianCode(F,{{0,1,a},{0,1,a}},{1+x+y,x*y})
│ │ │  
│ │ │ -o3 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                               }
│ │ │ +o3 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                        }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1   a+1 |
│ │ │ +                                             Code => image | 1   0   |
│ │ │ +                                                           | 0   0   |
│ │ │ +                                                           | 0   0   |
│ │ │                                                             | a+1 0   |
│ │ │                                                             | a+1 0   |
│ │ │                                                             | a   a   |
│ │ │ -                                                           | 1   0   |
│ │ │ +                                                           | 1   a+1 |
│ │ │                                                             | a   a   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                             GeneratorMatrix => | 1   a+1 a+1 a 1 a 0 0 1 |
│ │ │ -                                                                | a+1 0   0   a 0 a 0 0 1 |
│ │ │ -                                             Generators => {{1, a + 1, a + 1, a, 1, a, 0, 0, 1}, {a + 1, 0, 0, a, 0, a, 0, 0, 1}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 a   0 0 0 a+1 |
│ │ │ -                                                                  | 0 1 0 0 a+1 0 0 0 0   |
│ │ │ -                                                                  | 0 0 1 0 a+1 0 0 0 0   |
│ │ │ -                                                                  | 0 0 0 1 0   0 0 0 a   |
│ │ │ -                                                                  | 0 0 0 0 0   1 0 0 a   |
│ │ │ -                                                                  | 0 0 0 0 0   0 1 0 0   |
│ │ │ -                                                                  | 0 0 0 0 0   0 0 1 0   |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, a, 0, 0, 0, a + 1}, {0, 1, 0, 0, a + 1, 0, 0, 0, 0}, {0, 0, 1, 0, a + 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 0, 1, 0, 0, a}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ -                    Points => {{a, a}, {0, a}, {a, 0}, {a, 1}, {0, 0}, {1, a}, {1, 0}, {0, 1}, {1, 1}}
│ │ │ +                                             GeneratorMatrix => | 1 0 0 a+1 a+1 a 1   a 1 |
│ │ │ +                                                                | 0 0 0 0   0   a a+1 a 1 |
│ │ │ +                                             Generators => {{1, 0, 0, a + 1, a + 1, a, 1, a, 1}, {0, 0, 0, 0, 0, a, a + 1, a, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 a+1 0 a |
│ │ │ +                                                                  | 0 1 0 0 0 0 0   0 0 |
│ │ │ +                                                                  | 0 0 1 0 0 0 0   0 0 |
│ │ │ +                                                                  | 0 0 0 1 0 0 a   0 1 |
│ │ │ +                                                                  | 0 0 0 0 1 0 a   0 1 |
│ │ │ +                                                                  | 0 0 0 0 0 1 0   0 a |
│ │ │ +                                                                  | 0 0 0 0 0 0 0   1 a |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, a + 1, 0, a}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, a, 0, 1}, {0, 0, 0, 0, 1, 0, a, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, a}, {0, 0, 0, 0, 0, 0, 0, 1, a}}
│ │ │ +                    Points => {{0, 0}, {0, 1}, {1, 0}, {0, a}, {a, 0}, {a, 1}, {a, a}, {1, a}, {1, 1}}
│ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2         3           2
│ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a + 1)y  + a*y)
│ │ │  
│ │ │  o3 : EvaluationCode
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cartesian__Code.out
│ │ │ @@ -45,108 +45,108 @@
│ │ │  
│ │ │  i2 : F=GF(4);
│ │ │  
│ │ │  i3 : R=F[x,y];
│ │ │  
│ │ │  i4 : C=cartesianCode(F,{{0,1,a},{0,1,a}},{1+x+y,x*y})
│ │ │  
│ │ │ -o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                           }
│ │ │ +o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1   a+1 |
│ │ │ +                                             Code => image | a+1 0   |
│ │ │ +                                                           | a+1 0   |
│ │ │ +                                                           | a   a   |
│ │ │ +                                                           | a   a   |
│ │ │                                                             | 1   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │ +                                                           | 1   a+1 |
│ │ │                                                             | 1   1   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                                           | a   a   |
│ │ │ -                                                           | a   a   |
│ │ │ -                                             GeneratorMatrix => | 1   1 0 0 1 a+1 a+1 a a |
│ │ │ -                                                                | a+1 0 0 0 1 0   0   a a |
│ │ │ -                                             Generators => {{1, 1, 0, 0, 1, a + 1, a + 1, a, a}, {a + 1, 0, 0, 0, 1, 0, 0, a, a}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0 a+1 0 a   0 |
│ │ │ -                                                                  | 0 1 0 0 0 a   0 0   0 |
│ │ │ -                                                                  | 0 0 1 0 0 0   0 0   0 |
│ │ │ -                                                                  | 0 0 0 1 0 0   0 0   0 |
│ │ │ -                                                                  | 0 0 0 0 1 0   0 a+1 0 |
│ │ │ -                                                                  | 0 0 0 0 0 1   1 0   0 |
│ │ │ -                                                                  | 0 0 0 0 0 0   0 1   1 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, a, 0}, {0, 1, 0, 0, 0, a, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, a + 1, 0}, {0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ -                    Points => {{a, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, 0}, {0, a}, {a, 1}, {1, a}}
│ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1 0 0 1   1 |
│ │ │ +                                                                | 0   0   a a 0 0 0 a+1 1 |
│ │ │ +                                             Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, a + 1, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 a   1 |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 a   1 |
│ │ │ +                                                                  | 0 0 1 0 0 0 0 0   a |
│ │ │ +                                                                  | 0 0 0 1 0 0 0 0   a |
│ │ │ +                                                                  | 0 0 0 0 1 0 0 a+1 a |
│ │ │ +                                                                  | 0 0 0 0 0 1 0 0   0 |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0   0 |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, a, 1}, {0, 1, 0, 0, 0, 0, 0, a, 1}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, 0, 0, a + 1, a}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{a, 0}, {0, a}, {1, a}, {a, 1}, {0, 0}, {0, 1}, {1, 0}, {a, a}, {1, 1}}
│ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2         3           2
│ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a + 1)y  + a*y)
│ │ │  
│ │ │  o4 : EvaluationCode
│ │ │  
│ │ │  i5 : C.LinearCode
│ │ │  
│ │ │                                    9
│ │ │ -o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                  BaseField => F
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | 1   a+1 |
│ │ │ +                Code => image | a+1 0   |
│ │ │ +                              | a+1 0   |
│ │ │ +                              | a   a   |
│ │ │ +                              | a   a   |
│ │ │                                | 1   0   |
│ │ │                                | 0   0   |
│ │ │                                | 0   0   |
│ │ │ +                              | 1   a+1 |
│ │ │                                | 1   1   |
│ │ │ -                              | a+1 0   |
│ │ │ -                              | a+1 0   |
│ │ │ -                              | a   a   |
│ │ │ -                              | a   a   |
│ │ │ -                GeneratorMatrix => | 1   1 0 0 1 a+1 a+1 a a |
│ │ │ -                                   | a+1 0 0 0 1 0   0   a a |
│ │ │ -                Generators => {{1, 1, 0, 0, 1, a + 1, a + 1, a, a}, {a + 1, 0, 0, 0, 1, 0, 0, a, a}}
│ │ │ -                ParityCheckMatrix => | 1 0 0 0 0 a+1 0 a   0 |
│ │ │ -                                     | 0 1 0 0 0 a   0 0   0 |
│ │ │ -                                     | 0 0 1 0 0 0   0 0   0 |
│ │ │ -                                     | 0 0 0 1 0 0   0 0   0 |
│ │ │ -                                     | 0 0 0 0 1 0   0 a+1 0 |
│ │ │ -                                     | 0 0 0 0 0 1   1 0   0 |
│ │ │ -                                     | 0 0 0 0 0 0   0 1   1 |
│ │ │ -                ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, a, 0}, {0, 1, 0, 0, 0, a, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, a + 1, 0}, {0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ +                GeneratorMatrix => | a+1 a+1 a a 1 0 0 1   1 |
│ │ │ +                                   | 0   0   a a 0 0 0 a+1 1 |
│ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, a + 1, 1}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 0 0 a   1 |
│ │ │ +                                     | 0 1 0 0 0 0 0 a   1 |
│ │ │ +                                     | 0 0 1 0 0 0 0 0   a |
│ │ │ +                                     | 0 0 0 1 0 0 0 0   a |
│ │ │ +                                     | 0 0 0 0 1 0 0 a+1 a |
│ │ │ +                                     | 0 0 0 0 0 1 0 0   0 |
│ │ │ +                                     | 0 0 0 0 0 0 1 0   0 |
│ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, a, 1}, {0, 1, 0, 0, 0, 0, 0, a, 1}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, 0, 0, a + 1, a}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │  
│ │ │  o5 : LinearCode
│ │ │  
│ │ │  i6 : F=GF(4);
│ │ │  
│ │ │  i7 : R=F[x,y];
│ │ │  
│ │ │  i8 : C=cartesianCode(F,{{0,1,a},{0,1,a}},matrix{{1,2},{2,3}})
│ │ │  
│ │ │  o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | a+1 1   |
│ │ │ +                                             Code => image | a+1 1   |
│ │ │                                                             | a   a+1 |
│ │ │                                                             | 1   a+1 |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                             GeneratorMatrix => | 0 0 a+1 a   1   0 0 0 1 |
│ │ │ -                                                                | 0 0 1   a+1 a+1 0 0 0 1 |
│ │ │ -                                             Generators => {{0, 0, a + 1, a, 1, 0, 0, 0, 1}, {0, 0, 1, a + 1, a + 1, 0, 0, 0, 1}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0   0 0 0 0 |
│ │ │ -                                                                  | 0 1 0 0 0   0 0 0 0 |
│ │ │ -                                                                  | 0 0 1 0 1   0 0 0 a |
│ │ │ -                                                                  | 0 0 0 1 a+1 0 0 0 1 |
│ │ │ -                                                                  | 0 0 0 0 0   1 0 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0   0 1 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0   0 0 1 0 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 0, a}, {0, 0, 0, 1, a + 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ -                    Points => {{0, a}, {a, 0}, {1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0}, {1, 1}}
│ │ │ +                                                           | 0   0   |
│ │ │ +                                                           | 0   0   |
│ │ │ +                                             GeneratorMatrix => | a+1 a   1   0 0 0 1 0 0 |
│ │ │ +                                                                | 1   a+1 a+1 0 0 0 1 0 0 |
│ │ │ +                                             Generators => {{a + 1, a, 1, 0, 0, 0, 1, 0, 0}, {1, a + 1, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 1   0 0 0 a 0 0 |
│ │ │ +                                                                  | 0 1 a+1 0 0 0 1 0 0 |
│ │ │ +                                                                  | 0 0 0   1 0 0 0 0 0 |
│ │ │ +                                                                  | 0 0 0   0 1 0 0 0 0 |
│ │ │ +                                                                  | 0 0 0   0 0 1 0 0 0 |
│ │ │ +                                                                  | 0 0 0   0 0 0 0 1 0 |
│ │ │ +                                                                  | 0 0 0   0 0 0 0 0 1 |
│ │ │ +                                             ParityCheckRows => {{1, 0, 1, 0, 0, 0, a, 0, 0}, {0, 1, a + 1, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}}
│ │ │ +                    Points => {{1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0}, {1, 1}, {0, a}, {a, 0}}
│ │ │                                           2   2 3
│ │ │                      PolynomialSet => {t t , t t }
│ │ │                                         0 1   0 1
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2          3           2
│ │ │                      VanishingIdeal => ideal (t  + (a + 1)t  + a*t , t  + (a + 1)t  + a*t )
│ │ │                                                0           0      0   1           1      1
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_codewords.out
│ │ │ @@ -2,18 +2,18 @@
│ │ │  
│ │ │  i1 : F=GF(4,Variable=>a);
│ │ │  
│ │ │  i2 : C=linearCode(matrix{{1,a,0},{0,1,a}});
│ │ │  
│ │ │  i3 : codewords(C)
│ │ │  
│ │ │ -o3 = {{a, a, a}, {a + 1, a, 1}, {1, 1, 1}, {0, 1, a}, {0, a, a + 1}, {1, a +
│ │ │ +o3 = {{a, a, a}, {a + 1, a, 1}, {1, 1, 1}, {0, 1, a}, {a, 1, a + 1}, {0, a, a
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, a}, {0, a + 1, 1}, {a, 1, a + 1}, {1, a, 0}, {a + 1, a + 1, a + 1},
│ │ │ +     + 1}, {0, a + 1, 1}, {1, a + 1, a}, {a + 1, 0, a}, {a, 0, 1}, {a + 1, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {a, 0, 1}, {a + 1, 0, a}, {a + 1, 1, 0}, {1, 0, a + 1}, {a, a + 1, 0},
│ │ │ +     0}, {1, a, 0}, {a + 1, a + 1, a + 1}, {1, 0, a + 1}, {a, a + 1, 0}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 0, 0}}
│ │ │ +     0, 0}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_dim_lp__Linear__Code_rp.out
│ │ │ @@ -8,30 +8,30 @@
│ │ │  
│ │ │  i3 : H = hammingCode(2,3)
│ │ │  
│ │ │                                         7
│ │ │  o3 = LinearCode{AmbientModule => (GF 2)                                                                                   }
│ │ │                  BaseField => GF 2
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | 1 1 0 1 |
│ │ │ +                Code => image | 1 1 1 0 |
│ │ │                                | 1 0 1 1 |
│ │ │ -                              | 1 1 1 0 |
│ │ │ +                              | 1 1 0 1 |
│ │ │                                | 1 0 0 0 |
│ │ │                                | 0 1 0 0 |
│ │ │                                | 0 0 1 0 |
│ │ │                                | 0 0 0 1 |
│ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │                                     | 1 0 1 0 1 0 0 |
│ │ │ -                                   | 0 1 1 0 0 1 0 |
│ │ │ -                                   | 1 1 0 0 0 0 1 |
│ │ │ -                Generators => {{1, 1, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0}, {0, 1, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ +                                   | 1 1 0 0 0 1 0 |
│ │ │ +                                   | 0 1 1 0 0 0 1 |
│ │ │ +                Generators => {{1, 1, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0}, {1, 1, 0, 0, 0, 1, 0}, {0, 1, 1, 0, 0, 0, 1}}
│ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │ -                                     | 0 0 1 1 1 1 0 |
│ │ │ +                                     | 0 1 1 0 1 1 0 |
│ │ │                                       | 0 1 0 1 0 1 1 |
│ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0}, {0, 1, 0, 1, 0, 1, 1}}
│ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 1, 0}, {0, 1, 0, 1, 0, 1, 1}}
│ │ │  
│ │ │  o3 : LinearCode
│ │ │  
│ │ │  i4 : dim H
│ │ │  
│ │ │  o4 = 4
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_hamming__Code.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1730891173706535352
│ │ │  
│ │ │  i1 : C1 = hammingCode(2,3);
│ │ │  
│ │ │  i2 : C1.ParityCheckMatrix
│ │ │  
│ │ │  o2 = | 1 1 1 1 0 0 0 |
│ │ │ -     | 0 1 0 1 1 1 0 |
│ │ │ -     | 1 0 0 1 1 0 1 |
│ │ │ +     | 0 0 1 1 1 1 0 |
│ │ │ +     | 0 1 0 1 0 1 1 |
│ │ │  
│ │ │                    3           7
│ │ │  o2 : Matrix (GF 2)  <-- (GF 2)
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_messages.out
│ │ │ @@ -2,15 +2,15 @@
│ │ │  
│ │ │  i1 : F=GF(4,Variable=>a);
│ │ │  
│ │ │  i2 : R=linearCode(F,{{1,1,1}});
│ │ │  
│ │ │  i3 : messages R
│ │ │  
│ │ │ -o3 = {{1}, {a}, {a + 1}, {0}}
│ │ │ +o3 = {{0}, {1}, {a}, {a + 1}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : messages hammingCode(2,3)
│ │ │  
│ │ │  o4 = {{1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 0, 1, 1}, {1, 1, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_order__Code.out
│ │ │ @@ -10,39 +10,39 @@
│ │ │  
│ │ │  o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                                                                           }
│ │ │                      Points => {{0, 0}, {a, a}, {a + 1, a}, {1, a}, {a, a + 1}, {a + 1, a + 1}, {1, a + 1}, {0, 1}}
│ │ │                                                                 8
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                                                                    }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1 0   0   0   0 0   0   0   |
│ │ │ -                                                           | 1 a+1 1   a   1 a   a+1 a+1 |
│ │ │ -                                                           | 1 a   a+1 a   1 a+1 a+1 1   |
│ │ │ -                                                           | 1 1   a   a   1 1   a+1 a   |
│ │ │ -                                                           | 1 a+1 a   a+1 1 a   a   1   |
│ │ │ -                                                           | 1 a   1   a+1 1 a+1 a   a   |
│ │ │ -                                                           | 1 1   a+1 a+1 1 1   a   a+1 |
│ │ │ -                                                           | 1 0   0   1   0 0   1   0   |
│ │ │ -                                             GeneratorMatrix => | 1 1   1   1   1   1   1   1 |
│ │ │ +                                             Code => image | 0   1 0   0   0   0 0   0   |
│ │ │ +                                                           | a+1 1 a+1 1   a   1 a   a+1 |
│ │ │ +                                                           | 1   1 a   a+1 a   1 a+1 a+1 |
│ │ │ +                                                           | a   1 1   a   a   1 1   a+1 |
│ │ │ +                                                           | 1   1 a+1 a   a+1 1 a   a   |
│ │ │ +                                                           | a   1 a   1   a+1 1 a+1 a   |
│ │ │ +                                                           | a+1 1 1   a+1 a+1 1 1   a   |
│ │ │ +                                                           | 0   1 0   0   1   0 0   1   |
│ │ │ +                                             GeneratorMatrix => | 0 a+1 1   a   1   a   a+1 0 |
│ │ │ +                                                                | 1 1   1   1   1   1   1   1 |
│ │ │                                                                  | 0 a+1 a   1   a+1 a   1   0 |
│ │ │                                                                  | 0 1   a+1 a   a   1   a+1 0 |
│ │ │                                                                  | 0 a   a   a   a+1 a+1 a+1 1 |
│ │ │                                                                  | 0 1   1   1   1   1   1   0 |
│ │ │                                                                  | 0 a   a+1 1   a   a+1 1   0 |
│ │ │                                                                  | 0 a+1 a+1 a+1 a   a   a   1 |
│ │ │ -                                                                | 0 a+1 1   a   1   a   a+1 0 |
│ │ │ -                                             Generators => {{1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}, {0, a + 1, 1, a, 1, a, a + 1, 0}}
│ │ │ +                                             Generators => {{0, a + 1, 1, a, 1, a, a + 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}}
│ │ │                                               ParityCheckMatrix => | 1 1 1 1 1 1 1 1 |
│ │ │                                               ParityCheckRows => {{1, 1, 1, 1, 1, 1, 1, 1}}
│ │ │                                                  2               3    2        4
│ │ │                      VanishingIdeal => ideal (t t  + t t  + t , t  + t  + t , t  + t )
│ │ │                                                0 1    0 1    0   0    1    1   1    1
│ │ │ -                                          2   2         3       2
│ │ │ -                    PolynomialSet => {1, t , t t , t , t , t , t , t t }
│ │ │ -                                          0   0 1   1   0   0   1   0 1
│ │ │ +                                                2   2         3       2
│ │ │ +                    PolynomialSet => {t t , 1, t , t t , t , t , t , t }
│ │ │ +                                       0 1      0   0 1   1   0   0   1
│ │ │  
│ │ │  i5 : F = GF(4);
│ │ │  
│ │ │  i6 : R = F[x,y];
│ │ │  
│ │ │  i7 : I = ideal(x^3+y^2+y)
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_ring_lp__Linear__Code_rp.out
│ │ │ @@ -3,29 +3,29 @@
│ │ │  i1 : C = hammingCode(2, 3)
│ │ │  
│ │ │                                         7
│ │ │  o1 = LinearCode{AmbientModule => (GF 2)                                                                                   }
│ │ │                  BaseField => GF 2
│ │ │                  cache => CacheTable{}
│ │ │                  Code => image | 1 0 1 1 |
│ │ │ -                              | 1 1 0 1 |
│ │ │                                | 1 1 1 0 |
│ │ │ +                              | 1 1 0 1 |
│ │ │                                | 1 0 0 0 |
│ │ │                                | 0 1 0 0 |
│ │ │                                | 0 0 1 0 |
│ │ │                                | 0 0 0 1 |
│ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │                                     | 0 1 1 0 1 0 0 |
│ │ │ -                                   | 1 0 1 0 0 1 0 |
│ │ │ -                                   | 1 1 0 0 0 0 1 |
│ │ │ -                Generators => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0}, {1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ +                                   | 1 1 0 0 0 1 0 |
│ │ │ +                                   | 1 0 1 0 0 0 1 |
│ │ │ +                Generators => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0}, {1, 1, 0, 0, 0, 1, 0}, {1, 0, 1, 0, 0, 0, 1}}
│ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │ -                                     | 0 0 1 1 1 1 0 |
│ │ │ -                                     | 0 1 0 1 1 0 1 |
│ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0}, {0, 1, 0, 1, 1, 0, 1}}
│ │ │ +                                     | 0 1 0 1 1 1 0 |
│ │ │ +                                     | 0 1 1 0 0 1 1 |
│ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 0, 1, 1, 1, 0}, {0, 1, 1, 0, 0, 1, 1}}
│ │ │  
│ │ │  o1 : LinearCode
│ │ │  
│ │ │  i2 : ring(C)
│ │ │  
│ │ │  o2 = GF 2
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_syndrome__Decode.out
│ │ │ @@ -55,31 +55,31 @@
│ │ │                 | 0 |    | 0 |
│ │ │                 | 0 |    | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                 | 1 | => | 0 |
│ │ │ -               | 0 |    | 1 |
│ │ │ -               | 1 |    | 0 |
│ │ │ +               | 0 |    | 0 |
│ │ │ +               | 1 |    | 1 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                 | 1 | => | 0 |
│ │ │                 | 1 |    | 0 |
│ │ │ -               | 0 |    | 1 |
│ │ │ -                        | 0 |
│ │ │ +               | 0 |    | 0 |
│ │ │ +                        | 1 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                 | 1 | => | 0 |
│ │ │ +               | 1 |    | 1 |
│ │ │                 | 1 |    | 0 |
│ │ │ -               | 1 |    | 0 |
│ │ │ -                        | 1 |
│ │ │ +                        | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │  
│ │ │  o7 : HashTable
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_vector__Space.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 2210985853493542567
│ │ │  
│ │ │  i1 : H = hammingCode(2,3);
│ │ │  
│ │ │  i2 : vectorSpace H
│ │ │  
│ │ │ -o2 = image | 1 1 0 1 |
│ │ │ -           | 1 0 1 1 |
│ │ │ +o2 = image | 1 0 1 1 |
│ │ │ +           | 1 1 0 1 |
│ │ │             | 1 1 1 0 |
│ │ │             | 1 0 0 0 |
│ │ │             | 0 1 0 0 |
│ │ │             | 0 0 1 0 |
│ │ │             | 0 0 0 1 |
│ │ │  
│ │ │                                       7
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/___Sets.html
│ │ │ @@ -79,40 +79,40 @@
│ │ │  
    		          
                  
    i2 : R=F[x,y];
    
│ │ │      		          
                  
    i3 : C=cartesianCode(F,{{0,1,a},{0,1,a}},{1+x+y,x*y})
│ │ │  
│ │ │ -o3 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                               }
│ │ │ +o3 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                        }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1   a+1 |
│ │ │ +                                             Code => image | 1   0   |
│ │ │ +                                                           | 0   0   |
│ │ │ +                                                           | 0   0   |
│ │ │                                                             | a+1 0   |
│ │ │                                                             | a+1 0   |
│ │ │                                                             | a   a   |
│ │ │ -                                                           | 1   0   |
│ │ │ +                                                           | 1   a+1 |
│ │ │                                                             | a   a   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                             GeneratorMatrix => | 1   a+1 a+1 a 1 a 0 0 1 |
│ │ │ -                                                                | a+1 0   0   a 0 a 0 0 1 |
│ │ │ -                                             Generators => {{1, a + 1, a + 1, a, 1, a, 0, 0, 1}, {a + 1, 0, 0, a, 0, a, 0, 0, 1}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 a   0 0 0 a+1 |
│ │ │ -                                                                  | 0 1 0 0 a+1 0 0 0 0   |
│ │ │ -                                                                  | 0 0 1 0 a+1 0 0 0 0   |
│ │ │ -                                                                  | 0 0 0 1 0   0 0 0 a   |
│ │ │ -                                                                  | 0 0 0 0 0   1 0 0 a   |
│ │ │ -                                                                  | 0 0 0 0 0   0 1 0 0   |
│ │ │ -                                                                  | 0 0 0 0 0   0 0 1 0   |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, a, 0, 0, 0, a + 1}, {0, 1, 0, 0, a + 1, 0, 0, 0, 0}, {0, 0, 1, 0, a + 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 0, 1, 0, 0, a}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ -                    Points => {{a, a}, {0, a}, {a, 0}, {a, 1}, {0, 0}, {1, a}, {1, 0}, {0, 1}, {1, 1}}
│ │ │ +                                             GeneratorMatrix => | 1 0 0 a+1 a+1 a 1   a 1 |
│ │ │ +                                                                | 0 0 0 0   0   a a+1 a 1 |
│ │ │ +                                             Generators => {{1, 0, 0, a + 1, a + 1, a, 1, a, 1}, {0, 0, 0, 0, 0, a, a + 1, a, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 a+1 0 a |
│ │ │ +                                                                  | 0 1 0 0 0 0 0   0 0 |
│ │ │ +                                                                  | 0 0 1 0 0 0 0   0 0 |
│ │ │ +                                                                  | 0 0 0 1 0 0 a   0 1 |
│ │ │ +                                                                  | 0 0 0 0 1 0 a   0 1 |
│ │ │ +                                                                  | 0 0 0 0 0 1 0   0 a |
│ │ │ +                                                                  | 0 0 0 0 0 0 0   1 a |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, a + 1, 0, a}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, a, 0, 1}, {0, 0, 0, 0, 1, 0, a, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, a}, {0, 0, 0, 0, 0, 0, 0, 1, a}}
│ │ │ +                    Points => {{0, 0}, {0, 1}, {1, 0}, {0, a}, {a, 0}, {a, 1}, {a, a}, {1, a}, {1, 1}}
│ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2         3           2
│ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a + 1)y  + a*y)
│ │ │  
│ │ │  o3 : EvaluationCode
    
│ │ │      		          
                
                    
    i3 : R=F[x,y];
    
│ │ │      		            
                    
    i4 : C=cartesianCode(F,{{0,1,a},{0,1,a}},{1+x+y,x*y})
│ │ │  
│ │ │ -o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                           }
│ │ │ +o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1   a+1 |
│ │ │ +                                             Code => image | a+1 0   |
│ │ │ +                                                           | a+1 0   |
│ │ │ +                                                           | a   a   |
│ │ │ +                                                           | a   a   |
│ │ │                                                             | 1   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │ +                                                           | 1   a+1 |
│ │ │                                                             | 1   1   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                                           | a   a   |
│ │ │ -                                                           | a   a   |
│ │ │ -                                             GeneratorMatrix => | 1   1 0 0 1 a+1 a+1 a a |
│ │ │ -                                                                | a+1 0 0 0 1 0   0   a a |
│ │ │ -                                             Generators => {{1, 1, 0, 0, 1, a + 1, a + 1, a, a}, {a + 1, 0, 0, 0, 1, 0, 0, a, a}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0 a+1 0 a   0 |
│ │ │ -                                                                  | 0 1 0 0 0 a   0 0   0 |
│ │ │ -                                                                  | 0 0 1 0 0 0   0 0   0 |
│ │ │ -                                                                  | 0 0 0 1 0 0   0 0   0 |
│ │ │ -                                                                  | 0 0 0 0 1 0   0 a+1 0 |
│ │ │ -                                                                  | 0 0 0 0 0 1   1 0   0 |
│ │ │ -                                                                  | 0 0 0 0 0 0   0 1   1 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, a, 0}, {0, 1, 0, 0, 0, a, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, a + 1, 0}, {0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ -                    Points => {{a, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, 0}, {0, a}, {a, 1}, {1, a}}
│ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1 0 0 1   1 |
│ │ │ +                                                                | 0   0   a a 0 0 0 a+1 1 |
│ │ │ +                                             Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, a + 1, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 a   1 |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 a   1 |
│ │ │ +                                                                  | 0 0 1 0 0 0 0 0   a |
│ │ │ +                                                                  | 0 0 0 1 0 0 0 0   a |
│ │ │ +                                                                  | 0 0 0 0 1 0 0 a+1 a |
│ │ │ +                                                                  | 0 0 0 0 0 1 0 0   0 |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0   0 |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, a, 1}, {0, 1, 0, 0, 0, 0, 0, a, 1}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, 0, 0, a + 1, a}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{a, 0}, {0, a}, {1, a}, {a, 1}, {0, 0}, {0, 1}, {1, 0}, {a, a}, {1, 1}}
│ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2         3           2
│ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a + 1)y  + a*y)
│ │ │  
│ │ │  o4 : EvaluationCode
    
│ │ │      		            
                    
    i5 : C.LinearCode
│ │ │  
│ │ │                                    9
│ │ │ -o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                  BaseField => F
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | 1   a+1 |
│ │ │ +                Code => image | a+1 0   |
│ │ │ +                              | a+1 0   |
│ │ │ +                              | a   a   |
│ │ │ +                              | a   a   |
│ │ │                                | 1   0   |
│ │ │                                | 0   0   |
│ │ │                                | 0   0   |
│ │ │ +                              | 1   a+1 |
│ │ │                                | 1   1   |
│ │ │ -                              | a+1 0   |
│ │ │ -                              | a+1 0   |
│ │ │ -                              | a   a   |
│ │ │ -                              | a   a   |
│ │ │ -                GeneratorMatrix => | 1   1 0 0 1 a+1 a+1 a a |
│ │ │ -                                   | a+1 0 0 0 1 0   0   a a |
│ │ │ -                Generators => {{1, 1, 0, 0, 1, a + 1, a + 1, a, a}, {a + 1, 0, 0, 0, 1, 0, 0, a, a}}
│ │ │ -                ParityCheckMatrix => | 1 0 0 0 0 a+1 0 a   0 |
│ │ │ -                                     | 0 1 0 0 0 a   0 0   0 |
│ │ │ -                                     | 0 0 1 0 0 0   0 0   0 |
│ │ │ -                                     | 0 0 0 1 0 0   0 0   0 |
│ │ │ -                                     | 0 0 0 0 1 0   0 a+1 0 |
│ │ │ -                                     | 0 0 0 0 0 1   1 0   0 |
│ │ │ -                                     | 0 0 0 0 0 0   0 1   1 |
│ │ │ -                ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, a, 0}, {0, 1, 0, 0, 0, a, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, a + 1, 0}, {0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ +                GeneratorMatrix => | a+1 a+1 a a 1 0 0 1   1 |
│ │ │ +                                   | 0   0   a a 0 0 0 a+1 1 |
│ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, a + 1, 1}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 0 0 a   1 |
│ │ │ +                                     | 0 1 0 0 0 0 0 a   1 |
│ │ │ +                                     | 0 0 1 0 0 0 0 0   a |
│ │ │ +                                     | 0 0 0 1 0 0 0 0   a |
│ │ │ +                                     | 0 0 0 0 1 0 0 a+1 a |
│ │ │ +                                     | 0 0 0 0 0 1 0 0   0 |
│ │ │ +                                     | 0 0 0 0 0 0 1 0   0 |
│ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, a, 1}, {0, 1, 0, 0, 0, 0, 0, a, 1}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, 0, 0, a + 1, a}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │  
│ │ │  o5 : LinearCode
    
│ │ │      		            
    
│ │ │          
│ │ │          
    
│ │ │            
    a ring, a list and a Matrix are given
    
│ │ │ @@ -300,35 +300,35 @@
│ │ │                  
    i8 : C=cartesianCode(F,{{0,1,a},{0,1,a}},matrix{{1,2},{2,3}})
│ │ │  
│ │ │  o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | a+1 1   |
│ │ │ +                                             Code => image | a+1 1   |
│ │ │                                                             | a   a+1 |
│ │ │                                                             | 1   a+1 |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                             GeneratorMatrix => | 0 0 a+1 a   1   0 0 0 1 |
│ │ │ -                                                                | 0 0 1   a+1 a+1 0 0 0 1 |
│ │ │ -                                             Generators => {{0, 0, a + 1, a, 1, 0, 0, 0, 1}, {0, 0, 1, a + 1, a + 1, 0, 0, 0, 1}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0   0 0 0 0 |
│ │ │ -                                                                  | 0 1 0 0 0   0 0 0 0 |
│ │ │ -                                                                  | 0 0 1 0 1   0 0 0 a |
│ │ │ -                                                                  | 0 0 0 1 a+1 0 0 0 1 |
│ │ │ -                                                                  | 0 0 0 0 0   1 0 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0   0 1 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0   0 0 1 0 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 0, a}, {0, 0, 0, 1, a + 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ -                    Points => {{0, a}, {a, 0}, {1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0}, {1, 1}}
│ │ │ +                                                           | 0   0   |
│ │ │ +                                                           | 0   0   |
│ │ │ +                                             GeneratorMatrix => | a+1 a   1   0 0 0 1 0 0 |
│ │ │ +                                                                | 1   a+1 a+1 0 0 0 1 0 0 |
│ │ │ +                                             Generators => {{a + 1, a, 1, 0, 0, 0, 1, 0, 0}, {1, a + 1, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 1   0 0 0 a 0 0 |
│ │ │ +                                                                  | 0 1 a+1 0 0 0 1 0 0 |
│ │ │ +                                                                  | 0 0 0   1 0 0 0 0 0 |
│ │ │ +                                                                  | 0 0 0   0 1 0 0 0 0 |
│ │ │ +                                                                  | 0 0 0   0 0 1 0 0 0 |
│ │ │ +                                                                  | 0 0 0   0 0 0 0 1 0 |
│ │ │ +                                                                  | 0 0 0   0 0 0 0 0 1 |
│ │ │ +                                             ParityCheckRows => {{1, 0, 1, 0, 0, 0, a, 0, 0}, {0, 1, a + 1, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}}
│ │ │ +                    Points => {{1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0}, {1, 1}, {0, a}, {a, 0}}
│ │ │                                           2   2 3
│ │ │                      PolynomialSet => {t t , t t }
│ │ │                                         0 1   0 1
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2          3           2
│ │ │                      VanishingIdeal => ideal (t  + (a + 1)t  + a*t , t  + (a + 1)t  + a*t )
│ │ │                                                0           0      0   1           1      1
│ │ │ ├── html2text {}
│ │ │ │ @@ -124,49 +124,49 @@
│ │ │ │  o4 = EvaluationCode{cache => CacheTable{}
│ │ │ │  }
│ │ │ │                                                                 9
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │ -                                             Code => image | 1   a+1 |
│ │ │ │ +                                             Code => image | a+1 0   |
│ │ │ │ +                                                           | a+1 0   |
│ │ │ │ +                                                           | a   a   |
│ │ │ │ +                                                           | a   a   |
│ │ │ │                                                             | 1   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │ +                                                           | 1   a+1 |
│ │ │ │                                                             | 1   1   |
│ │ │ │ -                                                           | a+1 0   |
│ │ │ │ -                                                           | a+1 0   |
│ │ │ │ -                                                           | a   a   |
│ │ │ │ -                                                           | a   a   |
│ │ │ │ -                                             GeneratorMatrix => | 1   1 0 0 1
│ │ │ │ -a+1 a+1 a a |
│ │ │ │ -                                                                | a+1 0 0 0 1 0
│ │ │ │ -0   a a |
│ │ │ │ -                                             Generators => {{1, 1, 0, 0, 1, a +
│ │ │ │ -1, a + 1, a, a}, {a + 1, 0, 0, 0, 1, 0, 0, a, a}}
│ │ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0
│ │ │ │ -a+1 0 a   0 |
│ │ │ │ -                                                                  | 0 1 0 0 0 a
│ │ │ │ -0 0   0 |
│ │ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1
│ │ │ │ +0 0 1   1 |
│ │ │ │ +                                                                | 0   0   a a 0
│ │ │ │ +0 0 a+1 1 |
│ │ │ │ +                                             Generators => {{a + 1, a + 1, a,
│ │ │ │ +a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, a + 1, 1}}
│ │ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0
│ │ │ │ +0 a   1 |
│ │ │ │ +                                                                  | 0 1 0 0 0 0
│ │ │ │ +0 a   1 |
│ │ │ │                                                                    | 0 0 1 0 0 0
│ │ │ │ -0 0   0 |
│ │ │ │ +0 0   a |
│ │ │ │                                                                    | 0 0 0 1 0 0
│ │ │ │ -0 0   0 |
│ │ │ │ +0 0   a |
│ │ │ │                                                                    | 0 0 0 0 1 0
│ │ │ │ -0 a+1 0 |
│ │ │ │ +0 a+1 a |
│ │ │ │                                                                    | 0 0 0 0 0 1
│ │ │ │ -1 0   0 |
│ │ │ │ +0 0   0 |
│ │ │ │                                                                    | 0 0 0 0 0 0
│ │ │ │ -0 1   1 |
│ │ │ │ +1 0   0 |
│ │ │ │                                               ParityCheckRows => {{1, 0, 0, 0,
│ │ │ │ -0, a + 1, 0, a, 0}, {0, 1, 0, 0, 0, a, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0},
│ │ │ │ -{0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, a + 1, 0}, {0, 0, 0, 0, 0,
│ │ │ │ -1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ │ -                    Points => {{a, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, 0},
│ │ │ │ -{0, a}, {a, 1}, {1, a}}
│ │ │ │ +0, 0, 0, a, 1}, {0, 1, 0, 0, 0, 0, 0, a, 1}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0,
│ │ │ │ +0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, 0, 0, a + 1, a}, {0, 0, 0, 0, 0, 1, 0,
│ │ │ │ +0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ │ +                    Points => {{a, 0}, {0, a}, {1, a}, {a, 1}, {0, 0}, {0, 1},
│ │ │ │ +{1, 0}, {a, a}, {1, 1}}
│ │ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │ │                                                3           2         3
│ │ │ │  2
│ │ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a +
│ │ │ │  1)y  + a*y)
│ │ │ │  
│ │ │ │ @@ -174,38 +174,38 @@
│ │ │ │  i5 : C.LinearCode
│ │ │ │  
│ │ │ │                                    9
│ │ │ │  o5 = LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                  BaseField => F
│ │ │ │                  cache => CacheTable{}
│ │ │ │ -                Code => image | 1   a+1 |
│ │ │ │ +                Code => image | a+1 0   |
│ │ │ │ +                              | a+1 0   |
│ │ │ │ +                              | a   a   |
│ │ │ │ +                              | a   a   |
│ │ │ │                                | 1   0   |
│ │ │ │                                | 0   0   |
│ │ │ │                                | 0   0   |
│ │ │ │ +                              | 1   a+1 |
│ │ │ │                                | 1   1   |
│ │ │ │ -                              | a+1 0   |
│ │ │ │ -                              | a+1 0   |
│ │ │ │ -                              | a   a   |
│ │ │ │ -                              | a   a   |
│ │ │ │ -                GeneratorMatrix => | 1   1 0 0 1 a+1 a+1 a a |
│ │ │ │ -                                   | a+1 0 0 0 1 0   0   a a |
│ │ │ │ -                Generators => {{1, 1, 0, 0, 1, a + 1, a + 1, a, a}, {a + 1, 0,
│ │ │ │ -0, 0, 1, 0, 0, a, a}}
│ │ │ │ -                ParityCheckMatrix => | 1 0 0 0 0 a+1 0 a   0 |
│ │ │ │ -                                     | 0 1 0 0 0 a   0 0   0 |
│ │ │ │ -                                     | 0 0 1 0 0 0   0 0   0 |
│ │ │ │ -                                     | 0 0 0 1 0 0   0 0   0 |
│ │ │ │ -                                     | 0 0 0 0 1 0   0 a+1 0 |
│ │ │ │ -                                     | 0 0 0 0 0 1   1 0   0 |
│ │ │ │ -                                     | 0 0 0 0 0 0   0 1   1 |
│ │ │ │ -                ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, a, 0}, {0, 1, 0,
│ │ │ │ -0, 0, a, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0},
│ │ │ │ -{0, 0, 0, 0, 1, 0, 0, a + 1, 0}, {0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0,
│ │ │ │ -0, 0, 1, 1}}
│ │ │ │ +                GeneratorMatrix => | a+1 a+1 a a 1 0 0 1   1 |
│ │ │ │ +                                   | 0   0   a a 0 0 0 a+1 1 |
│ │ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a,
│ │ │ │ +a, 0, 0, 0, a + 1, 1}}
│ │ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 0 0 a   1 |
│ │ │ │ +                                     | 0 1 0 0 0 0 0 a   1 |
│ │ │ │ +                                     | 0 0 1 0 0 0 0 0   a |
│ │ │ │ +                                     | 0 0 0 1 0 0 0 0   a |
│ │ │ │ +                                     | 0 0 0 0 1 0 0 a+1 a |
│ │ │ │ +                                     | 0 0 0 0 0 1 0 0   0 |
│ │ │ │ +                                     | 0 0 0 0 0 0 1 0   0 |
│ │ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, a, 1}, {0, 1, 0, 0,
│ │ │ │ +0, 0, 0, a, 1}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0,
│ │ │ │ +0, 0, 0, 1, 0, 0, a + 1, a}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1,
│ │ │ │ +0, 0}}
│ │ │ │  
│ │ │ │  o5 : LinearCode
│ │ │ │  ********** aa rriinngg,, aa lliisstt aanndd aa MMaattrriixx aarree ggiivveenn **********
│ │ │ │      *   Usage:
│ │ │ │              cartesianCode(F, L, M)
│ │ │ │      * Inputs:
│ │ │ │            o F, a _r_i_n_g,
│ │ │ │ @@ -224,49 +224,49 @@
│ │ │ │  o8 = EvaluationCode{cache => CacheTable{}
│ │ │ │  }
│ │ │ │                                                                 9
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │ -                                             Code => image | 0   0   |
│ │ │ │ -                                                           | 0   0   |
│ │ │ │ -                                                           | a+1 1   |
│ │ │ │ +                                             Code => image | a+1 1   |
│ │ │ │                                                             | a   a+1 |
│ │ │ │                                                             | 1   a+1 |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 1   1   |
│ │ │ │ -                                             GeneratorMatrix => | 0 0 a+1 a   1
│ │ │ │ -0 0 0 1 |
│ │ │ │ -                                                                | 0 0 1   a+1
│ │ │ │ -a+1 0 0 0 1 |
│ │ │ │ -                                             Generators => {{0, 0, a + 1, a, 1,
│ │ │ │ -0, 0, 0, 1}, {0, 0, 1, a + 1, a + 1, 0, 0, 0, 1}}
│ │ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0
│ │ │ │ +                                                           | 0   0   |
│ │ │ │ +                                                           | 0   0   |
│ │ │ │ +                                             GeneratorMatrix => | a+1 a   1   0
│ │ │ │ +0 0 1 0 0 |
│ │ │ │ +                                                                | 1   a+1 a+1 0
│ │ │ │ +0 0 1 0 0 |
│ │ │ │ +                                             Generators => {{a + 1, a, 1, 0, 0,
│ │ │ │ +0, 1, 0, 0}, {1, a + 1, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ │ +                                             ParityCheckMatrix => | 1 0 1   0 0
│ │ │ │ +0 a 0 0 |
│ │ │ │ +                                                                  | 0 1 a+1 0 0
│ │ │ │ +0 1 0 0 |
│ │ │ │ +                                                                  | 0 0 0   1 0
│ │ │ │  0 0 0 0 |
│ │ │ │ -                                                                  | 0 1 0 0 0
│ │ │ │ +                                                                  | 0 0 0   0 1
│ │ │ │  0 0 0 0 |
│ │ │ │ -                                                                  | 0 0 1 0 1
│ │ │ │ -0 0 0 a |
│ │ │ │ -                                                                  | 0 0 0 1 a+1
│ │ │ │ -0 0 0 1 |
│ │ │ │ -                                                                  | 0 0 0 0 0
│ │ │ │ +                                                                  | 0 0 0   0 0
│ │ │ │  1 0 0 0 |
│ │ │ │ -                                                                  | 0 0 0 0 0
│ │ │ │ -0 1 0 0 |
│ │ │ │ -                                                                  | 0 0 0 0 0
│ │ │ │ +                                                                  | 0 0 0   0 0
│ │ │ │  0 0 1 0 |
│ │ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0,
│ │ │ │ -0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 0, a}, {0,
│ │ │ │ -0, 0, 1, a + 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1,
│ │ │ │ -0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ │ -                    Points => {{0, a}, {a, 0}, {1, a}, {a, 1}, {a, a}, {0, 0},
│ │ │ │ -{0, 1}, {1, 0}, {1, 1}}
│ │ │ │ +                                                                  | 0 0 0   0 0
│ │ │ │ +0 0 0 1 |
│ │ │ │ +                                             ParityCheckRows => {{1, 0, 1, 0,
│ │ │ │ +0, 0, a, 0, 0}, {0, 1, a + 1, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0},
│ │ │ │ +{0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0,
│ │ │ │ +1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}}
│ │ │ │ +                    Points => {{1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0},
│ │ │ │ +{1, 1}, {0, a}, {a, 0}}
│ │ │ │                                           2   2 3
│ │ │ │                      PolynomialSet => {t t , t t }
│ │ │ │                                         0 1   0 1
│ │ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │ │                                                3           2          3
│ │ │ │  2
│ │ │ │                      VanishingIdeal => ideal (t  + (a + 1)t  + a*t , t  + (a +
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_codewords.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │            
│ │ │            
│ │ │                
    i2 : C=linearCode(matrix{{1,a,0},{0,1,a}});
    
│ │ │            
│ │ │            
│ │ │                
    i3 : codewords(C)
│ │ │  
│ │ │ -o3 = {{a, a, a}, {a + 1, a, 1}, {1, 1, 1}, {0, 1, a}, {0, a, a + 1}, {1, a +
│ │ │ +o3 = {{a, a, a}, {a + 1, a, 1}, {1, 1, 1}, {0, 1, a}, {a, 1, a + 1}, {0, a, a
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, a}, {0, a + 1, 1}, {a, 1, a + 1}, {1, a, 0}, {a + 1, a + 1, a + 1},
│ │ │ +     + 1}, {0, a + 1, 1}, {1, a + 1, a}, {a + 1, 0, a}, {a, 0, 1}, {a + 1, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {a, 0, 1}, {a + 1, 0, a}, {a + 1, 1, 0}, {1, 0, a + 1}, {a, a + 1, 0},
│ │ │ +     0}, {1, a, 0}, {a + 1, a + 1, a + 1}, {1, 0, a + 1}, {a, a + 1, 0}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 0, 0}}
│ │ │ +     0, 0}}
│ │ │  
│ │ │  o3 : List
    
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    Ways to use codewords:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,20 +15,20 @@
│ │ │ │  Obtains all the codewords of a code C by multiplying all the elements of the
│ │ │ │  ambient space (obtained with the function messages) by the generator matrix of
│ │ │ │  C.
│ │ │ │  i1 : F=GF(4,Variable=>a);
│ │ │ │  i2 : C=linearCode(matrix{{1,a,0},{0,1,a}});
│ │ │ │  i3 : codewords(C)
│ │ │ │  
│ │ │ │ -o3 = {{a, a, a}, {a + 1, a, 1}, {1, 1, 1}, {0, 1, a}, {0, a, a + 1}, {1, a +
│ │ │ │ +o3 = {{a, a, a}, {a + 1, a, 1}, {1, 1, 1}, {0, 1, a}, {a, 1, a + 1}, {0, a, a
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, a}, {0, a + 1, 1}, {a, 1, a + 1}, {1, a, 0}, {a + 1, a + 1, a + 1},
│ │ │ │ +     + 1}, {0, a + 1, 1}, {1, a + 1, a}, {a + 1, 0, a}, {a, 0, 1}, {a + 1, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {a, 0, 1}, {a + 1, 0, a}, {a + 1, 1, 0}, {1, 0, a + 1}, {a, a + 1, 0},
│ │ │ │ +     0}, {1, a, 0}, {a + 1, a + 1, a + 1}, {1, 0, a + 1}, {a, a + 1, 0}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 0, 0}}
│ │ │ │ +     0, 0}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** WWaayyss ttoo uussee ccooddeewwoorrddss:: **********
│ │ │ │      * codewords(LinearCode)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_o_d_e_w_o_r_d_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_dim_lp__Linear__Code_rp.html
│ │ │ @@ -84,30 +84,30 @@
│ │ │            
│ │ │                
    i3 : H = hammingCode(2,3)
│ │ │  
│ │ │                                         7
│ │ │  o3 = LinearCode{AmbientModule => (GF 2)                                                                                   }
│ │ │                  BaseField => GF 2
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | 1 1 0 1 |
│ │ │ +                Code => image | 1 1 1 0 |
│ │ │                                | 1 0 1 1 |
│ │ │ -                              | 1 1 1 0 |
│ │ │ +                              | 1 1 0 1 |
│ │ │                                | 1 0 0 0 |
│ │ │                                | 0 1 0 0 |
│ │ │                                | 0 0 1 0 |
│ │ │                                | 0 0 0 1 |
│ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │                                     | 1 0 1 0 1 0 0 |
│ │ │ -                                   | 0 1 1 0 0 1 0 |
│ │ │ -                                   | 1 1 0 0 0 0 1 |
│ │ │ -                Generators => {{1, 1, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0}, {0, 1, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ +                                   | 1 1 0 0 0 1 0 |
│ │ │ +                                   | 0 1 1 0 0 0 1 |
│ │ │ +                Generators => {{1, 1, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0}, {1, 1, 0, 0, 0, 1, 0}, {0, 1, 1, 0, 0, 0, 1}}
│ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │ -                                     | 0 0 1 1 1 1 0 |
│ │ │ +                                     | 0 1 1 0 1 1 0 |
│ │ │                                       | 0 1 0 1 0 1 1 |
│ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0}, {0, 1, 0, 1, 0, 1, 1}}
│ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 1, 0}, {0, 1, 0, 1, 0, 1, 1}}
│ │ │  
│ │ │  o3 : LinearCode
    
│ │ │            
│ │ │            
│ │ │                
    i4 : dim H
│ │ │  
│ │ │  o4 = 4
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,31 +22,31 @@
│ │ │ │  i3 : H = hammingCode(2,3)
│ │ │ │  
│ │ │ │                                         7
│ │ │ │  o3 = LinearCode{AmbientModule => (GF 2)
│ │ │ │  }
│ │ │ │                  BaseField => GF 2
│ │ │ │                  cache => CacheTable{}
│ │ │ │ -                Code => image | 1 1 0 1 |
│ │ │ │ +                Code => image | 1 1 1 0 |
│ │ │ │                                | 1 0 1 1 |
│ │ │ │ -                              | 1 1 1 0 |
│ │ │ │ +                              | 1 1 0 1 |
│ │ │ │                                | 1 0 0 0 |
│ │ │ │                                | 0 1 0 0 |
│ │ │ │                                | 0 0 1 0 |
│ │ │ │                                | 0 0 0 1 |
│ │ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │ │                                     | 1 0 1 0 1 0 0 |
│ │ │ │ -                                   | 0 1 1 0 0 1 0 |
│ │ │ │ -                                   | 1 1 0 0 0 0 1 |
│ │ │ │ +                                   | 1 1 0 0 0 1 0 |
│ │ │ │ +                                   | 0 1 1 0 0 0 1 |
│ │ │ │                  Generators => {{1, 1, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0},
│ │ │ │ -{0, 1, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ │ +{1, 1, 0, 0, 0, 1, 0}, {0, 1, 1, 0, 0, 0, 1}}
│ │ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │ │ -                                     | 0 0 1 1 1 1 0 |
│ │ │ │ +                                     | 0 1 1 0 1 1 0 |
│ │ │ │                                       | 0 1 0 1 0 1 1 |
│ │ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1,
│ │ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 1,
│ │ │ │  0}, {0, 1, 0, 1, 0, 1, 1}}
│ │ │ │  
│ │ │ │  o3 : LinearCode
│ │ │ │  i4 : dim H
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_hamming__Code.html
│ │ │ @@ -76,16 +76,16 @@
│ │ │            
│ │ │                
    i1 : C1 = hammingCode(2,3);
    
│ │ │            
│ │ │            
│ │ │                
    i2 : C1.ParityCheckMatrix
│ │ │  
│ │ │  o2 = | 1 1 1 1 0 0 0 |
│ │ │ -     | 0 1 0 1 1 1 0 |
│ │ │ -     | 1 0 0 1 1 0 1 |
│ │ │ +     | 0 0 1 1 1 1 0 |
│ │ │ +     | 0 1 0 1 0 1 1 |
│ │ │  
│ │ │                    3           7
│ │ │  o2 : Matrix (GF 2)  <-- (GF 2)
    
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,16 +14,16 @@
│ │ │ │            o an instance of the type _L_i_n_e_a_r_C_o_d_e, $C$
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Returns the Hamming code $C$ over GF(q) whose dual has dimension s.
│ │ │ │  i1 : C1 = hammingCode(2,3);
│ │ │ │  i2 : C1.ParityCheckMatrix
│ │ │ │  
│ │ │ │  o2 = | 1 1 1 1 0 0 0 |
│ │ │ │ -     | 0 1 0 1 1 1 0 |
│ │ │ │ -     | 1 0 0 1 1 0 1 |
│ │ │ │ +     | 0 0 1 1 1 1 0 |
│ │ │ │ +     | 0 1 0 1 0 1 1 |
│ │ │ │  
│ │ │ │                    3           7
│ │ │ │  o2 : Matrix (GF 2)  <-- (GF 2)
│ │ │ │  ********** WWaayyss ttoo uussee hhaammmmiinnggCCooddee:: **********
│ │ │ │      * hammingCode(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _h_a_m_m_i_n_g_C_o_d_e is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_messages.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
│ │ │            
│ │ │                
    i2 : R=linearCode(F,{{1,1,1}});
    
│ │ │            
│ │ │            
│ │ │                
    i3 : messages R
│ │ │  
│ │ │ -o3 = {{1}, {a}, {a + 1}, {0}}
│ │ │ +o3 = {{0}, {1}, {a}, {a + 1}}
│ │ │  
│ │ │  o3 : List
    
│ │ │            
│ │ │          
│ │ │          
│ │ │            
│ │ │  
│ │ │            
                  
    i4 : messages hammingCode(2,3)
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  Given a code C of dimension $k$ over a finite field $F$, this function returns
│ │ │ │  the list that contains all the elements of $F^k$. Every element of the list can
│ │ │ │  be used to encode a message using the linear code C.
│ │ │ │  i1 : F=GF(4,Variable=>a);
│ │ │ │  i2 : R=linearCode(F,{{1,1,1}});
│ │ │ │  i3 : messages R
│ │ │ │  
│ │ │ │ -o3 = {{1}, {a}, {a + 1}, {0}}
│ │ │ │ +o3 = {{0}, {1}, {a}, {a + 1}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : messages hammingCode(2,3)
│ │ │ │  
│ │ │ │  o4 = {{1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 0, 1, 1}, {1, 1, 1, 0},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 1, 1, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}, {0, 1, 0, 0}, {0, 1, 1, 1},
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_order__Code.html
│ │ │ @@ -126,39 +126,39 @@
│ │ │  
│ │ │  o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                                                                           }
│ │ │                      Points => {{0, 0}, {a, a}, {a + 1, a}, {1, a}, {a, a + 1}, {a + 1, a + 1}, {1, a + 1}, {0, 1}}
│ │ │                                                                 8
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                                                                    }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1 0   0   0   0 0   0   0   |
│ │ │ -                                                           | 1 a+1 1   a   1 a   a+1 a+1 |
│ │ │ -                                                           | 1 a   a+1 a   1 a+1 a+1 1   |
│ │ │ -                                                           | 1 1   a   a   1 1   a+1 a   |
│ │ │ -                                                           | 1 a+1 a   a+1 1 a   a   1   |
│ │ │ -                                                           | 1 a   1   a+1 1 a+1 a   a   |
│ │ │ -                                                           | 1 1   a+1 a+1 1 1   a   a+1 |
│ │ │ -                                                           | 1 0   0   1   0 0   1   0   |
│ │ │ -                                             GeneratorMatrix => | 1 1   1   1   1   1   1   1 |
│ │ │ +                                             Code => image | 0   1 0   0   0   0 0   0   |
│ │ │ +                                                           | a+1 1 a+1 1   a   1 a   a+1 |
│ │ │ +                                                           | 1   1 a   a+1 a   1 a+1 a+1 |
│ │ │ +                                                           | a   1 1   a   a   1 1   a+1 |
│ │ │ +                                                           | 1   1 a+1 a   a+1 1 a   a   |
│ │ │ +                                                           | a   1 a   1   a+1 1 a+1 a   |
│ │ │ +                                                           | a+1 1 1   a+1 a+1 1 1   a   |
│ │ │ +                                                           | 0   1 0   0   1   0 0   1   |
│ │ │ +                                             GeneratorMatrix => | 0 a+1 1   a   1   a   a+1 0 |
│ │ │ +                                                                | 1 1   1   1   1   1   1   1 |
│ │ │                                                                  | 0 a+1 a   1   a+1 a   1   0 |
│ │ │                                                                  | 0 1   a+1 a   a   1   a+1 0 |
│ │ │                                                                  | 0 a   a   a   a+1 a+1 a+1 1 |
│ │ │                                                                  | 0 1   1   1   1   1   1   0 |
│ │ │                                                                  | 0 a   a+1 1   a   a+1 1   0 |
│ │ │                                                                  | 0 a+1 a+1 a+1 a   a   a   1 |
│ │ │ -                                                                | 0 a+1 1   a   1   a   a+1 0 |
│ │ │ -                                             Generators => {{1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}, {0, a + 1, 1, a, 1, a, a + 1, 0}}
│ │ │ +                                             Generators => {{0, a + 1, 1, a, 1, a, a + 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}}
│ │ │                                               ParityCheckMatrix => | 1 1 1 1 1 1 1 1 |
│ │ │                                               ParityCheckRows => {{1, 1, 1, 1, 1, 1, 1, 1}}
│ │ │                                                  2               3    2        4
│ │ │                      VanishingIdeal => ideal (t t  + t t  + t , t  + t  + t , t  + t )
│ │ │                                                0 1    0 1    0   0    1    1   1    1
│ │ │ -                                          2   2         3       2
│ │ │ -                    PolynomialSet => {1, t , t t , t , t , t , t , t t }
│ │ │ -                                          0   0 1   1   0   0   1   0 1
    
│ │ │ +                                                2   2         3       2
│ │ │ +                    PolynomialSet => {t t , 1, t , t t , t , t , t , t }
│ │ │ +                                       0 1      0   0 1   1   0   0   1
│ │ │      		            
    
│ │ │          
    
│ │ │          
    
│ │ │            
    given the ideal of the finite algebra associated to the order function and a list of points
    
│ │ │            
      
│ │ │              
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,63 +45,63 @@
│ │ │ │                      Points => {{0, 0}, {a, a}, {a + 1, a}, {1, a}, {a, a + 1},
│ │ │ │  {a + 1, a + 1}, {1, a + 1}, {0, 1}}
│ │ │ │                                                                 8
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │ -                                             Code => image | 1 0   0   0   0 0
│ │ │ │ -0   0   |
│ │ │ │ -                                                           | 1 a+1 1   a   1 a
│ │ │ │ -a+1 a+1 |
│ │ │ │ -                                                           | 1 a   a+1 a   1
│ │ │ │ -a+1 a+1 1   |
│ │ │ │ -                                                           | 1 1   a   a   1 1
│ │ │ │ -a+1 a   |
│ │ │ │ -                                                           | 1 a+1 a   a+1 1 a
│ │ │ │ -a   1   |
│ │ │ │ -                                                           | 1 a   1   a+1 1
│ │ │ │ -a+1 a   a   |
│ │ │ │ -                                                           | 1 1   a+1 a+1 1 1
│ │ │ │ -a   a+1 |
│ │ │ │ -                                                           | 1 0   0   1   0 0
│ │ │ │ -1   0   |
│ │ │ │ -                                             GeneratorMatrix => | 1 1   1   1
│ │ │ │ +                                             Code => image | 0   1 0   0   0
│ │ │ │ +0 0   0   |
│ │ │ │ +                                                           | a+1 1 a+1 1   a
│ │ │ │ +1 a   a+1 |
│ │ │ │ +                                                           | 1   1 a   a+1 a
│ │ │ │ +1 a+1 a+1 |
│ │ │ │ +                                                           | a   1 1   a   a
│ │ │ │ +1 1   a+1 |
│ │ │ │ +                                                           | 1   1 a+1 a   a+1
│ │ │ │ +1 a   a   |
│ │ │ │ +                                                           | a   1 a   1   a+1
│ │ │ │ +1 a+1 a   |
│ │ │ │ +                                                           | a+1 1 1   a+1 a+1
│ │ │ │ +1 1   a   |
│ │ │ │ +                                                           | 0   1 0   0   1
│ │ │ │ +0 0   1   |
│ │ │ │ +                                             GeneratorMatrix => | 0 a+1 1   a
│ │ │ │ +1   a   a+1 0 |
│ │ │ │ +                                                                | 1 1   1   1
│ │ │ │  1   1   1   1 |
│ │ │ │                                                                  | 0 a+1 a   1
│ │ │ │  a+1 a   1   0 |
│ │ │ │                                                                  | 0 1   a+1 a
│ │ │ │  a   1   a+1 0 |
│ │ │ │                                                                  | 0 a   a   a
│ │ │ │  a+1 a+1 a+1 1 |
│ │ │ │                                                                  | 0 1   1   1
│ │ │ │  1   1   1   0 |
│ │ │ │                                                                  | 0 a   a+1 1
│ │ │ │  a   a+1 1   0 |
│ │ │ │                                                                  | 0 a+1 a+1 a+1
│ │ │ │  a   a   a   1 |
│ │ │ │ -                                                                | 0 a+1 1   a
│ │ │ │ -1   a   a+1 0 |
│ │ │ │ -                                             Generators => {{1, 1, 1, 1, 1, 1,
│ │ │ │ -1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0,
│ │ │ │ -a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a,
│ │ │ │ -a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}, {0, a + 1, 1, a, 1, a, a +
│ │ │ │ -1, 0}}
│ │ │ │ +                                             Generators => {{0, a + 1, 1, a, 1,
│ │ │ │ +a, a + 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0,
│ │ │ │ +1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1,
│ │ │ │ +1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a,
│ │ │ │ +a, a, 1}}
│ │ │ │                                               ParityCheckMatrix => | 1 1 1 1 1 1
│ │ │ │  1 1 |
│ │ │ │                                               ParityCheckRows => {{1, 1, 1, 1,
│ │ │ │  1, 1, 1, 1}}
│ │ │ │                                                  2               3    2        4
│ │ │ │                      VanishingIdeal => ideal (t t  + t t  + t , t  + t  + t , t
│ │ │ │  + t )
│ │ │ │                                                0 1    0 1    0   0    1    1   1
│ │ │ │  1
│ │ │ │ -                                          2   2         3       2
│ │ │ │ -                    PolynomialSet => {1, t , t t , t , t , t , t , t t }
│ │ │ │ -                                          0   0 1   1   0   0   1   0 1
│ │ │ │ +                                                2   2         3       2
│ │ │ │ +                    PolynomialSet => {t t , 1, t , t t , t , t , t , t }
│ │ │ │ +                                       0 1      0   0 1   1   0   0   1
│ │ │ │  ********** ggiivveenn tthhee iiddeeaall ooff tthhee ffiinniittee aallggeebbrraa aassssoocciiaatteedd ttoo tthhee oorrddeerr ffuunnccttiioonn
│ │ │ │  aanndd aa lliisstt ooff ppooiinnttss **********
│ │ │ │      *   Usage:
│ │ │ │              orderCode(I,P,v,d)
│ │ │ │      * Inputs:
│ │ │ │            o I, an _i_d_e_a_l,
│ │ │ │            o P, a _l_i_s_t,
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_ring_lp__Linear__Code_rp.html
│ │ │ @@ -77,29 +77,29 @@
│ │ │                
      i1 : C = hammingCode(2, 3)
│ │ │  
│ │ │                                         7
│ │ │  o1 = LinearCode{AmbientModule => (GF 2)                                                                                   }
│ │ │                  BaseField => GF 2
│ │ │                  cache => CacheTable{}
│ │ │                  Code => image | 1 0 1 1 |
│ │ │ -                              | 1 1 0 1 |
│ │ │                                | 1 1 1 0 |
│ │ │ +                              | 1 1 0 1 |
│ │ │                                | 1 0 0 0 |
│ │ │                                | 0 1 0 0 |
│ │ │                                | 0 0 1 0 |
│ │ │                                | 0 0 0 1 |
│ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │                                     | 0 1 1 0 1 0 0 |
│ │ │ -                                   | 1 0 1 0 0 1 0 |
│ │ │ -                                   | 1 1 0 0 0 0 1 |
│ │ │ -                Generators => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0}, {1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ +                                   | 1 1 0 0 0 1 0 |
│ │ │ +                                   | 1 0 1 0 0 0 1 |
│ │ │ +                Generators => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0}, {1, 1, 0, 0, 0, 1, 0}, {1, 0, 1, 0, 0, 0, 1}}
│ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │ -                                     | 0 0 1 1 1 1 0 |
│ │ │ -                                     | 0 1 0 1 1 0 1 |
│ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0}, {0, 1, 0, 1, 1, 0, 1}}
│ │ │ +                                     | 0 1 0 1 1 1 0 |
│ │ │ +                                     | 0 1 1 0 0 1 1 |
│ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 0, 1, 1, 1, 0}, {0, 1, 1, 0, 0, 1, 1}}
│ │ │  
│ │ │  o1 : LinearCode
      
│ │ │            
│ │ │            
│ │ │                
      i2 : ring(C)
│ │ │  
│ │ │  o2 = GF 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,31 +19,31 @@
│ │ │ │  
│ │ │ │                                         7
│ │ │ │  o1 = LinearCode{AmbientModule => (GF 2)
│ │ │ │  }
│ │ │ │                  BaseField => GF 2
│ │ │ │                  cache => CacheTable{}
│ │ │ │                  Code => image | 1 0 1 1 |
│ │ │ │ -                              | 1 1 0 1 |
│ │ │ │                                | 1 1 1 0 |
│ │ │ │ +                              | 1 1 0 1 |
│ │ │ │                                | 1 0 0 0 |
│ │ │ │                                | 0 1 0 0 |
│ │ │ │                                | 0 0 1 0 |
│ │ │ │                                | 0 0 0 1 |
│ │ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │ │                                     | 0 1 1 0 1 0 0 |
│ │ │ │ -                                   | 1 0 1 0 0 1 0 |
│ │ │ │ -                                   | 1 1 0 0 0 0 1 |
│ │ │ │ +                                   | 1 1 0 0 0 1 0 |
│ │ │ │ +                                   | 1 0 1 0 0 0 1 |
│ │ │ │                  Generators => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0},
│ │ │ │ -{1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ │ +{1, 1, 0, 0, 0, 1, 0}, {1, 0, 1, 0, 0, 0, 1}}
│ │ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │ │ -                                     | 0 0 1 1 1 1 0 |
│ │ │ │ -                                     | 0 1 0 1 1 0 1 |
│ │ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1,
│ │ │ │ -0}, {0, 1, 0, 1, 1, 0, 1}}
│ │ │ │ +                                     | 0 1 0 1 1 1 0 |
│ │ │ │ +                                     | 0 1 1 0 0 1 1 |
│ │ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 0, 1, 1, 1,
│ │ │ │ +0}, {0, 1, 1, 0, 0, 1, 1}}
│ │ │ │  
│ │ │ │  o1 : LinearCode
│ │ │ │  i2 : ring(C)
│ │ │ │  
│ │ │ │  o2 = GF 2
│ │ │ │  
│ │ │ │  o2 : GaloisField
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_syndrome__Decode.html
│ │ │ @@ -137,31 +137,31 @@
│ │ │                 | 0 |    | 0 |
│ │ │                 | 0 |    | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                 | 1 | => | 0 |
│ │ │ -               | 0 |    | 1 |
│ │ │ -               | 1 |    | 0 |
│ │ │ +               | 0 |    | 0 |
│ │ │ +               | 1 |    | 1 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                 | 1 | => | 0 |
│ │ │                 | 1 |    | 0 |
│ │ │ -               | 0 |    | 1 |
│ │ │ -                        | 0 |
│ │ │ +               | 0 |    | 0 |
│ │ │ +                        | 1 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                 | 1 | => | 0 |
│ │ │ +               | 1 |    | 1 |
│ │ │                 | 1 |    | 0 |
│ │ │ -               | 1 |    | 0 |
│ │ │ -                        | 1 |
│ │ │ +                        | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │                          | 0 |
│ │ │  
│ │ │  o7 : HashTable
      
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -70,31 +70,31 @@
│ │ │ │                 | 0 |    | 0 |
│ │ │ │                 | 0 |    | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                 | 1 | => | 0 |
│ │ │ │ -               | 0 |    | 1 |
│ │ │ │ -               | 1 |    | 0 |
│ │ │ │ +               | 0 |    | 0 |
│ │ │ │ +               | 1 |    | 1 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                 | 1 | => | 0 |
│ │ │ │                 | 1 |    | 0 |
│ │ │ │ -               | 0 |    | 1 |
│ │ │ │ -                        | 0 |
│ │ │ │ +               | 0 |    | 0 |
│ │ │ │ +                        | 1 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                 | 1 | => | 0 |
│ │ │ │ +               | 1 |    | 1 |
│ │ │ │                 | 1 |    | 0 |
│ │ │ │ -               | 1 |    | 0 |
│ │ │ │ -                        | 1 |
│ │ │ │ +                        | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │                          | 0 |
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  ********** WWaayyss ttoo uussee ssyynnddrroommeeDDeeccooddee:: **********
│ │ │ │      * syndromeDecode(LinearCode,Matrix,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_vector__Space.html
│ │ │ @@ -73,16 +73,16 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
                    
      i1 : H = hammingCode(2,3);
      
│ │ │        		          
                    
      i2 : vectorSpace H
│ │ │  
│ │ │ -o2 = image | 1 1 0 1 |
│ │ │ -           | 1 0 1 1 |
│ │ │ +o2 = image | 1 0 1 1 |
│ │ │ +           | 1 1 0 1 |
│ │ │             | 1 1 1 0 |
│ │ │             | 1 0 0 0 |
│ │ │             | 0 1 0 0 |
│ │ │             | 0 0 1 0 |
│ │ │             | 0 0 0 1 |
│ │ │  
│ │ │                                       7
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,16 +13,16 @@
│ │ │ │            o a _m_o_d_u_l_e, $V$
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Given a linear code C, this function returns $V$, the vector space spanned by
│ │ │ │  the rows of a generator matrix of C.
│ │ │ │  i1 : H = hammingCode(2,3);
│ │ │ │  i2 : vectorSpace H
│ │ │ │  
│ │ │ │ -o2 = image | 1 1 0 1 |
│ │ │ │ -           | 1 0 1 1 |
│ │ │ │ +o2 = image | 1 0 1 1 |
│ │ │ │ +           | 1 1 0 1 |
│ │ │ │             | 1 1 1 0 |
│ │ │ │             | 1 0 0 0 |
│ │ │ │             | 0 1 0 0 |
│ │ │ │             | 0 0 1 0 |
│ │ │ │             | 0 0 0 1 |
│ │ │ │  
│ │ │ │                                       7
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  Y29ob21DYWxnKE5vcm1hbFRvcmljVmFyaWV0eSk=
│ │ │  #:len=2221
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibG9jYWxseSBzdGFzaGVkIGNvaG9tb2xv
│ │ │  Z3kgdmVjdG9ycyBmcm9tIENvaG9tQ2FsZyIsICJsaW5lbnVtIiA9PiAyODIsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.08307s elapsed
│ │ │ + -- 4.14296s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -265,45 +265,45 @@
│ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │  
│ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │  
│ │ │  o22 : List
│ │ │  
│ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .383663s elapsed
│ │ │ + -- .978784s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 13.5586s elapsed
│ │ │ + -- 11.5264s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
│ │ │  
│ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │  
│ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │  
│ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .404647s elapsed
│ │ │ + -- 1.03898s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .474485s elapsed
│ │ │ - -- .474531s elapsed
│ │ │ + -- .503137s elapsed
│ │ │ + -- .503185s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -263,15 +263,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
      
│ │ │        		          
                    
      i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.08307s elapsed
│ │ │ + -- 4.14296s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -347,23 +347,23 @@
│ │ │  
│ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │  
│ │ │  o22 : List
      
│ │ │        		          
                    
      i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .383663s elapsed
│ │ │ + -- .978784s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
      
│ │ │        		          
                    
      i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 13.5586s elapsed
│ │ │ + -- 11.5264s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
      
│ │ │        		          
                    
      i25 : assert(cohomvec1 == cohomvec2)
      
│ │ │ @@ -373,24 +373,24 @@
│ │ │  
│ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │  
│ │ │  o26 : List
│ │ │        		          
                    
      i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .404647s elapsed
│ │ │ + -- 1.03898s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
      
│ │ │        		          
                    
      i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .474485s elapsed
│ │ │ - -- .474531s elapsed
│ │ │ + -- .503137s elapsed
│ │ │ + -- .503185s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
      
│ │ │        		          
                    
      i29 : assert(cohomvec1 == cohomvec2)
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -182,15 +182,15 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, -1, -1}}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ │ - -- 3.08307s elapsed
│ │ │ │ + -- 4.14296s elapsed
│ │ │ │  
│ │ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -262,42 +262,42 @@
│ │ │ │                         {2, 2, 3, 1, -4, -6} => {{0, 1, 0, 0, 0}, {{1, 1x1*x2}}}
│ │ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │ │  
│ │ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │ │  
│ │ │ │  o22 : List
│ │ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ │ - -- .383663s elapsed
│ │ │ │ + -- .978784s elapsed
│ │ │ │  
│ │ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X
│ │ │ │  (0,0,1,2,0,-1))
│ │ │ │ - -- 13.5586s elapsed
│ │ │ │ + -- 11.5264s elapsed
│ │ │ │  
│ │ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o24 : List
│ │ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │ │  
│ │ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ │ - -- .404647s elapsed
│ │ │ │ + -- 1.03898s elapsed
│ │ │ │  
│ │ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j
│ │ │ │  (X, OO_X(0,0,1,2,-2,-1))
│ │ │ │ - -- .474485s elapsed
│ │ │ │ - -- .474531s elapsed
│ │ │ │ + -- .503137s elapsed
│ │ │ │ + -- .503185s elapsed
│ │ │ │  
│ │ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o28 : List
│ │ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ │ │  _c_o_h_o_m_C_a_l_g computes cohomology vectors by calling CohomCalg. It also stashes
│ │ │ │  it's results in the toric variety's cache table, so computations need not be
│ │ ├── ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  Y29tcGxleHJhbmsoUmluZ0VsZW1lbnQp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV4cmFuayxSaW5nRWxlbWVudCksImNvbXBs
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  RWlzZW5idWRTaGFtYXNoVG90YWwoTW9kdWxlKQ==
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTE2NSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRWlzZW5idWRTaGFtYXNoVG90YWwsTW9kdWxlKSwi
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : len = 10
│ │ │  
│ │ │  o6 = 10
│ │ │  
│ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 9.62565s (cpu); 5.0849s (thread); 0s (gc)
│ │ │ + -- used 13.6272s (cpu); 7.80357s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -140,37 +140,37 @@
│ │ │  i19 : R1 = R/ideal ff
│ │ │  
│ │ │  o19 = R1
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.159768s (cpu); 0.0880756s (thread); 0s (gc)
│ │ │ + -- used 0.163836s (cpu); 0.113938s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : ChainComplex
│ │ │  
│ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 1.59684s (cpu); 0.833997s (thread); 0s (gc)
│ │ │ + -- used 2.10129s (cpu); 1.2888s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │  
│ │ │  o21 : ChainComplex
│ │ │  
│ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 1.56294s (cpu); 0.779923s (thread); 0s (gc)
│ │ │ + -- used 1.96431s (cpu); 1.21054s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
│ │ │ @@ -1,17 +1,17 @@
│ │ │  -- -*- M2-comint -*- hash: 1731741365432311614
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.773258s (cpu); 0.449787s (thread); 0s (gc)
│ │ │ - -- used 0.33458s (cpu); 0.202346s (thread); 0s (gc)
│ │ │ - -- used 0.000140253s (cpu); 2.806e-06s (thread); 0s (gc)
│ │ │ + -- used 1.46856s (cpu); 0.546607s (thread); 0s (gc)
│ │ │ + -- used 0.531829s (cpu); 0.241852s (thread); 0s (gc)
│ │ │ + -- used 0.000118833s (cpu); 1.763e-06s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  4
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 1731741366884359753
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 1.25768s (cpu); 0.664277s (thread); 0s (gc)
│ │ │ + -- used 2.60149s (cpu); 1.17297s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.648202s (cpu); 0.372689s (thread); 0s (gc)
│ │ │ + -- used 1.81156s (cpu); 0.519884s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.26515s (cpu); 0.12613s (thread); 0s (gc)
│ │ │ + -- used 0.411676s (cpu); 0.250349s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │            
                    
      i6 : len = 10
│ │ │  
│ │ │  o6 = 10
      
│ │ │        		          
                    
      i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 9.62565s (cpu); 5.0849s (thread); 0s (gc)
│ │ │ + -- used 13.6272s (cpu); 7.80357s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -259,26 +259,26 @@
│ │ │  
│ │ │  o19 = R1
│ │ │  
│ │ │  o19 : QuotientRing
      
│ │ │        		          
                    
      i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.159768s (cpu); 0.0880756s (thread); 0s (gc)
│ │ │ + -- used 0.163836s (cpu); 0.113938s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : ChainComplex
      
│ │ │        		          
                    
      i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 1.59684s (cpu); 0.833997s (thread); 0s (gc)
│ │ │ + -- used 2.10129s (cpu); 1.2888s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -288,15 +288,15 @@
│ │ │          
      
│ │ │          
      
│ │ │            
      The function also deals correctly with complexes F where min F is not 0:
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 1.56294s (cpu); 0.779923s (thread); 0s (gc)
│ │ │ + -- used 1.96431s (cpu); 1.21054s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : ChainComplex
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o5 = R
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : len = 10
│ │ │ │  
│ │ │ │  o6 = 10
│ │ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ │ - -- used 9.62565s (cpu); 5.0849s (thread); 0s (gc)
│ │ │ │ + -- used 13.6272s (cpu); 7.80357s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S
│ │ │ │  \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /
│ │ │ │  S   \68     /    S   \76
│ │ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |-------
│ │ │ │  -|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |-
│ │ │ │  -------|   <-- |--------|
│ │ │ │ @@ -167,36 +167,36 @@
│ │ │ │  o18 : Matrix R  <-- R
│ │ │ │  i19 : R1 = R/ideal ff
│ │ │ │  
│ │ │ │  o19 = R1
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ │ - -- used 0.159768s (cpu); 0.0880756s (thread); 0s (gc)
│ │ │ │ + -- used 0.163836s (cpu); 0.113938s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        0       1       2        3        4
│ │ │ │  
│ │ │ │  o20 : ChainComplex
│ │ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ │ - -- used 1.59684s (cpu); 0.833997s (thread); 0s (gc)
│ │ │ │ + -- used 2.10129s (cpu); 1.2888s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │ │  
│ │ │ │        0         1         2          3          4
│ │ │ │  
│ │ │ │  o21 : ChainComplex
│ │ │ │  The function also deals correctly with complexes F where min F is not 0:
│ │ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ │ - -- used 1.56294s (cpu); 0.779923s (thread); 0s (gc)
│ │ │ │ + -- used 1.96431s (cpu); 1.21054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │            
                    
      i1 : setRandomSeed 0
│ │ │  
│ │ │  o1 = 0
      
│ │ │        		          
                    
      i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.773258s (cpu); 0.449787s (thread); 0s (gc)
│ │ │ - -- used 0.33458s (cpu); 0.202346s (thread); 0s (gc)
│ │ │ - -- used 0.000140253s (cpu); 2.806e-06s (thread); 0s (gc)
│ │ │ + -- used 1.46856s (cpu); 0.546607s (thread); 0s (gc)
│ │ │ + -- used 0.531829s (cpu); 0.241852s (thread); 0s (gc)
│ │ │ + -- used 0.000118833s (cpu); 1.763e-06s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  4
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,17 +18,17 @@
│ │ │ │  = S/(d-th powers of the variables), with full complexity (=c); that is, for an
│ │ │ │  appropriate syzygy M of M0 = R/(m1+m2) where m1 and m2 are monomials of the
│ │ │ │  same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ │ - -- used 0.773258s (cpu); 0.449787s (thread); 0s (gc)
│ │ │ │ - -- used 0.33458s (cpu); 0.202346s (thread); 0s (gc)
│ │ │ │ - -- used 0.000140253s (cpu); 2.806e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.46856s (cpu); 0.546607s (thread); 0s (gc)
│ │ │ │ + -- used 0.531829s (cpu); 0.241852s (thread); 0s (gc)
│ │ │ │ + -- used 0.000118833s (cpu); 1.763e-06s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │  4
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -82,25 +82,25 @@
│ │ │            
                    
      i1 : setRandomSeed 0
│ │ │  
│ │ │  o1 = 0
      
│ │ │        		          
                    
      i2 : twoMonomials(2,3)
│ │ │ - -- used 1.25768s (cpu); 0.664277s (thread); 0s (gc)
│ │ │ + -- used 2.60149s (cpu); 1.17297s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.648202s (cpu); 0.372689s (thread); 0s (gc)
│ │ │ + -- used 1.81156s (cpu); 0.519884s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.26515s (cpu); 0.12613s (thread); 0s (gc)
│ │ │ + -- used 0.411676s (cpu); 0.250349s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
      
│ │ │        		          
      
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,25 +20,25 @@
│ │ │ │  monomials in R = S/(d-th powers of the variables), with full complexity (=c);
│ │ │ │  that is, for an appropriate syzygy M of M0 = R/(m1, m2) where m1 and m2 are
│ │ │ │  monomials of the same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : twoMonomials(2,3)
│ │ │ │ - -- used 1.25768s (cpu); 0.664277s (thread); 0s (gc)
│ │ │ │ + -- used 2.60149s (cpu); 1.17297s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.648202s (cpu); 0.372689s (thread); 0s (gc)
│ │ │ │ + -- used 1.81156s (cpu); 0.519884s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.26515s (cpu); 0.12613s (thread); 0s (gc)
│ │ │ │ + -- used 0.411676s (cpu); 0.250349s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ttwwooMMoonnoommiiaallss:: **********
│ │ │ │      * twoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  SG9tKE1hdHJpeCxDb21wbGV4KQ==
│ │ │  #:len=295
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTI3MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoSG9tLE1hdHJpeCxDb21wbGV4KSwiSG9tKE1hdHJp
│ │ ├── ./usr/share/doc/Macaulay2/ConformalBlocks/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsRGl2aXNvck0wbmJhcg==
│ │ │  #:len=1172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY2xhc3Mgb2YgdGhl
│ │ │  IGNhbm9uaWNhbCBkaXZpc29yIG9uIHRoZSBtb2R1bGkgc3BhY2Ugb2Ygc3RhYmxlIG4tcG9pbnRl
│ │ ├── ./usr/share/doc/Macaulay2/ConvexInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  bUNvbnZleEh1bGxGYWNlcyguLi4sdG9GaWxlPT4uLi4p
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzU3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttQ29udmV4SHVsbEZhY2VzLHRvRmlsZV0sIm1Db252
│ │ ├── ./usr/share/doc/Macaulay2/ConwayPolynomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Y29ud2F5UG9seW5vbWlhbA==
│ │ │  #:len=1659
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvdmlkZSBhIENvbndheSBwb2x5bm9t
│ │ │  aWFsIiwgRGVzY3JpcHRpb24gPT4gKERJVntIRUFERVIyeyJTeW5vcHNpcyJ9LFVMe0xJe0RMeyJj
│ │ ├── ./usr/share/doc/Macaulay2/CorrespondenceScrolls/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=52
│ │ │  cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ==
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzcwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/CotangentSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  TGFiZWxz
│ │ │  #:len=183
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│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJMYWJlbHMi
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY3JpYmUoUmF0aW9uYWxNYXAp
│ │ │  #:len=1096
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGVzY3JpYmUgYSByYXRpb25hbCBtYXAi
│ │ │  LCAibGluZW51bSIgPT4gOTgzLCBJbnB1dHMgPT4ge1NQQU57VFR7InBoaSJ9LCIsICIsU1BBTnsi
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.59361s (cpu); 0.953677s (thread); 0s (gc)
│ │ │ + -- used 2.33275s (cpu); 1.5143s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ - -- used 1.14112s (cpu); 0.844219s (thread); 0s (gc)
│ │ │ + -- used 1.7595s (cpu); 1.10389s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │ @@ -62,26 +62,26 @@
│ │ │          0,2 1,3    0,1 2,3
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i9 : time ChernClass G
│ │ │ - -- used 0.303675s (cpu); 0.149817s (thread); 0s (gc)
│ │ │ + -- used 0.265939s (cpu); 0.207493s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
│ │ │  
│ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ - -- used 0.00727497s (cpu); 0.0100965s (thread); 0s (gc)
│ │ │ + -- used 0.342866s (cpu); 0.0600963s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
│ │ │ @@ -1,56 +1,56 @@
│ │ │  -- -*- M2-comint -*- hash: 10433409267944421825
│ │ │  
│ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │  
│ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00399965s (cpu); 0.00392251s (thread); 0s (gc)
│ │ │ + -- used 0.00400782s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │  
│ │ │              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
│ │ │  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 })
│ │ │            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
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
│ │ │  
│ │ │  i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.0437451s (cpu); 0.0440403s (thread); 0s (gc)
│ │ │ + -- used 0.0320008s (cpu); 0.0340255s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
│ │ │  
│ │ │  i4 : time degreeMap phi
│ │ │ - -- used 0.0829531s (cpu); 0.0341009s (thread); 0s (gc)
│ │ │ + -- used 0.124868s (cpu); 0.0674956s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.506956s (cpu); 0.382073s (thread); 0s (gc)
│ │ │ + -- used 0.45759s (cpu); 0.397078s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0565176s (cpu); 0.0573171s (thread); 0s (gc)
│ │ │ + -- used 0.0433452s (cpu); 0.0461662s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.000189065s (cpu); 0.00204299s (thread); 0s (gc)
│ │ │ + -- used 0.00105715s (cpu); 0.00171203s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              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
│ │ │  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 })
│ │ │            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
│ │ │                              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
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                                                             ------[x ..x ]
│ │ │                 ZZ                                                          300007  0   9
│ │ │  o7 : RingMap ------[t ..t ] <-- ----------------------------------------------------------------------------------------------------
│ │ │               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 )
│ │ │                                    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
│ │ │  
│ │ │  i8 : time psi = inverseMap phi
│ │ │ - -- used 0.566795s (cpu); 0.41755s (thread); 0s (gc)
│ │ │ + -- used 0.599868s (cpu); 0.4454s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       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
│ │ │  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 })
│ │ │            (x x  - x x  + x x , 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
│ │ │              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
│ │ │ @@ -76,32 +76,32 @@
│ │ │                                                          ------[x ..x ]
│ │ │                                                          300007  0   9                                                   ZZ
│ │ │  o8 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[t ..t ]
│ │ │               (x x  - x x  + x x , 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
│ │ │                 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
│ │ │  
│ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.0101323s (cpu); 0.00946356s (thread); 0s (gc)
│ │ │ + -- used 0.00798632s (cpu); 0.00712739s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.297917s (cpu); 0.233828s (thread); 0s (gc)
│ │ │ + -- used 0.255736s (cpu); 0.197823s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 4.99657s (cpu); 4.27726s (thread); 0s (gc)
│ │ │ + -- used 5.73993s (cpu); 5.35784s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.0011253s (cpu); 0.00197141s (thread); 0s (gc)
│ │ │ + -- used 0.00200211s (cpu); 0.00186545s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -147,15 +147,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │  
│ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.155471s (cpu); 0.0755467s (thread); 0s (gc)
│ │ │ + -- used 0.193117s (cpu); 0.0663768s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -217,15 +217,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i14 : time phi^(-1)
│ │ │ - -- used 0.504825s (cpu); 0.425036s (thread); 0s (gc)
│ │ │ + -- used 0.435332s (cpu); 0.435809s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -275,71 +275,71 @@
│ │ │                         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
│ │ │                          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
│ │ │                        }
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
│ │ │  
│ │ │  i15 : time degrees phi^(-1)
│ │ │ - -- used 0.355466s (cpu); 0.270355s (thread); 0s (gc)
│ │ │ + -- used 0.489651s (cpu); 0.36645s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.120907s (cpu); 0.0437002s (thread); 0s (gc)
│ │ │ + -- used 0.0734552s (cpu); 0.0218007s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00320772s (cpu); 0.00316311s (thread); 0s (gc)
│ │ │ + -- used 1.5298e-05s (cpu); 0.00351931s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i18 : time describe phi^(-1)
│ │ │ - -- used 0.00643296s (cpu); 0.00993896s (thread); 0s (gc)
│ │ │ + -- used 0.00846573s (cpu); 0.0123656s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │        number of minimal representatives: 1
│ │ │        dimension base locus: 4
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0084534s (cpu); 0.00881838s (thread); 0s (gc)
│ │ │ + -- used 0.0520925s (cpu); 0.052059s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i21 : time degrees f
│ │ │ - -- used 1.19108s (cpu); 0.901679s (thread); 0s (gc)
│ │ │ + -- used 1.50652s (cpu); 1.26241s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 0.000190698s (cpu); 1.3635e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000143619s (cpu); 1.1592e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.000926578s (cpu); 0.00142987s (thread); 0s (gc)
│ │ │ + -- used 0.000786354s (cpu); 0.00119327s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out
│ │ │ @@ -3,18 +3,18 @@
│ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │  
│ │ │                  ZZ
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i2 : time EulerCharacteristic I
│ │ │ - -- used 0.303459s (cpu); 0.15511s (thread); 0s (gc)
│ │ │ + -- used 0.30337s (cpu); 0.137229s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ - -- used 0.010188s (cpu); 0.011592s (thread); 0s (gc)
│ │ │ + -- used 0.0473119s (cpu); 0.0110623s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = 10
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i4 : time phi! ;
│ │ │ - -- used 0.135191s (cpu); 0.0762212s (thread); 0s (gc)
│ │ │ + -- used 0.514492s (cpu); 0.156149s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │  
│ │ │  i5 : describe phi
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │ @@ -37,15 +37,15 @@
│ │ │  
│ │ │  o8 = rational map defined by forms of degree 2
│ │ │       source variety: PP^4
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i9 : time phi! ;
│ │ │ - -- used 0.133287s (cpu); 0.0709123s (thread); 0s (gc)
│ │ │ + -- used 0.0936499s (cpu); 0.0318555s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i10 : describe phi
│ │ │  
│ │ │  o10 = rational map defined by forms of degree 2
│ │ │        source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │       - a*c + e         - b*c + f
│ │ │       ----------*v, x + ----------*v)
│ │ │        d*e - a*f         d*e - a*f
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │  
│ │ │  i6 : time phi^** q
│ │ │ - -- used 0.152268s (cpu); 0.150588s (thread); 0s (gc)
│ │ │ + -- used 0.289867s (cpu); 0.227233s (thread); 0s (gc)
│ │ │  
│ │ │                  -e        -d        -c        -b        -a
│ │ │  o6 = ideal (u + --*v, t + --*v, z + --*v, y + --*v, x + --*v)
│ │ │                   f         f         f         f         f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
│ │ │ @@ -47,49 +47,49 @@
│ │ │                                                                            P7
│ │ │  o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
│ │ │                 2 2                2 2                                        2 2                                                    2 2
│ │ │                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
│ │ │                 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
│ │ │  
│ │ │  i4 : time SegreClass X
│ │ │ - -- used 0.662143s (cpu); 0.428239s (thread); 0s (gc)
│ │ │ + -- used 0.70577s (cpu); 0.544501s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.426867s (cpu); 0.277592s (thread); 0s (gc)
│ │ │ + -- used 0.505052s (cpu); 0.447735s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ - -- used 0.0232568s (cpu); 0.0222927s (thread); 0s (gc)
│ │ │ + -- used 0.0738806s (cpu); 0.0177716s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ - -- used 0.0972585s (cpu); 0.0986145s (thread); 0s (gc)
│ │ │ + -- used 0.37345s (cpu); 0.205008s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │ @@ -105,15 +105,15 @@
│ │ │         ZZ
│ │ │  o9 = ------[x ..x ]
│ │ │       100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  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)
│ │ │ - -- used 0.0519939s (cpu); 0.0537215s (thread); 0s (gc)
│ │ │ + -- used 0.0961548s (cpu); 0.0958205s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  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 })
│ │ │             (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
│ │ │               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
│ │ │ @@ -122,15 +122,15 @@
│ │ │                                                           ------[y ..y ]
│ │ │                                                           100003  0   9                                                   ZZ
│ │ │  o10 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[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 )     100003  0   6
│ │ │                  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
│ │ │  
│ │ │  i11 : time SegreClass phi
│ │ │ - -- used 0.423874s (cpu); 0.279274s (thread); 0s (gc)
│ │ │ + -- used 0.441667s (cpu); 0.332601s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -150,27 +150,27 @@
│ │ │                                                            100003  0   9
│ │ │  o12 : Ideal of ----------------------------------------------------------------------------------------------------
│ │ │                 (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 )
│ │ │                   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
│ │ │  
│ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.423803s (cpu); 0.271486s (thread); 0s (gc)
│ │ │ + -- used 0.44219s (cpu); 0.386603s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
│ │ │  
│ │ │  i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.33445s (cpu); 0.978665s (thread); 0s (gc)
│ │ │ + -- used 1.4686s (cpu); 1.05555s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out
│ │ │ @@ -17,32 +17,32 @@
│ │ │  
│ │ │  o3 = QQ[u ..u ]
│ │ │           0   5
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.00118846s (cpu); 0.000368992s (thread); 0s (gc)
│ │ │ + -- used 0.00221773s (cpu); 0.000294312s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │  
│ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.299552s (cpu); 0.15716s (thread); 0s (gc)
│ │ │ + -- used 0.229326s (cpu); 0.169096s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.40278s (cpu); 0.331735s (thread); 0s (gc)
│ │ │ + -- used 0.430315s (cpu); 0.369564s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -113,48 +113,48 @@
│ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │                   ZZ
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
│ │ │  
│ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.151815s (cpu); 0.0652653s (thread); 0s (gc)
│ │ │ + -- used 0.0639987s (cpu); 0.0649795s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 3.0254s (cpu); 1.69288s (thread); 0s (gc)
│ │ │ + -- used 3.44941s (cpu); 1.95221s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 0.000199554s (cpu); 2.7712e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000124975s (cpu); 2.712e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 5.28753s (cpu); 2.90553s (thread); 0s (gc)
│ │ │ + -- used 6.23056s (cpu); 3.31739s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {28963, 31975, -30172, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {28963, 31975, -30172, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 4.10257s (cpu); 2.25715s (thread); 0s (gc)
│ │ │ + -- used 5.28201s (cpu); 3.46697s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │                        x x  - x x
│ │ │                         1 2    0 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
│ │ │  
│ │ │  i3 : time psi = approximateInverseMap phi
│ │ │ - -- used 0.218667s (cpu); 0.169055s (thread); 0s (gc)
│ │ │ + -- used 0.372312s (cpu); 0.254928s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 10
│ │ │  -- approximateInverseMap: step 2 of 10
│ │ │  -- approximateInverseMap: step 3 of 10
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ @@ -106,15 +106,15 @@
│ │ │                       }
│ │ │  
│ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ - -- used 0.218999s (cpu); 0.146713s (thread); 0s (gc)
│ │ │ + -- used 0.283764s (cpu); 0.228417s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │ @@ -189,15 +189,15 @@
│ │ │                         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
│ │ │                       }
│ │ │  
│ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │  
│ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ - -- used 2.28934s (cpu); 1.6958s (thread); 0s (gc)
│ │ │ + -- used 2.89445s (cpu); 2.60374s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ @@ -254,15 +254,15 @@
│ │ │  i9 : -- but...
│ │ │       phi * psi == 1
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ - -- used 3.34434s (cpu); 2.48164s (thread); 0s (gc)
│ │ │ + -- used 3.4615s (cpu); 3.05447s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
│ │ │ @@ -9,27 +9,27 @@
│ │ │                                   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      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
│ │ │  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 })
│ │ │                                   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
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │  
│ │ │  i5 : time degreeMap phi
│ │ │ - -- used 0.131475s (cpu); 0.0571066s (thread); 0s (gc)
│ │ │ + -- used 0.228985s (cpu); 0.0687556s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │  
│ │ │                                   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
│ │ │  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 })
│ │ │                                   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
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │  
│ │ │  i7 : time degreeMap phi'
│ │ │ - -- used 0.763454s (cpu); 0.537536s (thread); 0s (gc)
│ │ │ + -- used 0.716216s (cpu); 0.658864s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out
│ │ │ @@ -5,14 +5,14 @@
│ │ │  o2 : Ideal of P6
│ │ │  
│ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.00296865s (cpu); 0.000930424s (thread); 0s (gc)
│ │ │ + -- used 0.000591067s (cpu); 0.000562725s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │                        - x  + x x
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.108255s (cpu); 0.0343845s (thread); 0s (gc)
│ │ │ + -- used 0.0789011s (cpu); 0.0185917s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │                                    ZZ                                 ZZ
│ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
│ │ │                                  190181  0   1   2   3   4          190181  0   1   2   3   4   5
│ │ │ @@ -173,15 +173,15 @@
│ │ │  i8 : projectiveDegrees p2
│ │ │  
│ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : time g = graph p2;
│ │ │ - -- used 0.0283961s (cpu); 0.0275265s (thread); 0s (gc)
│ │ │ + -- used 0.0862843s (cpu); 0.0269329s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │  
│ │ │  i11 : g_1;
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │                        x  - x x
│ │ │                         1    0 3
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i3 : time ideal phi
│ │ │ - -- used 0.00399819s (cpu); 0.00324924s (thread); 0s (gc)
│ │ │ + -- used 0.00174627s (cpu); 0.00298329s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -108,15 +108,15 @@
│ │ │                        y
│ │ │                         4
│ │ │                       }
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
│ │ │  
│ │ │  i6 : time ideal phi'
│ │ │ - -- used 0.0900955s (cpu); 0.087187s (thread); 0s (gc)
│ │ │ + -- used 0.244771s (cpu); 0.180884s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │                        w w  - w w  + w w
│ │ │                         2 4    1 5    0 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │  
│ │ │  i2 : time psi = inverseMap phi
│ │ │ - -- used 0.158746s (cpu); 0.0885499s (thread); 0s (gc)
│ │ │ + -- used 0.333669s (cpu); 0.124609s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -158,15 +158,15 @@
│ │ │  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 })
│ │ │                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
│ │ │  
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ │  
│ │ │  i5 : time psi = inverseMap phi
│ │ │ - -- used 0.312892s (cpu); 0.187463s (thread); 0s (gc)
│ │ │ + -- used 0.465234s (cpu); 0.344834s (thread); 0s (gc)
│ │ │  
│ │ │  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  })
│ │ │                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
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │                        - -----x  - ---------x x  + -----------x x  + -----------x x  - --------x  - -------x x  + -----------x x x  + -----------x x x  + ----------x x  + --------x x  + --------x x x  - -----------x x  - -------x x  - ----------x x  - -------x  - ------x x  + ----------x x x  + -----------x x x  - ----------x x  + ---------x x x  + ----------x x x x  + -----------x x x  + ----------x x x  + ------------x x x  + ---------x x  - ---------x x  - -----------x x x  - -----------x x  - ----------x x x  + ----------x x x  - ---------x x  + -------x x  - --------x x  + ----------x x  + ------x  - ---------x x  - -----------x x x  + ------------x x x  + ----------x x  - -----------x x x  - -------------x x x x  + ------------x x x  - ----------x x x  - -----------x x x  - -------x x  + -----------x x x  + ------------x x x x  + ------------x x x  + -----------x x x x  + --------------x x x x  + ---------x x x  - ------------x x x  - ------------x x x  - ------------x x x  + ----------x x  - ----------x x  - ------------x x x  + -----------x x  - ----------x x x  - ------------x x x  - ---------x x  + ------------x x x  - --------x x x  + ------------x x x  + ----------x x  - --------x x  - ---------x x  - ---------x x  + ---------x x  - -----x
│ │ │                           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
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i3 : time inverse phi
│ │ │ - -- used 0.154532s (cpu); 0.0707467s (thread); 0s (gc)
│ │ │ + -- used 0.166999s (cpu); 0.109695s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out
│ │ │ @@ -40,18 +40,18 @@
│ │ │                        - t  + t t
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time isBirational phi
│ │ │ - -- used 0.0199621s (cpu); 0.0179823s (thread); 0s (gc)
│ │ │ + -- used 0.0806576s (cpu); 0.0265241s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ - -- used 0.0128589s (cpu); 0.0142427s (thread); 0s (gc)
│ │ │ + -- used 0.0406175s (cpu); 0.0115317s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : P8 = ZZ/101[x_0..x_8];
│ │ │  
│ │ │  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}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 2.25139s (cpu); 1.87155s (thread); 0s (gc)
│ │ │ + -- used 3.58107s (cpu); 3.34407s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │ @@ -20,13 +20,13 @@
│ │ │  o5 : Ideal of P7
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 4.07363s (cpu); 2.75303s (thread); 0s (gc)
│ │ │ + -- used 2.76432s (cpu); 2.14498s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
│ │ │ @@ -6,23 +6,23 @@
│ │ │  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 })
│ │ │                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
│ │ │  
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │  
│ │ │  i2 : time kernel(phi,1)
│ │ │ - -- used 0.115348s (cpu); 0.0365182s (thread); 0s (gc)
│ │ │ + -- used 0.0119988s (cpu); 0.0142586s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 0.474169s (cpu); 0.310819s (thread); 0s (gc)
│ │ │ + -- used 0.498382s (cpu); 0.439796s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │                8           9
│ │ │  
│ │ │                   ZZ
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i3 : time parametrize L
│ │ │ - -- used 0.00400026s (cpu); 0.00430712s (thread); 0s (gc)
│ │ │ + -- used 0.00400077s (cpu); 0.004189s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -116,15 +116,15 @@
│ │ │               5 9           6 9           7 9           8 9           9
│ │ │  
│ │ │                   ZZ
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i5 : time parametrize Q
│ │ │ - -- used 0.567237s (cpu); 0.347295s (thread); 0s (gc)
│ │ │ + -- used 0.730993s (cpu); 0.517449s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 3560583829489988690
│ │ │  
│ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │  
│ │ │  i2 : time p = point source f
│ │ │ - -- used 0.201188s (cpu); 0.137863s (thread); 0s (gc)
│ │ │ + -- used 0.218876s (cpu); 0.170155s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -20,12 +20,12 @@
│ │ │                                                             -----[y ..y  ]
│ │ │                                                             33331  0   11
│ │ │  o2 : Ideal of -------------------------------------------------------------------------------------------------------
│ │ │                (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 )
│ │ │                  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
│ │ │  
│ │ │  i3 : time p == f^* f p
│ │ │ - -- used 0.186193s (cpu); 0.119467s (thread); 0s (gc)
│ │ │ + -- used 0.22745s (cpu); 0.173947s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  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})
│ │ │                       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
│ │ │  
│ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │                          0   4                 0   5
│ │ │  
│ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │ - -- used 0.0182582s (cpu); 0.0155028s (thread); 0s (gc)
│ │ │ + -- used 0.0517608s (cpu); 0.0124244s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │ @@ -29,15 +29,15 @@
│ │ │                GF 109561[x ..x ]
│ │ │                           0   5
│ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │               x x  - x x  + x x                 0   4
│ │ │                2 3    1 4    0 5
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │ - -- used 0.00823188s (cpu); 0.0103648s (thread); 0s (gc)
│ │ │ + -- used 0.0593251s (cpu); 0.0112806s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │ @@ -48,21 +48,21 @@
│ │ │            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
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 0.00337059s (cpu); 4.1208e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00268412s (cpu); 2.9916e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 9.9857e-05s (cpu); 2.2512e-05s (thread); 0s (gc)
│ │ │ + -- used 7.5101e-05s (cpu); 2.0058e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  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);
│ │ │  
│ │ │                  ZZ
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │  
│ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.184482s (cpu); 0.107539s (thread); 0s (gc)
│ │ │ + -- used 0.115708s (cpu); 0.116323s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │                     0         1         2         3        4         5
│ │ │  
│ │ │  o4 : Ideal of X
│ │ │  
│ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │  
│ │ │  i6 : time phi = rationalMap D
│ │ │ - -- used 0.0267469s (cpu); 0.0261443s (thread); 0s (gc)
│ │ │ + -- used 0.0238955s (cpu); 0.02544s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -123,13 +123,13 @@
│ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │                       }
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │  
│ │ │  i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.26146s (cpu); 0.63888s (thread); 0s (gc)
│ │ │ + -- used 2.58621s (cpu); 0.717812s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330846641081
│ │ │  
│ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.18389s (cpu); 0.973906s (thread); 0s (gc)
│ │ │ + -- used 1.44067s (cpu); 1.20823s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1730018912715498288
│ │ │  
│ │ │  i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0895158s (cpu); 0.0880504s (thread); 0s (gc)
│ │ │ + -- used 0.119999s (cpu); 0.120048s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -62,15 +62,15 @@
│ │ │                        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
│ │ │                          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
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.0199392s (cpu); 0.018788s (thread); 0s (gc)
│ │ │ + -- used 0.0561777s (cpu); 0.0553741s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1729200582376678705
│ │ │  
│ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.067479s (cpu); 0.065041s (thread); 0s (gc)
│ │ │ + -- used 0.0519894s (cpu); 0.051045s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -50,15 +50,15 @@
│ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.00850034s (cpu); 0.00606449s (thread); 0s (gc)
│ │ │ + -- used 0.00333284s (cpu); 0.00519894s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out
│ │ │ @@ -7,34 +7,34 @@
│ │ │  i2 : str = toExternalString phi;
│ │ │  
│ │ │  i3 : #str
│ │ │  
│ │ │  o3 = 6927
│ │ │  
│ │ │  i4 : time phi' = value str;
│ │ │ - -- used 0.0200011s (cpu); 0.0206321s (thread); 0s (gc)
│ │ │ + -- used 0.291917s (cpu); 0.0799198s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00473485s (cpu); 0.00488245s (thread); 0s (gc)
│ │ │ + -- used 0.00300748s (cpu); 0.00441817s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │       number of minimal representatives: 1
│ │ │       dimension base locus: 1
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
│ │ │  
│ │ │  i6 : time describe inverse phi'
│ │ │ - -- used 0.00350015s (cpu); 0.00391244s (thread); 0s (gc)
│ │ │ + -- used 0.0022602s (cpu); 0.00354625s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -96,27 +96,27 @@
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.59361s (cpu); 0.953677s (thread); 0s (gc)
│ │ │ + -- used 2.33275s (cpu); 1.5143s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ - -- used 1.14112s (cpu); 0.844219s (thread); 0s (gc)
│ │ │ + -- used 1.7595s (cpu); 1.10389s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │ @@ -154,27 +154,27 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │            
│ │ │            
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.303675s (cpu); 0.149817s (thread); 0s (gc)
│ │ │ + -- used 0.265939s (cpu); 0.207493s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
    
│ │ │            
│ │ │            
│ │ │                
    i10 : time ChernClass(G,Certify=>true)
│ │ │ - -- used 0.00727497s (cpu); 0.0100965s (thread); 0s (gc)
│ │ │ + -- used 0.342866s (cpu); 0.0600963s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,25 +40,25 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 1.59361s (cpu); 0.953677s (thread); 0s (gc)
│ │ │ │ + -- used 2.33275s (cpu); 1.5143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │ - -- used 1.14112s (cpu); 0.844219s (thread); 0s (gc)
│ │ │ │ + -- used 1.7595s (cpu); 1.10389s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │ @@ -89,25 +89,25 @@
│ │ │ │          0,2 1,3    0,1 2,3
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i9 : time ChernClass G
│ │ │ │ - -- used 0.303675s (cpu); 0.149817s (thread); 0s (gc)
│ │ │ │ + -- used 0.265939s (cpu); 0.207493s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          9      8      7      6      5      4     3
│ │ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o9 : -----
│ │ │ │         10
│ │ │ │        H
│ │ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ │ - -- used 0.00727497s (cpu); 0.0100965s (thread); 0s (gc)
│ │ │ │ + -- used 0.342866s (cpu); 0.0600963s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -86,21 +86,21 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │            
│ │ │            
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.303459s (cpu); 0.15511s (thread); 0s (gc)
│ │ │ + -- used 0.30337s (cpu); 0.137229s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ - -- used 0.010188s (cpu); 0.011592s (thread); 0s (gc)
│ │ │ + -- used 0.0473119s (cpu); 0.0110623s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = 10
    
│ │ │            
│ │ │          
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,19 +32,19 @@
│ │ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i2 : time EulerCharacteristic I
│ │ │ │ - -- used 0.303459s (cpu); 0.15511s (thread); 0s (gc)
│ │ │ │ + -- used 0.30337s (cpu); 0.137229s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │ - -- used 0.010188s (cpu); 0.011592s (thread); 0s (gc)
│ │ │ │ + -- used 0.0473119s (cpu); 0.0110623s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    │ │ │ │ │ │ │ │ │ 
    i4 : time phi! ;
│ │ │ - -- used 0.135191s (cpu); 0.0762212s (thread); 0s (gc)
│ │ │ + -- used 0.514492s (cpu); 0.156149s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    │ │ │ │ │ │ │ │ │ 
    i5 : describe phi
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 2
│ │ │ @@ -115,15 +115,15 @@
│ │ │  o8 = rational map defined by forms of degree 2
│ │ │       source variety: PP^4
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    │ │ │ │ │ │ │ │ │ 
    i9 : time phi! ;
│ │ │ - -- used 0.133287s (cpu); 0.0709123s (thread); 0s (gc)
│ │ │ + -- used 0.0936499s (cpu); 0.0318555s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
    │ │ │ │ │ │ │ │ │ 
    i10 : describe phi
│ │ │  
│ │ │  o10 = rational map defined by forms of degree 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i3 : describe phi
│ │ │ │  
│ │ │ │  o3 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i4 : time phi! ;
│ │ │ │ - -- used 0.135191s (cpu); 0.0762212s (thread); 0s (gc)
│ │ │ │ + -- used 0.514492s (cpu); 0.156149s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │ │  i5 : describe phi
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  i8 : describe phi
│ │ │ │  
│ │ │ │  o8 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i9 : time phi! ;
│ │ │ │ - -- used 0.133287s (cpu); 0.0709123s (thread); 0s (gc)
│ │ │ │ + -- used 0.0936499s (cpu); 0.0318555s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │       ----------*v, x + ----------*v)
│ │ │        d*e - a*f         d*e - a*f
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    │ │ │ │ │ │ │ │ │ 
    i6 : time phi^** q
│ │ │ - -- used 0.152268s (cpu); 0.150588s (thread); 0s (gc)
│ │ │ + -- used 0.289867s (cpu); 0.227233s (thread); 0s (gc)
│ │ │  
│ │ │                  -e        -d        -c        -b        -a
│ │ │  o6 = ideal (u + --*v, t + --*v, z + --*v, y + --*v, x + --*v)
│ │ │                   f         f         f         f         f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -83,15 +83,15 @@ │ │ │ │ 2 2 │ │ │ │ - a*c + e - b*c + f │ │ │ │ ----------*v, x + ----------*v) │ │ │ │ d*e - a*f d*e - a*f │ │ │ │ │ │ │ │ o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v] │ │ │ │ i6 : time phi^** q │ │ │ │ - -- used 0.152268s (cpu); 0.150588s (thread); 0s (gc) │ │ │ │ + -- used 0.289867s (cpu); 0.227233s (thread); 0s (gc) │ │ │ │ │ │ │ │ -e -d -c -b -a │ │ │ │ o6 = ideal (u + --*v, t + --*v, z + --*v, y + --*v, x + --*v) │ │ │ │ f f f f f │ │ │ │ │ │ │ │ o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v] │ │ │ │ i7 : oo == p │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html │ │ │ @@ -131,52 +131,52 @@ │ │ │ o3 : Ideal of ------------------------------------------------------------------------------------------------------------------------- │ │ │ 2 2 2 2 2 2 2 2 │ │ │ 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 │ │ │ 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 │ │ │ │ │ │ │ │ │ 
    i4 : time SegreClass X
│ │ │ - -- used 0.662143s (cpu); 0.428239s (thread); 0s (gc)
│ │ │ + -- used 0.70577s (cpu); 0.544501s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
    │ │ │ │ │ │ │ │ │ 
    i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.426867s (cpu); 0.277592s (thread); 0s (gc)
│ │ │ + -- used 0.505052s (cpu); 0.447735s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
    │ │ │ │ │ │ │ │ │ 
    i6 : time SegreClass(X,Certify=>true)
│ │ │ - -- used 0.0232568s (cpu); 0.0222927s (thread); 0s (gc)
│ │ │ + -- used 0.0738806s (cpu); 0.0177716s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    │ │ │ │ │ │ │ │ │ 
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ - -- used 0.0972585s (cpu); 0.0986145s (thread); 0s (gc)
│ │ │ + -- used 0.37345s (cpu); 0.205008s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │ @@ -198,15 +198,15 @@
│ │ │  o9 = ------[x ..x ]
│ │ │       100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    │ │ │ │ │ │ │ │ │ 
    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)
│ │ │ - -- used 0.0519939s (cpu); 0.0537215s (thread); 0s (gc)
│ │ │ + -- used 0.0961548s (cpu); 0.0958205s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  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 })
│ │ │             (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
│ │ │               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
│ │ │ @@ -216,15 +216,15 @@
│ │ │                                                           100003  0   9                                                   ZZ
│ │ │  o10 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[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 )     100003  0   6
│ │ │                  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
    │ │ │ │ │ │ │ │ │ 
    i11 : time SegreClass phi
│ │ │ - -- used 0.423874s (cpu); 0.279274s (thread); 0s (gc)
│ │ │ + -- used 0.441667s (cpu); 0.332601s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -246,28 +246,28 @@
│ │ │  o12 : Ideal of ----------------------------------------------------------------------------------------------------
│ │ │                 (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 )
│ │ │                   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
    │ │ │ │ │ │ │ │ │ 
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.423803s (cpu); 0.271486s (thread); 0s (gc)
│ │ │ + -- used 0.44219s (cpu); 0.386603s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
    │ │ │ │ │ │ │ │ │ 
    i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.33445s (cpu); 0.978665s (thread); 0s (gc)
│ │ │ + -- used 1.4686s (cpu); 1.05555s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,46 +82,46 @@
│ │ │ │                 2 2                2 2                                        2
│ │ │ │  2                                                    2 2
│ │ │ │                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
│ │ │ │                 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
│ │ │ │  i4 : time SegreClass X
│ │ │ │ - -- used 0.662143s (cpu); 0.428239s (thread); 0s (gc)
│ │ │ │ + -- used 0.70577s (cpu); 0.544501s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ │ - -- used 0.426867s (cpu); 0.277592s (thread); 0s (gc)
│ │ │ │ + -- used 0.505052s (cpu); 0.447735s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o5 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ │ - -- used 0.0232568s (cpu); 0.0222927s (thread); 0s (gc)
│ │ │ │ + -- used 0.0738806s (cpu); 0.0177716s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │ - -- used 0.0972585s (cpu); 0.0986145s (thread); 0s (gc)
│ │ │ │ + -- used 0.37345s (cpu); 0.205008s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │ @@ -142,15 +142,15 @@
│ │ │ │         ZZ
│ │ │ │  o9 = ------[x ..x ]
│ │ │ │       100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  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)
│ │ │ │ - -- used 0.0519939s (cpu); 0.0537215s (thread); 0s (gc)
│ │ │ │ + -- used 0.0961548s (cpu); 0.0958205s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                          ZZ
│ │ │ │                                                        ------[y ..y ]
│ │ │ │                                                        100003  0   9
│ │ │ │  ZZ              2                              2
│ │ │ │  o10 = map (--------------------------------------------------------------------
│ │ │ │  --------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y
│ │ │ │ @@ -170,15 +170,15 @@
│ │ │ │  o10 : RingMap -----------------------------------------------------------------
│ │ │ │  ----------------------------------- <-- ------[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 )     100003  0   6
│ │ │ │                  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
│ │ │ │  i11 : time SegreClass phi
│ │ │ │ - -- used 0.423874s (cpu); 0.279274s (thread); 0s (gc)
│ │ │ │ + -- used 0.441667s (cpu); 0.332601s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │          10
│ │ │ │ @@ -199,26 +199,26 @@
│ │ │ │  ------------------------------------
│ │ │ │                 (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 )
│ │ │ │                   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
│ │ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │ │        time SegreClass B
│ │ │ │ - -- used 0.423803s (cpu); 0.271486s (thread); 0s (gc)
│ │ │ │ + -- used 0.44219s (cpu); 0.386603s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o13 : -----
│ │ │ │          10
│ │ │ │         H
│ │ │ │  i14 : -- Segre class of B in P^9
│ │ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ │ - -- used 1.33445s (cpu); 0.978665s (thread); 0s (gc)
│ │ │ │ + -- used 1.4686s (cpu); 1.05555s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 = QQ[u ..u ]
│ │ │           0   5
│ │ │  
│ │ │  o3 : PolynomialRing
    │ │ │ │ │ │ │ │ │ 
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.00118846s (cpu); 0.000368992s (thread); 0s (gc)
│ │ │ + -- used 0.00221773s (cpu); 0.000294312s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -113,21 +113,21 @@
│ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
    │ │ │ │ │ │ │ │ │

    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.299552s (cpu); 0.15716s (thread); 0s (gc)
│ │ │ + -- used 0.229326s (cpu); 0.169096s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │ 
    i6 : time rationalMap psi
│ │ │ - -- used 0.40278s (cpu); 0.331735s (thread); 0s (gc)
│ │ │ + -- used 0.430315s (cpu); 0.369564s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -211,15 +211,15 @@
│ │ │  
│ │ │                   ZZ
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    │ │ │ 
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.151815s (cpu); 0.0652653s (thread); 0s (gc)
│ │ │ + -- used 0.0639987s (cpu); 0.0649795s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -229,39 +229,39 @@
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
    │ │ │
    │ │ │

    The degree of the forms defining the abstract map T can be obtained by the following command:

    │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 3.0254s (cpu); 1.69288s (thread); 0s (gc)
│ │ │ + -- used 3.44941s (cpu); 1.95221s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    │ │ │
    │ │ │

    We verify that the composition of T with itself is defined by linear forms:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time T2 = T * T
│ │ │ - -- used 0.000199554s (cpu); 2.7712e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000124975s (cpu); 2.712e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    │ │ │ 
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 5.28753s (cpu); 2.90553s (thread); 0s (gc)
│ │ │ + -- used 6.23056s (cpu); 3.31739s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    │ │ │
    │ │ │

    We verify that the composition of T with itself leaves a random point fixed:

    │ │ │ │ │ │ │ │ │ @@ -286,15 +286,15 @@ │ │ │ o20 : List │ │ │ │ │ │
    │ │ │

    We now compute the concrete rational map corresponding to T:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -282,15 +282,15 @@ │ │ │ } │ │ │ │ │ │ o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time f = rationalMap T
│ │ │ - -- used 4.10257s (cpu); 2.25715s (thread); 0s (gc)
│ │ │ + -- used 5.28201s (cpu); 3.46697s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,32 +36,32 @@
│ │ │ │  i3 : P5 := QQ[u_0..u_5]
│ │ │ │  
│ │ │ │  o3 = QQ[u ..u ]
│ │ │ │           0   5
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ │ - -- used 0.00118846s (cpu); 0.000368992s (thread); 0s (gc)
│ │ │ │ + -- used 0.00221773s (cpu); 0.000294312s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: given by a function
│ │ │ │  
│ │ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │ │  Now we compute first the degree of the forms defining the abstract map psi and
│ │ │ │  then the corresponding concrete rational map.
│ │ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ │ - -- used 0.299552s (cpu); 0.15716s (thread); 0s (gc)
│ │ │ │ + -- used 0.229326s (cpu); 0.169096s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.40278s (cpu); 0.331735s (thread); 0s (gc)
│ │ │ │ + -- used 0.430315s (cpu); 0.369564s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: {
│ │ │ │ @@ -140,48 +140,48 @@
│ │ │ │  o13 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o13 : Ideal of -----[x ..x ]
│ │ │ │                 65521  0   3
│ │ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ │ - -- used 0.151815s (cpu); 0.0652653s (thread); 0s (gc)
│ │ │ │ + -- used 0.0639987s (cpu); 0.0649795s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  The degree of the forms defining the abstract map T can be obtained by the
│ │ │ │  following command:
│ │ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ │ - -- used 3.0254s (cpu); 1.69288s (thread); 0s (gc)
│ │ │ │ + -- used 3.44941s (cpu); 1.95221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 0.000199554s (cpu); 2.7712e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000124975s (cpu); 2.712e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 5.28753s (cpu); 2.90553s (thread); 0s (gc)
│ │ │ │ + -- used 6.23056s (cpu); 3.31739s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 1
│ │ │ │  We verify that the composition of T with itself leaves a random point fixed:
│ │ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │ │  
│ │ │ │  o18 = {28963, 31975, -30172, 1}
│ │ │ │  
│ │ │ │ @@ -194,15 +194,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {28963, 31975, -30172, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 4.10257s (cpu); 2.25715s (thread); 0s (gc)
│ │ │ │ + -- used 5.28201s (cpu); 3.46697s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │                         1 2    0 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
    │ │ │ 
    i3 : time psi = approximateInverseMap phi
│ │ │ - -- used 0.218667s (cpu); 0.169055s (thread); 0s (gc)
│ │ │ + -- used 0.372312s (cpu); 0.254928s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 10
│ │ │  -- approximateInverseMap: step 2 of 10
│ │ │  -- approximateInverseMap: step 3 of 10
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ @@ -193,15 +193,15 @@
│ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    │ │ │ 
    i4 : assert(phi * psi == 1 and psi * phi == 1)
    │ │ │ 
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ - -- used 0.218999s (cpu); 0.146713s (thread); 0s (gc)
│ │ │ + -- used 0.283764s (cpu); 0.228417s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    │ │ │ 
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ - -- used 2.28934s (cpu); 1.6958s (thread); 0s (gc)
│ │ │ + -- used 2.89445s (cpu); 2.60374s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ @@ -349,15 +349,15 @@
│ │ │       phi * psi == 1
│ │ │  
│ │ │  o9 = false
    │ │ │ 
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ - -- used 3.34434s (cpu); 2.48164s (thread); 0s (gc)
│ │ │ + -- used 3.4615s (cpu); 3.05447s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │ ├── html2text {}
│ │ │ │ @@ -126,15 +126,15 @@
│ │ │ │  
│ │ │ │                        x x  - x x
│ │ │ │                         1 2    0 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
│ │ │ │  i3 : time psi = approximateInverseMap phi
│ │ │ │ - -- used 0.218667s (cpu); 0.169055s (thread); 0s (gc)
│ │ │ │ + -- used 0.372312s (cpu); 0.254928s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 10
│ │ │ │  -- approximateInverseMap: step 2 of 10
│ │ │ │  -- approximateInverseMap: step 3 of 10
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │ @@ -250,15 +250,15 @@
│ │ │ │  0 6     3 6      6      0 7      1 7      3 7      4 7      6 7      7      0 8
│ │ │ │  1 8      2 8      3 8      4 8      5 8      6 8      7 8     8
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ │ - -- used 0.218999s (cpu); 0.146713s (thread); 0s (gc)
│ │ │ │ + -- used 0.283764s (cpu); 0.228417s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  
│ │ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i6 : assert(psi == psi')
│ │ │ │  A more complicated example is the following (here _i_n_v_e_r_s_e_M_a_p takes a lot of
│ │ │ │ @@ -416,15 +416,15 @@
│ │ │ │  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
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time
│ │ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ │ - -- used 2.28934s (cpu); 1.6958s (thread); 0s (gc)
│ │ │ │ + -- used 2.89445s (cpu); 2.60374s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │ @@ -523,15 +523,15 @@
│ │ │ │  o8 : RationalMap (quadratic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │ │  i9 : -- but...
│ │ │ │       phi * psi == 1
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │ - -- used 3.34434s (cpu); 2.48164s (thread); 0s (gc)
│ │ │ │ + -- used 3.4615s (cpu); 3.05447s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │  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 })
│ │ │                                   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
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    │ │ │ 
    i5 : time degreeMap phi
│ │ │ - -- used 0.131475s (cpu); 0.0571066s (thread); 0s (gc)
│ │ │ + -- used 0.228985s (cpu); 0.0687556s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    │ │ │ 
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │ @@ -108,15 +108,15 @@
│ │ │  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 })
│ │ │                                   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
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    │ │ │ 
    i7 : time degreeMap phi'
│ │ │ - -- used 0.763454s (cpu); 0.537536s (thread); 0s (gc)
│ │ │ + -- used 0.716216s (cpu); 0.658864s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
    │ │ │
    │ │ │ │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -267,15 +267,15 @@ │ │ │ │ 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 │ │ │ │ │ │ │ │ o4 : RingMap ringP8 <-- ringP14 │ │ │ │ i5 : time degreeMap phi │ │ │ │ - -- used 0.131475s (cpu); 0.0571066s (thread); 0s (gc) │ │ │ │ + -- used 0.228985s (cpu); 0.0687556s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 1 │ │ │ │ i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a │ │ │ │ general subspace of P^14 │ │ │ │ -- of dimension 5 (so that the composition phi':P^8--->P^8 must have │ │ │ │ degree equal to deg(G(1,5))=14) │ │ │ │ phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14)) │ │ │ │ @@ -419,15 +419,15 @@ │ │ │ │ 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 │ │ │ │ │ │ │ │ o6 : RingMap ringP8 <-- ringP8 │ │ │ │ i7 : time degreeMap phi' │ │ │ │ - -- used 0.763454s (cpu); 0.537536s (thread); 0s (gc) │ │ │ │ + -- used 0.716216s (cpu); 0.658864s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 14 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between │ │ │ │ projective varieties │ │ │ │ ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: ********** │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │ 
    i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    │ │ │ │ │ │ │ │ │ 
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.00296865s (cpu); 0.000930424s (thread); 0s (gc)
    │ │ │ + -- used 0.000591067s (cpu); 0.000562725s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ │ │ │ │ o2 : Ideal of P6 │ │ │ │ i3 : Phi = rationalMap(X,Dominant=>2); │ │ │ │ │ │ │ │ o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of │ │ │ │ PP^9) │ │ │ │ i4 : time forceImage(Phi,ideal 0_(target Phi)) │ │ │ │ - -- used 0.00296865s (cpu); 0.000930424s (thread); 0s (gc) │ │ │ │ + -- used 0.000591067s (cpu); 0.000562725s (thread); 0s (gc) │ │ │ │ i5 : Phi; │ │ │ │ │ │ │ │ o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional │ │ │ │ subvariety of PP^9) │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ If the declaration is false, nonsensical answers may result. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html │ │ │ @@ -113,15 +113,15 @@ │ │ │ 3 2 4 │ │ │ } │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ 
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.108255s (cpu); 0.0343845s (thread); 0s (gc)
    │ │ │ + -- used 0.0789011s (cpu); 0.0185917s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │                                    ZZ                                 ZZ
│ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
│ │ │ @@ -260,15 +260,15 @@
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │

    When the source of the rational map is a multi-projective variety, the method returns all the projections.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -51,15 +51,15 @@ │ │ │ │ - x + x x │ │ │ │ 3 2 4 │ │ │ │ } │ │ │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in │ │ │ │ PP^5) │ │ │ │ i3 : time (p1,p2) = graph phi; │ │ │ │ - -- used 0.108255s (cpu); 0.0343845s (thread); 0s (gc) │ │ │ │ + -- used 0.0789011s (cpu); 0.0185917s (thread); 0s (gc) │ │ │ │ i4 : p1 │ │ │ │ │ │ │ │ o4 = -- rational map -- │ │ │ │ ZZ ZZ │ │ │ │ source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y │ │ │ │ , y , y , y , y ]) defined by │ │ │ │ 190181 0 1 2 3 4 190181 0 │ │ │ │ @@ -193,15 +193,15 @@ │ │ │ │ │ │ │ │ o8 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ When the source of the rational map is a multi-projective variety, the method │ │ │ │ returns all the projections. │ │ │ │ i9 : time g = graph p2; │ │ │ │ - -- used 0.0283961s (cpu); 0.0275265s (thread); 0s (gc) │ │ │ │ + -- used 0.0862843s (cpu); 0.0269329s (thread); 0s (gc) │ │ │ │ i10 : g_0; │ │ │ │ │ │ │ │ o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety │ │ │ │ of PP^4 x PP^5 x PP^5 to PP^4) │ │ │ │ i11 : g_1; │ │ │ │ │ │ │ │ o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html │ │ │ @@ -107,15 +107,15 @@ │ │ │ 1 0 3 │ │ │ } │ │ │ │ │ │ o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ │ │ │ │ w w - w w + w w │ │ │ │ 2 4 1 5 0 6 │ │ │ │ } │ │ │ │ │ │ │ │ o1 : RationalMap (quadratic Cremona transformation of PP^20) │ │ │ │ i2 : time psi = inverseMap phi │ │ │ │ - -- used 0.158746s (cpu); 0.0885499s (thread); 0s (gc) │ │ │ │ + -- used 0.333669s (cpu); 0.124609s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = -- rational map -- │ │ │ │ source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w , w , w , w │ │ │ │ , w , w , w , w , w , w , w ]) │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 │ │ │ │ 14 15 16 17 18 19 20 │ │ │ │ target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w , w , w , w │ │ │ │ @@ -217,15 +217,15 @@ │ │ │ │ 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 │ │ │ │ │ │ │ │ o4 : RingMap QQ[w ..w ] <-- QQ[w ..w ] │ │ │ │ 0 26 0 26 │ │ │ │ i5 : time psi = inverseMap phi │ │ │ │ - -- used 0.312892s (cpu); 0.187463s (thread); 0s (gc) │ │ │ │ + -- used 0.465234s (cpu); 0.344834s (thread); 0s (gc) │ │ │ │ │ │ │ │ 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 │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 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 │ │ │ } │ │ │ │ │ │ o2 : RationalMap (rational map from PP^4 to PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time g = graph p2;
│ │ │ - -- used 0.0283961s (cpu); 0.0275265s (thread); 0s (gc)
    │ │ │ + -- used 0.0862843s (cpu); 0.0269329s (thread); 0s (gc) │ │ │ 
    i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
    │ │ │ 
    i3 : time ideal phi
│ │ │ - -- used 0.00399819s (cpu); 0.00324924s (thread); 0s (gc)
│ │ │ + -- used 0.00174627s (cpu); 0.00298329s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -185,15 +185,15 @@
│ │ │                         4
│ │ │                       }
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    │ │ │ 
    i6 : time ideal phi'
│ │ │ - -- used 0.0900955s (cpu); 0.087187s (thread); 0s (gc)
│ │ │ + -- used 0.244771s (cpu); 0.180884s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ │ ├── html2text {}
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │                         2
│ │ │ │                        x  - x x
│ │ │ │                         1    0 3
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i3 : time ideal phi
│ │ │ │ - -- used 0.00399819s (cpu); 0.00324924s (thread); 0s (gc)
│ │ │ │ + -- used 0.00174627s (cpu); 0.00298329s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2                                     2
│ │ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2
│ │ │ │       x x , x  - x x )
│ │ │ │ @@ -122,15 +122,15 @@
│ │ │ │                        y
│ │ │ │                         4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of
│ │ │ │  PP^5 x PP^4 to PP^4)
│ │ │ │  i6 : time ideal phi'
│ │ │ │ - -- used 0.0900955s (cpu); 0.087187s (thread); 0s (gc)
│ │ │ │ + -- used 0.244771s (cpu); 0.180884s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -154,15 +154,15 @@
│ │ │                         2 4    1 5    0 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    │ │ │ 
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.158746s (cpu); 0.0885499s (thread); 0s (gc)
│ │ │ + -- used 0.333669s (cpu); 0.124609s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -246,15 +246,15 @@
│ │ │                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
│ │ │  
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    │ │ │ 
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.312892s (cpu); 0.187463s (thread); 0s (gc)
│ │ │ + -- used 0.465234s (cpu); 0.344834s (thread); 0s (gc)
│ │ │  
│ │ │  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  })
│ │ │                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
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    │ │ │ 
    i3 : time inverse phi
│ │ │ - -- used 0.154532s (cpu); 0.0707467s (thread); 0s (gc)
│ │ │ + -- used 0.166999s (cpu); 0.109695s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ │ ├── html2text {}
│ │ │ │ @@ -282,15 +282,15 @@
│ │ │ │  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
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │ │  i3 : time inverse phi
│ │ │ │ - -- used 0.154532s (cpu); 0.0707467s (thread); 0s (gc)
│ │ │ │ + -- used 0.166999s (cpu); 0.109695s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -122,21 +122,21 @@
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    │ │ │ 
    i3 : time isBirational phi
│ │ │ - -- used 0.0199621s (cpu); 0.0179823s (thread); 0s (gc)
│ │ │ + -- used 0.0806576s (cpu); 0.0265241s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ 
    i4 : time isBirational(phi,Certify=>true)
│ │ │ - -- used 0.0128589s (cpu); 0.0142427s (thread); 0s (gc)
│ │ │ + -- used 0.0406175s (cpu); 0.0115317s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o4 = true
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -59,19 +59,19 @@ │ │ │ │ - t + t t │ │ │ │ 3 2 4 │ │ │ │ } │ │ │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in │ │ │ │ PP^5) │ │ │ │ i3 : time isBirational phi │ │ │ │ - -- used 0.0199621s (cpu); 0.0179823s (thread); 0s (gc) │ │ │ │ + -- used 0.0806576s (cpu); 0.0265241s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : time isBirational(phi,Certify=>true) │ │ │ │ - -- used 0.0128589s (cpu); 0.0142427s (thread); 0s (gc) │ │ │ │ + -- used 0.0406175s (cpu); 0.0115317s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant │ │ │ │ ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: ********** │ │ │ │ * isBirational(RationalMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ │ │ │ 
    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}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
    │ │ │ │ │ │ │ │ │ 
    i3 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 2.25139s (cpu); 1.87155s (thread); 0s (gc)
│ │ │ + -- used 3.58107s (cpu); 3.34407s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ 
    i4 : P7 = ZZ/101[x_0..x_7];
    │ │ │ │ │ │ @@ -104,15 +104,15 @@ │ │ │ │ │ │ 
    i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
    │ │ │ │ │ │ │ │ │ 
    i7 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 4.07363s (cpu); 2.75303s (thread); 0s (gc)
│ │ │ + -- used 2.76432s (cpu); 2.14498s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o7 = false
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ be to perform the command kernel map phi == 0. │ │ │ │ i1 : P8 = ZZ/101[x_0..x_8]; │ │ │ │ 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}}; │ │ │ │ │ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ i3 : time isDominant(phi,Certify=>true) │ │ │ │ - -- used 2.25139s (cpu); 1.87155s (thread); 0s (gc) │ │ │ │ + -- used 3.58107s (cpu); 3.34407s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : P7 = ZZ/101[x_0..x_7]; │ │ │ │ i5 : -- hyperelliptic curve of genus 3 │ │ │ │ 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- │ │ │ │ 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- │ │ │ │ @@ -65,15 +65,15 @@ │ │ │ │ 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); │ │ │ │ │ │ │ │ o5 : Ideal of P7 │ │ │ │ i6 : phi = rationalMap(C,3,2); │ │ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ i7 : time isDominant(phi,Certify=>true) │ │ │ │ - -- used 4.07363s (cpu); 2.75303s (thread); 0s (gc) │ │ │ │ + -- used 2.76432s (cpu); 2.14498s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o7 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational │ │ │ │ ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: ********** │ │ │ │ * isDominant(RationalMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ @@ -87,24 +87,24 @@ │ │ │ 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 │ │ │ │ │ │ o1 : RingMap QQ[x ..x ] <-- QQ[y ..y ] │ │ │ 0 8 0 11 │ │ │ │ │ │ │ │ │ 
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.115348s (cpu); 0.0365182s (thread); 0s (gc)
│ │ │ + -- used 0.0119988s (cpu); 0.0142586s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    │ │ │ │ │ │ │ │ │ 
    i3 : time kernel(phi,2)
│ │ │ - -- used 0.474169s (cpu); 0.310819s (thread); 0s (gc)
│ │ │ + -- used 0.498382s (cpu); 0.439796s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,22 +68,22 @@
│ │ │ │  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
│ │ │ │  
│ │ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │ │                   0   8          0   11
│ │ │ │  i2 : time kernel(phi,1)
│ │ │ │ - -- used 0.115348s (cpu); 0.0365182s (thread); 0s (gc)
│ │ │ │ + -- used 0.0119988s (cpu); 0.0142586s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 0.474169s (cpu); 0.310819s (thread); 0s (gc)
│ │ │ │ + -- used 0.498382s (cpu); 0.439796s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                             2
│ │ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │                   ZZ
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ 
    i3 : time parametrize L
│ │ │ - -- used 0.00400026s (cpu); 0.00430712s (thread); 0s (gc)
│ │ │ + -- used 0.00400077s (cpu); 0.004189s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │                   ZZ
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ 
    i5 : time parametrize Q
│ │ │ - -- used 0.567237s (cpu); 0.347295s (thread); 0s (gc)
│ │ │ + -- used 0.730993s (cpu); 0.517449s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00400026s (cpu); 0.00430712s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400077s (cpu); 0.004189s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -137,15 +137,15 @@
│ │ │ │       1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
│ │ │ │               5 9           6 9           7 9           8 9           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o4 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i5 : time parametrize Q
│ │ │ │ - -- used 0.567237s (cpu); 0.347295s (thread); 0s (gc)
│ │ │ │ + -- used 0.730993s (cpu); 0.517449s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
│ │ │                
    i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    
│ │ │            
│ │ │            
│ │ │                
    i2 : time p = point source f
│ │ │ - -- used 0.201188s (cpu); 0.137863s (thread); 0s (gc)
│ │ │ + -- used 0.218876s (cpu); 0.170155s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -96,15 +96,15 @@
│ │ │                                                             33331  0   11
│ │ │  o2 : Ideal of -------------------------------------------------------------------------------------------------------
│ │ │                (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 )
│ │ │                  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
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time p == f^* f p
│ │ │ - -- used 0.186193s (cpu); 0.119467s (thread); 0s (gc)
│ │ │ + -- used 0.22745s (cpu); 0.173947s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ documentation) , see _p_o_i_n_t_(_M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_). │ │ │ │ Below we verify the birationality of a rational map. │ │ │ │ i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331); │ │ │ │ │ │ │ │ o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to │ │ │ │ PP^8) │ │ │ │ i2 : time p = point source f │ │ │ │ - -- used 0.201188s (cpu); 0.137863s (thread); 0s (gc) │ │ │ │ + -- used 0.218876s (cpu); 0.170155s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = ideal (y - 9235y , y + 11075y , y - 5847y , y + 7396y , y + │ │ │ │ 10 11 9 11 8 11 7 11 6 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 13530y , y + 4359y , y - 2924y , y + 13040y , y + 6904y , y - │ │ │ │ 11 5 11 4 11 3 11 2 11 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ o2 : Ideal of ----------------------------------------------------------------- │ │ │ │ -------------------------------------- │ │ │ │ (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 ) │ │ │ │ 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 │ │ │ │ i3 : time p == f^* f p │ │ │ │ - -- used 0.186193s (cpu); 0.119467s (thread); 0s (gc) │ │ │ │ + -- used 0.22745s (cpu); 0.173947s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- Pick a random K rational point on the scheme X │ │ │ │ defined by I │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * point(PolynomialRing) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ 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 │ │ │ │ │ │ o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ] │ │ │ 0 4 0 5     │ │ │ │ │ │ │ │ │ 
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │ - -- used 0.0182582s (cpu); 0.0155028s (thread); 0s (gc)
│ │ │ + -- used 0.0517608s (cpu); 0.0124244s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ @@ -114,15 +114,15 @@ │ │ │ 0 5 │ │ │ o4 : RingMap ------------------ <-- GF 109561[t ..t ] │ │ │ x x - x x + x x 0 4 │ │ │ 2 3 1 4 0 5 │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │ - -- used 0.00823188s (cpu); 0.0103648s (thread); 0s (gc)
│ │ │ + -- used 0.0593251s (cpu); 0.0112806s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ @@ -137,23 +137,23 @@ │ │ │ │ │ │ ZZ ZZ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ 300007 0 6 300007 0 6 │ │ │ │ │ │ │ │ │ 
    i7 : time projectiveDegrees phi
│ │ │ - -- used 0.00337059s (cpu); 4.1208e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00268412s (cpu); 2.9916e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ 
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 9.9857e-05s (cpu); 2.2512e-05s (thread); 0s (gc)
│ │ │ + -- used 7.5101e-05s (cpu); 2.0058e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │

    Another way to use this method is by passing an integer i as second argument. However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and generally it is not faster.

    │ │ │ ├── html2text {} │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ t + t t , - t t + t t , - t t + t t , - t + t t , a}) │ │ │ │ 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 │ │ │ │ │ │ │ │ o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ] │ │ │ │ 0 4 0 5 │ │ │ │ i3 : time projectiveDegrees(phi,Certify=>true) │ │ │ │ - -- used 0.0182582s (cpu); 0.0155028s (thread); 0s (gc) │ │ │ │ + -- used 0.0517608s (cpu); 0.0124244s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o3 = {1, 2, 4, 4, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : psi=inverseMap(toMap(phi,Dominant=>infinity)) │ │ │ │ │ │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ │ │ │ │ GF 109561[x ..x ] │ │ │ │ 0 5 │ │ │ │ o4 : RingMap ------------------ <-- GF 109561[t ..t ] │ │ │ │ x x - x x + x x 0 4 │ │ │ │ 2 3 1 4 0 5 │ │ │ │ i5 : time projectiveDegrees(psi,Certify=>true) │ │ │ │ - -- used 0.00823188s (cpu); 0.0103648s (thread); 0s (gc) │ │ │ │ + -- used 0.0593251s (cpu); 0.0112806s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o5 = {2, 4, 4, 2, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : -- Cremona transformation of P^6 defined by the quadrics through a │ │ │ │ rational octic surface │ │ │ │ @@ -120,21 +120,21 @@ │ │ │ │ 4 5 5 0 6 1 6 2 6 3 6 4 6 │ │ │ │ 5 6 │ │ │ │ │ │ │ │ ZZ ZZ │ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ │ 300007 0 6 300007 0 6 │ │ │ │ i7 : time projectiveDegrees phi │ │ │ │ - -- used 0.00337059s (cpu); 4.1208e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00268412s (cpu); 2.9916e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 2, 4, 8, 8, 4, 1} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : time projectiveDegrees(phi,NumDegrees=>1) │ │ │ │ - -- used 9.9857e-05s (cpu); 2.2512e-05s (thread); 0s (gc) │ │ │ │ + -- used 7.5101e-05s (cpu); 2.0058e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = {4, 1} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ Another way to use this method is by passing an integer i as second argument. │ │ │ │ However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and │ │ │ │ generally it is not faster. │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ │ │ │ ZZ │ │ │ o2 : Ideal of -----[x ..x ] │ │ │ 33331 0 6 │ │ │ │ │ │ │ │ │ 
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.184482s (cpu); 0.107539s (thread); 0s (gc)
│ │ │ + -- used 0.115708s (cpu); 0.116323s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  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);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o2 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   6
│ │ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ │ - -- used 0.184482s (cpu); 0.107539s (thread); 0s (gc)
│ │ │ │ + -- used 0.115708s (cpu); 0.116323s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  o4 : Ideal of X
    │ │ │ │ │ │ │ │ │ 
    i5 : D = new Tally from {H => 2,C => 1};
    │ │ │ │ │ │ │ │ │ 
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.0267469s (cpu); 0.0261443s (thread); 0s (gc)
│ │ │ + -- used 0.0238955s (cpu); 0.02544s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -210,15 +210,15 @@
│ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │                       }
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    │ │ │ │ │ │ │ │ │ 
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.26146s (cpu); 0.63888s (thread); 0s (gc)
│ │ │ + -- used 2.58621s (cpu); 0.717812s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
    │ │ │ │ │ │ │ │ │

    See also the package Divisor, which provides general tools for working with divisors.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ │ │ │ │ o4 = ideal(- 32646x - 28377x + 26433x - 29566x + 3783x + 26696x ) │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o4 : Ideal of X │ │ │ │ i5 : D = new Tally from {H => 2,C => 1}; │ │ │ │ i6 : time phi = rationalMap D │ │ │ │ - -- used 0.0267469s (cpu); 0.0261443s (thread); 0s (gc) │ │ │ │ + -- used 0.0238955s (cpu); 0.02544s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = -- rational map -- │ │ │ │ ZZ │ │ │ │ source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by │ │ │ │ 65521 0 1 2 3 4 5 │ │ │ │ { │ │ │ │ 2 2 │ │ │ │ @@ -170,15 +170,15 @@ │ │ │ │ 2 2 │ │ │ │ x x x + x x x + x x x + x x + x x x - 2x x x + x x │ │ │ │ 0 1 5 0 2 5 1 2 5 2 5 1 4 5 2 4 5 4 5 │ │ │ │ } │ │ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20) │ │ │ │ i7 : time ? image(phi,"F4") │ │ │ │ - -- used 1.26146s (cpu); 0.63888s (thread); 0s (gc) │ │ │ │ + -- used 2.58621s (cpu); 0.717812s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153 │ │ │ │ hypersurfaces of degree 2 │ │ │ │ See also the package _D_i_v_i_s_o_r, which provides general tools for working with │ │ │ │ divisors. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html │ │ │ @@ -70,15 +70,15 @@ │ │ │ │ │ │
    │ │ │

    Description

    │ │ │

    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.18389s (cpu); 0.973906s (thread); 0s (gc)
│ │ │ + -- used 1.44067s (cpu); 1.20823s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │              K, according to the classification given in Table 1 of _S_p_e_c_i_a_l
│ │ │ │              _c_u_b_i_c_ _C_r_e_m_o_n_a_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _P_6_ _a_n_d_ _P_7.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A Cremona transformation is said to be special if the base locus scheme is
│ │ │ │  smooth and irreducible. To ensure this condition, the field K must be large
│ │ │ │  enough but no check is made.
│ │ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ │ - -- used 1.18389s (cpu); 0.973906s (thread); 0s (gc)
│ │ │ │ + -- used 1.44067s (cpu); 1.20823s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0895158s (cpu); 0.0880504s (thread); 0s (gc)
│ │ │ + -- used 0.119999s (cpu); 0.120048s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -136,15 +136,15 @@
│ │ │                          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
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │      		          
                  
    i2 : time describe oo
│ │ │ - -- used 0.0199392s (cpu); 0.018788s (thread); 0s (gc)
│ │ │ + -- used 0.0561777s (cpu); 0.0553741s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special cubic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 2 of _S_p_e_c_i_a_l_ _c_u_b_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _p_r_o_j_e_c_t_i_v_e
│ │ │ │              _s_p_a_c_e_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialCubicTransformation 9
│ │ │ │ - -- used 0.0895158s (cpu); 0.0880504s (thread); 0s (gc)
│ │ │ │ + -- used 0.119999s (cpu); 0.120048s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6
│ │ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -324,15 +324,15 @@
│ │ │ │  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
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.0199392s (cpu); 0.018788s (thread); 0s (gc)
│ │ │ │ + -- used 0.0561777s (cpu); 0.0553741s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.067479s (cpu); 0.065041s (thread); 0s (gc)
│ │ │ + -- used 0.0519894s (cpu); 0.051045s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -124,15 +124,15 @@
│ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │      		          
                  
    i2 : time describe oo
│ │ │ - -- used 0.00850034s (cpu); 0.00606449s (thread); 0s (gc)
│ │ │ + -- used 0.00333284s (cpu); 0.00519894s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special quadratic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 1 of _E_x_a_m_p_l_e_s_ _o_f_ _s_p_e_c_i_a_l_ _q_u_a_d_r_a_t_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s
│ │ │ │              _i_n_t_o_ _c_o_m_p_l_e_t_e_ _i_n_t_e_r_s_e_c_t_i_o_n_s_ _o_f_ _q_u_a_d_r_i_c_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ │ - -- used 0.067479s (cpu); 0.065041s (thread); 0s (gc)
│ │ │ │ + -- used 0.0519894s (cpu); 0.051045s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6   7   8
│ │ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │                                                     2
│ │ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.00850034s (cpu); 0.00606449s (thread); 0s (gc)
│ │ │ │ + -- used 0.00333284s (cpu); 0.00519894s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -82,36 +82,36 @@
│ │ │            
                  
    i3 : #str
│ │ │  
│ │ │  o3 = 6927
    
│ │ │      		          
                  
    i4 : time phi' = value str;
│ │ │ - -- used 0.0200011s (cpu); 0.0206321s (thread); 0s (gc)
│ │ │ + -- used 0.291917s (cpu); 0.0799198s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │      		          
                  
    i5 : time describe phi'
│ │ │ - -- used 0.00473485s (cpu); 0.00488245s (thread); 0s (gc)
│ │ │ + -- used 0.00300748s (cpu); 0.00441817s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │       number of minimal representatives: 1
│ │ │       dimension base locus: 1
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │      		          
                  
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00350015s (cpu); 0.00391244s (thread); 0s (gc)
│ │ │ + -- used 0.0022602s (cpu); 0.00354625s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i2 : str = toExternalString phi;
│ │ │ │  i3 : #str
│ │ │ │  
│ │ │ │  o3 = 6927
│ │ │ │  i4 : time phi' = value str;
│ │ │ │ - -- used 0.0200011s (cpu); 0.0206321s (thread); 0s (gc)
│ │ │ │ + -- used 0.291917s (cpu); 0.0799198s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00473485s (cpu); 0.00488245s (thread); 0s (gc)
│ │ │ │ + -- used 0.00300748s (cpu); 0.00441817s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^3
│ │ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ │       number of minimal representatives: 1
│ │ │ │       dimension base locus: 1
│ │ │ │       degree base locus: 5
│ │ │ │       coefficient ring: ZZ/33331
│ │ │ │  i6 : time describe inverse phi'
│ │ │ │ - -- used 0.00350015s (cpu); 0.00391244s (thread); 0s (gc)
│ │ │ │ + -- used 0.0022602s (cpu); 0.00354625s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -48,63 +48,63 @@
│ │ │          
    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 $(3,3)$.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │      		          
                  
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00399965s (cpu); 0.00392251s (thread); 0s (gc)
│ │ │ + -- used 0.00400782s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │  
│ │ │              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
│ │ │  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 })
│ │ │            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
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
    
│ │ │      		          
                  
    i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.0437451s (cpu); 0.0440403s (thread); 0s (gc)
│ │ │ + -- used 0.0320008s (cpu); 0.0340255s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │      		          
                  
    i4 : time degreeMap phi
│ │ │ - -- used 0.0829531s (cpu); 0.0341009s (thread); 0s (gc)
│ │ │ + -- used 0.124868s (cpu); 0.0674956s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │      		          
                  
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.506956s (cpu); 0.382073s (thread); 0s (gc)
│ │ │ + -- used 0.45759s (cpu); 0.397078s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │      		          
                  
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0565176s (cpu); 0.0573171s (thread); 0s (gc)
│ │ │ + -- used 0.0433452s (cpu); 0.0461662s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │      		          
                  
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.000189065s (cpu); 0.00204299s (thread); 0s (gc)
│ │ │ + -- used 0.00105715s (cpu); 0.00171203s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              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
│ │ │  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 })
│ │ │            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
│ │ │                              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
│ │ │ @@ -114,15 +114,15 @@
│ │ │                 ZZ                                                          300007  0   9
│ │ │  o7 : RingMap ------[t ..t ] <-- ----------------------------------------------------------------------------------------------------
│ │ │               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 )
│ │ │                                    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
    
│ │ │      		          
                  
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.566795s (cpu); 0.41755s (thread); 0s (gc)
│ │ │ + -- used 0.599868s (cpu); 0.4454s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       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
│ │ │  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 })
│ │ │            (x x  - x x  + x x , 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
│ │ │              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
│ │ │ @@ -132,38 +132,38 @@
│ │ │                                                          300007  0   9                                                   ZZ
│ │ │  o8 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[t ..t ]
│ │ │               (x x  - x x  + x x , 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
│ │ │                 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
    
│ │ │      		          
                  
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.0101323s (cpu); 0.00946356s (thread); 0s (gc)
│ │ │ + -- used 0.00798632s (cpu); 0.00712739s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │      		          
                  
    i10 : time degreeMap psi
│ │ │ - -- used 0.297917s (cpu); 0.233828s (thread); 0s (gc)
│ │ │ + -- used 0.255736s (cpu); 0.197823s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │      		          
                  
    i11 : time projectiveDegrees psi
│ │ │ - -- used 4.99657s (cpu); 4.27726s (thread); 0s (gc)
│ │ │ + -- used 5.73993s (cpu); 5.35784s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │      		          
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
                  
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.0011253s (cpu); 0.00197141s (thread); 0s (gc)
│ │ │ + -- used 0.00200211s (cpu); 0.00186545s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -210,15 +210,15 @@
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │      		          
                  
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.155471s (cpu); 0.0755467s (thread); 0s (gc)
│ │ │ + -- used 0.193117s (cpu); 0.0663768s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -281,15 +281,15 @@
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │      		          
                  
    i14 : time phi^(-1)
│ │ │ - -- used 0.504825s (cpu); 0.425036s (thread); 0s (gc)
│ │ │ + -- used 0.435332s (cpu); 0.435809s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -340,78 +340,78 @@
│ │ │                          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
│ │ │                        }
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │      		          
                  
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.355466s (cpu); 0.270355s (thread); 0s (gc)
│ │ │ + -- used 0.489651s (cpu); 0.36645s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │      		          
                  
    i16 : time degrees phi
│ │ │ - -- used 0.120907s (cpu); 0.0437002s (thread); 0s (gc)
│ │ │ + -- used 0.0734552s (cpu); 0.0218007s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │      		          
                  
    i17 : time describe phi
│ │ │ - -- used 0.00320772s (cpu); 0.00316311s (thread); 0s (gc)
│ │ │ + -- used 1.5298e-05s (cpu); 0.00351931s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
    
│ │ │      		          
                  
    i18 : time describe phi^(-1)
│ │ │ - -- used 0.00643296s (cpu); 0.00993896s (thread); 0s (gc)
│ │ │ + -- used 0.00846573s (cpu); 0.0123656s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │        number of minimal representatives: 1
│ │ │        dimension base locus: 4
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │      		          
                  
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0084534s (cpu); 0.00881838s (thread); 0s (gc)
│ │ │ + -- used 0.0520925s (cpu); 0.052059s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │      		          
                  
    i21 : time degrees f
│ │ │ - -- used 1.19108s (cpu); 0.901679s (thread); 0s (gc)
│ │ │ + -- used 1.50652s (cpu); 1.26241s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │      		          
                  
    i22 : time degree f
│ │ │ - -- used 0.000190698s (cpu); 1.3635e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000143619s (cpu); 1.1592e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │      		          
                  
    i23 : time describe f
│ │ │ - -- used 0.000926578s (cpu); 0.00142987s (thread); 0s (gc)
│ │ │ + -- used 0.000786354s (cpu); 0.00119327s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  map) from a list of $m+1$ homogeneous elements of the same degree in $K
│ │ │ │  [x_0,...,x_n]/I$.
│ │ │ │  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 $(3,3)$.
│ │ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00399965s (cpu); 0.00392251s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400782s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              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
│ │ │ │ @@ -52,43 +52,43 @@
│ │ │ │  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
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   9
│ │ │ │  i3 : time J = kernel(phi,2)
│ │ │ │ - -- used 0.0437451s (cpu); 0.0440403s (thread); 0s (gc)
│ │ │ │ + -- used 0.0320008s (cpu); 0.0340255s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of ------[x ..x ]
│ │ │ │                300007  0   9
│ │ │ │  i4 : time degreeMap phi
│ │ │ │ - -- used 0.0829531s (cpu); 0.0341009s (thread); 0s (gc)
│ │ │ │ + -- used 0.124868s (cpu); 0.0674956s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.506956s (cpu); 0.382073s (thread); 0s (gc)
│ │ │ │ + -- used 0.45759s (cpu); 0.397078s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.0565176s (cpu); 0.0573171s (thread); 0s (gc)
│ │ │ │ + -- used 0.0433452s (cpu); 0.0461662s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.000189065s (cpu); 0.00204299s (thread); 0s (gc)
│ │ │ │ + -- used 0.00105715s (cpu); 0.00171203s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                                         ZZ
│ │ │ │                                                                       ------[x
│ │ │ │  ..x ]
│ │ │ │              ZZ                                                       300007  0
│ │ │ │  9                                                  3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │ @@ -123,15 +123,15 @@
│ │ │ │  o7 : RingMap ------[t ..t ] <-- -----------------------------------------------
│ │ │ │  -----------------------------------------------------
│ │ │ │               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 )
│ │ │ │                                    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
│ │ │ │  i8 : time psi = inverseMap phi
│ │ │ │ - -- used 0.566795s (cpu); 0.41755s (thread); 0s (gc)
│ │ │ │ + -- used 0.599868s (cpu); 0.4454s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                         ZZ
│ │ │ │                                                       ------[x ..x ]
│ │ │ │                                                       300007  0   9
│ │ │ │  ZZ              3                2               2    2
│ │ │ │  2                          2     2        2                               2
│ │ │ │  2               2             2                       3
│ │ │ │ @@ -164,31 +164,31 @@
│ │ │ │  o8 : RingMap ------------------------------------------------------------------
│ │ │ │  ---------------------------------- <-- ------[t ..t ]
│ │ │ │               (x x  - x x  + x x , 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
│ │ │ │                 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
│ │ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ │ - -- used 0.0101323s (cpu); 0.00946356s (thread); 0s (gc)
│ │ │ │ + -- used 0.00798632s (cpu); 0.00712739s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.297917s (cpu); 0.233828s (thread); 0s (gc)
│ │ │ │ + -- used 0.255736s (cpu); 0.197823s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 4.99657s (cpu); 4.27726s (thread); 0s (gc)
│ │ │ │ + -- used 5.73993s (cpu); 5.35784s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  We repeat the example using the type _R_a_t_i_o_n_a_l_M_a_p and using deterministic
│ │ │ │  methods.
│ │ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.0011253s (cpu); 0.00197141s (thread); 0s (gc)
│ │ │ │ + -- used 0.00200211s (cpu); 0.00186545s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -233,15 +233,15 @@
│ │ │ │                            3                2    2
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ │ - -- used 0.155471s (cpu); 0.0755467s (thread); 0s (gc)
│ │ │ │ + -- used 0.193117s (cpu); 0.0663768s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                                     ZZ
│ │ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │ @@ -304,15 +304,15 @@
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i14 : time phi^(-1)
│ │ │ │ - -- used 0.504825s (cpu); 0.425036s (thread); 0s (gc)
│ │ │ │ + -- used 0.435332s (cpu); 0.435809s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                                     ZZ
│ │ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │  ]) defined by
│ │ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │ │                {
│ │ │ │ @@ -373,67 +373,67 @@
│ │ │ │                          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
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9
│ │ │ │  to PP^6)
│ │ │ │  i15 : time degrees phi^(-1)
│ │ │ │ - -- used 0.355466s (cpu); 0.270355s (thread); 0s (gc)
│ │ │ │ + -- used 0.489651s (cpu); 0.36645s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.120907s (cpu); 0.0437002s (thread); 0s (gc)
│ │ │ │ + -- used 0.0734552s (cpu); 0.0218007s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00320772s (cpu); 0.00316311s (thread); 0s (gc)
│ │ │ │ + -- used 1.5298e-05s (cpu); 0.00351931s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = rational map defined by forms of degree 3
│ │ │ │        source variety: PP^6
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i18 : time describe phi^(-1)
│ │ │ │ - -- used 0.00643296s (cpu); 0.00993896s (thread); 0s (gc)
│ │ │ │ + -- used 0.00846573s (cpu); 0.0123656s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 = rational map defined by forms of degree 3
│ │ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        target variety: PP^6
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ │        number of minimal representatives: 1
│ │ │ │        dimension base locus: 4
│ │ │ │        degree base locus: 24
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ │ - -- used 0.0084534s (cpu); 0.00881838s (thread); 0s (gc)
│ │ │ │ + -- used 0.0520925s (cpu); 0.052059s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety
│ │ │ │  of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │ │  i21 : time degrees f
│ │ │ │ - -- used 1.19108s (cpu); 0.901679s (thread); 0s (gc)
│ │ │ │ + -- used 1.50652s (cpu); 1.26241s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 0.000190698s (cpu); 1.3635e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000143619s (cpu); 1.1592e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.000926578s (cpu); 0.00142987s (thread); 0s (gc)
│ │ │ │ + -- used 0.000786354s (cpu); 0.00119327s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20
│ │ │ │  hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1,
│ │ │ │  1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2,
│ │ │ │  0},{2, 0})
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Q3ljbG90b21pYw==
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3ljbG90b21pYyBmaWVsZHMiLCBEZXNj
│ │ │  cmlwdGlvbiA9PiAoRU17IkN5Y2xvdG9taWMifSwiIGlzIGEgcGFja2FnZSBmb3IgY3ljbG90b21p
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c291cmNlKERHQWxnZWJyYU1hcCk=
│ │ │  #:len=659
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3V0cHV0cyB0aGUgc291cmNlIG9mIGEg
│ │ │  REdBbGdlYnJhTWFwIiwgImxpbmVudW0iID0+IDMxMDQsIElucHV0cyA9PiB7U1BBTntUVHsicGhp
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2     2     2       2 2     2 2      2 2      2 2     2 2        2 2       2 2        2       2       2
│ │ │         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 }
│ │ │                                      1     2     3         1       4        6        5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ - -- used 0.0133052s (cpu); 0.0129559s (thread); 0s (gc)
│ │ │ + -- used 0.0499711s (cpu); 0.013473s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S ..S ]
│ │ │                                 1   4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ - -- used 0.0166251s (cpu); 0.0162396s (thread); 0s (gc)
│ │ │ + -- used 0.131311s (cpu); 0.0232718s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a T }
│ │ │                                        1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ - -- used 0.0324166s (cpu); 0.0286597s (thread); 0s (gc)
│ │ │ + -- used 0.0725772s (cpu); 0.0157891s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -55,15 +55,15 @@
│ │ │                 {2} | 0 |
│ │ │                 {2} | 0 |
│ │ │                 {2} | 1 |
│ │ │  
│ │ │  o6 : ChainComplexMap
│ │ │  
│ │ │  i7 : HHg = HH g
│ │ │ - -- used 0.0133621s (cpu); 0.0130279s (thread); 0s (gc)
│ │ │ + -- used 0.123641s (cpu); 0.0204562s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -61,15 +61,15 @@
│ │ │                    {9} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |                
│ │ │  
│ │ │       4 : cokernel {12} | d c b a |                                       
│ │ │  
│ │ │  o6 : GradedModule
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ - -- used 0.0980879s (cpu); 0.0432325s (thread); 0s (gc)
│ │ │ + -- used 0.480558s (cpu); 0.0752885s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i8 : ideal HKR
│ │ │  
│ │ │ @@ -80,15 +80,15 @@
│ │ │  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}
│ │ │  
│ │ │  o9 = R'
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ - -- used 0.563938s (cpu); 0.492392s (thread); 0s (gc)
│ │ │ + -- used 0.692824s (cpu); 0.63252s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0213997s (cpu); 0.0168211s (thread); 0s (gc)
│ │ │ + -- used 0.0944368s (cpu); 0.0190203s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0166798s (cpu); 0.016755s (thread); 0s (gc)
│ │ │ + -- used 0.426659s (cpu); 0.091143s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  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}
│ │ │  
│ │ │ @@ -46,15 +46,15 @@
│ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.16452s (cpu); 0.0938128s (thread); 0s (gc)
│ │ │ + -- used 0.483422s (cpu); 0.155874s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -114,15 +114,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0534484s (cpu); 0.0489775s (thread); 0s (gc)
│ │ │ + -- used 0.479344s (cpu); 0.0864946s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -151,14 +151,14 @@
│ │ │         Underlying algebra => S[T ..T ]
│ │ │                                  1   4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ - -- used 0.0287422s (cpu); 0.0272575s (thread); 0s (gc)
│ │ │ + -- used 0.0882397s (cpu); 0.0153592s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          2
│ │ │  
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ - -- used 0.0150847s (cpu); 0.0139493s (thread); 0s (gc)
│ │ │ + -- used 0.0589662s (cpu); 0.0128478s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │  
│ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : ChainComplex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.105472s (cpu); 0.102342s (thread); 0s (gc)
│ │ │ + -- used 0.177312s (cpu); 0.116773s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T T T  - x x y T T T )
│ │ │               2 2 1 2 3    1 2 2 2 3 4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ - -- used 0.40381s (cpu); 0.400564s (thread); 0s (gc)
│ │ │ + -- used 0.68632s (cpu); 0.629492s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
│ │ ├── ./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
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1   4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ - -- used 0.148943s (cpu); 0.143147s (thread); 0s (gc)
│ │ │ + -- used 0.822595s (cpu); 0.402168s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.449036s (cpu); 0.378625s (thread); 0s (gc)
│ │ │ + -- used 1.0983s (cpu); 0.60554s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -249,15 +249,15 @@
│ │ │          
    
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
                  
    i17 : HHg = HH g
│ │ │ - -- used 0.0133052s (cpu); 0.0129559s (thread); 0s (gc)
│ │ │ + -- used 0.0499711s (cpu); 0.013473s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -210,15 +210,15 @@
│ │ │ │  a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
│ │ │ │                                      1     2     3         1       4        6
│ │ │ │  5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ - -- used 0.0133052s (cpu); 0.0129559s (thread); 0s (gc)
│ │ │ │ + -- used 0.0499711s (cpu); 0.013473s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │          
    
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1   4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ - -- used 0.0166251s (cpu); 0.0162396s (thread); 0s (gc)
│ │ │ │ + -- used 0.131311s (cpu); 0.0232718s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │        ZZ
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │                                 1   5
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a T }
│ │ │ │                                        1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ - -- used 0.0324166s (cpu); 0.0286597s (thread); 0s (gc)
│ │ │ │ + -- used 0.0725772s (cpu); 0.0157891s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i11 : C = killCycles(B)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │                 {2} | 0 |
│ │ │                 {2} | 1 |
│ │ │  
│ │ │  o6 : ChainComplexMap
│ │ │  
│ │ │            
│ │ │  
                  
    i5 : HB = HH B
│ │ │ - -- used 0.0166251s (cpu); 0.0162396s (thread); 0s (gc)
│ │ │ + -- used 0.131311s (cpu); 0.0232718s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │      		          
                  
    i6 : describe HB
│ │ │ @@ -148,15 +148,15 @@
│ │ │        Differential => {a, b, c, d, a T }
│ │ │                                        1
│ │ │  
│ │ │  o9 : DGAlgebra
    
│ │ │      		          
                  
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ - -- used 0.0324166s (cpu); 0.0286597s (thread); 0s (gc)
│ │ │ + -- used 0.0725772s (cpu); 0.0157891s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │      		          
              
                  
    i7 : HHg = HH g
│ │ │ - -- used 0.0133621s (cpu); 0.0130279s (thread); 0s (gc)
│ │ │ + -- used 0.123641s (cpu); 0.0204562s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │       2 : R  <------------- R  : 2
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 1 |
│ │ │ │  
│ │ │ │  o6 : ChainComplexMap
│ │ │ │  i7 : HHg = HH g
│ │ │ │ - -- used 0.0133621s (cpu); 0.0130279s (thread); 0s (gc)
│ │ │ │ + -- used 0.123641s (cpu); 0.0204562s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -130,15 +130,15 @@
│ │ │          
    
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i7 : HKR = HH KR
│ │ │ - -- used 0.0980879s (cpu); 0.0432325s (thread); 0s (gc)
│ │ │ + -- used 0.480558s (cpu); 0.0752885s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │      		          
                  
    i8 : ideal HKR
│ │ │ @@ -152,15 +152,15 @@
│ │ │  
│ │ │  o9 = R'
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │      		          
                  
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ - -- used 0.563938s (cpu); 0.492392s (thread); 0s (gc)
│ │ │ + -- used 0.692824s (cpu); 0.63252s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │      		          
                  
    i11 : numgens HKR'
│ │ │ ├── html2text {}
│ │ │ │ @@ -75,15 +75,15 @@
│ │ │ │  
│ │ │ │  o6 : GradedModule
│ │ │ │  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.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ - -- used 0.0980879s (cpu); 0.0432325s (thread); 0s (gc)
│ │ │ │ + -- used 0.480558s (cpu); 0.0752885s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │  o8 = ideal ()
│ │ │ │ @@ -92,15 +92,15 @@
│ │ │ │  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}
│ │ │ │  
│ │ │ │  o9 = R'
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ │ - -- used 0.563938s (cpu); 0.492392s (thread); 0s (gc)
│ │ │ │ + -- used 0.692824s (cpu); 0.63252s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ │ │  
│ │ │ │  o11 = 34
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
    
│ │ │      		          
                  
    i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0213997s (cpu); 0.0168211s (thread); 0s (gc)
│ │ │ + -- used 0.0944368s (cpu); 0.0190203s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │      		          
                  
    i5 : numgens HA
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.0213997s (cpu); 0.0168211s (thread); 0s (gc)
│ │ │ │ + -- used 0.0944368s (cpu); 0.0190203s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ │ │  
│ │ │ │  o5 = 4
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
    
│ │ │      		          
                  
    i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0166798s (cpu); 0.016755s (thread); 0s (gc)
│ │ │ + -- used 0.426659s (cpu); 0.091143s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │      		          
    
│ │ │          
    
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
    
│ │ │      		          
                  
    i8 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.16452s (cpu); 0.0938128s (thread); 0s (gc)
│ │ │ + -- used 0.483422s (cpu); 0.155874s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │      		          
                  
    i9 : numgens HA
│ │ │ @@ -215,15 +215,15 @@
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │      		          
                  
    i16 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0534484s (cpu); 0.0489775s (thread); 0s (gc)
│ │ │ + -- used 0.479344s (cpu); 0.0864946s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │      		          
    
│ │ │          
    
│ │ │ @@ -265,15 +265,15 @@
│ │ │                                  1   4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │      		          
                  
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ - -- used 0.0287422s (cpu); 0.0272575s (thread); 0s (gc)
│ │ │ + -- used 0.0882397s (cpu); 0.0153592s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │      		          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.0166798s (cpu); 0.016755s (thread); 0s (gc)
│ │ │ │ + -- used 0.426659s (cpu); 0.091143s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra)
│ │ │ │  since R is a complete intersection.
│ │ │ │  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}
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  o6 : DGAlgebra
│ │ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.16452s (cpu); 0.0938128s (thread); 0s (gc)
│ │ │ │ + -- used 0.483422s (cpu); 0.155874s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │  o9 = 19
│ │ │ │ @@ -121,15 +121,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.0534484s (cpu); 0.0489775s (thread); 0s (gc)
│ │ │ │ + -- used 0.479344s (cpu); 0.0864946s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │  options GenDegreeLimit and RelDegreeLimit.
│ │ │ │ @@ -156,15 +156,15 @@
│ │ │ │  o20 = {Ring => S                      }
│ │ │ │         Underlying algebra => S[T ..T ]
│ │ │ │                                  1   4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ - -- used 0.0287422s (cpu); 0.0272575s (thread); 0s (gc)
│ │ │ │ + -- used 0.0882397s (cpu); 0.0153592s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │          2
│ │ │  
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
    │ │ │ 
    i7 : H = HH(KR)
│ │ │ - -- used 0.0150847s (cpu); 0.0139493s (thread); 0s (gc)
│ │ │ + -- used 0.0589662s (cpu); 0.0128478s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    │ │ │ 
    i8 : homologyClass(KR,z1*z2)
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          2
│ │ │ │  
│ │ │ │  o6 : R[T ..T ]
│ │ │ │          1   3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ - -- used 0.0150847s (cpu); 0.0139493s (thread); 0s (gc)
│ │ │ │ + -- used 0.0589662s (cpu); 0.0128478s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ │ │  
│ │ │ │  o8 = X X
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ 
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.105472s (cpu); 0.102342s (thread); 0s (gc)
│ │ │ + -- used 0.177312s (cpu); 0.116773s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -55,15 +55,15 @@ │ │ │ │ 1 4 6 4 1 │ │ │ │ o5 = R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ │ │ i6 : HKR = HH(KR) │ │ │ │ - -- used 0.105472s (cpu); 0.102342s (thread); 0s (gc) │ │ │ │ + -- used 0.177312s (cpu); 0.116773s (thread); 0s (gc) │ │ │ │ Finding easy relations : │ │ │ │ o6 = HKR │ │ │ │ │ │ │ │ o6 : QuotientRing │ │ │ │ The following is the graded canonical module of R: │ │ │ │ i7 : degList = first entries vars Q / degree / first │ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product.html │ │ │ @@ -176,15 +176,15 @@ │ │ │
    │ │ │
    │ │ │

    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:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ - -- used 0.40381s (cpu); 0.400564s (thread); 0s (gc)
│ │ │ + -- used 0.68632s (cpu); 0.629492s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
    │ │ │ ├── html2text {} │ │ │ │ @@ -91,15 +91,15 @@ │ │ │ │ Note that the first return value of _g_e_t_B_o_u_n_d_a_r_y_P_r_e_i_m_a_g_e indicates that the │ │ │ │ inputs are indeed boundaries, and the second value is the lift of the boundary │ │ │ │ along the differential. │ │ │ │ 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: │ │ │ │ i10 : z123 = masseyTripleProduct(KR,z1,z2,z3) │ │ │ │ - -- used 0.40381s (cpu); 0.400564s (thread); 0s (gc) │ │ │ │ + -- used 0.68632s (cpu); 0.629492s (thread); 0s (gc) │ │ │ │ Finding easy relations : │ │ │ │ 2 │ │ │ │ o10 = x x y z T T T T │ │ │ │ 1 2 2 2 3 4 5 │ │ │ │ │ │ │ │ o10 : R[T ..T ] │ │ │ │ 1 5 │ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html │ │ │ @@ -114,15 +114,15 @@ │ │ │ Differential => {t , t , t , t } │ │ │ 1 2 3 4 │ │ │ │ │ │ o4 : DGAlgebra │ │ │ 
    i5 : H = HH(KR)
│ │ │ - -- used 0.148943s (cpu); 0.143147s (thread); 0s (gc)
│ │ │ + -- used 0.822595s (cpu); 0.402168s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    │ │ │ 
    i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │        Underlying algebra => R[T ..T ]
│ │ │ │                                 1   4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ - -- used 0.148943s (cpu); 0.143147s (thread); 0s (gc)
│ │ │ │ + -- used 0.822595s (cpu); 0.402168s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ │ │  
│ │ │ │                5       343
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
    │ │ │ 
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.449036s (cpu); 0.378625s (thread); 0s (gc)
│ │ │ + -- used 1.0983s (cpu); 0.60554s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    │ │ │ 
    i5 : numgens HB
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};
│ │ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ │ - -- used 0.449036s (cpu); 0.378625s (thread); 0s (gc)
│ │ │ │ + -- used 1.0983s (cpu); 0.60554s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ │ │  
│ │ │ │  o5 = 35
│ │ ├── ./usr/share/doc/Macaulay2/DecomposableSparseSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  c29sdmVEZWNvbXBvc2FibGVTeXN0ZW0oTGlzdCk=
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│ │ ├── ./usr/share/doc/Macaulay2/Depth/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  U2VlZA==
│ │ │  #:len=219
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/DeterminantalRepresentations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  Z2VuZXJhbGl6ZWRNaXhlZERpc2NyaW1pbmFudChMaXN0KQ==
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmFsaXplZE1peGVkRGlzY3JpbWluYW50LExp
│ │ ├── ./usr/share/doc/Macaulay2/DiffAlg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  bGluZWFyQ29tYg==
│ │ │  #:len=2390
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2VuZXJpYyBsaW5lYXIgY29tYmluYXRp
│ │ │  b24gb2YgZWxlbWVudHMiLCAibGluZW51bSIgPT4gODY0LCBJbnB1dHMgPT4ge1NQQU57VFR7Ikwi
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  Zmxvb3IoUldlaWxEaXZpc29yKQ==
│ │ │  #:len=260
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU1Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxvb3IsUldlaWxEaXZpc29yKSwiZmxvb3IoUldl
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/___Basic__Divisor_sp_pl_sp__Basic__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 11085051886200177329
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │ +o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │ +o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : D1 + D2
│ │ │  
│ │ │ -o4 = -4*Div(x) + 6*Div(y)
│ │ │ +o4 = 6*Div(y) + -4*Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 6*Div(x) + 4*Div(z)
│ │ │ +o5 = 4*Div(z) + 6*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : R = QQ[x,y];
│ │ │  
│ │ │  i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ @@ -64,30 +64,30 @@
│ │ │  
│ │ │  o12 : RWeilDivisor on R
│ │ │  
│ │ │  i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
│ │ │  
│ │ │  i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │  
│ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : -D
│ │ │  
│ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │  
│ │ │  o15 : WeilDivisor on R
│ │ │  
│ │ │  i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │  
│ │ │  o16 : WeilDivisor on R
│ │ │  
│ │ │  i17 : -E
│ │ │  
│ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/___Number_sp_st_sp__Basic__Divisor.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o3 = 1/2*Div(x) + -5/3*Div(y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, CoefficientType=>RR)
│ │ │  
│ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2)
│ │ │ +o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : 8*D
│ │ │  
│ │ │  o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │  
│ │ │ @@ -46,18 +46,18 @@
│ │ │  
│ │ │  o9 = 2.35667*Div(y) + -.707*Div(x)
│ │ │  
│ │ │  o9 : RWeilDivisor on R
│ │ │  
│ │ │  i10 : 6*F
│ │ │  
│ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2)
│ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x)
│ │ │  
│ │ │  o10 : RWeilDivisor on R
│ │ │  
│ │ │  i11 : (-3/2)*F
│ │ │  
│ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2)
│ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x)
│ │ │  
│ │ │  o11 : RWeilDivisor on R
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_apply__To__Coefficients.out
│ │ │ @@ -1,17 +1,17 @@
│ │ │  -- -*- M2-comint -*- hash: 14937934652040812889
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = -Div(z) + 2*Div(y) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + -Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : applyToCoefficients(D, u->5*u)
│ │ │  
│ │ │ -o3 = 10*Div(y) + -5*Div(z) + 5*Div(x)
│ │ │ +o3 = 5*Div(x) + 10*Div(y) + -5*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_dualize.out
│ │ │ @@ -44,51 +44,51 @@
│ │ │  i10 : J = m^9;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : M = J*R^1;
│ │ │  
│ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0442293s (cpu); 0.0454216s (thread); 0s (gc)
│ │ │ + -- used 0.0706306s (cpu); 0.0712794s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.386445s (cpu); 0.388413s (thread); 0s (gc)
│ │ │ + -- used 0.596945s (cpu); 0.599695s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.516665s (cpu); 0.447697s (thread); 0s (gc)
│ │ │ + -- used 0.676855s (cpu); 0.618389s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00277537s (cpu); 0.00285669s (thread); 0s (gc)
│ │ │ + -- used 0.042174s (cpu); 0.0427084s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 3.8001e-05s (cpu); 0.00208153s (thread); 0s (gc)
│ │ │ + -- used 8.4117e-05s (cpu); 0.0022917s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i18 : I = ideal(x,u);
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J = I^15;
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │  
│ │ │  i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.172064s (cpu); 0.095124s (thread); 0s (gc)
│ │ │ + -- used 0.163694s (cpu); 0.103046s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00560499s (cpu); 0.00590381s (thread); 0s (gc)
│ │ │ + -- used 0.00165973s (cpu); 0.00485289s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_is__Cartier.out
│ │ │ @@ -12,27 +12,27 @@
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : isCartier( D )
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : R = QQ[x, y, z];
│ │ │  
│ │ │  i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ -o8 = Div(x) + 2*Div(y)
│ │ │ +o8 = 2*Div(y) + Div(x)
│ │ │  
│ │ │  o8 : WeilDivisor on R
│ │ │  
│ │ │  i9 : isCartier( D )
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │ @@ -48,15 +48,15 @@
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_is__Linear__Equivalent.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 6019119347082811396
│ │ │  
│ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = Div(x, z) + 8*Div(y, z)
│ │ │ +o3 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = 8*Div(y, z) + Div(x, z)
│ │ │ +o7 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
│ │ │  
│ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │  
│ │ │  o8 = false
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_is__Q__Cartier.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 13719144060491348416
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isQCartier(10, D1)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │ @@ -44,21 +44,21 @@
│ │ │  
│ │ │  o10 = 0
│ │ │  
│ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = Div(x, z) + 2*Div(y, z)
│ │ │ +o12 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
│ │ │  
│ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ +o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
│ │ │  
│ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │  
│ │ │  o14 = 1
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_is__Q__Linear__Equivalent.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 13920959388108803216
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isQLinearEquivalent(10, D, E)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │ @@ -36,21 +36,21 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
│ │ │  
│ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
│ │ │  
│ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │  
│ │ │  o13 = true
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexify.out
│ │ │ @@ -103,104 +103,104 @@
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : J = I^21;
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time reflexify(J);
│ │ │ - -- used 0.263984s (cpu); 0.186105s (thread); 0s (gc)
│ │ │ + -- used 0.311119s (cpu); 0.216462s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.41735s (cpu); 0.34655s (thread); 0s (gc)
│ │ │ + -- used 0.456279s (cpu); 0.403964s (thread); 0s (gc)
│ │ │  
│ │ │  i25 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y);
│ │ │  
│ │ │  i26 : I = ideal(x-4*y, z);
│ │ │  
│ │ │  o26 : Ideal of R
│ │ │  
│ │ │  i27 : J = I^20;
│ │ │  
│ │ │  o27 : Ideal of R
│ │ │  
│ │ │  i28 : M = J*R^1;
│ │ │  
│ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.264868s (cpu); 0.125615s (thread); 0s (gc)
│ │ │ + -- used 0.30011s (cpu); 0.176381s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.16298s (cpu); 5.01923s (thread); 0s (gc)
│ │ │ + -- used 5.90236s (cpu); 4.60466s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
│ │ │  
│ │ │  i31 : J1 == J2
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.86731s (cpu); 4.55029s (thread); 0s (gc)
│ │ │ + -- used 5.94452s (cpu); 4.82164s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.579493s (cpu); 0.401292s (thread); 0s (gc)
│ │ │ + -- used 0.57163s (cpu); 0.431591s (thread); 0s (gc)
│ │ │  
│ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i35 : I = ideal(x,u);
│ │ │  
│ │ │  o35 : Ideal of R
│ │ │  
│ │ │  i36 : J = I^20;
│ │ │  
│ │ │  o36 : Ideal of R
│ │ │  
│ │ │  i37 : M = I^20*R^1;
│ │ │  
│ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.879462s (cpu); 0.331789s (thread); 0s (gc)
│ │ │ + -- used 1.26254s (cpu); 0.354247s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o38 : Ideal of R
│ │ │  
│ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.240056s (cpu); 0.0751777s (thread); 0s (gc)
│ │ │ + -- used 0.0093123s (cpu); 0.0121342s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o39 : Ideal of R
│ │ │  
│ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.0402565s (cpu); 0.0404667s (thread); 0s (gc)
│ │ │ + -- used 0.268245s (cpu); 0.129089s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00492897s (cpu); 0.00705635s (thread); 0s (gc)
│ │ │ + -- used 0.00322467s (cpu); 0.00538113s (thread); 0s (gc)
│ │ │  
│ │ │  i42 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i43 : I = ideal(x,y);
│ │ │  
│ │ │  o43 : Ideal of R
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexive__Power.out
│ │ │ @@ -23,44 +23,44 @@
│ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.114875s (cpu); 0.0601963s (thread); 0s (gc)
│ │ │ + -- used 0.0849408s (cpu); 0.0338104s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.123502s (cpu); 0.125616s (thread); 0s (gc)
│ │ │ + -- used 0.264096s (cpu); 0.265766s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : J20a == J20b
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0279996s (cpu); 0.0294503s (thread); 0s (gc)
│ │ │ + -- used 0.0400012s (cpu); 0.0390273s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.151204s (cpu); 0.0977003s (thread); 0s (gc)
│ │ │ + -- used 0.149731s (cpu); 0.0876416s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_to__R__Weil__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 12819564349892123361
│ │ │  
│ │ │  i1 : R = ZZ/5[x,y];
│ │ │  
│ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : F == 2*G
│ │ │  
│ │ │  o6 = true
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/___Basic__Divisor_sp_pl_sp__Basic__Divisor.html
│ │ │ @@ -79,36 +79,36 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : R = QQ[x, y, z];
    
│ │ │      		          
                  
    i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │ +o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │      		          
                  
    i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │ +o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │      		          
                  
    i4 : D1 + D2
│ │ │  
│ │ │ -o4 = -4*Div(x) + 6*Div(y)
│ │ │ +o4 = 6*Div(y) + -4*Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │      		          
                  
    i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 6*Div(x) + 4*Div(z)
│ │ │ +o5 = 4*Div(z) + 6*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    We can also add or subtract divisors with different coefficients.
    
│ │ │          
    
│ │ │ @@ -165,36 +165,36 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
    
│ │ │      		          
                  
    i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │  
│ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    
│ │ │      		          
                  
    i15 : -D
│ │ │  
│ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │  
│ │ │  o15 : WeilDivisor on R
    
│ │ │      		          
                  
    i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │  
│ │ │  o16 : WeilDivisor on R
    
│ │ │      		          
                  
    i17 : -E
│ │ │  
│ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    
│ │ │      		          
    
│ │ │        
│ │ │        
    
│ │ │          
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,30 +17,30 @@
│ │ │ │      * Outputs:
│ │ │ │            o an instance of the type _B_a_s_i_c_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  We can add or subtract two divisors:
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │ │ +o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │ │  
│ │ │ │ -o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │ │ +o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : D1 + D2
│ │ │ │  
│ │ │ │ -o4 = -4*Div(x) + 6*Div(y)
│ │ │ │ +o4 = 6*Div(y) + -4*Div(x)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : D1 - D2
│ │ │ │  
│ │ │ │ -o5 = 6*Div(x) + 4*Div(z)
│ │ │ │ +o5 = 4*Div(z) + 6*Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  We can also add or subtract divisors with different coefficients.
│ │ │ │  i6 : R = QQ[x,y];
│ │ │ │  i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)})
│ │ │ │  
│ │ │ │  o7 = 3*Div(x) + Div(y)
│ │ │ │ @@ -71,28 +71,28 @@
│ │ │ │  o12 = -Div(y) + 2.75*Div(x)
│ │ │ │  
│ │ │ │  o12 : RWeilDivisor on R
│ │ │ │  Finally, we can negate a divisor.
│ │ │ │  i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
│ │ │ │  i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │ │  
│ │ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : -D
│ │ │ │  
│ │ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │ │  
│ │ │ │  o15 : WeilDivisor on R
│ │ │ │  i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │ │  
│ │ │ │  o16 : WeilDivisor on R
│ │ │ │  i17 : -E
│ │ │ │  
│ │ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _B_a_s_i_c_D_i_v_i_s_o_r_ _+_ _B_a_s_i_c_D_i_v_i_s_o_r -- add or subtract two divisors, or negate a
│ │ │ │        divisor
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/___Number_sp_st_sp__Basic__Divisor.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  o3 = 1/2*Div(x) + -5/3*Div(y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    │ │ │ 
    i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, CoefficientType=>RR)
│ │ │  
│ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2)
│ │ │ +o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
    │ │ │ 
    i5 : 8*D
│ │ │  
│ │ │  o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │ @@ -131,22 +131,22 @@
│ │ │  o9 = 2.35667*Div(y) + -.707*Div(x)
│ │ │  
│ │ │  o9 : RWeilDivisor on R
    │ │ │ 
    i10 : 6*F
│ │ │  
│ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2)
│ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x)
│ │ │  
│ │ │  o10 : RWeilDivisor on R
    │ │ │ 
    i11 : (-3/2)*F
│ │ │  
│ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2)
│ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x)
│ │ │  
│ │ │  o11 : RWeilDivisor on R
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,15 +23,15 @@ │ │ │ │ │ │ │ │ o3 = 1/2*Div(x) + -5/3*Div(y) │ │ │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, │ │ │ │ CoefficientType=>RR) │ │ │ │ │ │ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2) │ │ │ │ +o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y) │ │ │ │ │ │ │ │ o4 : RWeilDivisor on R │ │ │ │ i5 : 8*D │ │ │ │ │ │ │ │ o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ @@ -53,20 +53,20 @@ │ │ │ │ i9 : (-1.414)*E │ │ │ │ │ │ │ │ o9 = 2.35667*Div(y) + -.707*Div(x) │ │ │ │ │ │ │ │ o9 : RWeilDivisor on R │ │ │ │ i10 : 6*F │ │ │ │ │ │ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2) │ │ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x) │ │ │ │ │ │ │ │ o10 : RWeilDivisor on R │ │ │ │ i11 : (-3/2)*F │ │ │ │ │ │ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2) │ │ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x) │ │ │ │ │ │ │ │ o11 : RWeilDivisor on R │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _N_u_m_b_e_r_ _*_ _B_a_s_i_c_D_i_v_i_s_o_r -- multiply a divisor by a number │ │ │ │ * QQ * RWeilDivisor │ │ │ │ * QQ * WeilDivisor │ │ │ │ * RR * QWeilDivisor │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_apply__To__Coefficients.html │ │ │ @@ -83,22 +83,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ 
    i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = -Div(z) + 2*Div(y) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + -Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ 
    i3 : applyToCoefficients(D, u->5*u)
│ │ │  
│ │ │ -o3 = 10*Div(y) + -5*Div(z) + 5*Div(x)
│ │ │ +o3 = 5*Div(x) + 10*Div(y) + -5*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -26,20 +26,20 @@ │ │ │ │ the output D is the same as the class of the input D1 (WeilDivisor, │ │ │ │ QWeilDivisor, RWeilDivisor, BasicDivisor). If Safe is set to true (the default │ │ │ │ is false), then the function will check to make sure the output is a valid │ │ │ │ divisor. │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ i2 : D = divisor(x*y^2/z) │ │ │ │ │ │ │ │ -o2 = -Div(z) + 2*Div(y) + Div(x) │ │ │ │ +o2 = Div(x) + 2*Div(y) + -Div(z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : applyToCoefficients(D, u->5*u) │ │ │ │ │ │ │ │ -o3 = 10*Div(y) + -5*Div(z) + 5*Div(x) │ │ │ │ +o3 = 5*Div(x) + 10*Div(y) + -5*Div(z) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _f_l_o_o_r_(_R_W_e_i_l_D_i_v_i_s_o_r_) -- produce a WeilDivisor whose coefficients are │ │ │ │ ceilings or floors of the divisor │ │ │ │ * _c_e_i_l_i_n_g_(_R_W_e_i_l_D_i_v_i_s_o_r_) -- produce a WeilDivisor whose coefficients are │ │ │ │ ceilings or floors of the divisor │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_dualize.html │ │ │ @@ -145,35 +145,35 @@ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ 
    i11 : M = J*R^1;
    │ │ │ │ │ │ │ │ │ 
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0442293s (cpu); 0.0454216s (thread); 0s (gc)
│ │ │ + -- used 0.0706306s (cpu); 0.0712794s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.386445s (cpu); 0.388413s (thread); 0s (gc)
│ │ │ + -- used 0.596945s (cpu); 0.599695s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.516665s (cpu); 0.447697s (thread); 0s (gc)
    │ │ │ + -- used 0.676855s (cpu); 0.618389s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00277537s (cpu); 0.00285669s (thread); 0s (gc)
    │ │ │ + -- used 0.042174s (cpu); 0.0427084s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 3.8001e-05s (cpu); 0.00208153s (thread); 0s (gc)
│ │ │ + -- used 8.4117e-05s (cpu); 0.0022917s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │
    │ │ │

    For monomial ideals in toric rings, frequently ModuleStrategy appears faster.

    │ │ │
    │ │ │ @@ -189,21 +189,21 @@ │ │ │ │ │ │ 
    i19 : J = I^15;
│ │ │  
│ │ │  o19 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.172064s (cpu); 0.095124s (thread); 0s (gc)
│ │ │ + -- used 0.163694s (cpu); 0.103046s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00560499s (cpu); 0.00590381s (thread); 0s (gc)
│ │ │ + -- used 0.00165973s (cpu); 0.00485289s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
    │ │ │ │ │ │ │ │ │
    │ │ │

    KnownDomain is an option for dualize. If it is false (default is true), then the computer will first check whether the ring is a domain, if it is not then it will revert to ModuleStrategy. If KnownDomain is set to true for a non-domain, then the function can return an incorrect answer.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -61,43 +61,43 @@ │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : J = m^9; │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : M = J*R^1; │ │ │ │ i12 : time dualize(J, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.0442293s (cpu); 0.0454216s (thread); 0s (gc) │ │ │ │ + -- used 0.0706306s (cpu); 0.0712794s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.386445s (cpu); 0.388413s (thread); 0s (gc) │ │ │ │ + -- used 0.596945s (cpu); 0.599695s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.516665s (cpu); 0.447697s (thread); 0s (gc) │ │ │ │ + -- used 0.676855s (cpu); 0.618389s (thread); 0s (gc) │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00277537s (cpu); 0.00285669s (thread); 0s (gc) │ │ │ │ + -- used 0.042174s (cpu); 0.0427084s (thread); 0s (gc) │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 3.8001e-05s (cpu); 0.00208153s (thread); 0s (gc) │ │ │ │ + -- used 8.4117e-05s (cpu); 0.0022917s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ For monomial ideals in toric rings, frequently ModuleStrategy appears faster. │ │ │ │ i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ i18 : I = ideal(x,u); │ │ │ │ │ │ │ │ o18 : Ideal of R │ │ │ │ i19 : J = I^15; │ │ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ i20 : time dualize(J, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.172064s (cpu); 0.095124s (thread); 0s (gc) │ │ │ │ + -- used 0.163694s (cpu); 0.103046s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00560499s (cpu); 0.00590381s (thread); 0s (gc) │ │ │ │ + -- used 0.00165973s (cpu); 0.00485289s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ KnownDomain is an option for dualize. If it is false (default is true), then │ │ │ │ the computer will first check whether the ring is a domain, if it is not then │ │ │ │ it will revert to ModuleStrategy. If KnownDomain is set to true for a non- │ │ │ │ domain, then the function can return an incorrect answer. │ │ │ │ i22 : R = QQ[x,y]/ideal(x*y); │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_is__Cartier.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    │ │ │ 
    i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    │ │ │ 
    i6 : isCartier( D )
│ │ │  
│ │ │  o6 = false
    │ │ │ @@ -119,15 +119,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : R = QQ[x, y, z];
    │ │ │ 
    i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ -o8 = Div(x) + 2*Div(y)
│ │ │ +o8 = 2*Div(y) + Div(x)
│ │ │  
│ │ │  o8 : WeilDivisor on R
    │ │ │ 
    i9 : isCartier( D )
│ │ │  
│ │ │  o9 = true
    │ │ │ @@ -156,15 +156,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
    │ │ │ 
    i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    │ │ │ 
    i15 : isCartier(D, IsGraded => true)
│ │ │  
│ │ │  o15 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -26,25 +26,25 @@ │ │ │ │ i3 : isCartier( D ) │ │ │ │ │ │ │ │ o3 = false │ │ │ │ Neither is this divisor. │ │ │ │ i4 : R = QQ[x, y, z] / ideal(x * y - z^2 ); │ │ │ │ i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o5 = Div(x, z) + 2*Div(y, z) │ │ │ │ +o5 = 2*Div(y, z) + Div(x, z) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ i6 : isCartier( D ) │ │ │ │ │ │ │ │ o6 = false │ │ │ │ Of course the next divisor is Cartier. │ │ │ │ i7 : R = QQ[x, y, z]; │ │ │ │ i8 : D = divisor({1, 2}, {ideal(x), ideal(y)}) │ │ │ │ │ │ │ │ -o8 = Div(x) + 2*Div(y) │ │ │ │ +o8 = 2*Div(y) + Div(x) │ │ │ │ │ │ │ │ o8 : WeilDivisor on R │ │ │ │ i9 : isCartier( D ) │ │ │ │ │ │ │ │ o9 = true │ │ │ │ If the option IsGraded is set to true (it is false by default), this will check │ │ │ │ as if D is a divisor on the $Proj$ of the ambient graded ring. │ │ │ │ @@ -56,15 +56,15 @@ │ │ │ │ o11 : WeilDivisor on R │ │ │ │ i12 : isCartier(D, IsGraded => true) │ │ │ │ │ │ │ │ o12 = true │ │ │ │ i13 : R = QQ[x, y, z] / ideal(x * y - z^2); │ │ │ │ i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o14 = Div(x, z) + 2*Div(y, z) │ │ │ │ +o14 = 2*Div(y, z) + Div(x, z) │ │ │ │ │ │ │ │ o14 : WeilDivisor on R │ │ │ │ i15 : isCartier(D, IsGraded => true) │ │ │ │ │ │ │ │ o15 = true │ │ │ │ The output value of this function is stored in the divisor's cache with the │ │ │ │ value of the last IsGraded option. If you change the IsGraded option, the value │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_is__Linear__Equivalent.html │ │ │ @@ -81,22 +81,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
    │ │ │ 
    i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ 
    i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = Div(x, z) + 8*Div(y, z)
│ │ │ +o3 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ 
    i4 : isLinearEquivalent(D1, D2)
│ │ │  
│ │ │  o4 = true
    │ │ │ @@ -108,22 +108,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
    │ │ │ 
    i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
    │ │ │ 
    i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = 8*Div(y, z) + Div(x, z)
│ │ │ +o7 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
    │ │ │ 
    i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │  
│ │ │  o8 = false
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,36 +18,36 @@ │ │ │ │ o flag, a _B_o_o_l_e_a_n_ _v_a_l_u_e, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Given two Weil divisors, this method checks whether they are linearly │ │ │ │ equivalent. │ │ │ │ i1 : R = QQ[x, y, z]/ ideal(x * y - z^2); │ │ │ │ i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o2 = 3*Div(x, z) + 8*Div(y, z) │ │ │ │ +o2 = 8*Div(y, z) + 3*Div(x, z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)}) │ │ │ │ │ │ │ │ -o3 = Div(x, z) + 8*Div(y, z) │ │ │ │ +o3 = 8*Div(y, z) + Div(x, z) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ i4 : isLinearEquivalent(D1, D2) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ If IsGraded is set to true (by default it is false), then it treats the │ │ │ │ divisors as divisors on the $Proj$ of their ambient ring. │ │ │ │ i5 : R = QQ[x, y, z]/ ideal(x * y - z^2); │ │ │ │ i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o6 = 8*Div(y, z) + 3*Div(x, z) │ │ │ │ +o6 = 3*Div(x, z) + 8*Div(y, z) │ │ │ │ │ │ │ │ o6 : WeilDivisor on R │ │ │ │ i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)}) │ │ │ │ │ │ │ │ -o7 = 8*Div(y, z) + Div(x, z) │ │ │ │ +o7 = Div(x, z) + 8*Div(y, z) │ │ │ │ │ │ │ │ o7 : WeilDivisor on R │ │ │ │ i8 : isLinearEquivalent(D1, D2, IsGraded => true) │ │ │ │ │ │ │ │ o8 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _O_O_ _R_W_e_i_l_D_i_v_i_s_o_r │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_is__Q__Cartier.html │ │ │ @@ -84,22 +84,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    │ │ │ 
    i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ 
    i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    │ │ │ 
    i4 : isQCartier(10, D1)
│ │ │  
│ │ │  o4 = 2
    │ │ │ @@ -145,22 +145,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -30,22 +30,22 @@ │ │ │ │ i3 : g = x^2+c*x+d │ │ │ │ │ │ │ │ 2 │ │ │ │ o3 = x + x*c + d │ │ │ │ │ │ │ │ o3 : R │ │ │ │ i4 : time eliminate(x,ideal(f,g)) │ │ │ │ - -- used 0.0056703s (cpu); 0.00402894s (thread); 0s (gc) │ │ │ │ + -- used 8.0591e-05s (cpu); 0.00209451s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 │ │ │ │ o4 = ideal(a*b*c - b*c - a d + a*c*d - b + 2b*d - d ) │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : time ideal resultant(f,g,x) │ │ │ │ - -- used 0.00111276s (cpu); 0.00238255s (thread); 0s (gc) │ │ │ │ + -- used 0.000242925s (cpu); 0.00121916s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 │ │ │ │ o5 = ideal(- a*b*c + b*c + a d - a*c*d + b - 2b*d + d ) │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : sylvesterMatrix(f,g,x) │ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html │ │ │ @@ -91,24 +91,24 @@ │ │ │ 2 │ │ │ o3 = x + x*c + d │ │ │ │ │ │ o3 : R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,22 +31,22 @@ │ │ │ │ i3 : g = x^2+c*x+d │ │ │ │ │ │ │ │ 2 │ │ │ │ o3 = x + x*c + d │ │ │ │ │ │ │ │ o3 : R │ │ │ │ i4 : time eliminate(x,ideal(f,g)) │ │ │ │ - -- used 0.0862086s (cpu); 0.0112134s (thread); 0s (gc) │ │ │ │ + -- used 0.0678867s (cpu); 0.010439s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 │ │ │ │ o4 = ideal(a*b*c - b*c - a d + a*c*d - b + 2b*d - d ) │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : time ideal resultant(f,g,x) │ │ │ │ - -- used 0.00047983s (cpu); 0.0014141s (thread); 0s (gc) │ │ │ │ + -- used 0.00035229s (cpu); 0.00120166s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 │ │ │ │ o5 = ideal(- a*b*c + b*c + a d - a*c*d + b - 2b*d + d ) │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : sylvesterMatrix(f,g,x) │ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html │ │ │ @@ -103,15 +103,15 @@ │ │ │ 8 5 │ │ │ o3 = x + x + x*c + d │ │ │ │ │ │ o3 : R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    │ │ │ 
    i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = Div(x, z) + 2*Div(y, z)
│ │ │ +o12 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
    │ │ │ 
    i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ +o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
    │ │ │ 
    i14 : isQCartier(10, D1, IsGraded => true)
│ │ │  
│ │ │  o14 = 1
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,21 +22,21 @@ │ │ │ │ Check whether $m$ times a Weil or Q-divisor $D$ is Cartier for each $m$ from 1 │ │ │ │ to a fixed positive integer n1 (if the divisor is a QWeilDivisor, it can search │ │ │ │ slightly higher than n1). If m * D1 is Cartier, it returns m. If it fails to │ │ │ │ find an m, it returns 0. │ │ │ │ i1 : R = QQ[x, y, z] / ideal(x * y - z^2 ); │ │ │ │ i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o2 = 2*Div(y, z) + Div(x, z) │ │ │ │ +o2 = Div(x, z) + 2*Div(y, z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => │ │ │ │ QQ) │ │ │ │ │ │ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z) │ │ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z) │ │ │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ i4 : isQCartier(10, D1) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : isQCartier(10, D2) │ │ │ │ │ │ │ │ @@ -60,21 +60,21 @@ │ │ │ │ │ │ │ │ o10 = 0 │ │ │ │ If the option IsGraded is set to true (by default it is false), then it treats │ │ │ │ the divisor as a divisor on the $Proj$ of their ambient ring. │ │ │ │ i11 : R = QQ[x, y, z] / ideal(x * y - z^2 ); │ │ │ │ i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o12 = Div(x, z) + 2*Div(y, z) │ │ │ │ +o12 = 2*Div(y, z) + Div(x, z) │ │ │ │ │ │ │ │ o12 : WeilDivisor on R │ │ │ │ i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => │ │ │ │ QQ) │ │ │ │ │ │ │ │ -o13 = 3/4*Div(x, z) + 1/2*Div(y, z) │ │ │ │ +o13 = 1/2*Div(y, z) + 3/4*Div(x, z) │ │ │ │ │ │ │ │ o13 : QWeilDivisor on R │ │ │ │ i14 : isQCartier(10, D1, IsGraded => true) │ │ │ │ │ │ │ │ o14 = 1 │ │ │ │ i15 : isQCartier(10, D2, IsGraded => true) │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_is__Q__Linear__Equivalent.html │ │ │ @@ -83,22 +83,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
    │ │ │ 
    i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    │ │ │ 
    i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    │ │ │ 
    i4 : isQLinearEquivalent(10, D, E)
│ │ │  
│ │ │  o4 = true
    │ │ │ @@ -138,22 +138,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
    │ │ │ 
    i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
    │ │ │ 
    i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
    │ │ │ 
    i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │  
│ │ │  o13 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,20 +20,20 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Given two rational divisors, this method returns true if they linearly │ │ │ │ equivalent after clearing denominators or if some further multiple up to n │ │ │ │ makes them linearly equivalent. Otherwise it returns false. │ │ │ │ i1 : R = QQ[x, y, z] / ideal(x * y - z^2); │ │ │ │ i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ) │ │ │ │ │ │ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z) │ │ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z) │ │ │ │ │ │ │ │ o2 : QWeilDivisor on R │ │ │ │ i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ) │ │ │ │ │ │ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z) │ │ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z) │ │ │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ i4 : isQLinearEquivalent(10, D, E) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ In the above ring, every pair of divisors is Q-linearly equivalent because the │ │ │ │ Weil divisor class group is isomorphic to Z/2. However, if we don't set n high │ │ │ │ @@ -53,21 +53,21 @@ │ │ │ │ o9 = true │ │ │ │ If IsGraded=>true (the default is false), then it treats the divisors as if │ │ │ │ they are divisors on the $Proj$ of their ambient ring. │ │ │ │ i10 : R = QQ[x, y, z] / ideal(x * y - z^2); │ │ │ │ i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => │ │ │ │ QQ) │ │ │ │ │ │ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z) │ │ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z) │ │ │ │ │ │ │ │ o11 : QWeilDivisor on R │ │ │ │ i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => │ │ │ │ QQ) │ │ │ │ │ │ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z) │ │ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z) │ │ │ │ │ │ │ │ o12 : QWeilDivisor on R │ │ │ │ i13 : isQLinearEquivalent(10, D, E, IsGraded => true) │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : isQLinearEquivalent(10, 3*D, E, IsGraded => true) │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_reflexify.html │ │ │ @@ -229,21 +229,21 @@ │ │ │ 
    i22 : J = I^21;
│ │ │  
│ │ │  o22 : Ideal of R
    │ │ │ 
    i23 : time reflexify(J);
│ │ │ - -- used 0.263984s (cpu); 0.186105s (thread); 0s (gc)
│ │ │ + -- used 0.311119s (cpu); 0.216462s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    │ │ │ 
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.41735s (cpu); 0.34655s (thread); 0s (gc)
    │ │ │ + -- used 0.456279s (cpu); 0.403964s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │

    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).

    │ │ │
    │ │ │
    │ │ │

    IdealStrategy. 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.

    │ │ │ @@ -269,42 +269,42 @@ │ │ │ o27 : Ideal of R │ │ │ 
    i28 : M = J*R^1;
    │ │ │ 
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.264868s (cpu); 0.125615s (thread); 0s (gc)
│ │ │ + -- used 0.30011s (cpu); 0.176381s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
    │ │ │ 
    i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.16298s (cpu); 5.01923s (thread); 0s (gc)
│ │ │ + -- used 5.90236s (cpu); 4.60466s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
    │ │ │ 
    i31 : J1 == J2
│ │ │  
│ │ │  o31 = true
    │ │ │ 
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.86731s (cpu); 4.55029s (thread); 0s (gc)
    │ │ │ + -- used 5.94452s (cpu); 4.82164s (thread); 0s (gc) │ │ │ 
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.579493s (cpu); 0.401292s (thread); 0s (gc)
    │ │ │ + -- used 0.57163s (cpu); 0.431591s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │

    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.

    │ │ │
    │ │ │ │ │ │ │ │ │ @@ -321,49 +321,49 @@ │ │ │ o36 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i37 : M = I^20*R^1;
    │ │ │ 
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.879462s (cpu); 0.331789s (thread); 0s (gc)
│ │ │ + -- used 1.26254s (cpu); 0.354247s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o38 : Ideal of R
    │ │ │ 
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.240056s (cpu); 0.0751777s (thread); 0s (gc)
│ │ │ + -- used 0.0093123s (cpu); 0.0121342s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o39 : Ideal of R
    │ │ │ 
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.0402565s (cpu); 0.0404667s (thread); 0s (gc)
    │ │ │ + -- used 0.268245s (cpu); 0.129089s (thread); 0s (gc) │ │ │ 
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00492897s (cpu); 0.00705635s (thread); 0s (gc)
    │ │ │ + -- used 0.00322467s (cpu); 0.00538113s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │

    For ideals, if KnownDomain is false (default value is true), 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.

    │ │ │
    │ │ │
    │ │ │

    Consider the following example showing the importance of making the correct assumption about the ring being a domain.

    │ │ │ ├── html2text {} │ │ │ │ @@ -115,19 +115,19 @@ │ │ │ │ i21 : I = ideal(x-z,y-2*z); │ │ │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ i22 : J = I^21; │ │ │ │ │ │ │ │ o22 : Ideal of R │ │ │ │ i23 : time reflexify(J); │ │ │ │ - -- used 0.263984s (cpu); 0.186105s (thread); 0s (gc) │ │ │ │ + -- used 0.311119s (cpu); 0.216462s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ i24 : time reflexify(J*R^1); │ │ │ │ - -- used 0.41735s (cpu); 0.34655s (thread); 0s (gc) │ │ │ │ + -- used 0.456279s (cpu); 0.403964s (thread); 0s (gc) │ │ │ │ Because of this, there are two strategies for computing a reflexification (at │ │ │ │ least if the module embeds as an ideal). │ │ │ │ IdealStrategy. 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. │ │ │ │ ModuleStrategy. This computes the reflexification simply by computing $Hom$ │ │ │ │ twice. │ │ │ │ ModuleStrategy is the default strategy for modules, IdealStrategy is the │ │ │ │ @@ -140,73 +140,73 @@ │ │ │ │ │ │ │ │ o26 : Ideal of R │ │ │ │ i27 : J = I^20; │ │ │ │ │ │ │ │ o27 : Ideal of R │ │ │ │ i28 : M = J*R^1; │ │ │ │ i29 : J1 = time reflexify( J, Strategy=>IdealStrategy ) │ │ │ │ - -- used 0.264868s (cpu); 0.125615s (thread); 0s (gc) │ │ │ │ + -- used 0.30011s (cpu); 0.176381s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 9 9 11 │ │ │ │ o29 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ │ │ o29 : Ideal of R │ │ │ │ i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ │ - -- used 7.16298s (cpu); 5.01923s (thread); 0s (gc) │ │ │ │ + -- used 5.90236s (cpu); 4.60466s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 9 9 11 │ │ │ │ o30 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : J1 == J2 │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ │ - -- used 5.86731s (cpu); 4.55029s (thread); 0s (gc) │ │ │ │ + -- used 5.94452s (cpu); 4.82164s (thread); 0s (gc) │ │ │ │ i33 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.579493s (cpu); 0.401292s (thread); 0s (gc) │ │ │ │ + -- used 0.57163s (cpu); 0.431591s (thread); 0s (gc) │ │ │ │ However, sometimes ModuleStrategy is faster, especially for Monomial ideals. │ │ │ │ i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ i35 : I = ideal(x,u); │ │ │ │ │ │ │ │ o35 : Ideal of R │ │ │ │ i36 : J = I^20; │ │ │ │ │ │ │ │ o36 : Ideal of R │ │ │ │ i37 : M = I^20*R^1; │ │ │ │ i38 : time reflexify( J, Strategy=>IdealStrategy ) │ │ │ │ - -- used 0.879462s (cpu); 0.331789s (thread); 0s (gc) │ │ │ │ + -- used 1.26254s (cpu); 0.354247s (thread); 0s (gc) │ │ │ │ │ │ │ │ 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 │ │ │ │ o38 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 │ │ │ │ x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 19 20 │ │ │ │ x u, x ) │ │ │ │ │ │ │ │ o38 : Ideal of R │ │ │ │ i39 : time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ │ - -- used 0.240056s (cpu); 0.0751777s (thread); 0s (gc) │ │ │ │ + -- used 0.0093123s (cpu); 0.0121342s (thread); 0s (gc) │ │ │ │ │ │ │ │ 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 │ │ │ │ o39 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 │ │ │ │ x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 19 20 │ │ │ │ x u, x ) │ │ │ │ │ │ │ │ o39 : Ideal of R │ │ │ │ i40 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ │ - -- used 0.0402565s (cpu); 0.0404667s (thread); 0s (gc) │ │ │ │ + -- used 0.268245s (cpu); 0.129089s (thread); 0s (gc) │ │ │ │ i41 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.00492897s (cpu); 0.00705635s (thread); 0s (gc) │ │ │ │ + -- used 0.00322467s (cpu); 0.00538113s (thread); 0s (gc) │ │ │ │ For ideals, if KnownDomain is false (default value is true), 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. │ │ │ │ Consider the following example showing the importance of making the correct │ │ │ │ assumption about the ring being a domain. │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_reflexive__Power.html │ │ │ @@ -114,26 +114,26 @@ │ │ │ 
    i6 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o6 : Ideal of R
    │ │ │ 
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.114875s (cpu); 0.0601963s (thread); 0s (gc)
│ │ │ + -- used 0.0849408s (cpu); 0.0338104s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ 
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    │ │ │ 
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.123502s (cpu); 0.125616s (thread); 0s (gc)
│ │ │ + -- used 0.264096s (cpu); 0.265766s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    │ │ │ 
    i10 : J20a == J20b
│ │ │  
│ │ │  o10 = true
    │ │ │ @@ -149,21 +149,21 @@ │ │ │ 
    i12 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o12 : Ideal of R
    │ │ │ 
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0279996s (cpu); 0.0294503s (thread); 0s (gc)
│ │ │ + -- used 0.0400012s (cpu); 0.0390273s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ 
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.151204s (cpu); 0.0977003s (thread); 0s (gc)
│ │ │ + -- used 0.149731s (cpu); 0.0876416s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
    │ │ │ 
    i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -41,39 +41,39 @@ │ │ │ │ of the generators of $I$. Consider the example of a cone over a point on an │ │ │ │ elliptic curve. │ │ │ │ i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3); │ │ │ │ i6 : I = ideal(x-z,y-2*z); │ │ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ i7 : time J20a = reflexivePower(20, I); │ │ │ │ - -- used 0.114875s (cpu); 0.0601963s (thread); 0s (gc) │ │ │ │ + -- used 0.0849408s (cpu); 0.0338104s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ i8 : I20 = I^20; │ │ │ │ │ │ │ │ o8 : Ideal of R │ │ │ │ i9 : time J20b = reflexify(I20); │ │ │ │ - -- used 0.123502s (cpu); 0.125616s (thread); 0s (gc) │ │ │ │ + -- used 0.264096s (cpu); 0.265766s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : J20a == J20b │ │ │ │ │ │ │ │ o10 = true │ │ │ │ This passes the Strategy option to a reflexify call. Valid options are │ │ │ │ IdealStrategy and ModuleStrategy. │ │ │ │ i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3); │ │ │ │ i12 : I = ideal(x-z,y-2*z); │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.0279996s (cpu); 0.0294503s (thread); 0s (gc) │ │ │ │ + -- used 0.0400012s (cpu); 0.0390273s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.151204s (cpu); 0.0977003s (thread); 0s (gc) │ │ │ │ + -- used 0.149731s (cpu); 0.0876416s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ i15 : J1 == J2 │ │ │ │ │ │ │ │ o15 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_f_l_e_x_i_f_y -- calculate the double dual of an ideal or module Hom(Hom(M, │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_to__R__Weil__Divisor.html │ │ │ @@ -78,36 +78,36 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ - │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -24,26 +24,18 @@ │ │ │ │ sizes using a random recursive algorithm. Limits can be placed on the both │ │ │ │ number of recursive steps taken (see _B_r_a_n_c_h_L_i_m_i_t) and on the time taken (see │ │ │ │ _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within │ │ │ │ the branch and time limits. │ │ │ │ i1 : R = QQ[x_1..x_5]; │ │ │ │ i2 : randomHyperGraph(R,{3,2,4}) │ │ │ │ i3 : randomHyperGraph(R,{3,2,4}) │ │ │ │ - │ │ │ │ -o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}} │ │ │ │ - 1 2 5 2 4 1 3 4 5 │ │ │ │ - "ring" => R │ │ │ │ - "vertices" => {x , x , x , x , x } │ │ │ │ - 1 2 3 4 5 │ │ │ │ - │ │ │ │ -o3 : HyperGraph │ │ │ │ i4 : randomHyperGraph(R,{3,2,4}) │ │ │ │ │ │ │ │ o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}} │ │ │ │ - 1 4 5 2 5 1 2 3 4 │ │ │ │ + 1 2 3 3 4 1 2 4 5 │ │ │ │ "ring" => R │ │ │ │ "vertices" => {x , x , x , x , x } │ │ │ │ 1 2 3 4 5 │ │ │ │ │ │ │ │ o4 : HyperGraph │ │ │ │ i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch │ │ │ │ limit reached │ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ emVyb0RpbVNvbHZlKElkZWFsKQ== │ │ │ #:len=254 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh6ZXJvRGltU29sdmUsSWRlYWwpLCJ6ZXJvRGltU29s │ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/example-output/___Eigen__Solver.out │ │ │ @@ -15,14 +15,14 @@ │ │ │ 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 + │ │ │ ------------------------------------------------------------------------ │ │ │ a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1) │ │ │ │ │ │ o2 : Ideal of QQ[a..f] │ │ │ │ │ │ i3 : elapsedTime sols = zeroDimSolve I; │ │ │ - -- .304758s elapsed │ │ │ + -- .409374s elapsed │ │ │ │ │ │ i4 : #sols -- 156 solutions │ │ │ │ │ │ o4 = 156 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html │ │ │ @@ -68,15 +68,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1) │ │ │ │ │ │ o2 : Ideal of QQ[a..f] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = ZZ/5[x,y];
    │ │ │ 
    i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ 
    i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    │ │ │ 
    i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
    │ │ │ 
    i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    │ │ │ 
    i6 : F == 2*G
│ │ │  
│ │ │  o6 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -16,30 +16,30 @@ │ │ │ │ o an instance of the type _R_W_e_i_l_D_i_v_i_s_o_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Turn a Weil divisor or a Q-divisor into a R-divisor (or do nothing to a R- │ │ │ │ divisor). │ │ │ │ i1 : R = ZZ/5[x,y]; │ │ │ │ i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)}) │ │ │ │ │ │ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y) │ │ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : E = (1/2)*D │ │ │ │ │ │ │ │ -o3 = -2*Div(x-y) + Div(x) │ │ │ │ +o3 = Div(x) + -2*Div(x-y) │ │ │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ i4 : F = toRWeilDivisor(D) │ │ │ │ │ │ │ │ -o4 = -4*Div(x-y) + 2*Div(x) │ │ │ │ +o4 = 2*Div(x) + -4*Div(x-y) │ │ │ │ │ │ │ │ o4 : RWeilDivisor on R │ │ │ │ i5 : G = toRWeilDivisor(E) │ │ │ │ │ │ │ │ -o5 = -2*Div(x-y) + Div(x) │ │ │ │ +o5 = Div(x) + -2*Div(x-y) │ │ │ │ │ │ │ │ o5 : RWeilDivisor on R │ │ │ │ i6 : F == 2*G │ │ │ │ │ │ │ │ o6 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_o_W_e_i_l_D_i_v_i_s_o_r -- create a Weil divisor from a Q or R-divisor │ │ ├── ./usr/share/doc/Macaulay2/Dmodules/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:38 2025 │ │ │ +# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ RG1vZHVsZXM= │ │ │ #:len=758 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRC1tb2R1bGVzIHBhY2thZ2UgY29sbGVj │ │ │ dGlvbiIsICJsaW5lbnVtIiA9PiAxNDIsICJmaWxlbmFtZSIgPT4gIkRtb2R1bGVzLm0yIiwgRGVz │ │ ├── ./usr/share/doc/Macaulay2/EagonResolution/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ VHJhbnNwb3Nl │ │ │ #:len=1366 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVHJhbnNwb3NlID0+IGZhbHNlLCBkZWZh │ │ │ dWx0IG9wdGlvbiBmb3IgcGljdHVyZSIsICJsaW5lbnVtIiA9PiAxMzUwLCBJbnB1dHMgPT4ge1NQ │ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ Y29tcGxlbWVudEdyYXBo │ │ │ #:len=2011 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY29tcGxlbWVudCBv │ │ │ ZiBhIGdyYXBoIG9yIGh5cGVyZ3JhcGgiLCAibGluZW51bSIgPT4gMjA4MCwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out │ │ │ @@ -2,26 +2,18 @@ │ │ │ │ │ │ i1 : R = QQ[x_1..x_5]; │ │ │ │ │ │ i2 : randomHyperGraph(R,{3,2,4}) │ │ │ │ │ │ i3 : randomHyperGraph(R,{3,2,4}) │ │ │ │ │ │ -o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}} │ │ │ - 1 2 5 2 4 1 3 4 5 │ │ │ - "ring" => R │ │ │ - "vertices" => {x , x , x , x , x } │ │ │ - 1 2 3 4 5 │ │ │ - │ │ │ -o3 : HyperGraph │ │ │ - │ │ │ i4 : randomHyperGraph(R,{3,2,4}) │ │ │ │ │ │ o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}} │ │ │ - 1 4 5 2 5 1 2 3 4 │ │ │ + 1 2 3 3 4 1 2 4 5 │ │ │ "ring" => R │ │ │ "vertices" => {x , x , x , x , x } │ │ │ 1 2 3 4 5 │ │ │ │ │ │ o4 : HyperGraph │ │ │ │ │ │ i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached │ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html │ │ │ @@ -84,29 +84,21 @@ │ │ │ 
    i1 : R = QQ[x_1..x_5];
    │ │ │ 
    i2 : randomHyperGraph(R,{3,2,4})
    │ │ │ 
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │ -
│ │ │ -o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   2   5     2   4     1   3   4   5
│ │ │ -                "ring" => R
│ │ │ -                "vertices" => {x , x , x , x , x }
│ │ │ -                                1   2   3   4   5
│ │ │ -
│ │ │ -o3 : HyperGraph
    │ │ │ +    		 
    i3 : randomHyperGraph(R,{3,2,4})
    │ │ │ 
    i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   4   5     2   5     1   2   3   4
│ │ │ +                              1   2   3     3   4     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    │ │ │ 
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .304758s elapsed
    │ │ │ + -- .409374s elapsed │ │ │ 
    i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 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 + │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1) │ │ │ │ │ │ │ │ o2 : Ideal of QQ[a..f] │ │ │ │ i3 : elapsedTime sols = zeroDimSolve I; │ │ │ │ - -- .304758s elapsed │ │ │ │ + -- .409374s elapsed │ │ │ │ i4 : #sols -- 156 solutions │ │ │ │ │ │ │ │ o4 = 156 │ │ │ │ The authors would like to acknowledge the June 2020 Macaulay2 workshop held │ │ │ │ virtually at Warwick, where this package was first developed. │ │ │ │ RReeffeerreenncceess: │ │ │ │ * [1] Sturmfels, Bernd. Solving systems of polynomial equations. No. 97. │ │ ├── ./usr/share/doc/Macaulay2/Elimination/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ ZWxpbWluYXRlKElkZWFsLExpc3Qp │ │ │ #:len=200 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgxLCAidW5kb2N1bWVudGVkIiA9PiB0 │ │ │ cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhlbGltaW5h │ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ @@ -17,23 +17,23 @@ │ │ │ │ │ │ 2 │ │ │ o3 = x + x*c + d │ │ │ │ │ │ o3 : R │ │ │ │ │ │ i4 : time eliminate(x,ideal(f,g)) │ │ │ - -- used 0.0056703s (cpu); 0.00402894s (thread); 0s (gc) │ │ │ + -- used 8.0591e-05s (cpu); 0.00209451s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 │ │ │ o4 = ideal(a*b*c - b*c - a d + a*c*d - b + 2b*d - d ) │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : time ideal resultant(f,g,x) │ │ │ - -- used 0.00111276s (cpu); 0.00238255s (thread); 0s (gc) │ │ │ + -- used 0.000242925s (cpu); 0.00121916s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 │ │ │ o5 = ideal(- a*b*c + b*c + a d - a*c*d + b - 2b*d + d ) │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ │ │ i6 : sylvesterMatrix(f,g,x) │ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out │ │ │ @@ -17,23 +17,23 @@ │ │ │ │ │ │ 2 │ │ │ o3 = x + x*c + d │ │ │ │ │ │ o3 : R │ │ │ │ │ │ i4 : time eliminate(x,ideal(f,g)) │ │ │ - -- used 0.0862086s (cpu); 0.0112134s (thread); 0s (gc) │ │ │ + -- used 0.0678867s (cpu); 0.010439s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 │ │ │ o4 = ideal(a*b*c - b*c - a d + a*c*d - b + 2b*d - d ) │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : time ideal resultant(f,g,x) │ │ │ - -- used 0.00047983s (cpu); 0.0014141s (thread); 0s (gc) │ │ │ + -- used 0.00035229s (cpu); 0.00120166s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 │ │ │ o5 = ideal(- a*b*c + b*c + a d - a*c*d + b - 2b*d + d ) │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ │ │ i6 : sylvesterMatrix(f,g,x) │ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ @@ -17,15 +17,15 @@ │ │ │ │ │ │ 8 5 │ │ │ o3 = x + x + x*c + d │ │ │ │ │ │ o3 : R │ │ │ │ │ │ i4 : time eliminate(ideal(f,g),x) │ │ │ - -- used 1.82282s (cpu); 1.48437s (thread); 0s (gc) │ │ │ + -- used 2.07211s (cpu); 1.90246s (thread); 0s (gc) │ │ │ │ │ │ 7 8 3 5 8 6 3 4 7 3 3 2 │ │ │ o4 = ideal(a b*c - a d + a b - b - 6a b*c - 18a b c + 7b c + 48a b c - │ │ │ ------------------------------------------------------------------------ │ │ │ 6 2 3 2 3 5 3 3 4 4 4 3 5 2 6 7 │ │ │ 21b c - 46a b c + 35b c + 15a b*c - 35b c + 21b c - 7b c + b*c + │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -73,15 +73,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ 3 4 4 │ │ │ - 216b*c*d + 2052a*d - 1944d ) │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : time ideal resultant(f,g,x) │ │ │ - -- used 0.0171485s (cpu); 0.0171405s (thread); 0s (gc) │ │ │ + -- used 0.0119978s (cpu); 0.0128444s (thread); 0s (gc) │ │ │ │ │ │ 7 8 3 5 8 6 3 4 7 3 3 2 │ │ │ o5 = ideal(- a b*c + a d - a b + b + 6a b*c + 18a b c - 7b c - 48a b c + │ │ │ ------------------------------------------------------------------------ │ │ │ 6 2 3 2 3 5 3 3 4 4 4 3 5 2 6 7 │ │ │ 21b c + 46a b c - 35b c - 15a b*c + 35b c - 21b c + 7b c - b*c - │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ │ │ 8 5 │ │ │ o4 = x + x + x*c + d │ │ │ │ │ │ o4 : R │ │ │ │ │ │ i5 : time eliminate(ideal(f,g),x) │ │ │ - -- used 1.83281s (cpu); 1.47452s (thread); 0s (gc) │ │ │ + -- used 1.96438s (cpu); 1.79449s (thread); 0s (gc) │ │ │ │ │ │ 7 8 3 5 8 6 3 4 7 3 3 2 │ │ │ o5 = ideal(a b*c - a d + a b - b - 6a b*c - 18a b c + 7b c + 48a b c - │ │ │ ------------------------------------------------------------------------ │ │ │ 6 2 3 2 3 5 3 3 4 4 4 3 5 2 6 7 │ │ │ 21b c - 46a b c + 35b c + 15a b*c - 35b c + 21b c - 7b c + b*c + │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -75,15 +75,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ 3 4 4 │ │ │ - 216b*c*d + 2052a*d - 1944d ) │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ │ │ i6 : time ideal resultant(f,g,x) │ │ │ - -- used 0.0159956s (cpu); 0.0178954s (thread); 0s (gc) │ │ │ + -- used 0.0159969s (cpu); 0.0130365s (thread); 0s (gc) │ │ │ │ │ │ 7 8 3 5 8 6 3 4 7 3 3 2 │ │ │ o6 = ideal(- a b*c + a d - a b + b + 6a b*c + 18a b c - 7b c - 48a b c + │ │ │ ------------------------------------------------------------------------ │ │ │ 6 2 3 2 3 5 3 3 4 4 4 3 5 2 6 7 │ │ │ 21b c + 46a b c - 35b c - 15a b*c + 35b c - 21b c + 7b c - b*c - │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ @@ -100,24 +100,24 @@ │ │ │ 2 │ │ │ o3 = x + x*c + d │ │ │ │ │ │ o3 : R │ │ │ 
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.0056703s (cpu); 0.00402894s (thread); 0s (gc)
│ │ │ + -- used 8.0591e-05s (cpu); 0.00209451s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │ 
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00111276s (cpu); 0.00238255s (thread); 0s (gc)
│ │ │ + -- used 0.000242925s (cpu); 0.00121916s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    │ │ │ 
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.0862086s (cpu); 0.0112134s (thread); 0s (gc)
│ │ │ + -- used 0.0678867s (cpu); 0.010439s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │ 
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00047983s (cpu); 0.0014141s (thread); 0s (gc)
│ │ │ + -- used 0.00035229s (cpu); 0.00120166s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    │ │ │ 
    i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.82282s (cpu); 1.48437s (thread); 0s (gc)
│ │ │ + -- used 2.07211s (cpu); 1.90246s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -160,15 +160,15 @@
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │ 
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0171485s (cpu); 0.0171405s (thread); 0s (gc)
│ │ │ + -- used 0.0119978s (cpu); 0.0128444s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  i3 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o3 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.82282s (cpu); 1.48437s (thread); 0s (gc)
│ │ │ │ + -- used 2.07211s (cpu); 1.90246s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0171485s (cpu); 0.0171405s (thread); 0s (gc)
│ │ │ │ + -- used 0.0119978s (cpu); 0.0128444s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │        8    5
│ │ │  o4 = x  + x  + x*c + d
│ │ │  
│ │ │  o4 : R
    │ │ │ 
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.83281s (cpu); 1.47452s (thread); 0s (gc)
│ │ │ + -- used 1.96438s (cpu); 1.79449s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -155,15 +155,15 @@
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o5 : Ideal of R
    │ │ │ 
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0159956s (cpu); 0.0178954s (thread); 0s (gc)
│ │ │ + -- used 0.0159969s (cpu); 0.0130365s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,15 +31,15 @@
│ │ │ │  i4 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o4 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o4 : R
│ │ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.83281s (cpu); 1.47452s (thread); 0s (gc)
│ │ │ │ + -- used 1.96438s (cpu); 1.79449s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0159956s (cpu); 0.0178954s (thread); 0s (gc)
│ │ │ │ + -- used 0.0159969s (cpu); 0.0130365s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  bWF4Q29sKE1hdHJpeCk=
│ │ │  #:len=257
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEyMCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF4Q29sLE1hdHJpeCksIm1heENvbChNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/EllipticCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  RWxsUG9pbnRXID09IEVsbFBvaW50Vw==
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzQ3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPT0sRWxsUG9pbnRXLEVsbFBvaW50Vyks
│ │ ├── ./usr/share/doc/Macaulay2/EllipticIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RWxsaXB0aWNJbnRlZ3JhbHM=
│ │ │  #:len=584
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMg
│ │ │  cGFja2FnZSBwcm92aWRlcyBzb21lIGZ1bmN0aW9ucyBmb3IgY29tcHV0aW5nIGVsbGlwdGljIGlu
│ │ ├── ./usr/share/doc/Macaulay2/EngineTests/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  RW5naW5lVGVzdHM=
│ │ │  #:len=327
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSB0ZXN0IHN1aXRlIGZvciB0aGUgTWFj
│ │ │  YXVsYXkyIGVuZ2luZSIsICJsaW5lbnVtIiA9PiA0NzcsICJmaWxlbmFtZSIgPT4gIkVuZ2luZVRl
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RW51bWVyYXRpb25DdXJ2ZXM=
│ │ │  #:len=747
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRW51bWVyYXRpb24gb2YgcmF0aW9uYWwg
│ │ │  Y3VydmVzIHZpYSB0b3J1cyBhY3Rpb25zIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7IiJ9
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1331975673177
│ │ │  
│ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0296402s (cpu); 0.0266657s (thread); 0s (gc)
│ │ │ + -- used 0.0199578s (cpu); 0.0216937s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out
│ │ │ @@ -37,83 +37,83 @@
│ │ │  i6 : rationalCurve(2) - rationalCurve(1)/8
│ │ │  
│ │ │  o6 = 609250
│ │ │  
│ │ │  o6 : QQ
│ │ │  
│ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.336645s (cpu); 0.292669s (thread); 0s (gc)
│ │ │ + -- used 0.444957s (cpu); 0.397796s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.196522s (cpu); 0.152451s (thread); 0s (gc)
│ │ │ + -- used 0.178508s (cpu); 0.18036s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 4.9158s (cpu); 4.16411s (thread); 0s (gc)
│ │ │ + -- used 6.19664s (cpu); 5.71291s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.194165s (cpu); 0.14128s (thread); 0s (gc)
│ │ │ + -- used 0.177237s (cpu); 0.180014s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 4.9366s (cpu); 4.18595s (thread); 0s (gc)
│ │ │ + -- used 6.32825s (cpu); 5.63986s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.6582s (cpu); 1.34963s (thread); 0s (gc)
│ │ │ + -- used 2.04936s (cpu); 1.64447s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.15585s (cpu); 5.41896s (thread); 0s (gc)
│ │ │ + -- used 8.19735s (cpu); 7.15957s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.5404s (cpu); 1.29761s (thread); 0s (gc)
│ │ │ + -- used 2.07382s (cpu); 1.93885s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.03999s (cpu); 5.27021s (thread); 0s (gc)
│ │ │ + -- used 8.97928s (cpu); 7.75984s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.56636s (cpu); 5.42334s (thread); 0s (gc)
│ │ │ + -- used 8.93705s (cpu); 7.26481s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │          
    
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.0296402s (cpu); 0.0266657s (thread); 0s (gc)
│ │ │ │ + -- used 0.0199578s (cpu); 0.0216937s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** WWaayyss ttoo uussee lliinneessHHyyppeerrssuurrffaaccee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -140,39 +140,39 @@
│ │ │          
    
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
    
│ │ │          
                  
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0296402s (cpu); 0.0266657s (thread); 0s (gc)
│ │ │ + -- used 0.0199578s (cpu); 0.0216937s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │          
                  
    i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.336645s (cpu); 0.292669s (thread); 0s (gc)
│ │ │ + -- used 0.444957s (cpu); 0.397796s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ @@ -180,89 +180,89 @@
│ │ │          
    
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
    
│ │ │          
                  
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.196522s (cpu); 0.152451s (thread); 0s (gc)
│ │ │ + -- used 0.178508s (cpu); 0.18036s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │      		          
                  
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 4.9158s (cpu); 4.16411s (thread); 0s (gc)
│ │ │ + -- used 6.19664s (cpu); 5.71291s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │          
                  
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.194165s (cpu); 0.14128s (thread); 0s (gc)
│ │ │ + -- used 0.177237s (cpu); 0.180014s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 4.9366s (cpu); 4.18595s (thread); 0s (gc)
│ │ │ + -- used 6.32825s (cpu); 5.63986s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.6582s (cpu); 1.34963s (thread); 0s (gc)
│ │ │ + -- used 2.04936s (cpu); 1.64447s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │      		          
                  
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.15585s (cpu); 5.41896s (thread); 0s (gc)
│ │ │ + -- used 8.19735s (cpu); 7.15957s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.5404s (cpu); 1.29761s (thread); 0s (gc)
│ │ │ + -- used 2.07382s (cpu); 1.93885s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    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:
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.03999s (cpu); 5.27021s (thread); 0s (gc)
│ │ │ + -- used 8.97928s (cpu); 7.75984s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │      		          
                  
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.56636s (cpu); 5.42334s (thread); 0s (gc)
│ │ │ + -- used 8.93705s (cpu); 7.26481s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
    
│ │ │      		          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,85 +60,85 @@
│ │ │ │  
│ │ │ │  o6 = 609250
│ │ │ │  
│ │ │ │  o6 : QQ
│ │ │ │  The numbers of conics on general complete intersection Calabi-Yau threefolds
│ │ │ │  can be computed as follows:
│ │ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ │ - -- used 0.336645s (cpu); 0.292669s (thread); 0s (gc)
│ │ │ │ + -- used 0.444957s (cpu); 0.397796s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.196522s (cpu); 0.152451s (thread); 0s (gc)
│ │ │ │ + -- used 0.178508s (cpu); 0.18036s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 4.9158s (cpu); 4.16411s (thread); 0s (gc)
│ │ │ │ + -- used 6.19664s (cpu); 5.71291s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        8564575000  422690816           4834592  11239424
│ │ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │ │            27          27                 3        27
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  The number of rational curves of degree 3 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ │ - -- used 0.194165s (cpu); 0.14128s (thread); 0s (gc)
│ │ │ │ + -- used 0.177237s (cpu); 0.180014s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 317206375
│ │ │ │  
│ │ │ │  o10 : QQ
│ │ │ │  The numbers of rational curves of degree 3 on general complete intersection
│ │ │ │  Calabi-Yau threefolds can be computed as follows:
│ │ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ │ - -- used 4.9366s (cpu); 4.18595s (thread); 0s (gc)
│ │ │ │ + -- used 6.32825s (cpu); 5.63986s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.6582s (cpu); 1.34963s (thread); 0s (gc)
│ │ │ │ + -- used 2.04936s (cpu); 1.64447s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.15585s (cpu); 5.41896s (thread); 0s (gc)
│ │ │ │ + -- used 8.19735s (cpu); 7.15957s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = 3883914084
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  The number of rational curves of degree 4 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ │ - -- used 1.5404s (cpu); 1.29761s (thread); 0s (gc)
│ │ │ │ + -- used 2.07382s (cpu); 1.93885s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  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:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 7.03999s (cpu); 5.27021s (thread); 0s (gc)
│ │ │ │ + -- used 8.97928s (cpu); 7.75984s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 7.56636s (cpu); 5.42334s (thread); 0s (gc)
│ │ │ │ + -- used 8.93705s (cpu); 7.26481s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee rraattiioonnaallCCuurrvvee:: **********
│ │ │ │      * rationalCurve(ZZ)
│ │ │ │      * rationalCurve(ZZ,List)
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  YnVpbGRFTW9ub21pYWxNYXAoUmluZyxSaW5nLExpc3Qp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIzNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYnVpbGRFTW9ub21pYWxNYXAsUmluZyxSaW5nLExp
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
│ │ │ @@ -10,34 +10,34 @@
│ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
│ │ │  
│ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00406175 seconds
│ │ │ +     -- used .00317789 seconds
│ │ │       -- used 0 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00315213 seconds
│ │ │ -     -- used .00397295 seconds
│ │ │ +     -- used .00333321 seconds
│ │ │ +     -- used .00400031 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00917258 seconds
│ │ │ -     -- used .023553 seconds
│ │ │ +     -- used .00337323 seconds
│ │ │ +     -- used .0318457 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0161827 seconds
│ │ │ -     -- used .187147 seconds
│ │ │ +     -- used .0159378 seconds
│ │ │ +     -- used .232856 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0342961 seconds
│ │ │ -     -- used .7618 seconds
│ │ │ +     -- used .0336027 seconds
│ │ │ +     -- used 1.10314 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -96,34 +96,34 @@
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
    │ │ │ 
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00406175 seconds
│ │ │ +     -- used .00317789 seconds
│ │ │       -- used 0 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00315213 seconds
│ │ │ -     -- used .00397295 seconds
│ │ │ +     -- used .00333321 seconds
│ │ │ +     -- used .00400031 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00917258 seconds
│ │ │ -     -- used .023553 seconds
│ │ │ +     -- used .00337323 seconds
│ │ │ +     -- used .0318457 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0161827 seconds
│ │ │ -     -- used .187147 seconds
│ │ │ +     -- used .0159378 seconds
│ │ │ +     -- used .232856 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0342961 seconds
│ │ │ -     -- used .7618 seconds
│ │ │ +     -- used .0336027 seconds
│ │ │ +     -- used 1.10314 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,34 +34,34 @@
│ │ │ │                    2               2
│ │ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │ │                    1   1 0   1 0   0
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- S
│ │ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │ │  3
│ │ │ │ -     -- used .00406175 seconds
│ │ │ │ +     -- used .00317789 seconds
│ │ │ │       -- used 0 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00315213 seconds
│ │ │ │ -     -- used .00397295 seconds
│ │ │ │ +     -- used .00333321 seconds
│ │ │ │ +     -- used .00400031 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00917258 seconds
│ │ │ │ -     -- used .023553 seconds
│ │ │ │ +     -- used .00337323 seconds
│ │ │ │ +     -- used .0318457 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0161827 seconds
│ │ │ │ -     -- used .187147 seconds
│ │ │ │ +     -- used .0159378 seconds
│ │ │ │ +     -- used .232856 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0342961 seconds
│ │ │ │ -     -- used .7618 seconds
│ │ │ │ +     -- used .0336027 seconds
│ │ │ │ +     -- used 1.10314 seconds
│ │ │ │  (49, 217)
│ │ │ │  
│ │ │ │                                     2
│ │ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/ExampleSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  aGVhcnQ=
│ │ │  #:len=1516
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│ │ │  Y3MgcHJvYmxlbSIsICJsaW5lbnVtIiA9PiA0OCwgSW5wdXRzID0+IHtTUEFOe1RUeyJrayJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c29sdmVNYWNhdWxheUV4cGFuc2lvbihMaXN0KQ==
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDMxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZU1hY2F1bGF5RXhwYW5zaW9uLExpc3QpLCJz
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorModules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  YWxtb3N0U3RhYmxlTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=292
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODAyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbG1vc3RTdGFibGVNb2R1bGUsTW9kdWxlKSwiYWxt
│ │ ├── ./usr/share/doc/Macaulay2/FGLM/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  ZmdsbShJZGVhbCxSaW5nKQ==
│ │ │  #:len=211
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzk2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmZ2xtLElkZWFsLFJpbmcpLCJmZ2xtKElkZWFsLFJp
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlTWlub3JzKFpaLE1hdHJpeCk=
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAyOCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVjdXJzaXZlTWlub3JzLFpaLE1hdHJpeCksInJl
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -462,50 +462,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.10715s (cpu); 0.107368s (thread); 0s (gc)
│ │ │ + -- used 0.187939s (cpu); 0.186654s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.33389s (cpu); 0.203875s (thread); 0s (gc)
│ │ │ + -- used 0.343333s (cpu); 0.243976s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.404835s (cpu); 0.275085s (thread); 0s (gc)
│ │ │ + -- used 0.516024s (cpu); 0.428512s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.314312s (cpu); 0.181456s (thread); 0s (gc)
│ │ │ + -- used 0.490338s (cpu); 0.252617s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.312529s (cpu); 0.1737s (thread); 0s (gc)
│ │ │ + -- used 0.332258s (cpu); 0.235725s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.348847s (cpu); 0.214069s (thread); 0s (gc)
│ │ │ + -- used 0.460208s (cpu); 0.358773s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.323316s (cpu); 0.192993s (thread); 0s (gc)
│ │ │ + -- used 0.365933s (cpu); 0.273051s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 13.4439s (cpu); 7.95676s (thread); 0s (gc)
│ │ │ + -- used 17.3469s (cpu); 11.9013s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -514,15 +514,15 @@
│ │ │                    LexSmallest => 16
│ │ │                    LexSmallestTerm => 16
│ │ │                    Points => 0
│ │ │                    Random => 16
│ │ │                    RandomNonzero => 16
│ │ │  
│ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.539994s (cpu); 0.365803s (thread); 0s (gc)
│ │ │ + -- used 0.600379s (cpu); 0.456404s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -582,15 +582,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.999209s (cpu); 0.602814s (thread); 0s (gc)
│ │ │ + -- used 1.23084s (cpu); 0.729775s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -605,47 +605,47 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.699354s (cpu); 0.37s (thread); 0s (gc)
│ │ │ + -- used 0.917925s (cpu); 0.525149s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 4.96948s (cpu); 3.56542s (thread); 0s (gc)
│ │ │ + -- used 6.32285s (cpu); 4.80086s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.03145s (cpu); 0.679038s (thread); 0s (gc)
│ │ │ + -- used 1.26344s (cpu); 0.939776s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.98086s (cpu); 2.51747s (thread); 0s (gc)
│ │ │ + -- used 4.20522s (cpu); 3.52377s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 3.40978s (cpu); 2.04641s (thread); 0s (gc)
│ │ │ + -- used 4.3251s (cpu); 2.8001s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 1.07836s (cpu); 0.777656s (thread); 0s (gc)
│ │ │ + -- used 1.62131s (cpu); 1.24061s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 4.00314s (cpu); 2.42639s (thread); 0s (gc)
│ │ │ + -- used 5.24785s (cpu); 3.29254s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 4.46603s (cpu); 2.88283s (thread); 0s (gc)
│ │ │ + -- used 6.44832s (cpu); 4.71612s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 14.8083s (cpu); 8.44014s (thread); 0s (gc)
│ │ │ + -- used 19.5055s (cpu); 12.6968s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 11.2852s (cpu); 6.42669s (thread); 0s (gc)
│ │ │ + -- used 15.5131s (cpu); 10.0397s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 0.993739s (cpu); 0.691947s (thread); 0s (gc)
│ │ │ + -- used 1.16092s (cpu); 0.922957s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 10.1165s (cpu); 7.01836s (thread); 0s (gc)
│ │ │ + -- used 13.42s (cpu); 10.3864s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -87,21 +87,21 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.33134s (cpu); 0.902944s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.80956s (cpu); 1.24704s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.174602s (cpu); 0.113061s (thread); 0s (gc)
│ │ │ + -- used 0.228062s (cpu); 0.178088s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -115,15 +115,15 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.157657s (cpu); 0.0982538s (thread); 0s (gc)
│ │ │ + -- used 0.300788s (cpu); 0.134525s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -137,15 +137,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.600684s (cpu); 0.416565s (thread); 0s (gc)
│ │ │ + -- used 0.690735s (cpu); 0.568017s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -165,15 +165,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.701715s (cpu); 0.449993s (thread); 0s (gc)
│ │ │ + -- used 0.829908s (cpu); 0.655906s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -214,15 +214,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 25, and computed = 23
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 25, and computed = 23.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.43376s (cpu); 0.25769s (thread); 0s (gc)
│ │ │ + -- used 0.601116s (cpu); 0.348032s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5509279875405941999
│ │ │  
│ │ │  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);
│ │ │  
│ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 1.65802s elapsed
│ │ │ + -- 2.0323s elapsed
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : peek StrategyDefault
│ │ │  
│ │ │  o3 = OptionTable{GRevLexLargest => 0      }
│ │ │                   GRevLexSmallest => 16
│ │ │ @@ -16,12 +16,12 @@
│ │ │                   LexSmallest => 16
│ │ │                   LexSmallestTerm => 16
│ │ │                   Points => 0
│ │ │                   Random => 16
│ │ │                   RandomNonzero => 16
│ │ │  
│ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.01214s elapsed
│ │ │ + -- 1.35087s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out
│ │ │ @@ -16,29 +16,29 @@
│ │ │  i5 : r = rank myDiff;
│ │ │  
│ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00400131s (cpu); 0.0024123s (thread); 0s (gc)
│ │ │ + -- used 0.0408105s (cpu); 0.0424731s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  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);
│ │ │  
│ │ │                 ZZ
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │  
│ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00265611s (cpu); 0.00233315s (thread); 0s (gc)
│ │ │ + -- used 0.00028781s (cpu); 0.00212849s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 5.4772e-05s (cpu); 0.0022959s (thread); 0s (gc)
│ │ │ + -- used 0.00158338s (cpu); 0.00207919s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : pdim(module I)
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.255808s (cpu); 0.130559s (thread); 0s (gc)
│ │ │ + -- used 0.280573s (cpu); 0.181159s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00883907s (cpu); 0.0114568s (thread); 0s (gc)
│ │ │ + -- used 0.00800158s (cpu); 0.0112099s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │  
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.437786s (cpu); 0.440787s (thread); 0s (gc)
│ │ │ + -- used 0.611988s (cpu); 0.610769s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.59111s (cpu); 1.34157s (thread); 0s (gc)
│ │ │ + -- used 2.22499s (cpu); 1.94741s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -17,44 +17,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 1.52067s (cpu); 1.10529s (thread); 0s (gc)
│ │ │ + -- used 2.24596s (cpu); 1.74857s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.53229s (cpu); 4.74206s (thread); 0s (gc)
│ │ │ + -- used 8.53673s (cpu); 6.38222s (thread); 0s (gc)
│ │ │  
│ │ │  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);
│ │ │  
│ │ │  i11 : dim(R)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199993s (cpu); 0.0195143s (thread); 0s (gc)
│ │ │ + -- used 0.0160025s (cpu); 0.0168s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.493224s (cpu); 0.310068s (thread); 0s (gc)
│ │ │ + -- used 0.688828s (cpu); 0.494292s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.852714s (cpu); 0.524319s (thread); 0s (gc)
│ │ │ + -- used 1.01499s (cpu); 0.776686s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.322571s (cpu); 0.208202s (thread); 0s (gc)
│ │ │ + -- used 0.380781s (cpu); 0.24463s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -386,15 +386,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ -internalChooseMinor: Choosing -- used 6.47455s (cpu); 4.61649s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Choosing -- used 9.50516s (cpu); 7.7463s (thread); 0s (gc)
│ │ │   LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -430,15 +430,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 186
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 186.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.54762s (cpu); 1.2674s (thread); 0s (gc)
│ │ │ + -- used 1.9773s (cpu); 1.57609s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ @@ -490,59 +490,59 @@
│ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.143625s (cpu); 0.0920942s (thread); 0s (gc)
│ │ │ + -- used 0.168031s (cpu); 0.125249s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.271628s (cpu); 0.149929s (thread); 0s (gc)
│ │ │ + -- used 0.29762s (cpu); 0.216079s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.725s (cpu); 1.31345s (thread); 0s (gc)
│ │ │ + -- used 2.04151s (cpu); 1.51965s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.13749s (cpu); 0.85499s (thread); 0s (gc)
│ │ │ + -- used 1.59786s (cpu); 1.21613s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.463182s (cpu); 0.273596s (thread); 0s (gc)
│ │ │ + -- used 0.596072s (cpu); 0.329815s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = true
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.17628s (cpu); 1.4555s (thread); 0s (gc)
│ │ │ + -- used 2.71977s (cpu); 2.17705s (thread); 0s (gc)
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.165936s (cpu); 0.105402s (thread); 0s (gc)
│ │ │ + -- used 0.165608s (cpu); 0.128097s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.211774s (cpu); 0.158759s (thread); 0s (gc)
│ │ │ + -- used 0.309078s (cpu); 0.261711s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.48716s (cpu); 1.11753s (thread); 0s (gc)
│ │ │ + -- used 2.27716s (cpu); 1.8885s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.52406s (cpu); 1.15802s (thread); 0s (gc)
│ │ │ + -- used 2.06232s (cpu); 1.60872s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -558,57 +558,57 @@
│ │ │
    │ │ │
    │ │ │

    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).

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.10715s (cpu); 0.107368s (thread); 0s (gc)
│ │ │ + -- used 0.187939s (cpu); 0.186654s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    │ │ │ 
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.33389s (cpu); 0.203875s (thread); 0s (gc)
│ │ │ + -- used 0.343333s (cpu); 0.243976s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    │ │ │ 
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.404835s (cpu); 0.275085s (thread); 0s (gc)
│ │ │ + -- used 0.516024s (cpu); 0.428512s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    │ │ │ 
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.314312s (cpu); 0.181456s (thread); 0s (gc)
│ │ │ + -- used 0.490338s (cpu); 0.252617s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    │ │ │ 
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.312529s (cpu); 0.1737s (thread); 0s (gc)
│ │ │ + -- used 0.332258s (cpu); 0.235725s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    │ │ │ 
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.348847s (cpu); 0.214069s (thread); 0s (gc)
│ │ │ + -- used 0.460208s (cpu); 0.358773s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    │ │ │ 
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.323316s (cpu); 0.192993s (thread); 0s (gc)
│ │ │ + -- used 0.365933s (cpu); 0.273051s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    │ │ │ 
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 13.4439s (cpu); 7.95676s (thread); 0s (gc)
│ │ │ + -- used 17.3469s (cpu); 11.9013s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
    │ │ │
    │ │ │
    │ │ │

    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, Points is almost always quite effective at finding valuable submatrices, but can be quite slow. In this particular example, we can see that LexSmallestTerm 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,

    │ │ │
    │ │ │ @@ -629,15 +629,15 @@ │ │ │
    │ │ │
    │ │ │

    says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest, LexSmallestTerm, Random, RandomNonzero all with equal probability (note RandomNonzero, 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:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.539994s (cpu); 0.365803s (thread); 0s (gc)
│ │ │ + -- used 0.600379s (cpu); 0.456404s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -728,15 +728,15 @@
│ │ │            
    i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
    │ │ │ 
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.999209s (cpu); 0.602814s (thread); 0s (gc)
│ │ │ + -- used 1.23084s (cpu); 0.729775s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    │ │ │ 
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ @@ -754,67 +754,67 @@
│ │ │            
    i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
    │ │ │ 
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.699354s (cpu); 0.37s (thread); 0s (gc)
│ │ │ + -- used 0.917925s (cpu); 0.525149s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    │ │ │
    │ │ │
    │ │ │

    Other options may also be passed to the RandomPoints package via the PointOptions option.

    │ │ │
    │ │ │
    │ │ │

    regularInCodimension: 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 regularInCodimension 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 codimCheckFunction. For more on that, see RegularInCodimensionTutorial and regularInCodimension. Let us finish running regularInCodimension on our example with several different strategies.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 4.96948s (cpu); 3.56542s (thread); 0s (gc)
    │ │ │ + -- used 6.32285s (cpu); 4.80086s (thread); 0s (gc) │ │ │ 
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.03145s (cpu); 0.679038s (thread); 0s (gc)
│ │ │ + -- used 1.26344s (cpu); 0.939776s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    │ │ │ 
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.98086s (cpu); 2.51747s (thread); 0s (gc)
    │ │ │ + -- used 4.20522s (cpu); 3.52377s (thread); 0s (gc) │ │ │ 
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 3.40978s (cpu); 2.04641s (thread); 0s (gc)
    │ │ │ + -- used 4.3251s (cpu); 2.8001s (thread); 0s (gc) │ │ │ 
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 1.07836s (cpu); 0.777656s (thread); 0s (gc)
│ │ │ + -- used 1.62131s (cpu); 1.24061s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    │ │ │ 
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 4.00314s (cpu); 2.42639s (thread); 0s (gc)
    │ │ │ + -- used 5.24785s (cpu); 3.29254s (thread); 0s (gc) │ │ │ 
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 4.46603s (cpu); 2.88283s (thread); 0s (gc)
    │ │ │ + -- used 6.44832s (cpu); 4.71612s (thread); 0s (gc) │ │ │ 
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 14.8083s (cpu); 8.44014s (thread); 0s (gc)
│ │ │ + -- used 19.5055s (cpu); 12.6968s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    │ │ │ 
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 11.2852s (cpu); 6.42669s (thread); 0s (gc)
│ │ │ + -- used 15.5131s (cpu); 10.0397s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    │ │ │
    │ │ │
    │ │ │

    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension. We set MaxMinors => 100 to keep it from running too long with an ineffective strategy. Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies. Note similar considerations also apply to projDim.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -486,44 +486,44 @@ │ │ │ │ o27 : Ideal of S │ │ │ │ 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). │ │ │ │ i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random)) │ │ │ │ - -- used 0.10715s (cpu); 0.107368s (thread); 0s (gc) │ │ │ │ + -- used 0.187939s (cpu); 0.186654s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = 2 │ │ │ │ i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest)) │ │ │ │ - -- used 0.33389s (cpu); 0.203875s (thread); 0s (gc) │ │ │ │ + -- used 0.343333s (cpu); 0.243976s (thread); 0s (gc) │ │ │ │ │ │ │ │ o29 = 3 │ │ │ │ i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm)) │ │ │ │ - -- used 0.404835s (cpu); 0.275085s (thread); 0s (gc) │ │ │ │ + -- used 0.516024s (cpu); 0.428512s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = 1 │ │ │ │ i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest)) │ │ │ │ - -- used 0.314312s (cpu); 0.181456s (thread); 0s (gc) │ │ │ │ + -- used 0.490338s (cpu); 0.252617s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = 2 │ │ │ │ i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest)) │ │ │ │ - -- used 0.312529s (cpu); 0.1737s (thread); 0s (gc) │ │ │ │ + -- used 0.332258s (cpu); 0.235725s (thread); 0s (gc) │ │ │ │ │ │ │ │ o32 = 3 │ │ │ │ i33 : time dim (J + chooseGoodMinors(8, 6, M, J, │ │ │ │ Strategy=>GRevLexSmallestTerm)) │ │ │ │ - -- used 0.348847s (cpu); 0.214069s (thread); 0s (gc) │ │ │ │ + -- used 0.460208s (cpu); 0.358773s (thread); 0s (gc) │ │ │ │ │ │ │ │ o33 = 3 │ │ │ │ i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest)) │ │ │ │ - -- used 0.323316s (cpu); 0.192993s (thread); 0s (gc) │ │ │ │ + -- used 0.365933s (cpu); 0.273051s (thread); 0s (gc) │ │ │ │ │ │ │ │ o34 = 3 │ │ │ │ i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points)) │ │ │ │ - -- used 13.4439s (cpu); 7.95676s (thread); 0s (gc) │ │ │ │ + -- used 17.3469s (cpu); 11.9013s (thread); 0s (gc) │ │ │ │ │ │ │ │ o35 = 1 │ │ │ │ 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, Points is almost always quite effective at │ │ │ │ finding valuable submatrices, but can be quite slow. In this particular │ │ │ │ example, we can see that LexSmallestTerm also performs very well (and does it │ │ │ │ @@ -544,15 +544,15 @@ │ │ │ │ says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest, │ │ │ │ LexSmallestTerm, Random, RandomNonzero all with equal probability (note │ │ │ │ RandomNonzero, 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: │ │ │ │ i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, │ │ │ │ Verbose=>true); │ │ │ │ - -- used 0.539994s (cpu); 0.365803s (thread); 0s (gc) │ │ │ │ + -- used 0.600379s (cpu); 0.456404s (thread); 0s (gc) │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing LexSmallest │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ │ internalChooseMinor: Choosing LexSmallest │ │ │ │ @@ -633,15 +633,15 @@ │ │ │ │ i41 : ptsStratGeometric = new OptionTable from (options │ │ │ │ chooseGoodMinors)#PointOptions; │ │ │ │ i42 : ptsStratGeometric#ExtendField --look at the default value │ │ │ │ │ │ │ │ o42 = true │ │ │ │ i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, │ │ │ │ PointOptions=>ptsStratGeometric)) │ │ │ │ - -- used 0.999209s (cpu); 0.602814s (thread); 0s (gc) │ │ │ │ + -- used 1.23084s (cpu); 0.729775s (thread); 0s (gc) │ │ │ │ │ │ │ │ o43 = 2 │ │ │ │ i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that │ │ │ │ value │ │ │ │ │ │ │ │ o44 = OptionTable{DecompositionStrategy => Decompose} │ │ │ │ DimensionFunction => dim │ │ │ │ @@ -655,58 +655,58 @@ │ │ │ │ │ │ │ │ o44 : OptionTable │ │ │ │ i45 : ptsStratRational.ExtendField --look at our changed value │ │ │ │ │ │ │ │ o45 = false │ │ │ │ i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, │ │ │ │ PointOptions=>ptsStratRational)) │ │ │ │ - -- used 0.699354s (cpu); 0.37s (thread); 0s (gc) │ │ │ │ + -- used 0.917925s (cpu); 0.525149s (thread); 0s (gc) │ │ │ │ │ │ │ │ o46 = 2 │ │ │ │ Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the │ │ │ │ _P_o_i_n_t_O_p_t_i_o_n_s option. │ │ │ │ rreegguullaarrIInnCCooddiimmeennssiioonn:: 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 regularInCodimension 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 codimCheckFunction. For more on │ │ │ │ that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish │ │ │ │ running regularInCodimension on our example with several different strategies. │ │ │ │ i47 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>StrategyDefault) │ │ │ │ - -- used 4.96948s (cpu); 3.56542s (thread); 0s (gc) │ │ │ │ + -- used 6.32285s (cpu); 4.80086s (thread); 0s (gc) │ │ │ │ i48 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>StrategyDefaultNonRandom) │ │ │ │ - -- used 1.03145s (cpu); 0.679038s (thread); 0s (gc) │ │ │ │ + -- used 1.26344s (cpu); 0.939776s (thread); 0s (gc) │ │ │ │ │ │ │ │ o48 = true │ │ │ │ i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random) │ │ │ │ - -- used 2.98086s (cpu); 2.51747s (thread); 0s (gc) │ │ │ │ + -- used 4.20522s (cpu); 3.52377s (thread); 0s (gc) │ │ │ │ i50 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>LexSmallest) │ │ │ │ - -- used 3.40978s (cpu); 2.04641s (thread); 0s (gc) │ │ │ │ + -- used 4.3251s (cpu); 2.8001s (thread); 0s (gc) │ │ │ │ i51 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>LexSmallestTerm) │ │ │ │ - -- used 1.07836s (cpu); 0.777656s (thread); 0s (gc) │ │ │ │ + -- used 1.62131s (cpu); 1.24061s (thread); 0s (gc) │ │ │ │ │ │ │ │ o51 = true │ │ │ │ i52 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>GRevLexSmallest) │ │ │ │ - -- used 4.00314s (cpu); 2.42639s (thread); 0s (gc) │ │ │ │ + -- used 5.24785s (cpu); 3.29254s (thread); 0s (gc) │ │ │ │ i53 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>GRevLexSmallestTerm) │ │ │ │ - -- used 4.46603s (cpu); 2.88283s (thread); 0s (gc) │ │ │ │ + -- used 6.44832s (cpu); 4.71612s (thread); 0s (gc) │ │ │ │ i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points) │ │ │ │ - -- used 14.8083s (cpu); 8.44014s (thread); 0s (gc) │ │ │ │ + -- used 19.5055s (cpu); 12.6968s (thread); 0s (gc) │ │ │ │ │ │ │ │ o54 = true │ │ │ │ i55 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>StrategyDefaultWithPoints) │ │ │ │ - -- used 11.2852s (cpu); 6.42669s (thread); 0s (gc) │ │ │ │ + -- used 15.5131s (cpu); 10.0397s (thread); 0s (gc) │ │ │ │ │ │ │ │ o55 = true │ │ │ │ If regularInCodimension outputs nothing, then it couldn't verify that the ring │ │ │ │ was regular in that codimension. We set MaxMinors => 100 to keep it from │ │ │ │ running too long with an ineffective strategy. Again, even though │ │ │ │ GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular │ │ │ │ example, in others they perform better than other strategies. Note similar │ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ @@ -67,21 +67,21 @@ │ │ │
    │ │ │
    │ │ │

    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.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 0.993739s (cpu); 0.691947s (thread); 0s (gc)
│ │ │ + -- used 1.16092s (cpu); 0.922957s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    │ │ │ 
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 10.1165s (cpu); 7.01836s (thread); 0s (gc)
    │ │ │ + -- used 13.42s (cpu); 10.3864s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │

    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. regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null 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 Verbose option. We go through the output and explain what each line is telling us.

    │ │ │
    │ │ │ │ │ │ │ │ │ @@ -154,27 +154,27 @@ │ │ │ internalChooseMinor: Choosing LexSmallest │ │ │ internalChooseMinor: Choosing LexSmallestTerm │ │ │ internalChooseMinor: Choosing LexSmallest │ │ │ internalChooseMinor: Choosing Random │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 49, and computed = 39 │ │ │ regularInCodimension: singularLocus dimension verified by isCodimAtLeast │ │ │ regularInCodimension: partial singular locus dimension computed, = 2 │ │ │ -regularInCodimension: Loop completed, submatrices considered = 49, and compute -- used 1.33134s (cpu); 0.902944s (thread); 0s (gc) │ │ │ +regularInCodimension: Loop completed, submatrices considered = 49, and compute -- used 1.80956s (cpu); 1.24704s (thread); 0s (gc) │ │ │ d = 39. singular locus dimension appears to be = 2 │ │ │ │ │ │ o6 = true │ │ │ │ │ │
    │ │ │
    │ │ │

    MaxMinors. The first output says that we will compute up to 452.9 minors before giving up. We can control that by setting the option MaxMinors.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.174602s (cpu); 0.113061s (thread); 0s (gc)
│ │ │ + -- used 0.228062s (cpu); 0.178088s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -197,15 +197,15 @@
│ │ │          
│ │ │          
    
│ │ │            
    Selecting submatrices of the Jacobian.  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:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.157657s (cpu); 0.0982538s (thread); 0s (gc)
│ │ │ + -- used 0.300788s (cpu); 0.134525s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -228,15 +228,15 @@
│ │ │          
│ │ │          
    
│ │ │            
    Computing minors vs considering the dimension of what has been computed.  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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.600684s (cpu); 0.416565s (thread); 0s (gc)
│ │ │ + -- used 0.690735s (cpu); 0.568017s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -265,15 +265,15 @@
│ │ │          
│ │ │          
    
│ │ │            
    CodimCheckFunction. The option CodimCheckFunction controls how frequently the dimension of the partial ideal of minors is computed.  For instance, setting CodimCheckFunction => t -> t/5 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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
                  
    i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.701715s (cpu); 0.449993s (thread); 0s (gc)
│ │ │ + -- used 0.829908s (cpu); 0.655906s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -320,15 +320,15 @@
│ │ │          
    
│ │ │          
    
│ │ │            
    isCodimAtLeast and dim.  We see the lines about the ``isCodimAtLeast failed''.  This means that isCodimAtLeast 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 isCodimAtLeast is called can be controlled via the options SPairsFunction and PairLimit, which are simply passed to isCodimAtLeast.  You can force the function to only use isCodimAtLeast and not call dimension by setting UseOnlyFastCodim => true.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.43376s (cpu); 0.25769s (thread); 0s (gc)
│ │ │ + -- used 0.601116s (cpu); 0.348032s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  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.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 0.993739s (cpu); 0.691947s (thread); 0s (gc)
│ │ │ │ + -- used 1.16092s (cpu); 0.922957s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 10.1165s (cpu); 7.01836s (thread); 0s (gc)
│ │ │ │ + -- used 13.42s (cpu); 10.3864s (thread); 0s (gc)
│ │ │ │  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. regularInCodimension returns true
│ │ │ │  if it verified that the ring is regular in codim 1 or 2 (respectively) and null
│ │ │ │  if not. Because of the randomness that exists in terms of selecting minors, the
│ │ │ │ @@ -121,22 +121,22 @@
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 49, and computed = 39
│ │ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ │  regularInCodimension:  Loop completed, submatrices considered = 49, and compute
│ │ │ │ --- used 1.33134s (cpu); 0.902944s (thread); 0s (gc)
│ │ │ │ +-- used 1.80956s (cpu); 1.24704s (thread); 0s (gc)
│ │ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  MMaaxxMMiinnoorrss.. The first output says that we will compute up to 452.9 minors before
│ │ │ │  giving up. We can control that by setting the option MaxMinors.
│ │ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ │ - -- used 0.174602s (cpu); 0.113061s (thread); 0s (gc)
│ │ │ │ + -- used 0.228062s (cpu); 0.178088s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -159,15 +159,15 @@
│ │ │ │  There are other finer ways to control the MaxMinors option, but they will not
│ │ │ │  be discussed in this tutorial. See _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n.
│ │ │ │  SSeelleeccttiinngg ssuubbmmaattrriicceess ooff tthhee JJaaccoobbiiaann.. 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:
│ │ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom,
│ │ │ │  Verbose=>true)
│ │ │ │ - -- used 0.157657s (cpu); 0.0982538s (thread); 0s (gc)
│ │ │ │ + -- used 0.300788s (cpu); 0.134525s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -197,15 +197,15 @@
│ │ │ │  CCoommppuuttiinngg mmiinnoorrss vvss ccoonnssiiddeerriinngg tthhee ddiimmeennssiioonn ooff wwhhaatt hhaass bbeeeenn ccoommppuutteedd..
│ │ │ │  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.
│ │ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t-
│ │ │ │  >3, Verbose=>true)
│ │ │ │ - -- used 0.600684s (cpu); 0.416565s (thread); 0s (gc)
│ │ │ │ + -- used 0.690735s (cpu); 0.568017s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │ @@ -243,15 +243,15 @@
│ │ │ │  dimension of the partial ideal of minors is computed. For instance, setting
│ │ │ │  CodimCheckFunction => t -> t/5 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.
│ │ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t-
│ │ │ │  >t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ │ - -- used 0.701715s (cpu); 0.449993s (thread); 0s (gc)
│ │ │ │ + -- used 0.829908s (cpu); 0.655906s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  dimension computed'' indicates we did a complete dimension computation of the
│ │ │ │  partial ideal defining the singular locus. How isCodimAtLeast is called can be
│ │ │ │  controlled via the options SPairsFunction and PairLimit, which are simply
│ │ │ │  passed to _i_s_C_o_d_i_m_A_t_L_e_a_s_t. You can force the function to only use isCodimAtLeast
│ │ │ │  and not call dimension by setting UseOnlyFastCodim => true.
│ │ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim =>
│ │ │ │  true, Verbose=>true)
│ │ │ │ - -- used 0.43376s (cpu); 0.25769s (thread); 0s (gc)
│ │ │ │ + -- used 0.601116s (cpu); 0.348032s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │          
│ │ │  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.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    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);
    
│ │ │      		          
                  
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 1.65802s elapsed
│ │ │ + -- 2.0323s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │      		          
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • 
│ │ │  StrategyDefault: 1.65 seconds          
      • 
│ │ │ @@ -137,15 +137,15 @@
│ │ │  StrategyPoints: choose all submatrices via Points.          
│ │ │            
    • 
│ │ │  StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.          
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.01214s elapsed
│ │ │ + -- 1.35087s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │      		          
    
│ │ │        
│ │ │        
    
│ │ │          
    For the programmer
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  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);
│ │ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ │ - -- 1.65802s elapsed
│ │ │ │ + -- 2.0323s elapsed
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  In this particular example, on one machine, we list average time to completion
│ │ │ │  of each of the above strategies after 100 runs.
│ │ │ │      * StrategyDefault: 1.65 seconds
│ │ │ │      * StrategyRandom: 8.32 seconds
│ │ │ │      * StrategyDefaultNonRandom: 0.99 seconds
│ │ │ │ @@ -135,12 +135,12 @@
│ │ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported. It
│ │ │ │  begins as the default strategy, but the user can modify it.
│ │ │ │  
│ │ │ │  UUssiinngg aa ssiinnggllee hheeuurriissttiicc Alternatively, if the user only wants to use say
│ │ │ │  LexSmallestTerm they can set, Strategy to point to that symbol, instead of a
│ │ │ │  creating a custom strategy HashTable. For example:
│ │ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ │ - -- 1.01214s elapsed
│ │ │ │ + -- 1.35087s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │            
                  
    i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │      		          
                  
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00400131s (cpu); 0.0024123s (thread); 0s (gc)
│ │ │ + -- used 0.0408105s (cpu); 0.0424731s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    The function works by computing gb(I, PairLimit=>f(i)) for successive values of i.  Here f(i) is a function that takes t, 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 \{1\}).  And i is a counting variable. You can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>( (i) -> f(i) ), the default function is SPairsFunction=>i->ceiling(1.5^i)   Perhaps more commonly however, the user may want to instead tell the function to compute for larger values of i.  This is done via the option PairLimit.  This is the max value of i to be plugged into SPairsFunction before the function gives up.  In other words, PairLimit=>5 will tell the function to check codimension 5 times.
    
│ │ │          
    
│ │ │ @@ -124,22 +124,22 @@
│ │ │  
│ │ │                 ZZ
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
    
│ │ │      		          
                  
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00265611s (cpu); 0.00233315s (thread); 0s (gc)
│ │ │ + -- used 0.00028781s (cpu); 0.00212849s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │      		          
                  
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 5.4772e-05s (cpu); 0.0022959s (thread); 0s (gc)
│ │ │ + -- used 0.00158338s (cpu); 0.00207919s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    Notice in the first case the function returned null, because the depth of search was not high enough.  It only computed codim 5 times.  The second returned true, but it did so as soon as the answer was found (and before we hit the PairLimit limit).
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │               30      12
│ │ │ │  o4 : Matrix R   <-- R
│ │ │ │  i5 : r = rank myDiff;
│ │ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ │ - -- used 0.00400131s (cpu); 0.0024123s (thread); 0s (gc)
│ │ │ │ + -- used 0.0408105s (cpu); 0.0424731s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  The function works by computing gb(I, PairLimit=>f(i)) for successive values of
│ │ │ │  i. Here f(i) is a function that takes t, 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 \{1\}). And i is a counting variable. You
│ │ │ │  can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>
│ │ │ │ @@ -73,20 +73,20 @@
│ │ │ │  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);
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ │ - -- used 0.00265611s (cpu); 0.00233315s (thread); 0s (gc)
│ │ │ │ + -- used 0.00028781s (cpu); 0.00212849s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 5.4772e-05s (cpu); 0.0022959s (thread); 0s (gc)
│ │ │ │ + -- used 0.00158338s (cpu); 0.00207919s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  Notice in the first case the function returned null, because the depth of
│ │ │ │  search was not high enough. It only computed codim 5 times. The second returned
│ │ │ │  true, but it did so as soon as the answer was found (and before we hit the
│ │ │ │  PairLimit limit).
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │            
                  
    i3 : pdim(module I)
│ │ │  
│ │ │  o3 = 2
    
│ │ │      		          
                  
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.255808s (cpu); 0.130559s (thread); 0s (gc)
│ │ │ + -- used 0.280573s (cpu); 0.181159s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │      		          
                  
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00883907s (cpu); 0.0114568s (thread); 0s (gc)
│ │ │ + -- used 0.00800158s (cpu); 0.0112099s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    The option MaxMinors 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 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors => ZZ 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 MaxMinors => List.  In this case, the list specifies how many minors to be computed at each step, (working backwards). Finally, you can also set MaxMinors to be a custom function of the dimension $d$ of the polynomial ring and the maximum number of minors.
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,19 +45,19 @@
│ │ │ │  i2 : I = ideal((x^3+y)^2, (x^2+y^2)^2, (x+y^3)^2, (x*y)^2);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : pdim(module I)
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.255808s (cpu); 0.130559s (thread); 0s (gc)
│ │ │ │ + -- used 0.280573s (cpu); 0.181159s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.00883907s (cpu); 0.0114568s (thread); 0s (gc)
│ │ │ │ + -- used 0.00800158s (cpu); 0.0112099s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  The option MaxMinors 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 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors
│ │ │ │  => ZZ to specify that a fixed integer is used for each step. Alternatively, the
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html
│ │ │ @@ -94,21 +94,21 @@
│ │ │  
    i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │  
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
    │ │ │ 
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.437786s (cpu); 0.440787s (thread); 0s (gc)
│ │ │ + -- used 0.611988s (cpu); 0.610769s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ 
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.59111s (cpu); 1.34157s (thread); 0s (gc)
│ │ │ + -- used 2.22499s (cpu); 1.94741s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │ 
    i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,19 +28,19 @@ │ │ │ │ strategy for minors │ │ │ │ i1 : R = QQ[x,y]; │ │ │ │ i2 : M = random(R^{5,5,5,5,5,5}, R^7); │ │ │ │ │ │ │ │ 6 7 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : time I2 = recursiveMinors(4, M, Threads=>0); │ │ │ │ - -- used 0.437786s (cpu); 0.440787s (thread); 0s (gc) │ │ │ │ + -- used 0.611988s (cpu); 0.610769s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : time I1 = minors(4, M, Strategy=>Cofactor); │ │ │ │ - -- used 1.59111s (cpu); 1.34157s (thread); 0s (gc) │ │ │ │ + -- used 2.22499s (cpu); 1.94741s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : I1 == I2 │ │ │ │ │ │ │ │ o5 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_i_n_o_r_s -- ideal generated by minors │ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ @@ -129,21 +129,21 @@ │ │ │ 
    i7 : dim S
│ │ │  
│ │ │  o7 = 3
    │ │ │ 
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 1.52067s (cpu); 1.10529s (thread); 0s (gc)
│ │ │ + -- used 2.24596s (cpu); 1.74857s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ 
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.53229s (cpu); 4.74206s (thread); 0s (gc)
    │ │ │ + -- used 8.53673s (cpu); 6.38222s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │

    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.

    │ │ │
    │ │ │
    │ │ │

    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 dim singularLocus call takes an enormous amount of time, otherwise the running times of dim singularLocus and our function are frequently about the same.

    │ │ │ @@ -155,33 +155,33 @@ │ │ │ 
    i11 : dim(R)
│ │ │  
│ │ │  o11 = 2
    │ │ │ 
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199993s (cpu); 0.0195143s (thread); 0s (gc)
│ │ │ + -- used 0.0160025s (cpu); 0.0168s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    │ │ │ 
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.493224s (cpu); 0.310068s (thread); 0s (gc)
│ │ │ + -- used 0.688828s (cpu); 0.494292s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    │ │ │ 
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.852714s (cpu); 0.524319s (thread); 0s (gc)
│ │ │ + -- used 1.01499s (cpu); 0.776686s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    │ │ │ 
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.322571s (cpu); 0.208202s (thread); 0s (gc)
│ │ │ + -- used 0.380781s (cpu); 0.24463s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
    │ │ │
    │ │ │
    │ │ │

    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 Verbose can be used to see this in action.

    │ │ │
    │ │ │ @@ -519,15 +519,15 @@ │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ internalChooseMinor: Choosing LexSmallest │ │ │ internalChooseMinor: Choosing Random │ │ │ internalChooseMinor: Choosing Random │ │ │ internalChooseMinor: Choosing LexSmallestTerm │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -internalChooseMinor: Choosing -- used 6.47455s (cpu); 4.61649s (thread); 0s (gc) │ │ │ +internalChooseMinor: Choosing -- used 9.50516s (cpu); 7.7463s (thread); 0s (gc) │ │ │ LexSmallestTerm │ │ │ internalChooseMinor: Choosing Random │ │ │ internalChooseMinor: Choosing Random │ │ │ internalChooseMinor: Choosing GRevLexSmallest │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ internalChooseMinor: Choosing GRevLexSmallest │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ @@ -569,15 +569,15 @@ │ │ │
    │ │ │
    │ │ │

    The maximum number of minors considered can be controlled by the option MaxMinors. Alternatively, it can be controlled in a more precise way by passing a function to the option MaxMinors. 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 regularInCodimension does not recompute determinants, so MaxMinors or is only an upper bound on the number of minors computed.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.54762s (cpu); 1.2674s (thread); 0s (gc)
│ │ │ + -- used 1.9773s (cpu); 1.57609s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ @@ -638,73 +638,73 @@
│ │ │  
    i19 : StrategyCurrent#LexSmallest = 100;
    │ │ │ 
    i20 : StrategyCurrent#LexSmallestTerm = 0;
    │ │ │ 
    i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.143625s (cpu); 0.0920942s (thread); 0s (gc)
│ │ │ + -- used 0.168031s (cpu); 0.125249s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    │ │ │ 
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.271628s (cpu); 0.149929s (thread); 0s (gc)
│ │ │ + -- used 0.29762s (cpu); 0.216079s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    │ │ │ 
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.725s (cpu); 1.31345s (thread); 0s (gc)
│ │ │ + -- used 2.04151s (cpu); 1.51965s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    │ │ │ 
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.13749s (cpu); 0.85499s (thread); 0s (gc)
│ │ │ + -- used 1.59786s (cpu); 1.21613s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    │ │ │ 
    i25 : StrategyCurrent#LexSmallest = 0;
    │ │ │ 
    i26 : StrategyCurrent#LexSmallestTerm = 100;
    │ │ │ 
    i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.463182s (cpu); 0.273596s (thread); 0s (gc)
│ │ │ + -- used 0.596072s (cpu); 0.329815s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = true
    │ │ │ 
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.17628s (cpu); 1.4555s (thread); 0s (gc)
    │ │ │ + -- used 2.71977s (cpu); 2.17705s (thread); 0s (gc) │ │ │ 
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.165936s (cpu); 0.105402s (thread); 0s (gc)
│ │ │ + -- used 0.165608s (cpu); 0.128097s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    │ │ │ 
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.211774s (cpu); 0.158759s (thread); 0s (gc)
│ │ │ + -- used 0.309078s (cpu); 0.261711s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    │ │ │ 
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.48716s (cpu); 1.11753s (thread); 0s (gc)
│ │ │ + -- used 2.27716s (cpu); 1.8885s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    │ │ │ 
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.52406s (cpu); 1.15802s (thread); 0s (gc)
│ │ │ + -- used 2.06232s (cpu); 1.60872s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    │ │ │
    │ │ │
    │ │ │

    The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction. This is should be a function of a single variable, the number of minors computed. Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed. By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -73,19 +73,19 @@ │ │ │ │ │ │ │ │ o5 : Ideal of T │ │ │ │ i6 : S = T/I; │ │ │ │ i7 : dim S │ │ │ │ │ │ │ │ o7 = 3 │ │ │ │ i8 : time regularInCodimension(1, S) │ │ │ │ - -- used 1.52067s (cpu); 1.10529s (thread); 0s (gc) │ │ │ │ + -- used 2.24596s (cpu); 1.74857s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : time regularInCodimension(2, S) │ │ │ │ - -- used 6.53229s (cpu); 4.74206s (thread); 0s (gc) │ │ │ │ + -- used 8.53673s (cpu); 6.38222s (thread); 0s (gc) │ │ │ │ There are numerous examples where regularInCodimension is several orders of │ │ │ │ magnitude faster that calls of dim singularLocus. │ │ │ │ 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 dim singularLocus call takes an enormous amount │ │ │ │ of time, otherwise the running times of dim singularLocus and our function are │ │ │ │ frequently about the same. │ │ │ │ @@ -93,27 +93,27 @@ │ │ │ │ (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); │ │ │ │ i11 : dim(R) │ │ │ │ │ │ │ │ o11 = 2 │ │ │ │ i12 : time (dim singularLocus (R)) │ │ │ │ - -- used 0.0199993s (cpu); 0.0195143s (thread); 0s (gc) │ │ │ │ + -- used 0.0160025s (cpu); 0.0168s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = -1 │ │ │ │ i13 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.493224s (cpu); 0.310068s (thread); 0s (gc) │ │ │ │ + -- used 0.688828s (cpu); 0.494292s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.852714s (cpu); 0.524319s (thread); 0s (gc) │ │ │ │ + -- used 1.01499s (cpu); 0.776686s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.322571s (cpu); 0.208202s (thread); 0s (gc) │ │ │ │ + -- used 0.380781s (cpu); 0.24463s (thread); 0s (gc) │ │ │ │ │ │ │ │ o15 = true │ │ │ │ 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 Verbose can │ │ │ │ be used to see this in action. │ │ │ │ i16 : time regularInCodimension(2, S, Verbose=>true) │ │ │ │ @@ -462,16 +462,15 @@ │ │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ │ internalChooseMinor: Choosing LexSmallest │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing LexSmallestTerm │ │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -internalChooseMinor: Choosing -- used 6.47455s (cpu); 4.61649s (thread); 0s │ │ │ │ -(gc) │ │ │ │ +internalChooseMinor: Choosing -- used 9.50516s (cpu); 7.7463s (thread); 0s (gc) │ │ │ │ LexSmallestTerm │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ │ internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ @@ -517,15 +516,15 @@ │ │ │ │ a function to the option MaxMinors. 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 regularInCodimension does not recompute │ │ │ │ determinants, so MaxMinors or is only an upper bound on the number of minors │ │ │ │ computed. │ │ │ │ i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30) │ │ │ │ - -- used 1.54762s (cpu); 1.2674s (thread); 0s (gc) │ │ │ │ + -- used 1.9773s (cpu); 1.57609s (thread); 0s (gc) │ │ │ │ regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 │ │ │ │ minors, we will compute up to 30 of them. │ │ │ │ regularInCodimension: About to enter loop │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ │ internalChooseMinor: Choosing LexSmallestTerm │ │ │ │ @@ -592,51 +591,51 @@ │ │ │ │ because there are a small number of entries with nonzero constant terms, which │ │ │ │ are selected repeatedly). However, in our first example, the LexSmallestTerm is │ │ │ │ much faster, and Random does not perform well at all. │ │ │ │ i18 : StrategyCurrent#Random = 0; │ │ │ │ i19 : StrategyCurrent#LexSmallest = 100; │ │ │ │ i20 : StrategyCurrent#LexSmallestTerm = 0; │ │ │ │ i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.143625s (cpu); 0.0920942s (thread); 0s (gc) │ │ │ │ + -- used 0.168031s (cpu); 0.125249s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = true │ │ │ │ i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.271628s (cpu); 0.149929s (thread); 0s (gc) │ │ │ │ + -- used 0.29762s (cpu); 0.216079s (thread); 0s (gc) │ │ │ │ │ │ │ │ o22 = true │ │ │ │ i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 1.725s (cpu); 1.31345s (thread); 0s (gc) │ │ │ │ + -- used 2.04151s (cpu); 1.51965s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = true │ │ │ │ i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 1.13749s (cpu); 0.85499s (thread); 0s (gc) │ │ │ │ + -- used 1.59786s (cpu); 1.21613s (thread); 0s (gc) │ │ │ │ │ │ │ │ o24 = true │ │ │ │ i25 : StrategyCurrent#LexSmallest = 0; │ │ │ │ i26 : StrategyCurrent#LexSmallestTerm = 100; │ │ │ │ i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.463182s (cpu); 0.273596s (thread); 0s (gc) │ │ │ │ + -- used 0.596072s (cpu); 0.329815s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = true │ │ │ │ i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 2.17628s (cpu); 1.4555s (thread); 0s (gc) │ │ │ │ + -- used 2.71977s (cpu); 2.17705s (thread); 0s (gc) │ │ │ │ i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.165936s (cpu); 0.105402s (thread); 0s (gc) │ │ │ │ + -- used 0.165608s (cpu); 0.128097s (thread); 0s (gc) │ │ │ │ │ │ │ │ o29 = true │ │ │ │ i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.211774s (cpu); 0.158759s (thread); 0s (gc) │ │ │ │ + -- used 0.309078s (cpu); 0.261711s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = true │ │ │ │ i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.48716s (cpu); 1.11753s (thread); 0s (gc) │ │ │ │ + -- used 2.27716s (cpu); 1.8885s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.52406s (cpu); 1.15802s (thread); 0s (gc) │ │ │ │ + -- used 2.06232s (cpu); 1.60872s (thread); 0s (gc) │ │ │ │ │ │ │ │ o32 = true │ │ │ │ The minimum number of minors computed before checking the codimension can also │ │ │ │ be controlled by an option MinMinorsFunction. This is should be a function of a │ │ │ │ single variable, the number of minors computed. Finally, via the option │ │ │ │ CodimCheckFunction, you can pass the regularInCodimension a function which │ │ │ │ controls how frequently the codimension of the partial Jacobian ideal is │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bmV4dERlZ3JlZShNYXRyaXgsWlosUmluZyk= │ │ │ #:len=288 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhuZXh0RGVncmVlLE1hdHJpeCxaWixSaW5nKSwibmV4 │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ @@ -81,23 +81,23 @@ │ │ │ │ │ │ i14 : K3=nextDegree(gens ker Q2,2,S); │ │ │ │ │ │ 8 8 │ │ │ o14 : Matrix R <-- R │ │ │ │ │ │ i15 : time I=co1Fitting(K3) │ │ │ - -- used 0.000134933s (cpu); 0.00237563s (thread); 0s (gc) │ │ │ + -- used 0.0012091s (cpu); 0.00204597s (thread); 0s (gc) │ │ │ │ │ │ o15 = ideal (a a + a - a , a a - a , a a + a - a , a a - a ) │ │ │ 9 11 5 12 3 11 6 9 10 4 11 3 10 5 │ │ │ │ │ │ o15 : Ideal of R │ │ │ │ │ │ i16 : time J=fittingIdeal(2-1,coker K3); │ │ │ - -- used 0.00561582s (cpu); 0.0061178s (thread); 0s (gc) │ │ │ + -- used 0.00474641s (cpu); 0.00498312s (thread); 0s (gc) │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ │ │ i17 : I==J │ │ │ │ │ │ o17 = true │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ @@ -166,24 +166,24 @@ │ │ │ 
    i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │  
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
    │ │ │ 
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.000134933s (cpu); 0.00237563s (thread); 0s (gc)
│ │ │ + -- used 0.0012091s (cpu); 0.00204597s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
    │ │ │ 
    i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.00561582s (cpu); 0.0061178s (thread); 0s (gc)
│ │ │ + -- used 0.00474641s (cpu); 0.00498312s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ 
    i17 : I==J
│ │ │  
│ │ │  o17 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -95,22 +95,22 @@ │ │ │ │ 2 6 │ │ │ │ o13 : Matrix R <-- R │ │ │ │ i14 : K3=nextDegree(gens ker Q2,2,S); │ │ │ │ │ │ │ │ 8 8 │ │ │ │ o14 : Matrix R <-- R │ │ │ │ i15 : time I=co1Fitting(K3) │ │ │ │ - -- used 0.000134933s (cpu); 0.00237563s (thread); 0s (gc) │ │ │ │ + -- used 0.0012091s (cpu); 0.00204597s (thread); 0s (gc) │ │ │ │ │ │ │ │ o15 = ideal (a a + a - a , a a - a , a a + a - a , a a - a ) │ │ │ │ 9 11 5 12 3 11 6 9 10 4 11 3 10 5 │ │ │ │ │ │ │ │ o15 : Ideal of R │ │ │ │ i16 : time J=fittingIdeal(2-1,coker K3); │ │ │ │ - -- used 0.00561582s (cpu); 0.0061178s (thread); 0s (gc) │ │ │ │ + -- used 0.00474641s (cpu); 0.00498312s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ i17 : I==J │ │ │ │ │ │ │ │ o17 = true │ │ │ │ Note that our method is a bit faster for this small example, and for rank 2 │ │ │ │ quotients of S^3=\mathbb{Z}[x,y]^3 the time difference is massive. │ │ ├── ./usr/share/doc/Macaulay2/FirstPackage/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ Rmlyc3RQYWNrYWdl │ │ │ #:len=509 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZXhhbXBsZSBNYWNhdWxheTIgcGFj │ │ │ a2FnZSIsICJsaW5lbnVtIiA9PiA1MywgImZpbGVuYW1lIiA9PiAiRmlyc3RQYWNrYWdlLm0yIiwg │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ dmFsdWUodWludDE2KQ== │ │ │ #:len=273 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY5Miwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodmFsdWUsdWludDE2KSwidmFsdWUodWludDE2KSIs │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out │ │ │ @@ -4,19 +4,19 @@ │ │ │ │ │ │ o1 = 5 │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ i2 : peek x │ │ │ │ │ │ -o2 = int32{Address => 0x7f5daa523ec0} │ │ │ +o2 = int32{Address => 0x7f0d842307a0} │ │ │ │ │ │ i3 : address x │ │ │ │ │ │ -o3 = 0x7f5daa523ec0 │ │ │ +o3 = 0x7f0d842307a0 │ │ │ │ │ │ o3 : Pointer │ │ │ │ │ │ i4 : class x │ │ │ │ │ │ o4 = int32 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ │ │ │ o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog} │ │ │ │ │ │ o2 : ForeignObject of type char** │ │ │ │ │ │ i3 : voidstarstar {address int 0, address int 1, address int 2} │ │ │ │ │ │ -o3 = {0x7fce2ba5fcd0, 0x7fce2ba5fcc0, 0x7fce2ba5fcb0} │ │ │ +o3 = {0x7f09585cb500, 0x7f09585cb4f0, 0x7f09585cb4e0} │ │ │ │ │ │ o3 : ForeignObject of type void** │ │ │ │ │ │ i4 : x = charstarstar {"foo", "bar", "baz"} │ │ │ │ │ │ o4 = {foo, bar, baz} │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type_sp__Visible__List.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = {foo, bar} │ │ │ │ │ │ o1 : ForeignObject of type char** │ │ │ │ │ │ i2 : voidstarstar {address int 0, address int 1, address int 2} │ │ │ │ │ │ -o2 = {0x7f9bfd1e7ae0, 0x7f9bfd1e7ad0, 0x7f9bfd1e7ac0} │ │ │ +o2 = {0x7f223b05ef30, 0x7f223b05ef20, 0x7f223b05ef10} │ │ │ │ │ │ o2 : ForeignObject of type void** │ │ │ │ │ │ i3 : int2star = foreignPointerArrayType(2 * int) │ │ │ │ │ │ o3 = int32[2]* │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Type_sp__Pointer.out │ │ │ @@ -1,15 +1,15 @@ │ │ │ -- -*- M2-comint -*- hash: 1730835169888399450 │ │ │ │ │ │ i1 : ptr = address int 0 │ │ │ │ │ │ -o1 = 0x7f3cb62b2350 │ │ │ +o1 = 0x7f7b2de308b0 │ │ │ │ │ │ o1 : Pointer │ │ │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ -o2 = 0x7f3cb62b2350 │ │ │ +o2 = 0x7f7b2de308b0 │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = 5 │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ i2 : ptr = address x │ │ │ │ │ │ -o2 = 0x7fa785fa61e0 │ │ │ +o2 = 0x7f7169e30890 │ │ │ │ │ │ o2 : Pointer │ │ │ │ │ │ i3 : int ptr │ │ │ │ │ │ o3 = 5 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out │ │ │ @@ -1,12 +1,12 @@ │ │ │ -- -*- M2-comint -*- hash: 1731230829183683930 │ │ │ │ │ │ i1 : ptr = voidstar address int 5 │ │ │ │ │ │ -o1 = 0x7f14108492a0 │ │ │ +o1 = 0x7f594da30770 │ │ │ │ │ │ o1 : ForeignObject of type void* │ │ │ │ │ │ i2 : int * ptr │ │ │ │ │ │ o2 = 5 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = myunion │ │ │ │ │ │ o1 : ForeignUnionType │ │ │ │ │ │ i2 : myunion 27 │ │ │ │ │ │ -o2 = HashTable{"bar" => 6.94487e-310} │ │ │ +o2 = HashTable{"bar" => 6.9037e-310} │ │ │ "foo" => 27 │ │ │ │ │ │ o2 : ForeignObject of type myunion │ │ │ │ │ │ i3 : myunion pi │ │ │ │ │ │ o3 = HashTable{"bar" => 3.14159 } │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out │ │ │ @@ -4,28 +4,28 @@ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ i2 : peek x │ │ │ │ │ │ -o2 = int32{Address => 0x7f9390dc9080} │ │ │ +o2 = int32{Address => 0x7f1e080d57e0} │ │ │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ -o3 = 0x7f9390dc9080 │ │ │ +o3 = 0x7f1e080d57e0 │ │ │ │ │ │ o3 : Pointer │ │ │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ -o4 = 0x7f9390dc9085 │ │ │ +o4 = 0x7f1e080d57e5 │ │ │ │ │ │ o4 : Pointer │ │ │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ -o5 = 0x7f9390dc907d │ │ │ +o5 = 0x7f1e080d57dd │ │ │ │ │ │ o5 : Pointer │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out │ │ │ @@ -4,10 +4,10 @@ │ │ │ │ │ │ o1 = mpfr │ │ │ │ │ │ o1 : SharedLibrary │ │ │ │ │ │ i2 : peek mpfr │ │ │ │ │ │ -o2 = SharedLibrary{0x7f2afa172550, mpfr} │ │ │ +o2 = SharedLibrary{0x7f3a0dd50550, mpfr} │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = 5 │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ i2 : ptr = address x │ │ │ │ │ │ -o2 = 0x7f7a7a509710 │ │ │ +o2 = 0x7f85a447e100 │ │ │ │ │ │ o2 : Pointer │ │ │ │ │ │ i3 : *ptr = int 6 │ │ │ │ │ │ o3 = 6 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out │ │ │ @@ -1,15 +1,15 @@ │ │ │ -- -*- M2-comint -*- hash: 1730181884377373595 │ │ │ │ │ │ i1 : address int │ │ │ │ │ │ -o1 = 0x563f3b92bbc0 │ │ │ +o1 = 0x555df68ecbc0 │ │ │ │ │ │ o1 : Pointer │ │ │ │ │ │ i2 : address int 5 │ │ │ │ │ │ -o2 = 0x7f0602c6c0f0 │ │ │ +o2 = 0x7f18bce67b90 │ │ │ │ │ │ o2 : Pointer │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out │ │ │ @@ -78,14 +78,14 @@ │ │ │ │ │ │ o16 = free │ │ │ │ │ │ o16 : ForeignFunction │ │ │ │ │ │ i17 : x = malloc 8 │ │ │ │ │ │ -o17 = 0x7f01340639d0 │ │ │ +o17 = 0x7fa8540639d0 │ │ │ │ │ │ o17 : ForeignObject of type void* │ │ │ │ │ │ i18 : registerFinalizer(x, free) │ │ │ │ │ │ i19 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_get__Memory.out │ │ │ @@ -1,21 +1,21 @@ │ │ │ -- -*- M2-comint -*- hash: 10647988412767280310 │ │ │ │ │ │ i1 : ptr = getMemory 8 │ │ │ │ │ │ -o1 = 0x7f39cc2e59a0 │ │ │ +o1 = 0x7feb5bfbe9a0 │ │ │ │ │ │ o1 : ForeignObject of type void* │ │ │ │ │ │ i2 : ptr = getMemory(8, Atomic => true) │ │ │ │ │ │ -o2 = 0x7f39d4990d40 │ │ │ +o2 = 0x7feb58d85cd0 │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ │ │ i3 : ptr = getMemory int │ │ │ │ │ │ -o3 = 0x7f39d4990c50 │ │ │ +o3 = 0x7feb58d85be0 │ │ │ │ │ │ o3 : ForeignObject of type void* │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.out │ │ │ @@ -17,18 +17,18 @@ │ │ │ o3 = finalizer │ │ │ │ │ │ o3 : FunctionClosure │ │ │ │ │ │ i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer)) │ │ │ │ │ │ i5 : collectGarbage() │ │ │ -freeing memory at 0x7f114006a210 │ │ │ -freeing memory at 0x7f114005f3f0 │ │ │ -freeing memory at 0x7f114006a230 │ │ │ -freeing memory at 0x7f1140069180 │ │ │ -freeing memory at 0x7f11400639d0 │ │ │ -freeing memory at 0x7f1140063bf0 │ │ │ -freeing memory at 0x7f1140066c50 │ │ │ -freeing memory at 0x7f1140063bc0 │ │ │ -freeing memory at 0x7f11400639f0 │ │ │ +freeing memory at 0x7f534005f3f0 │ │ │ +freeing memory at 0x7f534006a210 │ │ │ +freeing memory at 0x7f53400639f0 │ │ │ +freeing memory at 0x7f5340063bf0 │ │ │ +freeing memory at 0x7f53400639d0 │ │ │ +freeing memory at 0x7f5340066c50 │ │ │ +freeing memory at 0x7f5340069180 │ │ │ +freeing memory at 0x7f5340063bc0 │ │ │ +freeing memory at 0x7f534006a230 │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_value_lp__Foreign__Object_rp.out │ │ │ @@ -20,21 +20,21 @@ │ │ │ │ │ │ o4 = 5 │ │ │ │ │ │ o4 : RR (of precision 53) │ │ │ │ │ │ i5 : x = voidstar address int 5 │ │ │ │ │ │ -o5 = 0x7f769ebd8d90 │ │ │ +o5 = 0x7f2d9e7c3010 │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ │ │ i6 : value x │ │ │ │ │ │ -o6 = 0x7f769ebd8d90 │ │ │ +o6 = 0x7f2d9e7c3010 │ │ │ │ │ │ o6 : Pointer │ │ │ │ │ │ i7 : x = charstar "Hello, world!" │ │ │ │ │ │ o7 = Hello, world! │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html │ │ │ @@ -54,25 +54,25 @@ │ │ │ o1 = 5 │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f5daa523ec0}
    │ │ │ +o2 = int32{Address => 0x7f0d842307a0} │ │ │
    │ │ │
    │ │ │

    To get this, use address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f5daa523ec0
│ │ │ +o3 = 0x7f0d842307a0
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Use class to determine the type of the object.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,19 +10,19 @@ │ │ │ │ i1 : x = int 5 │ │ │ │ │ │ │ │ o1 = 5 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7f5daa523ec0} │ │ │ │ +o2 = int32{Address => 0x7f0d842307a0} │ │ │ │ To get this, use _a_d_d_r_e_s_s. │ │ │ │ i3 : address x │ │ │ │ │ │ │ │ -o3 = 0x7f5daa523ec0 │ │ │ │ +o3 = 0x7f0d842307a0 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Use _c_l_a_s_s to determine the type of the object. │ │ │ │ i4 : class x │ │ │ │ │ │ │ │ o4 = int32 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html │ │ │ @@ -62,15 +62,15 @@ │ │ │ o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog} │ │ │ │ │ │ o2 : ForeignObject of type char** │ │ │ 
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7fce2ba5fcd0, 0x7fce2ba5fcc0, 0x7fce2ba5fcb0}
│ │ │ +o3 = {0x7f09585cb500, 0x7f09585cb4f0, 0x7f09585cb4e0}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
    │ │ │
    │ │ │
    │ │ │

    Foreign pointer arrays may be subscripted using _.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ "lazy", "dog"} │ │ │ │ │ │ │ │ o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog} │ │ │ │ │ │ │ │ o2 : ForeignObject of type char** │ │ │ │ i3 : voidstarstar {address int 0, address int 1, address int 2} │ │ │ │ │ │ │ │ -o3 = {0x7fce2ba5fcd0, 0x7fce2ba5fcc0, 0x7fce2ba5fcb0} │ │ │ │ +o3 = {0x7f09585cb500, 0x7f09585cb4f0, 0x7f09585cb4e0} │ │ │ │ │ │ │ │ o3 : ForeignObject of type void** │ │ │ │ Foreign pointer arrays may be subscripted using __. │ │ │ │ i4 : x = charstarstar {"foo", "bar", "baz"} │ │ │ │ │ │ │ │ o4 = {foo, bar, baz} │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ o1 = {foo, bar} │ │ │ │ │ │ o1 : ForeignObject of type char** │ │ │ │ │ │ │ │ │ 
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f9bfd1e7ae0, 0x7f9bfd1e7ad0, 0x7f9bfd1e7ac0}
│ │ │ +o2 = {0x7f223b05ef30, 0x7f223b05ef20, 0x7f223b05ef10}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
    │ │ │ │ │ │ │ │ │ 
    i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │  
│ │ │  o3 = int32[2]*
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i1 : charstarstar {"foo", "bar"}
│ │ │ │  
│ │ │ │  o1 = {foo, bar}
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type char**
│ │ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o2 = {0x7f9bfd1e7ae0, 0x7f9bfd1e7ad0, 0x7f9bfd1e7ac0}
│ │ │ │ +o2 = {0x7f223b05ef30, 0x7f223b05ef20, 0x7f223b05ef10}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │          
    
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f3cb62b2350
│ │ │ +o1 = 0x7f7b2de308b0
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │      		          
                  
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f3cb62b2350
│ │ │ +o2 = 0x7f7b2de308b0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    
│ │ │      		          
    
│ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -16,18 +16,18 @@ │ │ │ │ * Outputs: │ │ │ │ o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ To cast a Macaulay2 pointer to a foreign object with a pointer type, give the │ │ │ │ type followed by the pointer. │ │ │ │ i1 : ptr = address int 0 │ │ │ │ │ │ │ │ -o1 = 0x7f3cb62b2350 │ │ │ │ +o1 = 0x7f7b2de308b0 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ │ │ -o2 = 0x7f3cb62b2350 │ │ │ │ +o2 = 0x7f7b2de308b0 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign │ │ │ │ pointer │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ o1 = 5 │ │ │ │ │ │ o1 : ForeignObject of type int32     │ │ │ │ │ │ │ │ │ 
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7fa785fa61e0
│ │ │ +o2 = 0x7f7169e30890
│ │ │  
│ │ │  o2 : Pointer
    │ │ │ │ │ │ │ │ │ 
    i3 : int ptr
│ │ │  
│ │ │  o3 = 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7fa785fa61e0
│ │ │ │ +o2 = 0x7f7169e30890
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f14108492a0
│ │ │ +o1 = 0x7f594da30770
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    
│ │ │      		          
                  
    i2 : int * ptr
│ │ │  
│ │ │  o2 = 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, of type T;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This is syntactic sugar for T value ptr (see _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r) for
│ │ │ │  dereferencing pointers.
│ │ │ │  i1 : ptr = voidstar address int 5
│ │ │ │  
│ │ │ │ -o1 = 0x7f14108492a0
│ │ │ │ +o1 = 0x7f594da30770
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │  o1 = myunion
│ │ │  
│ │ │  o1 : ForeignUnionType
    
│ │ │      		          
                  
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.94487e-310}
│ │ │ +o2 = HashTable{"bar" => 6.9037e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
    
│ │ │      		          
                  
    i3 : myunion pi
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i1 : myunion = foreignUnionType("myunion", {"foo" => int, "bar" => double})
│ │ │ │  
│ │ │ │  o1 = myunion
│ │ │ │  
│ │ │ │  o1 : ForeignUnionType
│ │ │ │  i2 : myunion 27
│ │ │ │  
│ │ │ │ -o2 = HashTable{"bar" => 6.94487e-310}
│ │ │ │ +o2 = HashTable{"bar" => 6.9037e-310}
│ │ │ │                 "foo" => 27
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type myunion
│ │ │ │  i3 : myunion pi
│ │ │ │  
│ │ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ │ │                 "foo" => 1413754136
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html
│ │ │ @@ -54,44 +54,44 @@
│ │ │  o1 = 20
│ │ │  
│ │ │  o1 : ForeignObject of type int32
    
│ │ │      		          
                  
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f9390dc9080}
    
│ │ │ +o2 = int32{Address => 0x7f1e080d57e0}
│ │ │      		          
    
│ │ │          
    
│ │ │            
    These pointers can be accessed using address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f9390dc9080
│ │ │ +o3 = 0x7f1e080d57e0
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    Simple arithmetic can be performed on pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f9390dc9085
│ │ │ +o4 = 0x7f1e080d57e5
│ │ │  
│ │ │  o4 : Pointer
    
│ │ │      		          
                  
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f9390dc907d
│ │ │ +o5 = 0x7f1e080d57dd
│ │ │  
│ │ │  o5 : Pointer
    
│ │ │      		          
    
│ │ │
    │ │ │
    │ │ │

    Functions and methods returning a pointer:

    │ │ │ ├── html2text {} │ │ │ │ @@ -10,30 +10,30 @@ │ │ │ │ i1 : x = int 20 │ │ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7f9390dc9080} │ │ │ │ +o2 = int32{Address => 0x7f1e080d57e0} │ │ │ │ These pointers can be accessed using _a_d_d_r_e_s_s. │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ │ │ -o3 = 0x7f9390dc9080 │ │ │ │ +o3 = 0x7f1e080d57e0 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Simple arithmetic can be performed on pointers. │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ │ │ -o4 = 0x7f9390dc9085 │ │ │ │ +o4 = 0x7f1e080d57e5 │ │ │ │ │ │ │ │ o4 : Pointer │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ │ │ -o5 = 0x7f9390dc907d │ │ │ │ +o5 = 0x7f1e080d57dd │ │ │ │ │ │ │ │ o5 : Pointer │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: ********** │ │ │ │ * _a_d_d_r_e_s_s -- pointer to type or object │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa ppooiinntteerr:: ********** │ │ │ │ * * Pointer = Thing -- see _*_ _v_o_i_d_s_t_a_r_ _=_ _T_h_i_n_g -- assign value to object at │ │ │ │ address │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ @@ -54,15 +54,15 @@ │ │ │ o1 = mpfr │ │ │ │ │ │ o1 : SharedLibrary     │ │ │ │ │ │ │ │ │ 
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f2afa172550, mpfr}
    │ │ │ +o2 = SharedLibrary{0x7f3a0dd50550, mpfr} │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Functions and methods returning a shared library:

    │ │ │
    │ │ │
    │ │ │

    Ways to use address:

    │ │ │ ├── html2text {} │ │ │ │ @@ -12,22 +12,22 @@ │ │ │ │ * Outputs: │ │ │ │ o a _p_o_i_n_t_e_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ If x is a foreign type, then this returns the address to the ffi_type struct │ │ │ │ used by libffi to identify the type. │ │ │ │ i1 : address int │ │ │ │ │ │ │ │ -o1 = 0x563f3b92bbc0 │ │ │ │ +o1 = 0x555df68ecbc0 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ If x is a foreign object, then this returns the address to the object. It │ │ │ │ behaves like the & "address-of" operator in C. │ │ │ │ i2 : address int 5 │ │ │ │ │ │ │ │ -o2 = 0x7f0602c6c0f0 │ │ │ │ +o2 = 0x7f18bce67b90 │ │ │ │ │ │ │ │ o2 : Pointer │ │ │ │ ********** WWaayyss ttoo uussee aaddddrreessss:: ********** │ │ │ │ * address(ForeignObject) │ │ │ │ * address(ForeignType) │ │ │ │ * address(Nothing) (missing documentation) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html │ │ │ @@ -204,15 +204,15 @@ │ │ │ o16 = free │ │ │ │ │ │ o16 : ForeignFunction     │ │ │ │ │ │ │ │ │ 
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f01340639d0
│ │ │ +o17 = 0x7fa8540639d0
│ │ │  
│ │ │  o17 : ForeignObject of type void*
    │ │ │ │ │ │ │ │ │ 
    i18 : registerFinalizer(x, free)
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ i16 : free = foreignFunction("free", void, voidstar) │ │ │ │ │ │ │ │ o16 = free │ │ │ │ │ │ │ │ o16 : ForeignFunction │ │ │ │ i17 : x = malloc 8 │ │ │ │ │ │ │ │ -o17 = 0x7f01340639d0 │ │ │ │ +o17 = 0x7fa8540639d0 │ │ │ │ │ │ │ │ o17 : ForeignObject of type void* │ │ │ │ i18 : registerFinalizer(x, free) │ │ │ │ ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: ********** │ │ │ │ * foreignFunction(Pointer,String,ForeignType,VisibleList) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,VisibleList) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html │ │ │ @@ -79,39 +79,39 @@ │ │ │
    │ │ │

    Allocate n bytes of memory using the GC garbage collector.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f39cc2e59a0
│ │ │ +o1 = 0x7feb5bfbe9a0
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │

    If the memory will not contain any pointers, then set the Atomic option to true.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f39d4990d40
│ │ │ +o2 = 0x7feb58d85cd0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │

    Alternatively, a foreign object type T may be specified. In this case, the number of bytes and whether the Atomic option should be set will be determined automatically.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f39d4990c50
│ │ │ +o3 = 0x7feb58d85be0
│ │ │  
│ │ │  o3 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -15,30 +15,30 @@ │ │ │ │ o Atomic => ..., default value false │ │ │ │ * Outputs: │ │ │ │ o an instance of the type _v_o_i_d_s_t_a_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Allocate n bytes of memory using the _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r. │ │ │ │ i1 : ptr = getMemory 8 │ │ │ │ │ │ │ │ -o1 = 0x7f39cc2e59a0 │ │ │ │ +o1 = 0x7feb5bfbe9a0 │ │ │ │ │ │ │ │ o1 : ForeignObject of type void* │ │ │ │ If the memory will not contain any pointers, then set the Atomic option to │ │ │ │ _t_r_u_e. │ │ │ │ i2 : ptr = getMemory(8, Atomic => true) │ │ │ │ │ │ │ │ -o2 = 0x7f39d4990d40 │ │ │ │ +o2 = 0x7feb58d85cd0 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ Alternatively, a foreign object type T may be specified. In this case, the │ │ │ │ number of bytes and whether the Atomic option should be set will be determined │ │ │ │ automatically. │ │ │ │ i3 : ptr = getMemory int │ │ │ │ │ │ │ │ -o3 = 0x7f39d4990c50 │ │ │ │ +o3 = 0x7feb58d85be0 │ │ │ │ │ │ │ │ o3 : ForeignObject of type void* │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a │ │ │ │ foreign object │ │ │ │ ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: ********** │ │ │ │ * getMemory(ForeignType) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html │ │ │ @@ -91,23 +91,23 @@ │ │ │ o3 : FunctionClosure │ │ │ │ │ │ │ │ │ 
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    │ │ │ │ │ │ │ │ │ 
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f114006a210
│ │ │ -freeing memory at 0x7f114005f3f0
│ │ │ -freeing memory at 0x7f114006a230
│ │ │ -freeing memory at 0x7f1140069180
│ │ │ -freeing memory at 0x7f11400639d0
│ │ │ -freeing memory at 0x7f1140063bf0
│ │ │ -freeing memory at 0x7f1140066c50
│ │ │ -freeing memory at 0x7f1140063bc0
│ │ │ -freeing memory at 0x7f11400639f0
    │ │ │ +freeing memory at 0x7f534005f3f0 │ │ │ +freeing memory at 0x7f534006a210 │ │ │ +freeing memory at 0x7f53400639f0 │ │ │ +freeing memory at 0x7f5340063bf0 │ │ │ +freeing memory at 0x7f53400639d0 │ │ │ +freeing memory at 0x7f5340066c50 │ │ │ +freeing memory at 0x7f5340069180 │ │ │ +freeing memory at 0x7f5340063bc0 │ │ │ +freeing memory at 0x7f534006a230 │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -242,21 +242,21 @@ │ │ │ │ o26 = | 7/4 1/2 7 6/7 | │ │ │ │ | 7/9 7/10 3/7 2/3 | │ │ │ │ | 7/10 7/3 5/2 1 | │ │ │ │ │ │ │ │ 3 4 │ │ │ │ o26 : Matrix QQ <-- QQ │ │ │ │ i27 : time C = orbitClosure(X,Mat) │ │ │ │ - -- used 0.602865s (cpu); 0.309627s (thread); 0s (gc) │ │ │ │ + -- used 9.5535s (cpu); 1.6112s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o27 : KClass │ │ │ │ i28 : time C = orbitClosure(X,Mat, RREFMethod => true) │ │ │ │ - -- used 1.63569s (cpu); 0.773805s (thread); 0s (gc) │ │ │ │ + -- used 23.9633s (cpu); 4.26127s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o28 : KClass │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_l_i_z_e_d_F_l_a_g_V_a_r_i_e_t_y -- makes a generalized flag variety as a GKM │ │ │ │ variety │ │ ├── ./usr/share/doc/Macaulay2/GenericInitialIdeal/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ Z2luKElkZWFsKQ== │ │ │ #:len=250 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGdlbmVyaWMgaW5pdGlhbCBpZGVh │ │ │ bCIsIERlc2NyaXB0aW9uID0+IHt9LCAibGluZW51bSIgPT4gMTc2LCBLZXkgPT4gKGdpbixJZGVh │ │ ├── ./usr/share/doc/Macaulay2/GeometricDecomposability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ SXNJZGVhbFVubWl4ZWQ= │ │ │ #:len=1529 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3BlY2lmeSB3aGV0aGVyIGFuIGlkZWFs │ │ │ IGlzIHVubWl4ZWQiLCAibGluZW51bSIgPT4gMTg0NiwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsi │ │ ├── ./usr/share/doc/Macaulay2/GradedLieAlgebras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ WlogXyBFeHRBbGdlYnJh │ │ │ #:len=932 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSB6ZXJvIGVsZW1lbnQiLCBE │ │ │ ZXNjcmlwdGlvbiA9PiAoRElWe0hFQURFUjJ7IlN5bm9wc2lzIn0sVUx7TEl7REx7ImNsYXNzIiA9 │ │ ├── ./usr/share/doc/Macaulay2/GraphicalModels/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ VmFyaWFibGVOYW1l │ │ │ #:len=616 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl │ │ │ IGluZGV0ZXJtaW5hdGUgbmFtZSBpbiBtYXJrb3ZSaW5nIiwgImxpbmVudW0iID0+IDE4ODksIFNl │ │ ├── ./usr/share/doc/Macaulay2/GraphicalModelsMLE/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c29sdmVyTUxFKExpc3QsR3JhcGgp │ │ │ #:len=274 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk1OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc29sdmVyTUxFLExpc3QsR3JhcGgpLCJzb2x2ZXJN │ │ ├── ./usr/share/doc/Macaulay2/Graphics/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ cGljdHVyZVpvbmUoU3BoZXJlKQ== │ │ │ #:len=747 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCB0aGUgem9uZSB0aGF0IGNvbnRh │ │ │ aW5zIHRoZSBzcGhlcmUiLCAibGluZW51bSIgPT4gMTExNywgSW5wdXRzID0+IHtTUEFOe1RUeyJz │ │ ├── ./usr/share/doc/Macaulay2/Graphs/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ cmV2ZXJzZUJyZWFkdGhGaXJzdFNlYXJjaA== │ │ │ #:len=1565 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicnVucyBhIHJldmVyc2UgYnJlYWR0aCBm │ │ │ aXJzdCBzZWFyY2ggb24gdGhlIGRpZ3JhcGggc3RhcnRpbmcgYXQgYSBzcGVjaWZpZWQgbm9kZSIs │ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out │ │ │ @@ -32,12 +32,12 @@ │ │ │ 5 => {6} │ │ │ 6 => {} │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ i3 : keys H │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ +o3 = {map, newDigraph, digraph} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ 6 => {} │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ │ │ │ 
    i3 : keys H
│ │ │  
│ │ │ -o3 = {digraph, map, newDigraph}
│ │ │ +o3 = {map, newDigraph, digraph}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 4 => {} │ │ │ │ 5 => {6} │ │ │ │ 6 => {} │ │ │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ i3 : keys H │ │ │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ │ +o3 = {map, newDigraph, digraph} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_o_p_S_o_r_t -- topologically sort the vertices of a digraph │ │ │ │ * _S_o_r_t_e_d_D_i_g_r_a_p_h -- hashtable used in topSort │ │ │ │ * _t_o_p_o_l_o_g_i_c_a_l_S_o_r_t -- outputs a list of vertices in a topologically sorted │ │ │ │ order of a DAG. │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ ZmluZFdlaWdodENvbnN0cmFpbnRz │ │ │ #:len=2630 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyBhIG1hdHJpeCBvZiB3ZWln │ │ │ aHQgY29uc3RyYWludHMiLCAibGluZW51bSIgPT4gNjcyLCBJbnB1dHMgPT4ge1NQQU57VFR7Ik0i │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ @@ -243,67 +243,67 @@ │ │ │ | 2 2 2 2 2 2 2 2 2 2 2 2 | │ │ │ |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 | │ │ │ | 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| │ │ │ +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o12 = | -13 -21 -15 -5 36 46 -6 -25 12 -22 -33 37 24 -36 -36 -30 -29 -8 19 │ │ │ +o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13 │ │ │ ----------------------------------------------------------------------- │ │ │ - -13 19 -29 -29 -10 | │ │ │ + -36 -30 -29 -10 | │ │ │ │ │ │ 1 24 │ │ │ o12 : Matrix kk <-- kk │ │ │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o13 = | 50 -8 -9 9 0 5 -15 38 0 -42 -18 28 45 39 16 -46 34 19 -16 -24 -38 0 │ │ │ +o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39 │ │ │ ----------------------------------------------------------------------- │ │ │ - -47 21 | │ │ │ + -16 0 -47 21 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ 2 2 2 │ │ │ -o14 = ideal (a + 12b*c - 5c - 22a*d + 36b*d - 21c*d - 13d , a*b - 36b*c - │ │ │ +o14 = ideal (a + 2b*c - 32c - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 33c - 30a*d + 37b*d + 46c*d - 15d , b + 19b*c - 29c - 29a*d - 8b*d + │ │ │ + 2 2 2 2 │ │ │ + 30c - 29a*d - 23b*d + 30c*d + 41d , b - 36b*c - 8c - 30a*d - 22b*d + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 24c*d - 6d , a*c - 29b*c + 19c - 10a*d - 13b*d - 36c*d - 25d ) │ │ │ + 2 2 2 │ │ │ + 19c*d + 48d , a*c - 29b*c + 24c - 10a*d - 13b*d + 19c*d + 36d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 2 │ │ │ -o15 = ideal (a + 9c - 42a*d - 8c*d + 50d , a*b + 16b*c - 18c - 46a*d + │ │ │ + 2 2 2 │ │ │ +o15 = ideal (a - 31b*c + 26c - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 28b*d + 5c*d - 9d , b - 38b*c + 34c + 19b*d + 45c*d - 15d , a*c - │ │ │ + 2 2 2 2 │ │ │ + 16c - 17a*d - 7b*d + 31c*d + 11d , b - 16b*c + 19c + 34b*d - 41c*d + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 │ │ │ - 47b*c - 16c + 21a*d - 24b*d + 39c*d + 38d ) │ │ │ + 2 2 2 │ │ │ + 38d , a*c - 47b*c - 38c + 21a*d + 39b*d - 24c*d + 24d ) │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = {ideal (a - 29b + 19c - 48d, b + 19b*c - 29c - 41b*d - 31c*d + 16d ), │ │ │ + 2 2 2 │ │ │ +o16 = {ideal (a - 29b + 24c - 44d, b - 36b*c - 8c + 17b*d + 32c*d + 41d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 10d, b + 3d, a - d)} │ │ │ + ideal (c - 10d, b + 33d, a + 10d)} │ │ │ │ │ │ o16 : List │ │ │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ -o17 = {ideal (b - 42c - 19d, a + 30c + 33d), ideal (b + 4c + 38d, a - 30c + │ │ │ +o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 26d)} │ │ │ + 29d)} │ │ │ │ │ │ o17 : List │ │ │ │ │ │ i18 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ @@ -103,46 +103,46 @@ │ │ │ │ │ │ i13 : #compsJ │ │ │ │ │ │ o13 = 2 │ │ │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o14 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21 │ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 │ │ │ ----------------------------------------------------------------------- │ │ │ - -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 | │ │ │ + 40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 | │ │ │ │ │ │ 1 36 │ │ │ o14 : Matrix kk <-- kk │ │ │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7 │ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 | │ │ │ + 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ │ │ │ 1 36 │ │ │ o15 : Matrix kk <-- kk │ │ │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = ideal (a - 40b*c + 21c + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c - │ │ │ + 2 2 2 │ │ │ +o16 = ideal (a + 25b*c - 43c + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 50c - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c - 29a*d + 30b*d - │ │ │ + 2 2 2 │ │ │ + 46c + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c + 39a*d - 20b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - 30c*d + 38d , b + 21b*c + 19c + 19a*d - 38b*d + 10c*d + 47d , b*c + │ │ │ + 2 2 2 2 2 │ │ │ + - 22c*d - 2d , b + 24b*c + 19c - 10a*d + 21b*d - 16c*d - 35d , b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 │ │ │ - 24b*c*d - 16c d + 39a*d - 27b*d + 8c*d + 7d , c - 22b*c*d - 24c d - │ │ │ + 2 2 2 2 3 3 2 │ │ │ + 29b*c*d - 30c d - 36a*d + 34b*d + 48c*d - 23d , c - 24b*c*d - 16c d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 3 │ │ │ - 29a*d - 29b*d - 10c*d + 48d ) │ │ │ + 2 2 2 3 │ │ │ + + 19a*d - 38b*d - 29c*d - 26d ) │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ 0 1 2 3 │ │ │ o17 = total: 1 6 8 3 │ │ │ @@ -150,28 +150,28 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o17 : BettiTally │ │ │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o18 = ideal (a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c + │ │ │ + 2 2 2 │ │ │ +o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 33c + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c + 14a*d - 43b*d - │ │ │ + 2 2 2 │ │ │ + 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , b*c + 34b*c*d + │ │ │ + 2 2 2 2 2 │ │ │ + - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ - 2c d + 16a*d + 45b*d - 15c*d + 10d , c - 28b*c*d + 38c d - 39a*d - │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ + - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 3 │ │ │ - 47b*d - 13c*d + 4d ) │ │ │ + 2 2 3 │ │ │ + + 38b*d - 39c*d - d ) │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ 0 1 2 3 │ │ │ o19 = total: 1 6 8 3 │ │ │ @@ -179,26 +179,30 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o19 : BettiTally │ │ │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ - +----------------------------------------------------------------------------------------------------------+ │ │ │ -o20 = |ideal (c - 16d, b + 31d, a + 12d) | │ │ │ - +----------------------------------------------------------------------------------------------------------+ │ │ │ - |ideal (c - 29d, b + 29d, a - 27d) | │ │ │ - +----------------------------------------------------------------------------------------------------------+ │ │ │ - |ideal (c + 41d, b + 35d, a - 25d) | │ │ │ - +----------------------------------------------------------------------------------------------------------+ │ │ │ - | 2 2 2 2 2 | │ │ │ - |ideal (a - 8b - 36c + 37d, c - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d - 9d , b - 17b*d + 21c*d - 34d )| │ │ │ - +----------------------------------------------------------------------------------------------------------+ │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d) | │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 3 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - 24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i21 : netList decompose F2 │ │ │ │ │ │ - +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ - | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ -o21 = |ideal (a*c - 15b*c + 19c + 14a*d - 43b*d - 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c + a*d + 42b*d + 37c*d + 13d , a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , c - 28b*c*d + 38c d - 39a*d - 47b*d - 13c*d + 4d , b*c + 34b*c*d + 2c d + 16a*d + 45b*d - 15c*d + 10d )| │ │ │ - +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + +-------------------------------------------------------+ │ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 24d, b - 49d, a) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ + +-------------------------------------------------------+ │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ 2 │ │ │ c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : pt = randomPointOnRationalVariety I │ │ │ │ │ │ -o4 = | -25 20 -30 -16 24 -36 | │ │ │ +o4 = | 1 49 24 -23 -36 -30 | │ │ │ │ │ │ 1 6 │ │ │ o4 : Matrix kk <-- kk │ │ │ │ │ │ i5 : sub(I, pt) == 0 │ │ │ │ │ │ o5 = true │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ @@ -206,48 +206,48 @@ │ │ │ 
    i13 : #compsJ
│ │ │  
│ │ │  o13 = 2
    │ │ │ │ │ │ │ │ │ 
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21
│ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 |
│ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ 
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7
│ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 |
│ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ 
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o16 = ideal (a  - 40b*c + 21c  + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c -
│ │ │ +              2              2                             2               
│ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                  2                  
│ │ │ -      50c  - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c  - 29a*d + 30b*d -
│ │ │ +         2                              2                  2                
│ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2     2  
│ │ │ -      30c*d + 38d , b  + 21b*c + 19c  + 19a*d - 38b*d + 10c*d + 47d , b*c  +
│ │ │ +                  2   2              2                              2     2  
│ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2       2     3   3                2   
│ │ │ -      24b*c*d - 16c d + 39a*d  - 27b*d  + 8c*d  + 7d , c  - 22b*c*d - 24c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2        2        2      3
│ │ │ -      29a*d  - 29b*d  - 10c*d  + 48d )
│ │ │ +             2        2        2      3
│ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │  
│ │ │  o16 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i17 : betti res F1
│ │ │  
│ │ │               0 1 2 3
│ │ │ @@ -257,28 +257,28 @@
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o17 : BettiTally
    │ │ │ │ │ │ │ │ │ 
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o18 = ideal (a  - 35b*c + 33c  + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c +
│ │ │ +              2             2                              2               
│ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                            2                   2                  
│ │ │ -      33c  + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c  + 14a*d - 43b*d -
│ │ │ +        2                              2                   2                
│ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                     2     2            
│ │ │ -      47c*d + 27d , b  + 22b*c - 18c  + 7b*d + 23c*d + 13d , b*c  + 34b*c*d +
│ │ │ +                 2   2              2                      2     2          
│ │ │ +      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2         2        2        2      3   3                2         2  
│ │ │ -      2c d + 16a*d  + 45b*d  - 15c*d  + 10d , c  - 28b*c*d + 38c d - 39a*d  -
│ │ │ +           2         2        2       2      3   3               2         2
│ │ │ +      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2        2     3
│ │ │ -      47b*d  - 13c*d  + 4d )
│ │ │ +             2        2    3
│ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │  
│ │ │  o18 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i19 : betti res F2
│ │ │  
│ │ │               0 1 2 3
│ │ │ @@ -293,32 +293,36 @@
│ │ │          
    
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 16d, b + 31d, a + 12d)                                                                         |
│ │ │ -      +----------------------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c - 29d, b + 29d, a - 27d)                                                                         |
│ │ │ -      +----------------------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 41d, b + 35d, a - 25d)                                                                         |
│ │ │ -      +----------------------------------------------------------------------------------------------------------+
│ │ │ -      |                            2                     2                          2   2                      2 |
│ │ │ -      |ideal (a - 8b - 36c + 37d, c  - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d - 9d , b  - 17b*d + 21c*d - 34d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
│ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │      		          
                  
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                        2                              2   2              2                     2                   2                            2   2              2                              2   3                2         2        2        2     3     2               2         2        2        2      3 |
│ │ │ -o21 = |ideal (a*c - 15b*c + 19c  + 14a*d - 43b*d - 47c*d + 27d , b  + 22b*c - 18c  + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c  + a*d + 42b*d + 37c*d + 13d , a  - 35b*c + 33c  + 46a*d - 33b*d - 48c*d + 18d , c  - 28b*c*d + 38c d - 39a*d  - 47b*d  - 13c*d  + 4d , b*c  + 34b*c*d + 2c d + 16a*d  + 45b*d  - 15c*d  + 10d )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +-------------------------------------------------------+
│ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 43d, b - 47d, a - 27d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 24d, b - 49d, a)                            |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 14d, b + 31d, a - 16d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |                                      2              2 |
│ │ │ +      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
│ │ │ +      +-------------------------------------------------------+
│ │ │      		          
    
│ │ │          
    
│ │ │            
    We can determine what these represent.  One should be a set of 6 points, where 5 lie on a plane.  The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.
    
│ │ │          
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -171,71 +171,71 @@ │ │ │ │ 32 13 21 33 19 31 │ │ │ │ i12 : compsJ = decompose J; │ │ │ │ i13 : #compsJ │ │ │ │ │ │ │ │ o13 = 2 │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o14 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21 │ │ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 | │ │ │ │ + 40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o14 : Matrix kk <-- kk │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7 │ │ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 | │ │ │ │ + 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o15 : Matrix kk <-- kk │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = ideal (a - 40b*c + 21c + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o16 = ideal (a + 25b*c - 43c + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 50c - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c - 29a*d + 30b*d - │ │ │ │ + 2 2 2 │ │ │ │ + 46c + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c + 39a*d - 20b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - 30c*d + 38d , b + 21b*c + 19c + 19a*d - 38b*d + 10c*d + 47d , b*c + │ │ │ │ + 2 2 2 2 2 │ │ │ │ + - 22c*d - 2d , b + 24b*c + 19c - 10a*d + 21b*d - 16c*d - 35d , b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 24b*c*d - 16c d + 39a*d - 27b*d + 8c*d + 7d , c - 22b*c*d - 24c d - │ │ │ │ + 2 2 2 2 3 3 2 │ │ │ │ + 29b*c*d - 30c d - 36a*d + 34b*d + 48c*d - 23d , c - 24b*c*d - 16c d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - 29a*d - 29b*d - 10c*d + 48d ) │ │ │ │ + 2 2 2 3 │ │ │ │ + + 19a*d - 38b*d - 29c*d - 26d ) │ │ │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o17 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o17 : BettiTally │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o18 = ideal (a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 33c + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c + 14a*d - 43b*d - │ │ │ │ + 2 2 2 │ │ │ │ + 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , b*c + 34b*c*d + │ │ │ │ + 2 2 2 2 2 │ │ │ │ + - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ │ - 2c d + 16a*d + 45b*d - 15c*d + 10d , c - 28b*c*d + 38c d - 39a*d - │ │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ │ + - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 3 │ │ │ │ - 47b*d - 13c*d + 4d ) │ │ │ │ + 2 2 3 │ │ │ │ + + 38b*d - 39c*d - d ) │ │ │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o19 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ @@ -243,54 +243,43 @@ │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o19 : BettiTally │ │ │ │ What are the ideals F1 and F2? │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -----------------------------------+ │ │ │ │ -o20 = |ideal (c - 16d, b + 31d, a + 12d) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------------------------+ │ │ │ │ - |ideal (c - 29d, b + 29d, a - 27d) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------------------------+ │ │ │ │ - |ideal (c + 41d, b + 35d, a - 25d) │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +--+ │ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -----------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 | │ │ │ │ - |ideal (a - 8b - 36c + 37d, c - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d │ │ │ │ -- 9d , b - 17b*d + 21c*d - 34d )| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------------------------+ │ │ │ │ -i21 : netList decompose F2 │ │ │ │ - │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ -+ │ │ │ │ - | 2 2 2 │ │ │ │ -2 2 2 2 2 │ │ │ │ -2 2 3 2 2 2 │ │ │ │ -2 3 2 2 2 2 2 3 | │ │ │ │ -o21 = |ideal (a*c - 15b*c + 19c + 14a*d - 43b*d - 47c*d + 27d , b + 22b*c - │ │ │ │ -18c + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c + a*d + 42b*d + 37c*d + 13d , a │ │ │ │ -- 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , c - 28b*c*d + 38c d - 39a*d - │ │ │ │ -47b*d - 13c*d + 4d , b*c + 34b*c*d + 2c d + 16a*d + 45b*d - 15c*d + 10d │ │ │ │ +--+ │ │ │ │ + | 2 2 2 3 │ │ │ │ +2 2 2 3 2 2 2 3 | │ │ │ │ + |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - │ │ │ │ +24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d │ │ │ │ )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ -+ │ │ │ │ +--+ │ │ │ │ +i21 : netList decompose F2 │ │ │ │ + │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + |ideal (c + 24d, b - 49d, a) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + | 2 2 | │ │ │ │ + |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ We can determine what these represent. One should be a set of 6 points, where 5 │ │ │ │ lie on a plane. The other should be 6 points with 3 points on one line, and the │ │ │ │ other 3 points on a skew line. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can │ │ │ │ be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ 
    i4 : pt = randomPointOnRationalVariety I
│ │ │  
│ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │  
│ │ │                1       6
│ │ │  o4 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ 
    i5 : sub(I, pt) == 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                               2
│ │ │ │       c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : pt = randomPointOnRationalVariety I
│ │ │ │  
│ │ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │ │  
│ │ │ │                1       6
│ │ │ │  o4 : Matrix kk  <-- kk
│ │ │ │  i5 : sub(I, pt) == 0
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : S = kk[a..d];
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html
│ │ │ @@ -320,75 +320,75 @@
│ │ │          
    
│ │ │            
    We can find random points on each component, since these components are rational.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o12 = | -13 -21 -15 -5 36 46 -6 -25 12 -22 -33 37 24 -36 -36 -30 -29 -8 19
│ │ │ +o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -13 19 -29 -29 -10 |
│ │ │ +      -36 -30 -29 -10 |
│ │ │  
│ │ │                 1       24
│ │ │  o12 : Matrix kk  <-- kk
    
│ │ │      		          
                  
    i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o13 = | 50 -8 -9 9 0 5 -15 38 0 -42 -18 28 45 39 16 -46 34 19 -16 -24 -38 0
│ │ │ +o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 21 |
│ │ │ +      -16 0 -47 21 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    
│ │ │      		          
                  
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │                2             2                              2               
│ │ │ -o14 = ideal (a  + 12b*c - 5c  - 22a*d + 36b*d - 21c*d - 13d , a*b - 36b*c -
│ │ │ +o14 = ideal (a  + 2b*c - 32c  - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                 
│ │ │ -      33c  - 30a*d + 37b*d + 46c*d - 15d , b  + 19b*c - 29c  - 29a*d - 8b*d +
│ │ │ +         2                              2   2             2                  
│ │ │ +      30c  - 29a*d - 23b*d + 30c*d + 41d , b  - 36b*c - 8c  - 30a*d - 22b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2                   2                              2
│ │ │ -      24c*d - 6d , a*c - 29b*c + 19c  - 10a*d - 13b*d - 36c*d - 25d )
│ │ │ +                 2                   2                              2
│ │ │ +      19c*d + 48d , a*c - 29b*c + 24c  - 10a*d - 13b*d + 19c*d + 36d )
│ │ │  
│ │ │  o14 : Ideal of S
    
│ │ │      		          
                  
    i15 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2     2                     2                   2          
│ │ │ -o15 = ideal (a  + 9c  - 42a*d - 8c*d + 50d , a*b + 16b*c - 18c  - 46a*d +
│ │ │ +              2              2                              2               
│ │ │ +o15 = ideal (a  - 31b*c + 26c  - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                       2   2              2                      2       
│ │ │ -      28b*d + 5c*d - 9d , b  - 38b*c + 34c  + 19b*d + 45c*d - 15d , a*c -
│ │ │ +         2                             2   2              2                  
│ │ │ +      16c  - 17a*d - 7b*d + 31c*d + 11d , b  - 16b*c + 19c  + 34b*d - 41c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                              2
│ │ │ -      47b*c - 16c  + 21a*d - 24b*d + 39c*d + 38d )
│ │ │ +         2                   2                              2
│ │ │ +      38d , a*c - 47b*c - 38c  + 21a*d + 39b*d - 24c*d + 24d )
│ │ │  
│ │ │  o15 : Ideal of S
    
│ │ │      		          
                  
    i16 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o16 = {ideal (a - 29b + 19c - 48d, b  + 19b*c - 29c  - 41b*d - 31c*d + 16d ),
│ │ │ +                                    2             2                      2
│ │ │ +o16 = {ideal (a - 29b + 24c - 44d, b  - 36b*c - 8c  + 17b*d + 32c*d + 41d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 10d, b + 3d, a - d)}
│ │ │ +      ideal (c - 10d, b + 33d, a + 10d)}
│ │ │  
│ │ │  o16 : List
    
│ │ │      		          
                  
    i17 : decompose F2
│ │ │  
│ │ │ -o17 = {ideal (b - 42c - 19d, a + 30c + 33d), ideal (b + 4c + 38d, a - 30c +
│ │ │ +o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +      29d)}
│ │ │  
│ │ │  o17 : List
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    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.
    
│ │ │            
    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.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -424,65 +424,65 @@
│ │ │ │  -----------------------------------------------------------+
│ │ │ │  This tells us that there are 2 components (at least over the given field).
│ │ │ │  Their dimensions are 11, 8.
│ │ │ │  We can find random points on each component, since these components are
│ │ │ │  rational.
│ │ │ │  i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o12 = | -13 -21 -15 -5 36 46 -6 -25 12 -22 -33 37 24 -36 -36 -30 -29 -8 19
│ │ │ │ +o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -13 19 -29 -29 -10 |
│ │ │ │ +      -36 -30 -29 -10 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o12 : Matrix kk  <-- kk
│ │ │ │  i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o13 = | 50 -8 -9 9 0 5 -15 38 0 -42 -18 28 45 39 16 -46 34 19 -16 -24 -38 0
│ │ │ │ +o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -47 21 |
│ │ │ │ +      -16 0 -47 21 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o13 : Matrix kk  <-- kk
│ │ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │                2             2                              2
│ │ │ │ -o14 = ideal (a  + 12b*c - 5c  - 22a*d + 36b*d - 21c*d - 13d , a*b - 36b*c -
│ │ │ │ +o14 = ideal (a  + 2b*c - 32c  - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2   2              2
│ │ │ │ -      33c  - 30a*d + 37b*d + 46c*d - 15d , b  + 19b*c - 29c  - 29a*d - 8b*d +
│ │ │ │ +         2                              2   2             2
│ │ │ │ +      30c  - 29a*d - 23b*d + 30c*d + 41d , b  - 36b*c - 8c  - 30a*d - 22b*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                2                   2                              2
│ │ │ │ -      24c*d - 6d , a*c - 29b*c + 19c  - 10a*d - 13b*d - 36c*d - 25d )
│ │ │ │ +                 2                   2                              2
│ │ │ │ +      19c*d + 48d , a*c - 29b*c + 24c  - 10a*d - 13b*d + 19c*d + 36d )
│ │ │ │  
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : F2 = sub(F, (vars S)|pt2)
│ │ │ │  
│ │ │ │ -              2     2                     2                   2
│ │ │ │ -o15 = ideal (a  + 9c  - 42a*d - 8c*d + 50d , a*b + 16b*c - 18c  - 46a*d +
│ │ │ │ +              2              2                              2
│ │ │ │ +o15 = ideal (a  - 31b*c + 26c  - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                       2   2              2                      2
│ │ │ │ -      28b*d + 5c*d - 9d , b  - 38b*c + 34c  + 19b*d + 45c*d - 15d , a*c -
│ │ │ │ +         2                             2   2              2
│ │ │ │ +      16c  - 17a*d - 7b*d + 31c*d + 11d , b  - 16b*c + 19c  + 34b*d - 41c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                              2
│ │ │ │ -      47b*c - 16c  + 21a*d - 24b*d + 39c*d + 38d )
│ │ │ │ +         2                   2                              2
│ │ │ │ +      38d , a*c - 47b*c - 38c  + 21a*d + 39b*d - 24c*d + 24d )
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : decompose F1
│ │ │ │  
│ │ │ │ -                                    2              2                      2
│ │ │ │ -o16 = {ideal (a - 29b + 19c - 48d, b  + 19b*c - 29c  - 41b*d - 31c*d + 16d ),
│ │ │ │ +                                    2             2                      2
│ │ │ │ +o16 = {ideal (a - 29b + 24c - 44d, b  - 36b*c - 8c  + 17b*d + 32c*d + 41d ),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ideal (c - 10d, b + 3d, a - d)}
│ │ │ │ +      ideal (c - 10d, b + 33d, a + 10d)}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : decompose F2
│ │ │ │  
│ │ │ │ -o17 = {ideal (b - 42c - 19d, a + 30c + 33d), ideal (b + 4c + 38d, a - 30c +
│ │ │ │ +o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      26d)}
│ │ │ │ +      29d)}
│ │ │ │  
│ │ │ │  o17 : List
│ │ │ │  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.
│ │ │ │  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
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  c2V0V2Fsa1RyYWNlKFpaKQ==
│ │ │  #:len=251
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzZXRXYWxrVHJhY2UsWlopLCJzZXRXYWxrVHJhY2Uo
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out
│ │ │ @@ -11,21 +11,21 @@
│ │ │  i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];
│ │ │  
│ │ │  i4 : I2 = sub(I1, R2);
│ │ │  
│ │ │  o4 : Ideal of R2
│ │ │  
│ │ │  i5 : elapsedTime gb I2
│ │ │ - -- 3.77668s elapsed
│ │ │ + -- 2.70794s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
│ │ │  
│ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.99521s elapsed
│ │ │ + -- 2.47807s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html
│ │ │ @@ -76,28 +76,28 @@
│ │ │            
│ │ │                
    i4 : I2 = sub(I1, R2);
│ │ │  
│ │ │  o4 : Ideal of R2
    
│ │ │            
│ │ │            
│ │ │                
    i5 : elapsedTime gb I2
│ │ │ - -- 3.77668s elapsed
│ │ │ + -- 2.70794s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
    
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    but it is faster to compute directly in the first order and then use the Groebner walk.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.99521s elapsed
│ │ │ + -- 2.47807s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
    
│ │ │      		          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,23 +38,23 @@
│ │ │ │  using a different weight vector and then graded reverse lexicographic we could
│ │ │ │  substitute and compute directly,
│ │ │ │  i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];
│ │ │ │  i4 : I2 = sub(I1, R2);
│ │ │ │  
│ │ │ │  o4 : Ideal of R2
│ │ │ │  i5 : elapsedTime gb I2
│ │ │ │ - -- 3.77668s elapsed
│ │ │ │ + -- 2.70794s elapsed
│ │ │ │  
│ │ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │ │  
│ │ │ │  o5 : GroebnerBasis
│ │ │ │  but it is faster to compute directly in the first order and then use the
│ │ │ │  Groebner walk.
│ │ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ │ - -- 2.99521s elapsed
│ │ │ │ + -- 2.47807s elapsed
│ │ │ │  
│ │ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │ │  
│ │ │ │  o6 : GroebnerBasis
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The target ring must be the same ring as the ring of the starting ideal, except
│ │ │ │  with different monomial order. The ring must be a polynomial ring over a field.
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  aWRlYWxPZlByb2plY3RpdmVQb2ludHM=
│ │ │  #:len=1240
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGlkZWFsIG9mIHNl
│ │ │  dCBvZiBwb2ludHMiLCAibGluZW51bSIgPT4gNDI0LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi
│ │ ├── ./usr/share/doc/Macaulay2/HigherCIOperators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Y2lPcGVyYXRvclJlc29sdXRpb24=
│ │ │  #:len=2668
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiXCJsaWZ0IHJlc29sdXRpb24gZnJvbSBj
│ │ │  b21wbGV0ZSBpbnRlcnNlY3Rpb24gdXNpbmcgaGlnaGVyIGNpLW9wZXJhdG9yc1wiIiwgImxpbmVu
│ │ ├── ./usr/share/doc/Macaulay2/HighestWeights/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  R3JvdXBBY3Rpbmc=
│ │ │  #:len=621
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3RvcmVzIHRoZSBEeW5raW4gdHlwZSBv
│ │ │  ZiB0aGUgZ3JvdXAgYWN0aW5nIG9uIGEgcmluZyIsICJsaW5lbnVtIiA9PiA4MywgU2VlQWxzbyA9
│ │ ├── ./usr/share/doc/Macaulay2/HodgeIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  a2FwcGE=
│ │ │  #:len=1396
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTWlsbGVyLU1vcml0YS1NdW1mb3JkIGNs
│ │ │  YXNzZXMiLCAibGluZW51bSIgPT4gNzM4LCBJbnB1dHMgPT4ge1NQQU57VFR7ImEifSwiLCAiLFNQ
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  ZXVsZXJPcGVyYXRvcnM=
│ │ │  #:len=1959
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXVsZXIgT3BlcmF0b3JzIiwgImxpbmVu
│ │ │  dW0iID0+IDE0MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJBIn0sIiwgIixTUEFOeyJhICIsVE8ye25l
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
│ │ │ @@ -42,19 +42,19 @@
│ │ │  i5 : w = {9,1,99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ - -- .000004138s elapsed
│ │ │ - -- .000002986s elapsed
│ │ │ - -- .000003447s elapsed
│ │ │ - -- .00000562s elapsed
│ │ │ - -- .000002555s elapsed
│ │ │ + -- .000003166s elapsed
│ │ │ + -- .000003125s elapsed
│ │ │ + -- .000003596s elapsed
│ │ │ + -- .000003567s elapsed
│ │ │ + -- .000003196s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[t_1..t_5];
│ │ │  
│ │ │  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);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004579s elapsed
│ │ │ + -- .000003286s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -24,15 +24,15 @@
│ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │  
│ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .000004359s elapsed
│ │ │ + -- .000002425s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
│ │ │ @@ -122,19 +122,19 @@
│ │ │  
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ 
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ - -- .000004138s elapsed
│ │ │ - -- .000002986s elapsed
│ │ │ - -- .000003447s elapsed
│ │ │ - -- .00000562s elapsed
│ │ │ - -- .000002555s elapsed
│ │ │ + -- .000003166s elapsed
│ │ │ + -- .000003125s elapsed
│ │ │ + -- .000003596s elapsed
│ │ │ + -- .000003567s elapsed
│ │ │ + -- .000003196s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,19 +55,19 @@
│ │ │ │                    1   6   1   6
│ │ │ │  i5 : w = {9,1,99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │ - -- .000004138s elapsed
│ │ │ │ - -- .000002986s elapsed
│ │ │ │ - -- .000003447s elapsed
│ │ │ │ - -- .00000562s elapsed
│ │ │ │ - -- .000002555s elapsed
│ │ │ │ + -- .000003166s elapsed
│ │ │ │ + -- .000003125s elapsed
│ │ │ │ + -- .000003596s elapsed
│ │ │ │ + -- .000003567s elapsed
│ │ │ │ + -- .000003196s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │            
│ │ │                
    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);
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │            
│ │ │            
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004579s elapsed
│ │ │ + -- .000003286s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -104,15 +104,15 @@
│ │ │          
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
    
│ │ │      		          
                  
    i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .000004359s elapsed
│ │ │ + -- .000002425s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Here is [_S_S_T, Example 2.3.16]:
│ │ │ │  i1 : R = QQ[t_1..t_5];
│ │ │ │  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);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │ - -- .000004579s elapsed
│ │ │ │ + -- .000003286s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │         1             1             1             3                 2
│ │ │ │       - -logX logX  - -logX logX  - -logX logX  - -logX logX  + logX }
│ │ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │ - -- .000004359s elapsed
│ │ │ │ + -- .000002425s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  YWxsZ2VucyhER0FsZ2VicmEsWlop
│ │ │  #:len=269
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbGxnZW5zLERHQWxnZWJyYSxaWiksImFsbGdlbnMo
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out
│ │ │ @@ -85,82 +85,82 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T  }, - T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T ,
│ │ │ -          5   10      5 10      18      4   6    4 6    1 10      20      4 
│ │ │ +o15 = {({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  -
│ │ │ +          1   8      3 6    5 7    1 8      12      14      4   9    4 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   -
│ │ │ -       7      1 6    4 7      11      2   6    2 6    3 8    4 9      14  
│ │ │ +      T T   - z*T   + z*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T }, -
│ │ │ +       5 10      17      19      2   8      2 8    5 9      15      5   7    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │ -         17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │ +      T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  + T T  + T T  +
│ │ │ +       3 6    5 7    1 8      12      14      4   9    2 6    3 8    4 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │ -       3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │ +      y*T   - z*T  ), ({T , T  }, T T  + T T   - z*T   + y*T  ), ({T , T },
│ │ │ +         14      17      3   10    5 6    3 10      18      20      5   6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ -       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      T T  + T T   - z*T   + y*T  ), ({T , T  }, - T T  - T T   + z*T   +
│ │ │ +       5 6    3 10      18      20      4   10      5 7    4 10      12  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   +
│ │ │ -         13      2   9      2 9      16      1   9      5 6    1 9      14  
│ │ │ +      z*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  -
│ │ │ +         20      1   6      1 6    4 7      11      3   9    5 8    3 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T  ), ({T , T }, - T T  + z*T  ), ({T , T  }, T T  - T T   + x*T  ),
│ │ │ -         17      4   7      4 7      13      1   10    4 6    1 10      20  
│ │ │ +      z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T  }, -
│ │ │ +         15      16      4   6    4 6    3 7      11      13      5   10    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  + z*T  ), ({T , T }, T T  - z*T   + x*T  ), ({T ,
│ │ │ -         5   9      5 9      16      3   7    3 7      11      12      5 
│ │ │ +      T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T , T }, - T T  -
│ │ │ +       5 10      18      4   6    4 6    1 10      20      4   7      1 6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ -       6      5 6    1 9      14      17      5   8    5 8    3 9      15  
│ │ │ +      T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ +       4 7      11      2   6    2 6    3 8    4 9      14      17      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, T T  -
│ │ │ -         16      4   8      2 7    4 8      12      14      3   9    3 9  
│ │ │ +      T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  + T T  + T T  +
│ │ │ +       10    4 9    5 10      17      19      3   8    2 6    3 8    4 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   + y*T  ), ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T ,
│ │ │ -         15      17      3   6      3 6    5 7    1 8      12      14      1 
│ │ │ +      y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T , T }, - T T  -
│ │ │ +         14      17      3   6    3 6      11      12      5   7      5 7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  + x*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T  }, -
│ │ │ -       7      1 7      13      5   9      2 8    5 9      15      3   10    
│ │ │ +      T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ), ({T ,
│ │ │ +       4 10      12      20      3   7    4 6    3 7      11      13      2 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T   - z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   +
│ │ │ -       4 8    3 10      18      19      2   7      2 7    4 8      12  
│ │ │ +      T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T ,
│ │ │ +       9      2 9      16      1   9      5 6    1 9      14      17      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T  ), ({T , T }, - T T  + T T   - z*T   + x*T  ), ({T , T  }, - T T 
│ │ │ -         14      4   8      4 8    3 10      18      19      2   10      5 8
│ │ │ +      T }, - T T  + z*T  ), ({T , T  }, T T  - T T   + x*T  ), ({T , T }, -
│ │ │ +       7      4 7      13      1   10    4 6    1 10      20      5   9    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T T   + y*T  ), ({T , T  }, - T T   + x*T  ), ({T , T }, - T T  -
│ │ │ -         2 10      19      4   10      4 10      18      5   8      5 8  
│ │ │ +      T T  + z*T  ), ({T , T }, T T  - z*T   + x*T  ), ({T , T }, - T T  -
│ │ │ +       5 9      16      3   7    3 7      11      12      5   6      5 6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T   + y*T  ), ({T , T }, T T  + x*T   - z*T  ), ({T , T }, - T T  -
│ │ │ -       2 10      19      3   8    3 8      15      17      1   8      3 6  
│ │ │ +      T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T ,
│ │ │ +       1 9      14      17      5   8    5 8    3 9      15      16      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  - T T  + z*T   + x*T  ), ({T , T }, T T  - T T   - z*T   + z*T  ),
│ │ │ -       5 7    1 8      12      14      4   9    4 9    5 10      17      19  
│ │ │ +      T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, T T  - z*T   + y*T  ),
│ │ │ +       8      2 7    4 8      12      14      3   9    3 9      15      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  + y*T  ), ({T , T }, - T T  - T T  - T T  +
│ │ │ -         2   8      2 8    5 9      15      5   7      3 6    5 7    1 8  
│ │ │ +      ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, - T T  +
│ │ │ +         3   6      3 6    5 7    1 8      12      14      1   7      1 7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ -         12      14      4   9    2 6    3 8    4 9      14      17      3 
│ │ │ +      x*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T  }, - T T  + T T   -
│ │ │ +         13      5   9      2 8    5 9      15      3   10      4 8    3 10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T  }, T T  + T T   - z*T   + y*T  ), ({T , T }, T T  + T T   - z*T   +
│ │ │ -       10    5 6    3 10      18      20      5   6    5 6    3 10      18  
│ │ │ +      z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, -
│ │ │ +         18      19      2   7      2 7    4 8      12      14      4   8    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T  }, - T T  - T T   + z*T   + z*T  ), ({T , T }, - T T 
│ │ │ -         20      4   10      5 7    4 10      12      20      1   6      1 6
│ │ │ +      T T  + T T   - z*T   + x*T  ), ({T , T  }, - T T  - T T   + y*T  ),
│ │ │ +       4 8    3 10      18      19      2   10      5 8    2 10      19  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T T  + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T , T },
│ │ │ -         4 7      11      3   9    5 8    3 9      15      16      4   6  
│ │ │ +      ({T , T  }, - T T   + x*T  ), ({T , T }, - T T  - T T   + y*T  ), ({T ,
│ │ │ +         4   10      4 10      18      5   8      5 8    2 10      19      3 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  - z*T   + y*T  )}
│ │ │ -       4 6    3 7      11      13
│ │ │ +      T }, T T  + x*T   - z*T  )}
│ │ │ +       8    3 8      15      17
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │  o16 = -1
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -191,82 +191,82 @@
│ │ │  
    		          
                  
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │      		          
                  
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T  }, - T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T ,
│ │ │ -          5   10      5 10      18      4   6    4 6    1 10      20      4 
│ │ │ +o15 = {({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  -
│ │ │ +          1   8      3 6    5 7    1 8      12      14      4   9    4 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   -
│ │ │ -       7      1 6    4 7      11      2   6    2 6    3 8    4 9      14  
│ │ │ +      T T   - z*T   + z*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T }, -
│ │ │ +       5 10      17      19      2   8      2 8    5 9      15      5   7    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │ -         17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │ +      T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  + T T  + T T  +
│ │ │ +       3 6    5 7    1 8      12      14      4   9    2 6    3 8    4 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │ -       3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │ +      y*T   - z*T  ), ({T , T  }, T T  + T T   - z*T   + y*T  ), ({T , T },
│ │ │ +         14      17      3   10    5 6    3 10      18      20      5   6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ -       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      T T  + T T   - z*T   + y*T  ), ({T , T  }, - T T  - T T   + z*T   +
│ │ │ +       5 6    3 10      18      20      4   10      5 7    4 10      12  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   +
│ │ │ -         13      2   9      2 9      16      1   9      5 6    1 9      14  
│ │ │ +      z*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  -
│ │ │ +         20      1   6      1 6    4 7      11      3   9    5 8    3 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T  ), ({T , T }, - T T  + z*T  ), ({T , T  }, T T  - T T   + x*T  ),
│ │ │ -         17      4   7      4 7      13      1   10    4 6    1 10      20  
│ │ │ +      z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T  }, -
│ │ │ +         15      16      4   6    4 6    3 7      11      13      5   10    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  + z*T  ), ({T , T }, T T  - z*T   + x*T  ), ({T ,
│ │ │ -         5   9      5 9      16      3   7    3 7      11      12      5 
│ │ │ +      T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T , T }, - T T  -
│ │ │ +       5 10      18      4   6    4 6    1 10      20      4   7      1 6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ -       6      5 6    1 9      14      17      5   8    5 8    3 9      15  
│ │ │ +      T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ +       4 7      11      2   6    2 6    3 8    4 9      14      17      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, T T  -
│ │ │ -         16      4   8      2 7    4 8      12      14      3   9    3 9  
│ │ │ +      T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  + T T  + T T  +
│ │ │ +       10    4 9    5 10      17      19      3   8    2 6    3 8    4 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   + y*T  ), ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T ,
│ │ │ -         15      17      3   6      3 6    5 7    1 8      12      14      1 
│ │ │ +      y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T , T }, - T T  -
│ │ │ +         14      17      3   6    3 6      11      12      5   7      5 7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  + x*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T  }, -
│ │ │ -       7      1 7      13      5   9      2 8    5 9      15      3   10    
│ │ │ +      T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ), ({T ,
│ │ │ +       4 10      12      20      3   7    4 6    3 7      11      13      2 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T   - z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   +
│ │ │ -       4 8    3 10      18      19      2   7      2 7    4 8      12  
│ │ │ +      T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T ,
│ │ │ +       9      2 9      16      1   9      5 6    1 9      14      17      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T  ), ({T , T }, - T T  + T T   - z*T   + x*T  ), ({T , T  }, - T T 
│ │ │ -         14      4   8      4 8    3 10      18      19      2   10      5 8
│ │ │ +      T }, - T T  + z*T  ), ({T , T  }, T T  - T T   + x*T  ), ({T , T }, -
│ │ │ +       7      4 7      13      1   10    4 6    1 10      20      5   9    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T T   + y*T  ), ({T , T  }, - T T   + x*T  ), ({T , T }, - T T  -
│ │ │ -         2 10      19      4   10      4 10      18      5   8      5 8  
│ │ │ +      T T  + z*T  ), ({T , T }, T T  - z*T   + x*T  ), ({T , T }, - T T  -
│ │ │ +       5 9      16      3   7    3 7      11      12      5   6      5 6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T   + y*T  ), ({T , T }, T T  + x*T   - z*T  ), ({T , T }, - T T  -
│ │ │ -       2 10      19      3   8    3 8      15      17      1   8      3 6  
│ │ │ +      T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T ,
│ │ │ +       1 9      14      17      5   8    5 8    3 9      15      16      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  - T T  + z*T   + x*T  ), ({T , T }, T T  - T T   - z*T   + z*T  ),
│ │ │ -       5 7    1 8      12      14      4   9    4 9    5 10      17      19  
│ │ │ +      T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, T T  - z*T   + y*T  ),
│ │ │ +       8      2 7    4 8      12      14      3   9    3 9      15      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  + y*T  ), ({T , T }, - T T  - T T  - T T  +
│ │ │ -         2   8      2 8    5 9      15      5   7      3 6    5 7    1 8  
│ │ │ +      ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, - T T  +
│ │ │ +         3   6      3 6    5 7    1 8      12      14      1   7      1 7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ -         12      14      4   9    2 6    3 8    4 9      14      17      3 
│ │ │ +      x*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T  }, - T T  + T T   -
│ │ │ +         13      5   9      2 8    5 9      15      3   10      4 8    3 10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T  }, T T  + T T   - z*T   + y*T  ), ({T , T }, T T  + T T   - z*T   +
│ │ │ -       10    5 6    3 10      18      20      5   6    5 6    3 10      18  
│ │ │ +      z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, -
│ │ │ +         18      19      2   7      2 7    4 8      12      14      4   8    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T  }, - T T  - T T   + z*T   + z*T  ), ({T , T }, - T T 
│ │ │ -         20      4   10      5 7    4 10      12      20      1   6      1 6
│ │ │ +      T T  + T T   - z*T   + x*T  ), ({T , T  }, - T T  - T T   + y*T  ),
│ │ │ +       4 8    3 10      18      19      2   10      5 8    2 10      19  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T T  + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T , T },
│ │ │ -         4 7      11      3   9    5 8    3 9      15      16      4   6  
│ │ │ +      ({T , T  }, - T T   + x*T  ), ({T , T }, - T T  - T T   + y*T  ), ({T ,
│ │ │ +         4   10      4 10      18      5   8      5 8    2 10      19      3 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  - z*T   + y*T  )}
│ │ │ -       4 6    3 7      11      13
│ │ │ +      T }, T T  + x*T   - z*T  )}
│ │ │ +       8    3 8      15      17
│ │ │  
│ │ │  o15 : List
    
│ │ │      		          
                  
    i16 : H#(H'_0)
│ │ │  
│ │ │  o16 = -1
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,82 +119,82 @@
│ │ │ │  37   38   39   40   41   42   43   44
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T , T  }, - T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T ,
│ │ │ │ -          5   10      5 10      18      4   6    4 6    1 10      20      4
│ │ │ │ +o15 = {({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  -
│ │ │ │ +          1   8      3 6    5 7    1 8      12      14      4   9    4 9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   -
│ │ │ │ -       7      1 6    4 7      11      2   6    2 6    3 8    4 9      14
│ │ │ │ +      T T   - z*T   + z*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T }, -
│ │ │ │ +       5 10      17      19      2   8      2 8    5 9      15      5   7
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │ │ -         17      5   10    4 9    5 10      17      19      3   8    2 6
│ │ │ │ +      T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  + T T  + T T  +
│ │ │ │ +       3 6    5 7    1 8      12      14      4   9    2 6    3 8    4 9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │ │ -       3 8    4 9      14      17      3   6    3 6      11      12      5
│ │ │ │ +      y*T   - z*T  ), ({T , T  }, T T  + T T   - z*T   + y*T  ), ({T , T },
│ │ │ │ +         14      17      3   10    5 6    3 10      18      20      5   6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ │ -       7      5 7    4 10      12      20      3   7    4 6    3 7      11
│ │ │ │ +      T T  + T T   - z*T   + y*T  ), ({T , T  }, - T T  - T T   + z*T   +
│ │ │ │ +       5 6    3 10      18      20      4   10      5 7    4 10      12
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T  ), ({T , T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   +
│ │ │ │ -         13      2   9      2 9      16      1   9      5 6    1 9      14
│ │ │ │ +      z*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  -
│ │ │ │ +         20      1   6      1 6    4 7      11      3   9    5 8    3 9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T  ), ({T , T }, - T T  + z*T  ), ({T , T  }, T T  - T T   + x*T  ),
│ │ │ │ -         17      4   7      4 7      13      1   10    4 6    1 10      20
│ │ │ │ +      z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T  }, -
│ │ │ │ +         15      16      4   6    4 6    3 7      11      13      5   10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  + z*T  ), ({T , T }, T T  - z*T   + x*T  ), ({T ,
│ │ │ │ -         5   9      5 9      16      3   7    3 7      11      12      5
│ │ │ │ +      T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T , T }, - T T  -
│ │ │ │ +       5 10      18      4   6    4 6    1 10      20      4   7      1 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ │ -       6      5 6    1 9      14      17      5   8    5 8    3 9      15
│ │ │ │ +      T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ │ +       4 7      11      2   6    2 6    3 8    4 9      14      17      5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, T T  -
│ │ │ │ -         16      4   8      2 7    4 8      12      14      3   9    3 9
│ │ │ │ +      T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  + T T  + T T  +
│ │ │ │ +       10    4 9    5 10      17      19      3   8    2 6    3 8    4 9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   + y*T  ), ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T ,
│ │ │ │ -         15      17      3   6      3 6    5 7    1 8      12      14      1
│ │ │ │ +      y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T , T }, - T T  -
│ │ │ │ +         14      17      3   6    3 6      11      12      5   7      5 7
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  + x*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T  }, -
│ │ │ │ -       7      1 7      13      5   9      2 8    5 9      15      3   10
│ │ │ │ +      T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ), ({T ,
│ │ │ │ +       4 10      12      20      3   7    4 6    3 7      11      13      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  + T T   - z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   +
│ │ │ │ -       4 8    3 10      18      19      2   7      2 7    4 8      12
│ │ │ │ +      T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T ,
│ │ │ │ +       9      2 9      16      1   9      5 6    1 9      14      17      4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T  ), ({T , T }, - T T  + T T   - z*T   + x*T  ), ({T , T  }, - T T
│ │ │ │ -         14      4   8      4 8    3 10      18      19      2   10      5 8
│ │ │ │ +      T }, - T T  + z*T  ), ({T , T  }, T T  - T T   + x*T  ), ({T , T }, -
│ │ │ │ +       7      4 7      13      1   10    4 6    1 10      20      5   9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      - T T   + y*T  ), ({T , T  }, - T T   + x*T  ), ({T , T }, - T T  -
│ │ │ │ -         2 10      19      4   10      4 10      18      5   8      5 8
│ │ │ │ +      T T  + z*T  ), ({T , T }, T T  - z*T   + x*T  ), ({T , T }, - T T  -
│ │ │ │ +       5 9      16      3   7    3 7      11      12      5   6      5 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T   + y*T  ), ({T , T }, T T  + x*T   - z*T  ), ({T , T }, - T T  -
│ │ │ │ -       2 10      19      3   8    3 8      15      17      1   8      3 6
│ │ │ │ +      T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T ,
│ │ │ │ +       1 9      14      17      5   8    5 8    3 9      15      16      4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  - T T  + z*T   + x*T  ), ({T , T }, T T  - T T   - z*T   + z*T  ),
│ │ │ │ -       5 7    1 8      12      14      4   9    4 9    5 10      17      19
│ │ │ │ +      T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, T T  - z*T   + y*T  ),
│ │ │ │ +       8      2 7    4 8      12      14      3   9    3 9      15      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  + y*T  ), ({T , T }, - T T  - T T  - T T  +
│ │ │ │ -         2   8      2 8    5 9      15      5   7      3 6    5 7    1 8
│ │ │ │ +      ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, - T T  +
│ │ │ │ +         3   6      3 6    5 7    1 8      12      14      1   7      1 7
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ │ -         12      14      4   9    2 6    3 8    4 9      14      17      3
│ │ │ │ +      x*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T  }, - T T  + T T   -
│ │ │ │ +         13      5   9      2 8    5 9      15      3   10      4 8    3 10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T  }, T T  + T T   - z*T   + y*T  ), ({T , T }, T T  + T T   - z*T   +
│ │ │ │ -       10    5 6    3 10      18      20      5   6    5 6    3 10      18
│ │ │ │ +      z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, -
│ │ │ │ +         18      19      2   7      2 7    4 8      12      14      4   8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T  ), ({T , T  }, - T T  - T T   + z*T   + z*T  ), ({T , T }, - T T
│ │ │ │ -         20      4   10      5 7    4 10      12      20      1   6      1 6
│ │ │ │ +      T T  + T T   - z*T   + x*T  ), ({T , T  }, - T T  - T T   + y*T  ),
│ │ │ │ +       4 8    3 10      18      19      2   10      5 8    2 10      19
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      - T T  + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T , T },
│ │ │ │ -         4 7      11      3   9    5 8    3 9      15      16      4   6
│ │ │ │ +      ({T , T  }, - T T   + x*T  ), ({T , T }, - T T  - T T   + y*T  ), ({T ,
│ │ │ │ +         4   10      4 10      18      5   8      5 8    2 10      19      3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  + T T  - z*T   + y*T  )}
│ │ │ │ -       4 6    3 7      11      13
│ │ │ │ +      T }, T T  + x*T   - z*T  )}
│ │ │ │ +       8    3 8      15      17
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │  o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T ..T  ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  cmVzdHJpY3Rpb24oQXJyYW5nZW1lbnQsTGlzdCk=
│ │ │  #:len=343
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVzdHJpY3Rpb24sQXJyYW5nZW1lbnQsTGlzdCks
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aW50ZWdyYWxDbG9zdXJlKC4uLixMaW1pdD0+Li4uKQ==
│ │ │  #:len=1120
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZG8gYSBwYXJ0aWFsIGludGVncmFsIGNs
│ │ │  b3N1cmUiLCAibGluZW51bSIgPT4gMTQyOCwgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  i3 : R = S/f
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : time R' = integralClosure R
│ │ │ - -- used 0.691658s (cpu); 0.373649s (thread); 0s (gc)
│ │ │ + -- used 0.887862s (cpu); 0.532821s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -83,15 +83,15 @@
│ │ │  i9 : R = S/f
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.599848s (cpu); 0.317903s (thread); 0s (gc)
│ │ │ + -- used 0.833052s (cpu); 0.454312s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -150,15 +150,15 @@
│ │ │  i15 : R = S/f
│ │ │  
│ │ │  o15 = R
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │  
│ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.692691s (cpu); 0.349777s (thread); 0s (gc)
│ │ │ + -- used 0.769119s (cpu); 0.488883s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -208,15 +208,15 @@
│ │ │  i20 : R = S/f
│ │ │  
│ │ │  o20 = R
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │  
│ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.876864s (cpu); 0.442627s (thread); 0s (gc)
│ │ │ + -- used 1.13444s (cpu); 0.670078s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.23059s (cpu); 0.60134s (thread); 0s (gc)
│ │ │ + -- used 1.58482s (cpu); 0.939924s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
│ │ │  
│ │ │  i27 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -324,15 +324,15 @@
│ │ │  i30 : R = S/f
│ │ │  
│ │ │  o30 = R
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │  
│ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.817042s (cpu); 0.367945s (thread); 0s (gc)
│ │ │ + -- used 1.04905s (cpu); 0.502139s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
│ │ │  
│ │ │  i32 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -382,15 +382,15 @@
│ │ │  i35 : R = S/f
│ │ │  
│ │ │  o35 = R
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │  
│ │ │  i36 : time R' = integralClosure R
│ │ │ - -- used 0.139028s (cpu); 0.0626647s (thread); 0s (gc)
│ │ │ + -- used 0.158146s (cpu); 0.0932663s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
│ │ │  
│ │ │  i37 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -432,15 +432,15 @@
│ │ │  i40 : R = S/I
│ │ │  
│ │ │  o40 = R
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │  
│ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.137365s (cpu); 0.0615872s (thread); 0s (gc)
│ │ │ + -- used 0.150705s (cpu); 0.0908567s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
│ │ │  
│ │ │  i42 : icFractions R
│ │ │  
│ │ │ @@ -467,15 +467,15 @@
│ │ │  i45 : R = S/I
│ │ │  
│ │ │  o45 = R
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │  
│ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.167495s (cpu); 0.0877408s (thread); 0s (gc)
│ │ │ + -- used 0.13204s (cpu); 0.0681967s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
│ │ │  
│ │ │  i47 : icFractions R
│ │ │  
│ │ │ @@ -501,15 +501,15 @@
│ │ │  i50 : R = S/I
│ │ │  
│ │ │  o50 = R
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │  
│ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.139863s (cpu); 0.0633843s (thread); 0s (gc)
│ │ │ + -- used 0.186921s (cpu); 0.119854s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
│ │ │  
│ │ │  i52 : icFractions R
│ │ │  
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i55 : R = S/I
│ │ │  
│ │ │  o55 = R
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │  
│ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.145503s (cpu); 0.060919s (thread); 0s (gc)
│ │ │ + -- used 0.21552s (cpu); 0.154823s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -633,15 +633,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.177738s (cpu); 0.0898242s (thread); 0s (gc)
│ │ │ + -- used 0.224831s (cpu); 0.159891s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -722,15 +722,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500907s (cpu); 0.357025s (thread); 0s (gc)
│ │ │ + -- used 0.62228s (cpu); 0.499672s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -750,15 +750,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.413444s (cpu); 0.259042s (thread); 0s (gc)
│ │ │ + -- used 0.731709s (cpu); 0.40774s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -778,20 +778,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time 0 sec #minors 4]
│ │ │ + [jacobian time .00215067 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0923998 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.313375s (cpu); 0.230372s (thread); 0s (gc)
│ │ │ -  time .216976 sec  #fractions 6]
│ │ │ + [step 0:   time .191864 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.724639s (cpu); 0.409971s (thread); 0s (gc)
│ │ │ +  time .524771 sec  #fractions 6]
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -814,17 +814,17 @@
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .213809 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.494127s (cpu); 0.320163s (thread); 0s (gc)
│ │ │ -  time .276321 sec  #fractions 6]
│ │ │ + [step 0:   time .574994 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.952548s (cpu); 0.446174s (thread); 0s (gc)
│ │ │ +  time .373652 sec  #fractions 6]
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -847,17 +847,17 @@
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .230136 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.748048s (cpu); 0.513758s (thread); 0s (gc)
│ │ │ -  time .513497 sec  #fractions 6]
│ │ │ + [step 0:   time .327969 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.956361s (cpu); 0.733282s (thread); 0s (gc)
│ │ │ +  time .624615 sec  #fractions 6]
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
│ │ │  
│ │ │  i95 : icFractions R
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
│ │ │ @@ -1,50 +1,50 @@
│ │ │  -- -*- M2-comint -*- hash: 13177954069434615273
│ │ │  
│ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │  
│ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .00369747 sec #minors 3]
│ │ │ + [jacobian time 0 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │        radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .00796539 seconds
│ │ │ -      idlizer2:  .0130686 seconds
│ │ │ -      minpres:   .011545 seconds
│ │ │ -  time .12558 sec  #fractions 4]
│ │ │ +      idlizer1:  .00798892 seconds
│ │ │ +      idlizer2:  .00800094 seconds
│ │ │ +      minpres:   .00799965 seconds
│ │ │ +  time .0981859 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00400043 seconds
│ │ │ -      idlizer1:  .016817 seconds
│ │ │ -      idlizer2:  .0846016 seconds
│ │ │ -      minpres:   .0119995 seconds
│ │ │ -  time .131259 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .0039999 seconds
│ │ │ +      idlizer1:  .060909 seconds
│ │ │ +      idlizer2:  .0701711 seconds
│ │ │ +      minpres:   .00799908 seconds
│ │ │ +  time .151081 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │        radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0119982 seconds
│ │ │ -      idlizer2:  .0120027 seconds
│ │ │ -      minpres:   .0120025 seconds
│ │ │ -  time .126433 sec  #fractions 5]
│ │ │ +      idlizer1:  .011998 seconds
│ │ │ +      idlizer2:  .00800113 seconds
│ │ │ +      minpres:   .0568902 seconds
│ │ │ +  time .205486 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0120008 seconds
│ │ │ -      idlizer2:  .0159986 seconds
│ │ │ -      minpres:   .102383 seconds
│ │ │ -  time .146386 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00400008 seconds
│ │ │ +      idlizer1:  .0120016 seconds
│ │ │ +      idlizer2:  .0137195 seconds
│ │ │ +      minpres:   .0790716 seconds
│ │ │ +  time .119053 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .00800125 seconds
│ │ │ -      idlizer2:  .0159992 seconds
│ │ │ -      minpres:   .0119989 seconds
│ │ │ -  time .0559994 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00400077 seconds
│ │ │ +      idlizer1:  .0093878 seconds
│ │ │ +      idlizer2:  .0639988 seconds
│ │ │ +      minpres:   .0120029 seconds
│ │ │ +  time .0973903 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00399864 seconds
│ │ │ -      idlizer1:   -- used 0.692546s (cpu); 0.376483s (thread); 0s (gc)
│ │ │ -.00665112 seconds
│ │ │ -  time .100933 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0409253 seconds
│ │ │ +      idlizer1:   -- used 0.788996s (cpu); 0.51748s (thread); 0s (gc)
│ │ │ +.00400008 seconds
│ │ │ +  time .1138 sec  #fractions 5]
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 1.61913s (cpu); 0.70672s (thread); 0s (gc)
│ │ │ + -- used 3.78342s (cpu); 1.1197s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.940472s (cpu); 0.475156s (thread); 0s (gc)
│ │ │ + -- used 2.58346s (cpu); 0.795824s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │      		          
                  
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.691658s (cpu); 0.373649s (thread); 0s (gc)
│ │ │ + -- used 0.887862s (cpu); 0.532821s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │      		          
                  
    i5 : netList (ideal R')_*
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │      		          
                  
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.599848s (cpu); 0.317903s (thread); 0s (gc)
│ │ │ + -- used 0.833052s (cpu); 0.454312s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │      		          
                  
    i11 : netList (ideal R')_*
│ │ │ @@ -240,15 +240,15 @@
│ │ │  
│ │ │  o15 = R
│ │ │  
│ │ │  o15 : QuotientRing
    
│ │ │      		          
                  
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.692691s (cpu); 0.349777s (thread); 0s (gc)
│ │ │ + -- used 0.769119s (cpu); 0.488883s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │      		          
                  
    i17 : netList (ideal R')_*
│ │ │ @@ -305,15 +305,15 @@
│ │ │  
│ │ │  o20 = R
│ │ │  
│ │ │  o20 : QuotientRing
    
│ │ │      		          
                  
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.876864s (cpu); 0.442627s (thread); 0s (gc)
│ │ │ + -- used 1.13444s (cpu); 0.670078s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    
│ │ │      		          
                  
    i22 : netList (ideal R')_*
│ │ │ @@ -370,15 +370,15 @@
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
    
│ │ │      		          
                  
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.23059s (cpu); 0.60134s (thread); 0s (gc)
│ │ │ + -- used 1.58482s (cpu); 0.939924s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    
│ │ │      		          
                  
    i27 : netList (ideal R')_*
│ │ │ @@ -435,15 +435,15 @@
│ │ │  
│ │ │  o30 = R
│ │ │  
│ │ │  o30 : QuotientRing
    
│ │ │      		          
                  
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.817042s (cpu); 0.367945s (thread); 0s (gc)
│ │ │ + -- used 1.04905s (cpu); 0.502139s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    
│ │ │      		          
                  
    i32 : netList (ideal R')_*
│ │ │ @@ -500,15 +500,15 @@
│ │ │  
│ │ │  o35 = R
│ │ │  
│ │ │  o35 : QuotientRing
    
│ │ │      		          
                  
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.139028s (cpu); 0.0626647s (thread); 0s (gc)
│ │ │ + -- used 0.158146s (cpu); 0.0932663s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    
│ │ │      		          
                  
    i37 : netList (ideal R')_*
│ │ │ @@ -560,15 +560,15 @@
│ │ │  
│ │ │  o40 = R
│ │ │  
│ │ │  o40 : QuotientRing
    
│ │ │      		          
                  
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.137365s (cpu); 0.0615872s (thread); 0s (gc)
│ │ │ + -- used 0.150705s (cpu); 0.0908567s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    
│ │ │      		          
                  
    i42 : icFractions R
│ │ │ @@ -602,15 +602,15 @@
│ │ │  
│ │ │  o45 = R
│ │ │  
│ │ │  o45 : QuotientRing
    
│ │ │      		          
                  
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.167495s (cpu); 0.0877408s (thread); 0s (gc)
│ │ │ + -- used 0.13204s (cpu); 0.0681967s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    
│ │ │      		          
                  
    i47 : icFractions R
│ │ │ @@ -643,15 +643,15 @@
│ │ │  
│ │ │  o50 = R
│ │ │  
│ │ │  o50 : QuotientRing
    
│ │ │      		          
                  
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.139863s (cpu); 0.0633843s (thread); 0s (gc)
│ │ │ + -- used 0.186921s (cpu); 0.119854s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    
│ │ │      		          
                  
    i52 : icFractions R
│ │ │ @@ -685,15 +685,15 @@
│ │ │  
│ │ │  o55 = R
│ │ │  
│ │ │  o55 : QuotientRing
    
│ │ │      		          
                  
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.145503s (cpu); 0.060919s (thread); 0s (gc)
│ │ │ + -- used 0.21552s (cpu); 0.154823s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    
│ │ │      		          
                  
    i57 : icFractions R
│ │ │ @@ -798,15 +798,15 @@
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
    
│ │ │      		          
                  
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.177738s (cpu); 0.0898242s (thread); 0s (gc)
│ │ │ + -- used 0.224831s (cpu); 0.159891s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    
│ │ │      		          
                  
    i68 : icFractions R
│ │ │ @@ -900,15 +900,15 @@
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
    
│ │ │      		          
                  
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500907s (cpu); 0.357025s (thread); 0s (gc)
│ │ │ + -- used 0.62228s (cpu); 0.499672s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    
│ │ │      		          
                  
    i79 : icFractions R
│ │ │ @@ -934,15 +934,15 @@
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
    
│ │ │      		          
                  
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.413444s (cpu); 0.259042s (thread); 0s (gc)
│ │ │ + -- used 0.731709s (cpu); 0.40774s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    
│ │ │      		          
                  
    i83 : icFractions R
│ │ │ @@ -968,20 +968,20 @@
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
    
│ │ │      		          
                  
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time 0 sec #minors 4]
│ │ │ + [jacobian time .00215067 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0923998 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.313375s (cpu); 0.230372s (thread); 0s (gc)
│ │ │ -  time .216976 sec  #fractions 6]
│ │ │ + [step 0:   time .191864 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.724639s (cpu); 0.409971s (thread); 0s (gc)
│ │ │ +  time .524771 sec  #fractions 6]
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    
│ │ │      		          
                  
    i87 : icFractions R
│ │ │ @@ -1010,17 +1010,17 @@
│ │ │  o89 : QuotientRing
    
│ │ │      		          
                  
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .213809 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.494127s (cpu); 0.320163s (thread); 0s (gc)
│ │ │ -  time .276321 sec  #fractions 6]
│ │ │ + [step 0:   time .574994 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.952548s (cpu); 0.446174s (thread); 0s (gc)
│ │ │ +  time .373652 sec  #fractions 6]
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    
│ │ │      		          
                  
    i91 : icFractions R
│ │ │ @@ -1052,17 +1052,17 @@
│ │ │  o93 : QuotientRing
    
│ │ │      		          
                  
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .230136 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.748048s (cpu); 0.513758s (thread); 0s (gc)
│ │ │ -  time .513497 sec  #fractions 6]
│ │ │ + [step 0:   time .327969 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.956361s (cpu); 0.733282s (thread); 0s (gc)
│ │ │ +  time .624615 sec  #fractions 6]
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
    
│ │ │      		          
                  
    i95 : icFractions R
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o2 : Ideal of S
│ │ │ │  i3 : R = S/f
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : time R' = integralClosure R
│ │ │ │ - -- used 0.691658s (cpu); 0.373649s (thread); 0s (gc)
│ │ │ │ + -- used 0.887862s (cpu); 0.532821s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = R'
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : netList (ideal R')_*
│ │ │ │  
│ │ │ │       +------------------------------------------------------------------------+
│ │ │ │ @@ -110,15 +110,15 @@
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : R = S/f
│ │ │ │  
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.599848s (cpu); 0.317903s (thread); 0s (gc)
│ │ │ │ + -- used 0.833052s (cpu); 0.454312s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = R'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -200,15 +200,15 @@
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : R = S/f
│ │ │ │  
│ │ │ │  o15 = R
│ │ │ │  
│ │ │ │  o15 : QuotientRing
│ │ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.692691s (cpu); 0.349777s (thread); 0s (gc)
│ │ │ │ + -- used 0.769119s (cpu); 0.488883s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = R'
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  i17 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -282,15 +282,15 @@
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : R = S/f
│ │ │ │  
│ │ │ │  o20 = R
│ │ │ │  
│ │ │ │  o20 : QuotientRing
│ │ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ │ - -- used 0.876864s (cpu); 0.442627s (thread); 0s (gc)
│ │ │ │ + -- used 1.13444s (cpu); 0.670078s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = R'
│ │ │ │  
│ │ │ │  o21 : QuotientRing
│ │ │ │  i22 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -364,15 +364,15 @@
│ │ │ │  o24 : Ideal of S
│ │ │ │  i25 : R = S/f
│ │ │ │  
│ │ │ │  o25 = R
│ │ │ │  
│ │ │ │  o25 : QuotientRing
│ │ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 1.23059s (cpu); 0.60134s (thread); 0s (gc)
│ │ │ │ + -- used 1.58482s (cpu); 0.939924s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o26 = R'
│ │ │ │  
│ │ │ │  o26 : QuotientRing
│ │ │ │  i27 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -446,15 +446,15 @@
│ │ │ │  o29 : Ideal of S
│ │ │ │  i30 : R = S/f
│ │ │ │  
│ │ │ │  o30 = R
│ │ │ │  
│ │ │ │  o30 : QuotientRing
│ │ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.817042s (cpu); 0.367945s (thread); 0s (gc)
│ │ │ │ + -- used 1.04905s (cpu); 0.502139s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = R'
│ │ │ │  
│ │ │ │  o31 : QuotientRing
│ │ │ │  i32 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -528,15 +528,15 @@
│ │ │ │  o34 : Ideal of S
│ │ │ │  i35 : R = S/f
│ │ │ │  
│ │ │ │  o35 = R
│ │ │ │  
│ │ │ │  o35 : QuotientRing
│ │ │ │  i36 : time R' = integralClosure R
│ │ │ │ - -- used 0.139028s (cpu); 0.0626647s (thread); 0s (gc)
│ │ │ │ + -- used 0.158146s (cpu); 0.0932663s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o36 = R'
│ │ │ │  
│ │ │ │  o36 : QuotientRing
│ │ │ │  i37 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +-----------+
│ │ │ │ @@ -574,15 +574,15 @@
│ │ │ │  o39 : Ideal of S
│ │ │ │  i40 : R = S/I
│ │ │ │  
│ │ │ │  o40 = R
│ │ │ │  
│ │ │ │  o40 : QuotientRing
│ │ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.137365s (cpu); 0.0615872s (thread); 0s (gc)
│ │ │ │ + -- used 0.150705s (cpu); 0.0908567s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o41 = R'
│ │ │ │  
│ │ │ │  o41 : QuotientRing
│ │ │ │  i42 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -604,15 +604,15 @@
│ │ │ │  o44 : Ideal of S
│ │ │ │  i45 : R = S/I
│ │ │ │  
│ │ │ │  o45 = R
│ │ │ │  
│ │ │ │  o45 : QuotientRing
│ │ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.167495s (cpu); 0.0877408s (thread); 0s (gc)
│ │ │ │ + -- used 0.13204s (cpu); 0.0681967s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = R'
│ │ │ │  
│ │ │ │  o46 : QuotientRing
│ │ │ │  i47 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -633,15 +633,15 @@
│ │ │ │  o49 : Ideal of S
│ │ │ │  i50 : R = S/I
│ │ │ │  
│ │ │ │  o50 = R
│ │ │ │  
│ │ │ │  o50 : QuotientRing
│ │ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 0.139863s (cpu); 0.0633843s (thread); 0s (gc)
│ │ │ │ + -- used 0.186921s (cpu); 0.119854s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = R'
│ │ │ │  
│ │ │ │  o51 : QuotientRing
│ │ │ │  i52 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -663,15 +663,15 @@
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : R = S/I
│ │ │ │  
│ │ │ │  o55 = R
│ │ │ │  
│ │ │ │  o55 : QuotientRing
│ │ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.145503s (cpu); 0.060919s (thread); 0s (gc)
│ │ │ │ + -- used 0.21552s (cpu); 0.154823s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o56 = R'
│ │ │ │  
│ │ │ │  o56 : QuotientRing
│ │ │ │  i57 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -756,15 +756,15 @@
│ │ │ │  o65 : BettiTally
│ │ │ │  i66 : R = S/I
│ │ │ │  
│ │ │ │  o66 = R
│ │ │ │  
│ │ │ │  o66 : QuotientRing
│ │ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.177738s (cpu); 0.0898242s (thread); 0s (gc)
│ │ │ │ + -- used 0.224831s (cpu); 0.159891s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o67 = R'
│ │ │ │  
│ │ │ │  o67 : QuotientRing
│ │ │ │  i68 : icFractions R
│ │ │ │  
│ │ │ │          2    2
│ │ │ │ @@ -840,15 +840,15 @@
│ │ │ │  o76 : BettiTally
│ │ │ │  i77 : R = S/I
│ │ │ │  
│ │ │ │  o77 = R
│ │ │ │  
│ │ │ │  o77 : QuotientRing
│ │ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.500907s (cpu); 0.357025s (thread); 0s (gc)
│ │ │ │ + -- used 0.62228s (cpu); 0.499672s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o78 = R'
│ │ │ │  
│ │ │ │  o78 : QuotientRing
│ │ │ │  i79 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -864,15 +864,15 @@
│ │ │ │  o80 : PolynomialRing
│ │ │ │  i81 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o81 = R
│ │ │ │  
│ │ │ │  o81 : QuotientRing
│ │ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.413444s (cpu); 0.259042s (thread); 0s (gc)
│ │ │ │ + -- used 0.731709s (cpu); 0.40774s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o82 = R'
│ │ │ │  
│ │ │ │  o82 : QuotientRing
│ │ │ │  i83 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -888,20 +888,20 @@
│ │ │ │  o84 : PolynomialRing
│ │ │ │  i85 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o85 = R
│ │ │ │  
│ │ │ │  o85 : QuotientRing
│ │ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ │ - [jacobian time 0 sec #minors 4]
│ │ │ │ + [jacobian time .00215067 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .0923998 sec  #fractions 6]
│ │ │ │ - [step 1:  -- used 0.313375s (cpu); 0.230372s (thread); 0s (gc)
│ │ │ │ -  time .216976 sec  #fractions 6]
│ │ │ │ + [step 0:   time .191864 sec  #fractions 6]
│ │ │ │ + [step 1:  -- used 0.724639s (cpu); 0.409971s (thread); 0s (gc)
│ │ │ │ +  time .524771 sec  #fractions 6]
│ │ │ │  
│ │ │ │  o86 = R'
│ │ │ │  
│ │ │ │  o86 : QuotientRing
│ │ │ │  i87 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -920,17 +920,17 @@
│ │ │ │  o89 = R
│ │ │ │  
│ │ │ │  o89 : QuotientRing
│ │ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ │   [jacobian time 0 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .213809 sec  #fractions 6]
│ │ │ │ - [step 1:  -- used 0.494127s (cpu); 0.320163s (thread); 0s (gc)
│ │ │ │ -  time .276321 sec  #fractions 6]
│ │ │ │ + [step 0:   time .574994 sec  #fractions 6]
│ │ │ │ + [step 1:  -- used 0.952548s (cpu); 0.446174s (thread); 0s (gc)
│ │ │ │ +  time .373652 sec  #fractions 6]
│ │ │ │  
│ │ │ │  o90 = R'
│ │ │ │  
│ │ │ │  o90 : QuotientRing
│ │ │ │  i91 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -952,17 +952,17 @@
│ │ │ │  
│ │ │ │  o93 : QuotientRing
│ │ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos,
│ │ │ │  StartWithOneMinor}, Verbosity => 1)
│ │ │ │   [jacobian time 0 sec #minors 1]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .230136 sec  #fractions 6]
│ │ │ │ - [step 1:  -- used 0.748048s (cpu); 0.513758s (thread); 0s (gc)
│ │ │ │ -  time .513497 sec  #fractions 6]
│ │ │ │ + [step 0:   time .327969 sec  #fractions 6]
│ │ │ │ + [step 1:  -- used 0.956361s (cpu); 0.733282s (thread); 0s (gc)
│ │ │ │ +  time .624615 sec  #fractions 6]
│ │ │ │  
│ │ │ │  o94 = R'
│ │ │ │  
│ │ │ │  o94 : QuotientRing
│ │ │ │  i95 : icFractions R
│ │ │ │  
│ │ │ │           2     2          2   2     3      2
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html
│ │ │ @@ -66,52 +66,52 @@
│ │ │          
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    
│ │ │      		          
                  
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .00369747 sec #minors 3]
│ │ │ + [jacobian time 0 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │        radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .00796539 seconds
│ │ │ -      idlizer2:  .0130686 seconds
│ │ │ -      minpres:   .011545 seconds
│ │ │ -  time .12558 sec  #fractions 4]
│ │ │ +      idlizer1:  .00798892 seconds
│ │ │ +      idlizer2:  .00800094 seconds
│ │ │ +      minpres:   .00799965 seconds
│ │ │ +  time .0981859 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00400043 seconds
│ │ │ -      idlizer1:  .016817 seconds
│ │ │ -      idlizer2:  .0846016 seconds
│ │ │ -      minpres:   .0119995 seconds
│ │ │ -  time .131259 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .0039999 seconds
│ │ │ +      idlizer1:  .060909 seconds
│ │ │ +      idlizer2:  .0701711 seconds
│ │ │ +      minpres:   .00799908 seconds
│ │ │ +  time .151081 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │        radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0119982 seconds
│ │ │ -      idlizer2:  .0120027 seconds
│ │ │ -      minpres:   .0120025 seconds
│ │ │ -  time .126433 sec  #fractions 5]
│ │ │ +      idlizer1:  .011998 seconds
│ │ │ +      idlizer2:  .00800113 seconds
│ │ │ +      minpres:   .0568902 seconds
│ │ │ +  time .205486 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0120008 seconds
│ │ │ -      idlizer2:  .0159986 seconds
│ │ │ -      minpres:   .102383 seconds
│ │ │ -  time .146386 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00400008 seconds
│ │ │ +      idlizer1:  .0120016 seconds
│ │ │ +      idlizer2:  .0137195 seconds
│ │ │ +      minpres:   .0790716 seconds
│ │ │ +  time .119053 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .00800125 seconds
│ │ │ -      idlizer2:  .0159992 seconds
│ │ │ -      minpres:   .0119989 seconds
│ │ │ -  time .0559994 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00400077 seconds
│ │ │ +      idlizer1:  .0093878 seconds
│ │ │ +      idlizer2:  .0639988 seconds
│ │ │ +      minpres:   .0120029 seconds
│ │ │ +  time .0973903 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00399864 seconds
│ │ │ -      idlizer1:   -- used 0.692546s (cpu); 0.376483s (thread); 0s (gc)
│ │ │ -.00665112 seconds
│ │ │ -  time .100933 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0409253 seconds
│ │ │ +      idlizer1:   -- used 0.788996s (cpu); 0.51748s (thread); 0s (gc)
│ │ │ +.00400008 seconds
│ │ │ +  time .1138 sec  #fractions 5]
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    
│ │ │      		          
                  
    i3 : trim ideal R'
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,52 +13,52 @@
│ │ │ │              displayed. A value of 0 means: keep quiet.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  When the computation takes a considerable time, this function can be used to
│ │ │ │  decide if it will ever finish, or to get a feel for what is happening during
│ │ │ │  the computation.
│ │ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ │ - [jacobian time .00369747 sec #minors 3]
│ │ │ │ + [jacobian time 0 sec #minors 3]
│ │ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │   [step 0:
│ │ │ │        radical (use minprimes) 0 seconds
│ │ │ │ -      idlizer1:  .00796539 seconds
│ │ │ │ -      idlizer2:  .0130686 seconds
│ │ │ │ -      minpres:   .011545 seconds
│ │ │ │ -  time .12558 sec  #fractions 4]
│ │ │ │ +      idlizer1:  .00798892 seconds
│ │ │ │ +      idlizer2:  .00800094 seconds
│ │ │ │ +      minpres:   .00799965 seconds
│ │ │ │ +  time .0981859 sec  #fractions 4]
│ │ │ │   [step 1:
│ │ │ │ -      radical (use minprimes) .00400043 seconds
│ │ │ │ -      idlizer1:  .016817 seconds
│ │ │ │ -      idlizer2:  .0846016 seconds
│ │ │ │ -      minpres:   .0119995 seconds
│ │ │ │ -  time .131259 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .0039999 seconds
│ │ │ │ +      idlizer1:  .060909 seconds
│ │ │ │ +      idlizer2:  .0701711 seconds
│ │ │ │ +      minpres:   .00799908 seconds
│ │ │ │ +  time .151081 sec  #fractions 4]
│ │ │ │   [step 2:
│ │ │ │        radical (use minprimes) 0 seconds
│ │ │ │ -      idlizer1:  .0119982 seconds
│ │ │ │ -      idlizer2:  .0120027 seconds
│ │ │ │ -      minpres:   .0120025 seconds
│ │ │ │ -  time .126433 sec  #fractions 5]
│ │ │ │ +      idlizer1:  .011998 seconds
│ │ │ │ +      idlizer2:  .00800113 seconds
│ │ │ │ +      minpres:   .0568902 seconds
│ │ │ │ +  time .205486 sec  #fractions 5]
│ │ │ │   [step 3:
│ │ │ │ -      radical (use minprimes) 0 seconds
│ │ │ │ -      idlizer1:  .0120008 seconds
│ │ │ │ -      idlizer2:  .0159986 seconds
│ │ │ │ -      minpres:   .102383 seconds
│ │ │ │ -  time .146386 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00400008 seconds
│ │ │ │ +      idlizer1:  .0120016 seconds
│ │ │ │ +      idlizer2:  .0137195 seconds
│ │ │ │ +      minpres:   .0790716 seconds
│ │ │ │ +  time .119053 sec  #fractions 5]
│ │ │ │   [step 4:
│ │ │ │ -      radical (use minprimes) 0 seconds
│ │ │ │ -      idlizer1:  .00800125 seconds
│ │ │ │ -      idlizer2:  .0159992 seconds
│ │ │ │ -      minpres:   .0119989 seconds
│ │ │ │ -  time .0559994 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00400077 seconds
│ │ │ │ +      idlizer1:  .0093878 seconds
│ │ │ │ +      idlizer2:  .0639988 seconds
│ │ │ │ +      minpres:   .0120029 seconds
│ │ │ │ +  time .0973903 sec  #fractions 5]
│ │ │ │   [step 5:
│ │ │ │ -      radical (use minprimes) .00399864 seconds
│ │ │ │ -      idlizer1:   -- used 0.692546s (cpu); 0.376483s (thread); 0s (gc)
│ │ │ │ -.00665112 seconds
│ │ │ │ -  time .100933 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .0409253 seconds
│ │ │ │ +      idlizer1:   -- used 0.788996s (cpu); 0.51748s (thread); 0s (gc)
│ │ │ │ +.00400008 seconds
│ │ │ │ +  time .1138 sec  #fractions 5]
│ │ │ │  
│ │ │ │  o2 = R'
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : trim ideal R'
│ │ │ │  
│ │ │ │                       3   2                     2 2    4           4
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html
│ │ │ @@ -111,27 +111,27 @@
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
    
│ │ │      		          
                  
    i4 : time integralClosure J
│ │ │ - -- used 1.61913s (cpu); 0.70672s (thread); 0s (gc)
│ │ │ + -- used 3.78342s (cpu); 1.1197s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │      		          
                  
    i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.940472s (cpu); 0.475156s (thread); 0s (gc)
│ │ │ + -- used 2.58346s (cpu); 0.795824s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ ├── html2text {}
│ │ │ │ @@ -47,25 +47,25 @@
│ │ │ │  i3 : J = ideal jacobian ideal F
│ │ │ │  
│ │ │ │                  2      2    2        2   2 2     2
│ │ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time integralClosure J
│ │ │ │ - -- used 1.61913s (cpu); 0.70672s (thread); 0s (gc)
│ │ │ │ + -- used 3.78342s (cpu); 1.1197s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ │ - -- used 0.940472s (cpu); 0.475156s (thread); 0s (gc)
│ │ │ │ + -- used 2.58346s (cpu); 0.795824s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c2NocmVpZXJHcmFwaA==
│ │ │  #:len=1712
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NocmVpZXIgZ3JhcGggb2YgYSBmaW5p
│ │ │  dGUgZ3JvdXAiLCAibGluZW51bSIgPT4gMjYxLCBJbnB1dHMgPT4ge1NQQU57VFR7IkcifSwiLCAi
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00215919s elapsed
│ │ │ + -- .00198847s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000634819s elapsed
│ │ │ + -- .000356685s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
│ │ │ @@ -23,23 +23,23 @@
│ │ │  o3 = QQ[x..z] <- {| 0 -1 0  |, | 0 -1 0 |}
│ │ │                    | 1 0  0  |  | 1 0  0 |
│ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.703448s (cpu); 0.455373s (thread); 0s (gc)
│ │ │ + -- used 1.03322s (cpu); 0.836639s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   3       3
│ │ │ -o4 = {z , x  + y , x y - x*y }
│ │ │ +       2    2   2   3       3
│ │ │ +o4 = {x  + y , z , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.639755s (cpu); 0.336773s (thread); 0s (gc)
│ │ │ + -- used 1.04482s (cpu); 0.702253s (thread); 0s (gc)
│ │ │  
│ │ │             8         7          6 2         5 3         4 4         3 5  
│ │ │  o5 = {4096x  + 24576x y + 38912x y  - 18432x y  - 65280x y  + 18432x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │             2 6           7        8         6 2         5   2         4 2 2  
│ │ │       38912x y  - 24576x*y  + 4096y  - 23040x z  - 69120x y*z  - 28800x y z  -
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -81,23 +81,23 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 2 6             2 6            8
│ │ │       348625296x z  - 348625296y z  + 43046721z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.019971s (cpu); 0.0214392s (thread); 0s (gc)
│ │ │ + -- used 0.0199973s (cpu); 0.0211987s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.83382s (cpu); 1.16479s (thread); 0s (gc)
│ │ │ + -- used 3.19982s (cpu); 2.26863s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .6125s elapsed
│ │ │ + -- .828077s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ - -- .634295s elapsed
│ │ │ + -- .675065s elapsed
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .629561s elapsed
│ │ │ + -- .740044s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .0410553s elapsed
│ │ │ + -- .0781188s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
                  
    i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
    
│ │ │      		          
                  
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00215919s elapsed
│ │ │ + -- .00198847s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -104,15 +104,15 @@
│ │ │            
                  
    i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
    
│ │ │      		          
                  
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000634819s elapsed
    
│ │ │ + -- .000356685s elapsed
│ │ │      		          
    
│ │ │        
│ │ │        
    
│ │ │          
    For the programmer
    
│ │ │          
    The object equivariantHilbert is a symbol.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │       | 0  -1 1 |
│ │ │ │  
│ │ │ │  o3 : DiagonalAction
│ │ │ │  i4 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ │ - -- .00215919s elapsed
│ │ │ │ + -- .00198847s elapsed
│ │ │ │  
│ │ │ │                    -1    -1       2 2              -2    -1 -1    -2  2
│ │ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │ │             0 1    1     0        0 1    0    1    1     0  1     0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T
│ │ │ │ @@ -54,10 +54,10 @@
│ │ │ │  
│ │ │ │  o5 : ZZ[z ..z ][T]
│ │ │ │           0   1
│ │ │ │  i6 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ │ - -- .000634819s elapsed
│ │ │ │ + -- .000356685s elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l.
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │  o3 : FiniteGroupAction
    
│ │ │      		          
    
│ │ │          
    The two algorithms used in primaryInvariants are timed. One sees that the Dade algorithm is faster, however the primary invariants output are all of degree 8 and have ugly coefficients.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.703448s (cpu); 0.455373s (thread); 0s (gc)
│ │ │ + -- used 1.03322s (cpu); 0.836639s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   3       3
│ │ │ -o4 = {z , x  + y , x y - x*y }
│ │ │ +       2    2   2   3       3
│ │ │ +o4 = {x  + y , z , x y - x*y }
│ │ │  
│ │ │  o4 : List
    
│ │ │      		          
                  
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.639755s (cpu); 0.336773s (thread); 0s (gc)
│ │ │ + -- used 1.04482s (cpu); 0.702253s (thread); 0s (gc)
│ │ │  
│ │ │             8         7          6 2         5 3         4 4         3 5  
│ │ │  o5 = {4096x  + 24576x y + 38912x y  - 18432x y  - 65280x y  + 18432x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │             2 6           7        8         6 2         5   2         4 2 2  
│ │ │       38912x y  - 24576x*y  + 4096y  - 23040x z  - 69120x y*z  - 28800x y z  -
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -141,24 +141,24 @@
│ │ │  o5 : List
    
│ │ │      		          
    
│ │ │          
    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.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.019971s (cpu); 0.0214392s (thread); 0s (gc)
│ │ │ + -- used 0.0199973s (cpu); 0.0211987s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    
│ │ │      		          
                  
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.83382s (cpu); 1.16479s (thread); 0s (gc)
│ │ │ + -- used 3.19982s (cpu); 2.26863s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  The two algorithms used in _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s are timed. One sees that the Dade
│ │ │ │  algorithm is faster, however the primary invariants output are all of degree 8
│ │ │ │  and have ugly coefficients.
│ │ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ │ - -- used 0.703448s (cpu); 0.455373s (thread); 0s (gc)
│ │ │ │ + -- used 1.03322s (cpu); 0.836639s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -       2   2    2   3       3
│ │ │ │ -o4 = {z , x  + y , x y - x*y }
│ │ │ │ +       2    2   2   3       3
│ │ │ │ +o4 = {x  + y , z , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.639755s (cpu); 0.336773s (thread); 0s (gc)
│ │ │ │ + -- used 1.04482s (cpu); 0.702253s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             8         7          6 2         5 3         4 4         3 5
│ │ │ │  o5 = {4096x  + 24576x y + 38912x y  - 18432x y  - 65280x y  + 18432x y  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2 6           7        8         6 2         5   2         4 2 2
│ │ │ │       38912x y  - 24576x*y  + 4096y  - 23040x z  - 69120x y*z  - 28800x y z  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -129,22 +129,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  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.
│ │ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ │ - -- used 0.019971s (cpu); 0.0214392s (thread); 0s (gc)
│ │ │ │ + -- used 0.0199973s (cpu); 0.0211987s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            4    4
│ │ │ │  o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 1.83382s (cpu); 1.16479s (thread); 0s (gc)
│ │ │ │ + -- used 3.19982s (cpu); 2.26863s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -89,29 +89,29 @@
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │      		          
                  
    i4 : elapsedTime invariants S4
│ │ │ - -- .6125s elapsed
│ │ │ + -- .828077s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
    
│ │ │      		          
                  
    i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ - -- .634295s elapsed
│ │ │ + -- .675065s elapsed
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,27 +34,27 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .6125s elapsed
│ │ │ │ + -- .828077s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ │ - -- .634295s elapsed
│ │ │ │ + -- .675065s elapsed
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html
│ │ │ @@ -89,29 +89,29 @@
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │      		          
                  
    i4 : elapsedTime invariants S4
│ │ │ - -- .629561s elapsed
│ │ │ + -- .740044s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
    
│ │ │      		          
                  
    i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .0410553s elapsed
│ │ │ + -- .0781188s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,27 +36,27 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .629561s elapsed
│ │ │ │ + -- .740044s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ │ - -- .0410553s elapsed
│ │ │ │ + -- .0781188s elapsed
│ │ │ │  
│ │ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InverseSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  ZnJvbUR1YWwoTWF0cml4KQ==
│ │ │  #:len=249
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmcm9tRHVhbCxNYXRyaXgpLCJmcm9tRHVhbChNYXRy
│ │ ├── ./usr/share/doc/Macaulay2/InvolutiveBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  aW52UmVkdWNlKFJpbmdFbGVtZW50LEludm9sdXRpdmVCYXNpcyk=
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaW52UmVkdWNlLFJpbmdFbGVtZW50LEludm9sdXRp
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  aXNJc29tb3JwaGljKE1hdHJpeCxNYXRyaXgp
│ │ │  #:len=270
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDA2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0lzb21vcnBoaWMsTWF0cml4LE1hdHJpeCksImlz
│ │ ├── ./usr/share/doc/Macaulay2/JSON/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  TmFtZVNlcGFyYXRvcg==
│ │ │  #:len=210
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJOYW1lU2VwYXJhdG9yIiwiTmFtZVNlcGFyYXRvciIs
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -39,19 +39,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-134606-0/0.json
│ │ │ +o9 = /tmp/M2-247428-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-134606-0/0.json
│ │ │ +o10 = /tmp/M2-247428-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -150,20 +150,20 @@
│ │ │          
    
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
                  
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-134606-0/0.json
    
│ │ │ +o9 = /tmp/M2-247428-0/0.json
│ │ │      		          
                  
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-134606-0/0.json
│ │ │ +o10 = /tmp/M2-247428-0/0.json
│ │ │  
│ │ │  o10 : File
    
│ │ │      		          
                  
    i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,18 +54,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-134606-0/0.json
│ │ │ │ +o9 = /tmp/M2-247428-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-134606-0/0.json
│ │ │ │ +o10 = /tmp/M2-247428-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  amV0cyhaWixBZmZpbmVWYXJpZXR5KQ==
│ │ │  #:len=2469
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGpldHMgb2YgYW4gYWZmaW5lIHZh
│ │ │  cmlldHkiLCAibGluZW51bSIgPT4gMTY2NywgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
│ │ │ @@ -17,24 +17,24 @@
│ │ │  o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00195448s elapsed
│ │ │ + -- .00173645s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i5 : elapsedTime radical J2I
│ │ │ - -- .31263s elapsed
│ │ │ + -- .595297s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : I.cache.?jet
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 : elapsedTime jets(3,I)
│ │ │ - -- .00789879s elapsed
│ │ │ + -- .0074075s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i9 : I.cache.?jet
│ │ │ @@ -53,23 +53,23 @@
│ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │                                 | 2x0x1-y1       |
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i11 : elapsedTime jets(3,I)
│ │ │ - -- .00214164s elapsed
│ │ │ + -- .00196759s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i12 : elapsedTime jets(2,I)
│ │ │ - -- .00194612s elapsed
│ │ │ + -- .00176346s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │  
│ │ │  i13 : Q = R/I
│ │ │ @@ -148,15 +148,15 @@
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : f.cache.?jet
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : elapsedTime jets(3,f)
│ │ │ - -- .0110379s elapsed
│ │ │ + -- .0097507s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │                                 | t2 2t0t2+t1^2  |
│ │ │                                 | t1 2t0t1       |
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i27 : elapsedTime jets(2,f)
│ │ │ - -- .000765543s elapsed
│ │ │ + -- .000528717s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -73,25 +73,25 @@
│ │ │          
    
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  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.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .00195448s elapsed
│ │ │ │ + -- .00173645s elapsed
│ │ │ │  
│ │ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │ │  
│ │ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  i5 : elapsedTime radical J2I
│ │ │ │ - -- .31263s elapsed
│ │ │ │ + -- .595297s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ @@ -118,24 +118,24 @@
│ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │                                 | 2x0x1-y1       |
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00195448s elapsed
│ │ │ + -- .00173645s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │      		          
                  
    i5 : elapsedTime radical J2I
│ │ │ - -- .31263s elapsed
│ │ │ + -- .595297s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │      		          
                  
    i7 : I.cache.?jet
│ │ │  
│ │ │  o7 = false
    
│ │ │      		          
                  
    i8 : elapsedTime jets(3,I)
│ │ │ - -- .00789879s elapsed
│ │ │ + -- .0074075s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │      		          
              
                  
    i11 : elapsedTime jets(3,I)
│ │ │ - -- .00214164s elapsed
│ │ │ + -- .00196759s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │      		          
                  
    i12 : elapsedTime jets(2,I)
│ │ │ - -- .00194612s elapsed
│ │ │ + -- .00176346s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
    
│ │ │      		          
    
│ │ │ @@ -236,15 +236,15 @@
│ │ │            
                  
    i23 : f.cache.?jet
│ │ │  
│ │ │  o23 = false
    
│ │ │      		          
                  
    i24 : elapsedTime jets(3,f)
│ │ │ - -- .0110379s elapsed
│ │ │ + -- .0097507s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -264,15 +264,15 @@
│ │ │                                 | t2 2t0t2+t1^2  |
│ │ │                                 | t1 2t0t1       |
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │      		          
                  
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000765543s elapsed
│ │ │ + -- .000528717s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o6 = ideal(x  - y)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : I.cache.?jet
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  i8 : elapsedTime jets(3,I)
│ │ │ │ - -- .00789879s elapsed
│ │ │ │ + -- .0074075s elapsed
│ │ │ │  
│ │ │ │                                                    2                 2
│ │ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i9 : I.cache.?jet
│ │ │ │  
│ │ │ │ @@ -58,22 +58,22 @@
│ │ │ │  
│ │ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │ │                                 | 2x0x1-y1       |
│ │ │ │                                 | x0^2-y0        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i11 : elapsedTime jets(3,I)
│ │ │ │ - -- .00214164s elapsed
│ │ │ │ + -- .00196759s elapsed
│ │ │ │  
│ │ │ │                                                     2                 2
│ │ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i12 : elapsedTime jets(2,I)
│ │ │ │ - -- .00194612s elapsed
│ │ │ │ + -- .00176346s elapsed
│ │ │ │  
│ │ │ │                               2                 2
│ │ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  For quotient rings, data is stored under *.jet. Each jets order gives rise to a
│ │ │ │  different quotient that is stored separately under *.jet.jetsRing (order zero
│ │ │ │ @@ -153,15 +153,15 @@
│ │ │ │  i22 : isWellDefined f
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : f.cache.?jet
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : elapsedTime jets(3,f)
│ │ │ │ - -- .0110379s elapsed
│ │ │ │ + -- .0097507s elapsed
│ │ │ │  
│ │ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3,
│ │ │ │  y3]                                                      2                    2
│ │ │ │  o24 = map (QQ[t0][t1][t2][t3], ------------------------------------------------
│ │ │ │  ----------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                                                        2
│ │ │ │  2
│ │ │ │ @@ -183,15 +183,15 @@
│ │ │ │  
│ │ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │ │                                 | t2 2t0t2+t1^2  |
│ │ │ │                                 | t1 2t0t1       |
│ │ │ │                                 | t0 t0^2        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i27 : elapsedTime jets(2,f)
│ │ │ │ - -- .000765543s elapsed
│ │ │ │ + -- .000528717s elapsed
│ │ │ │  
│ │ │ │                                     QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  2                    2
│ │ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2,
│ │ │ │  2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                              2                 2
│ │ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  Y2Fub25pY2FsSG9tb3RvcGllcyguLi4sRmluZUdyYWRpbmc9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU4NSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2Fub25pY2FsSG9tb3RvcGllcyxGaW5lR3JhZGlu
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │        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         
│ │ │                                                                                                                                                                                                                                                                                                                       10
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0229344s elapsed
│ │ │ + -- .0605774s elapsed
│ │ │  
│ │ │  o5 = 350
│ │ │  
│ │ │  i6 : betti F_a, betti F
│ │ │  
│ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ @@ -46,19 +46,19 @@
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │  
│ │ │  o9 = 14
│ │ │  
│ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00203535s elapsed
│ │ │ - -- .00582659s elapsed
│ │ │ - -- .0231181s elapsed
│ │ │ - -- .0314129s elapsed
│ │ │ - -- .00364306s elapsed
│ │ │ + -- .00179454s elapsed
│ │ │ + -- .00497143s elapsed
│ │ │ + -- .060344s elapsed
│ │ │ + -- .00900955s elapsed
│ │ │ + -- .00320145s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out
│ │ │ @@ -3,20 +3,20 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00224859s elapsed
│ │ │ - -- .00607912s elapsed
│ │ │ - -- .0220902s elapsed
│ │ │ - -- .0100536s elapsed
│ │ │ - -- .00340846s elapsed
│ │ │ - -- .337028s elapsed
│ │ │ + -- .00195235s elapsed
│ │ │ + -- .00524863s elapsed
│ │ │ + -- .0193959s elapsed
│ │ │ + -- .00958441s elapsed
│ │ │ + -- .00279801s elapsed
│ │ │ + -- .395539s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -26,15 +26,15 @@
│ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .219311s elapsed
│ │ │ + -- .274046s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,22 +48,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00425743s elapsed
│ │ │ - -- .0173426s elapsed
│ │ │ - -- .0943482s elapsed
│ │ │ - -- 1.07453s elapsed
│ │ │ - -- .306247s elapsed
│ │ │ - -- .0523164s elapsed
│ │ │ - -- .00604333s elapsed
│ │ │ - -- 4.92691s elapsed
│ │ │ + -- .0540031s elapsed
│ │ │ + -- .0145816s elapsed
│ │ │ + -- .154782s elapsed
│ │ │ + -- 1.31986s elapsed
│ │ │ + -- .391793s elapsed
│ │ │ + -- .0779048s elapsed
│ │ │ + -- .0054453s elapsed
│ │ │ + -- 5.90773s elapsed
│ │ │  
│ │ │  i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out
│ │ │ @@ -3,19 +3,19 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00219401s elapsed
│ │ │ - -- .00591613s elapsed
│ │ │ - -- .0530805s elapsed
│ │ │ - -- .00969284s elapsed
│ │ │ - -- .0188432s elapsed
│ │ │ + -- .00196056s elapsed
│ │ │ + -- .00517971s elapsed
│ │ │ + -- .0195997s elapsed
│ │ │ + -- .00818808s elapsed
│ │ │ + -- .00281309s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,15 +48,15 @@
│ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .165128s elapsed
│ │ │ + -- .250066s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -70,22 +70,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o6 : BettiTally
│ │ │  
│ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00418592s elapsed
│ │ │ - -- .041455s elapsed
│ │ │ - -- .106786s elapsed
│ │ │ - -- 1.12243s elapsed
│ │ │ - -- .306161s elapsed
│ │ │ - -- .0378051s elapsed
│ │ │ - -- .00649705s elapsed
│ │ │ - -- 4.94679s elapsed
│ │ │ + -- .00402656s elapsed
│ │ │ + -- .0150707s elapsed
│ │ │ + -- .187955s elapsed
│ │ │ + -- 1.38855s elapsed
│ │ │ + -- .405881s elapsed
│ │ │ + -- .0750955s elapsed
│ │ │ + -- .00550418s elapsed
│ │ │ + -- 6.20885s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,42 +3,42 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .023887s elapsed
│ │ │ - -- .0224611s elapsed
│ │ │ - -- .000281838s elapsed
│ │ │ - -- .000134322s elapsed
│ │ │ - -- .000117361s elapsed
│ │ │ - -- .000113723s elapsed
│ │ │ - -- .000115275s elapsed
│ │ │ - -- .000120876s elapsed
│ │ │ - -- .000143789s elapsed
│ │ │ - -- .000447297s elapsed
│ │ │ - -- .000368159s elapsed
│ │ │ - -- .000119123s elapsed
│ │ │ - -- .000112329s elapsed
│ │ │ - -- .000118642s elapsed
│ │ │ - -- .000107462s elapsed
│ │ │ - -- .000108013s elapsed
│ │ │ - -- .000114595s elapsed
│ │ │ - -- .000106309s elapsed
│ │ │ - -- .000123922s elapsed
│ │ │ - -- .000128231s elapsed
│ │ │ - -- .000131386s elapsed
│ │ │ - -- .000124613s elapsed
│ │ │ - -- .000131696s elapsed
│ │ │ - -- .000118632s elapsed
│ │ │ - -- .00010717s elapsed
│ │ │ - -- .00012855s elapsed
│ │ │ - -- .0001177s elapsed
│ │ │ - -- .000111488s elapsed
│ │ │ + -- .00813004s elapsed
│ │ │ + -- .0151821s elapsed
│ │ │ + -- .000210683s elapsed
│ │ │ + -- .000145201s elapsed
│ │ │ + -- .000110266s elapsed
│ │ │ + -- .000113962s elapsed
│ │ │ + -- .00010155s elapsed
│ │ │ + -- .000299048s elapsed
│ │ │ + -- .000148066s elapsed
│ │ │ + -- .000139039s elapsed
│ │ │ + -- .000254766s elapsed
│ │ │ + -- .000133519s elapsed
│ │ │ + -- .000153757s elapsed
│ │ │ + -- .000117559s elapsed
│ │ │ + -- .000106218s elapsed
│ │ │ + -- .000103673s elapsed
│ │ │ + -- .000108984s elapsed
│ │ │ + -- .000103645s elapsed
│ │ │ + -- .000117719s elapsed
│ │ │ + -- .000123641s elapsed
│ │ │ + -- .000136404s elapsed
│ │ │ + -- .000165299s elapsed
│ │ │ + -- .000132147s elapsed
│ │ │ + -- .00010642s elapsed
│ │ │ + -- .000111127s elapsed
│ │ │ + -- .000108342s elapsed
│ │ │ + -- .00011782s elapsed
│ │ │ + -- .000103855s elapsed
│ │ │  (number Of blocks, 26)
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  2
│ │ │   2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
│ │ │ @@ -3,17 +3,17 @@
│ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : computeBound 3
│ │ │ - -- .231161s elapsed
│ │ │ - -- .127287s elapsed
│ │ │ - -- .192592s elapsed
│ │ │ - -- .221279s elapsed
│ │ │ - -- .151329s elapsed
│ │ │ - -- .27634s elapsed
│ │ │ + -- .310819s elapsed
│ │ │ + -- .196029s elapsed
│ │ │ + -- .288433s elapsed
│ │ │ + -- .295623s elapsed
│ │ │ + -- .220784s elapsed
│ │ │ + -- .288721s elapsed
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out
│ │ │ @@ -9,19 +9,19 @@
│ │ │  i2 : e=(-1,5)
│ │ │  
│ │ │  o2 = (-1, 5)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00231647s elapsed
│ │ │ - -- .00572909s elapsed
│ │ │ - -- .0420116s elapsed
│ │ │ - -- .0100077s elapsed
│ │ │ - -- .00326168s elapsed
│ │ │ + -- .00231515s elapsed
│ │ │ + -- .00492044s elapsed
│ │ │ + -- .0706832s elapsed
│ │ │ + -- .0079248s elapsed
│ │ │ + -- .00298573s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i4 : keys h
│ │ │  
│ │ │  o4 = {0, 2, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .190761s elapsed
│ │ │ + -- .313112s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -77,19 +77,19 @@
│ │ │  i7 : e=(-1,5^2)
│ │ │  
│ │ │  o7 = (-1, 25)
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00238839s elapsed
│ │ │ - -- .00636492s elapsed
│ │ │ - -- .0233021s elapsed
│ │ │ - -- .00920894s elapsed
│ │ │ - -- .00338048s elapsed
│ │ │ + -- .00225391s elapsed
│ │ │ + -- .00567321s elapsed
│ │ │ + -- .0722083s elapsed
│ │ │ + -- .00821501s elapsed
│ │ │ + -- .00301775s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
│ │ │ @@ -1,33 +1,33 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0384115s elapsed
│ │ │ - -- .000040386s elapsed
│ │ │ - -- .000105137s elapsed
│ │ │ - -- .000089628s elapsed
│ │ │ - -- .000109706s elapsed
│ │ │ - -- .000203892s elapsed
│ │ │ - -- .000115807s elapsed
│ │ │ - -- .00003236s elapsed
│ │ │ - -- .0639071s elapsed
│ │ │ - -- .000145333s elapsed
│ │ │ - -- .000126558s elapsed
│ │ │ - -- .000122309s elapsed
│ │ │ - -- .000122129s elapsed
│ │ │ - -- .000105167s elapsed
│ │ │ - -- .000094497s elapsed
│ │ │ - -- .000122089s elapsed
│ │ │ - -- .000033253s elapsed
│ │ │ - -- .000093946s elapsed
│ │ │ - -- .000030117s elapsed
│ │ │ + -- .0104189s elapsed
│ │ │ + -- .000033212s elapsed
│ │ │ + -- .000073056s elapsed
│ │ │ + -- .000055284s elapsed
│ │ │ + -- .000062507s elapsed
│ │ │ + -- .000068509s elapsed
│ │ │ + -- .00006456s elapsed
│ │ │ + -- .000019516s elapsed
│ │ │ + -- .0330295s elapsed
│ │ │ + -- .000080479s elapsed
│ │ │ + -- .000069519s elapsed
│ │ │ + -- .000103393s elapsed
│ │ │ + -- .000100337s elapsed
│ │ │ + -- .000089156s elapsed
│ │ │ + -- .000073146s elapsed
│ │ │ + -- .00008012s elapsed
│ │ │ + -- .000025957s elapsed
│ │ │ + -- .000058098s elapsed
│ │ │ + -- .000027381s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │                                                                                                                                                                                                                                                                                                                       10
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │      		          
                  
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0229344s elapsed
│ │ │ + -- .0605774s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │      		          
                  
    i6 : betti F_a, betti F
│ │ │  
│ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │ @@ -127,19 +127,19 @@
│ │ │            
                  
    i8 : L3 = select(L,c->c%3==0); #L3
│ │ │  
│ │ │  o9 = 14
    
│ │ │      		          
                  
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00203535s elapsed
│ │ │ - -- .00582659s elapsed
│ │ │ - -- .0231181s elapsed
│ │ │ - -- .0314129s elapsed
│ │ │ - -- .00364306s elapsed
│ │ │ + -- .00179454s elapsed
│ │ │ + -- .00497143s elapsed
│ │ │ + -- .060344s elapsed
│ │ │ + -- .00900955s elapsed
│ │ │ + -- .00320145s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │       0                            1                             2
│ │ │ │  3                              4                              5
│ │ │ │  6                              7                              8
│ │ │ │  9
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ │ - -- .0229344s elapsed
│ │ │ │ + -- .0605774s elapsed
│ │ │ │  
│ │ │ │  o5 = 350
│ │ │ │  i6 : betti F_a, betti F
│ │ │ │  
│ │ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ │            6: 350      0: 1  .   .   .   .   .   .   .  . .
│ │ │ │ @@ -73,19 +73,19 @@
│ │ │ │  o7 = 2   3
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │ │  
│ │ │ │  o9 = 14
│ │ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ │ - -- .00203535s elapsed
│ │ │ │ - -- .00582659s elapsed
│ │ │ │ - -- .0231181s elapsed
│ │ │ │ - -- .0314129s elapsed
│ │ │ │ - -- .00364306s elapsed
│ │ │ │ + -- .00179454s elapsed
│ │ │ │ + -- .00497143s elapsed
│ │ │ │ + -- .060344s elapsed
│ │ │ │ + -- .00900955s elapsed
│ │ │ │ + -- .00320145s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -83,20 +83,20 @@
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │      		          
                  
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00224859s elapsed
│ │ │ - -- .00607912s elapsed
│ │ │ - -- .0220902s elapsed
│ │ │ - -- .0100536s elapsed
│ │ │ - -- .00340846s elapsed
│ │ │ - -- .337028s elapsed
│ │ │ + -- .00195235s elapsed
│ │ │ + -- .00524863s elapsed
│ │ │ + -- .0193959s elapsed
│ │ │ + -- .00958441s elapsed
│ │ │ + -- .00279801s elapsed
│ │ │ + -- .395539s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │                ZZ
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │      		          
                  
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .219311s elapsed
│ │ │ + -- .274046s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -132,22 +132,22 @@
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │      		          
                  
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00425743s elapsed
│ │ │ - -- .0173426s elapsed
│ │ │ - -- .0943482s elapsed
│ │ │ - -- 1.07453s elapsed
│ │ │ - -- .306247s elapsed
│ │ │ - -- .0523164s elapsed
│ │ │ - -- .00604333s elapsed
│ │ │ - -- 4.92691s elapsed
    
│ │ │ + -- .0540031s elapsed
│ │ │ + -- .0145816s elapsed
│ │ │ + -- .154782s elapsed
│ │ │ + -- 1.31986s elapsed
│ │ │ + -- .391793s elapsed
│ │ │ + -- .0779048s elapsed
│ │ │ + -- .0054453s elapsed
│ │ │ + -- 5.90773s elapsed
│ │ │      		          
                  
    i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,20 +26,20 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ │ - -- .00224859s elapsed
│ │ │ │ - -- .00607912s elapsed
│ │ │ │ - -- .0220902s elapsed
│ │ │ │ - -- .0100536s elapsed
│ │ │ │ - -- .00340846s elapsed
│ │ │ │ - -- .337028s elapsed
│ │ │ │ + -- .00195235s elapsed
│ │ │ │ + -- .00524863s elapsed
│ │ │ │ + -- .0193959s elapsed
│ │ │ │ + -- .00958441s elapsed
│ │ │ │ + -- .00279801s elapsed
│ │ │ │ + -- .395539s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  o2 : BettiTally
│ │ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .219311s elapsed
│ │ │ │ + -- .274046s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -67,22 +67,22 @@
│ │ │ │  o5 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00425743s elapsed
│ │ │ │ - -- .0173426s elapsed
│ │ │ │ - -- .0943482s elapsed
│ │ │ │ - -- 1.07453s elapsed
│ │ │ │ - -- .306247s elapsed
│ │ │ │ - -- .0523164s elapsed
│ │ │ │ - -- .00604333s elapsed
│ │ │ │ - -- 4.92691s elapsed
│ │ │ │ + -- .0540031s elapsed
│ │ │ │ + -- .0145816s elapsed
│ │ │ │ + -- .154782s elapsed
│ │ │ │ + -- 1.31986s elapsed
│ │ │ │ + -- .391793s elapsed
│ │ │ │ + -- .0779048s elapsed
│ │ │ │ + -- .0054453s elapsed
│ │ │ │ + -- 5.90773s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -78,19 +78,19 @@
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │      		          
                  
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00219401s elapsed
│ │ │ - -- .00591613s elapsed
│ │ │ - -- .0530805s elapsed
│ │ │ - -- .00969284s elapsed
│ │ │ - -- .0188432s elapsed
│ │ │ + -- .00196056s elapsed
│ │ │ + -- .00517971s elapsed
│ │ │ + -- .0195997s elapsed
│ │ │ + -- .00818808s elapsed
│ │ │ + -- .00281309s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │                ZZ
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │      		          
                  
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .165128s elapsed
│ │ │ + -- .250066s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -150,22 +150,22 @@
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │      		          
                  
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00418592s elapsed
│ │ │ - -- .041455s elapsed
│ │ │ - -- .106786s elapsed
│ │ │ - -- 1.12243s elapsed
│ │ │ - -- .306161s elapsed
│ │ │ - -- .0378051s elapsed
│ │ │ - -- .00649705s elapsed
│ │ │ - -- 4.94679s elapsed
    
│ │ │ + -- .00402656s elapsed
│ │ │ + -- .0150707s elapsed
│ │ │ + -- .187955s elapsed
│ │ │ + -- 1.38855s elapsed
│ │ │ + -- .405881s elapsed
│ │ │ + -- .0750955s elapsed
│ │ │ + -- .00550418s elapsed
│ │ │ + -- 6.20885s elapsed
│ │ │      		          
                  
    i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,19 +22,19 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ │ - -- .00219401s elapsed
│ │ │ │ - -- .00591613s elapsed
│ │ │ │ - -- .0530805s elapsed
│ │ │ │ - -- .00969284s elapsed
│ │ │ │ - -- .0188432s elapsed
│ │ │ │ + -- .00196056s elapsed
│ │ │ │ + -- .00517971s elapsed
│ │ │ │ + -- .0195997s elapsed
│ │ │ │ + -- .00818808s elapsed
│ │ │ │ + -- .00281309s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .165128s elapsed
│ │ │ │ + -- .250066s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -84,22 +84,22 @@
│ │ │ │  o6 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o6 : BettiTally
│ │ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00418592s elapsed
│ │ │ │ - -- .041455s elapsed
│ │ │ │ - -- .106786s elapsed
│ │ │ │ - -- 1.12243s elapsed
│ │ │ │ - -- .306161s elapsed
│ │ │ │ - -- .0378051s elapsed
│ │ │ │ - -- .00649705s elapsed
│ │ │ │ - -- 4.94679s elapsed
│ │ │ │ + -- .00402656s elapsed
│ │ │ │ + -- .0150707s elapsed
│ │ │ │ + -- .187955s elapsed
│ │ │ │ + -- 1.38855s elapsed
│ │ │ │ + -- .405881s elapsed
│ │ │ │ + -- .0750955s elapsed
│ │ │ │ + -- .00550418s elapsed
│ │ │ │ + -- 6.20885s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -78,42 +78,42 @@
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │      		          
                  
    i2 : d=carpetDet(a,b)
│ │ │ - -- .023887s elapsed
│ │ │ - -- .0224611s elapsed
│ │ │ - -- .000281838s elapsed
│ │ │ - -- .000134322s elapsed
│ │ │ - -- .000117361s elapsed
│ │ │ - -- .000113723s elapsed
│ │ │ - -- .000115275s elapsed
│ │ │ - -- .000120876s elapsed
│ │ │ - -- .000143789s elapsed
│ │ │ - -- .000447297s elapsed
│ │ │ - -- .000368159s elapsed
│ │ │ - -- .000119123s elapsed
│ │ │ - -- .000112329s elapsed
│ │ │ - -- .000118642s elapsed
│ │ │ - -- .000107462s elapsed
│ │ │ - -- .000108013s elapsed
│ │ │ - -- .000114595s elapsed
│ │ │ - -- .000106309s elapsed
│ │ │ - -- .000123922s elapsed
│ │ │ - -- .000128231s elapsed
│ │ │ - -- .000131386s elapsed
│ │ │ - -- .000124613s elapsed
│ │ │ - -- .000131696s elapsed
│ │ │ - -- .000118632s elapsed
│ │ │ - -- .00010717s elapsed
│ │ │ - -- .00012855s elapsed
│ │ │ - -- .0001177s elapsed
│ │ │ - -- .000111488s elapsed
│ │ │ + -- .00813004s elapsed
│ │ │ + -- .0151821s elapsed
│ │ │ + -- .000210683s elapsed
│ │ │ + -- .000145201s elapsed
│ │ │ + -- .000110266s elapsed
│ │ │ + -- .000113962s elapsed
│ │ │ + -- .00010155s elapsed
│ │ │ + -- .000299048s elapsed
│ │ │ + -- .000148066s elapsed
│ │ │ + -- .000139039s elapsed
│ │ │ + -- .000254766s elapsed
│ │ │ + -- .000133519s elapsed
│ │ │ + -- .000153757s elapsed
│ │ │ + -- .000117559s elapsed
│ │ │ + -- .000106218s elapsed
│ │ │ + -- .000103673s elapsed
│ │ │ + -- .000108984s elapsed
│ │ │ + -- .000103645s elapsed
│ │ │ + -- .000117719s elapsed
│ │ │ + -- .000123641s elapsed
│ │ │ + -- .000136404s elapsed
│ │ │ + -- .000165299s elapsed
│ │ │ + -- .000132147s elapsed
│ │ │ + -- .00010642s elapsed
│ │ │ + -- .000111127s elapsed
│ │ │ + -- .000108342s elapsed
│ │ │ + -- .00011782s elapsed
│ │ │ + -- .000103855s elapsed
│ │ │  (number Of blocks, 26)
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  2
│ │ │   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,42 +20,42 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .023887s elapsed
│ │ │ │ - -- .0224611s elapsed
│ │ │ │ - -- .000281838s elapsed
│ │ │ │ - -- .000134322s elapsed
│ │ │ │ - -- .000117361s elapsed
│ │ │ │ - -- .000113723s elapsed
│ │ │ │ - -- .000115275s elapsed
│ │ │ │ - -- .000120876s elapsed
│ │ │ │ - -- .000143789s elapsed
│ │ │ │ - -- .000447297s elapsed
│ │ │ │ - -- .000368159s elapsed
│ │ │ │ - -- .000119123s elapsed
│ │ │ │ - -- .000112329s elapsed
│ │ │ │ - -- .000118642s elapsed
│ │ │ │ - -- .000107462s elapsed
│ │ │ │ - -- .000108013s elapsed
│ │ │ │ - -- .000114595s elapsed
│ │ │ │ - -- .000106309s elapsed
│ │ │ │ - -- .000123922s elapsed
│ │ │ │ - -- .000128231s elapsed
│ │ │ │ - -- .000131386s elapsed
│ │ │ │ - -- .000124613s elapsed
│ │ │ │ - -- .000131696s elapsed
│ │ │ │ - -- .000118632s elapsed
│ │ │ │ - -- .00010717s elapsed
│ │ │ │ - -- .00012855s elapsed
│ │ │ │ - -- .0001177s elapsed
│ │ │ │ - -- .000111488s elapsed
│ │ │ │ + -- .00813004s elapsed
│ │ │ │ + -- .0151821s elapsed
│ │ │ │ + -- .000210683s elapsed
│ │ │ │ + -- .000145201s elapsed
│ │ │ │ + -- .000110266s elapsed
│ │ │ │ + -- .000113962s elapsed
│ │ │ │ + -- .00010155s elapsed
│ │ │ │ + -- .000299048s elapsed
│ │ │ │ + -- .000148066s elapsed
│ │ │ │ + -- .000139039s elapsed
│ │ │ │ + -- .000254766s elapsed
│ │ │ │ + -- .000133519s elapsed
│ │ │ │ + -- .000153757s elapsed
│ │ │ │ + -- .000117559s elapsed
│ │ │ │ + -- .000106218s elapsed
│ │ │ │ + -- .000103673s elapsed
│ │ │ │ + -- .000108984s elapsed
│ │ │ │ + -- .000103645s elapsed
│ │ │ │ + -- .000117719s elapsed
│ │ │ │ + -- .000123641s elapsed
│ │ │ │ + -- .000136404s elapsed
│ │ │ │ + -- .000165299s elapsed
│ │ │ │ + -- .000132147s elapsed
│ │ │ │ + -- .00010642s elapsed
│ │ │ │ + -- .000111127s elapsed
│ │ │ │ + -- .000108342s elapsed
│ │ │ │ + -- .00011782s elapsed
│ │ │ │ + -- .000103855s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │  1
│ │ │ │  1
│ │ │ │  1
│ │ │ │  1
│ │ │ │  2
│ │ │ │   2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -86,20 +86,20 @@
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │      		          
                  
    i2 : computeBound 3
│ │ │ - -- .231161s elapsed
│ │ │ - -- .127287s elapsed
│ │ │ - -- .192592s elapsed
│ │ │ - -- .221279s elapsed
│ │ │ - -- .151329s elapsed
│ │ │ - -- .27634s elapsed
│ │ │ + -- .310819s elapsed
│ │ │ + -- .196029s elapsed
│ │ │ + -- .288433s elapsed
│ │ │ + -- .295623s elapsed
│ │ │ + -- .220784s elapsed
│ │ │ + -- .288721s elapsed
│ │ │  
│ │ │  o2 = 6
    
│ │ │      		          
    
│ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -26,20 +26,20 @@ │ │ │ │ classes mod k. We conjecture that c=k^2-k. │ │ │ │ i1 : (a,b)=computeBound(6,4,3) │ │ │ │ │ │ │ │ o1 = (9, 7) │ │ │ │ │ │ │ │ o1 : Sequence │ │ │ │ i2 : computeBound 3 │ │ │ │ - -- .231161s elapsed │ │ │ │ - -- .127287s elapsed │ │ │ │ - -- .192592s elapsed │ │ │ │ - -- .221279s elapsed │ │ │ │ - -- .151329s elapsed │ │ │ │ - -- .27634s elapsed │ │ │ │ + -- .310819s elapsed │ │ │ │ + -- .196029s elapsed │ │ │ │ + -- .288433s elapsed │ │ │ │ + -- .295623s elapsed │ │ │ │ + -- .220784s elapsed │ │ │ │ + -- .288721s elapsed │ │ │ │ │ │ │ │ o2 = 6 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics │ │ │ │ ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: ********** │ │ │ │ * computeBound(ZZ) │ │ │ │ * computeBound(ZZ,ZZ,ZZ) │ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ @@ -87,19 +87,19 @@ │ │ │ │ │ │ o2 = (-1, 5) │ │ │ │ │ │ o2 : Sequence     │ │ │ │ │ │ │ │ │ 
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00231647s elapsed
│ │ │ - -- .00572909s elapsed
│ │ │ - -- .0420116s elapsed
│ │ │ - -- .0100077s elapsed
│ │ │ - -- .00326168s elapsed
│ │ │ + -- .00231515s elapsed
│ │ │ + -- .00492044s elapsed
│ │ │ + -- .0706832s elapsed
│ │ │ + -- .0079248s elapsed
│ │ │ + -- .00298573s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o4 = {0, 2, 3, 5}
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .190761s elapsed
│ │ │ + -- .313112s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -165,19 +165,19 @@
│ │ │  
│ │ │  o7 = (-1, 25)
│ │ │  
│ │ │  o7 : Sequence
    │ │ │ │ │ │ │ │ │ 
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00238839s elapsed
│ │ │ - -- .00636492s elapsed
│ │ │ - -- .0233021s elapsed
│ │ │ - -- .00920894s elapsed
│ │ │ - -- .00338048s elapsed
│ │ │ + -- .00225391s elapsed
│ │ │ + -- .00567321s elapsed
│ │ │ + -- .0722083s elapsed
│ │ │ + -- .00821501s elapsed
│ │ │ + -- .00301775s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,19 +28,19 @@
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : e=(-1,5)
│ │ │ │  
│ │ │ │  o2 = (-1, 5)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00231647s elapsed
│ │ │ │ - -- .00572909s elapsed
│ │ │ │ - -- .0420116s elapsed
│ │ │ │ - -- .0100077s elapsed
│ │ │ │ - -- .00326168s elapsed
│ │ │ │ + -- .00231515s elapsed
│ │ │ │ + -- .00492044s elapsed
│ │ │ │ + -- .0706832s elapsed
│ │ │ │ + -- .0079248s elapsed
│ │ │ │ + -- .00298573s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -66,15 +66,15 @@
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : keys h
│ │ │ │  
│ │ │ │  o4 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ │ - -- .190761s elapsed
│ │ │ │ + -- .313112s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -95,19 +95,19 @@
│ │ │ │  these mistakes.
│ │ │ │  i7 : e=(-1,5^2)
│ │ │ │  
│ │ │ │  o7 = (-1, 25)
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00238839s elapsed
│ │ │ │ - -- .00636492s elapsed
│ │ │ │ - -- .0233021s elapsed
│ │ │ │ - -- .00920894s elapsed
│ │ │ │ - -- .00338048s elapsed
│ │ │ │ + -- .00225391s elapsed
│ │ │ │ + -- .00567321s elapsed
│ │ │ │ + -- .0722083s elapsed
│ │ │ │ + -- .00821501s elapsed
│ │ │ │ + -- .00301775s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -76,33 +76,33 @@
│ │ │            
│ │ │                
    i1 : a=4
│ │ │  
│ │ │  o1 = 4
    
│ │ │            
│ │ │            
│ │ │                
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0384115s elapsed
│ │ │ - -- .000040386s elapsed
│ │ │ - -- .000105137s elapsed
│ │ │ - -- .000089628s elapsed
│ │ │ - -- .000109706s elapsed
│ │ │ - -- .000203892s elapsed
│ │ │ - -- .000115807s elapsed
│ │ │ - -- .00003236s elapsed
│ │ │ - -- .0639071s elapsed
│ │ │ - -- .000145333s elapsed
│ │ │ - -- .000126558s elapsed
│ │ │ - -- .000122309s elapsed
│ │ │ - -- .000122129s elapsed
│ │ │ - -- .000105167s elapsed
│ │ │ - -- .000094497s elapsed
│ │ │ - -- .000122089s elapsed
│ │ │ - -- .000033253s elapsed
│ │ │ - -- .000093946s elapsed
│ │ │ - -- .000030117s elapsed
│ │ │ + -- .0104189s elapsed
│ │ │ + -- .000033212s elapsed
│ │ │ + -- .000073056s elapsed
│ │ │ + -- .000055284s elapsed
│ │ │ + -- .000062507s elapsed
│ │ │ + -- .000068509s elapsed
│ │ │ + -- .00006456s elapsed
│ │ │ + -- .000019516s elapsed
│ │ │ + -- .0330295s elapsed
│ │ │ + -- .000080479s elapsed
│ │ │ + -- .000069519s elapsed
│ │ │ + -- .000103393s elapsed
│ │ │ + -- .000100337s elapsed
│ │ │ + -- .000089156s elapsed
│ │ │ + -- .000073146s elapsed
│ │ │ + -- .00008012s elapsed
│ │ │ + -- .000025957s elapsed
│ │ │ + -- .000058098s elapsed
│ │ │ + -- .000027381s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,33 +20,33 @@
│ │ │ │  grading. Viewed as a resolution over QQ(e_1,e_2), this resolution is non-
│ │ │ │  minimal and carries further gradings. We decompose the crucial map of the a-th
│ │ │ │  strand into blocks, compute their determinants, and factor the product.
│ │ │ │  i1 : a=4
│ │ │ │  
│ │ │ │  o1 = 4
│ │ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ │ - -- .0384115s elapsed
│ │ │ │ - -- .000040386s elapsed
│ │ │ │ - -- .000105137s elapsed
│ │ │ │ - -- .000089628s elapsed
│ │ │ │ - -- .000109706s elapsed
│ │ │ │ - -- .000203892s elapsed
│ │ │ │ - -- .000115807s elapsed
│ │ │ │ - -- .00003236s elapsed
│ │ │ │ - -- .0639071s elapsed
│ │ │ │ - -- .000145333s elapsed
│ │ │ │ - -- .000126558s elapsed
│ │ │ │ - -- .000122309s elapsed
│ │ │ │ - -- .000122129s elapsed
│ │ │ │ - -- .000105167s elapsed
│ │ │ │ - -- .000094497s elapsed
│ │ │ │ - -- .000122089s elapsed
│ │ │ │ - -- .000033253s elapsed
│ │ │ │ - -- .000093946s elapsed
│ │ │ │ - -- .000030117s elapsed
│ │ │ │ + -- .0104189s elapsed
│ │ │ │ + -- .000033212s elapsed
│ │ │ │ + -- .000073056s elapsed
│ │ │ │ + -- .000055284s elapsed
│ │ │ │ + -- .000062507s elapsed
│ │ │ │ + -- .000068509s elapsed
│ │ │ │ + -- .00006456s elapsed
│ │ │ │ + -- .000019516s elapsed
│ │ │ │ + -- .0330295s elapsed
│ │ │ │ + -- .000080479s elapsed
│ │ │ │ + -- .000069519s elapsed
│ │ │ │ + -- .000103393s elapsed
│ │ │ │ + -- .000100337s elapsed
│ │ │ │ + -- .000089156s elapsed
│ │ │ │ + -- .000073146s elapsed
│ │ │ │ + -- .00008012s elapsed
│ │ │ │ + -- .000025957s elapsed
│ │ │ │ + -- .000058098s elapsed
│ │ │ │ + -- .000027381s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Surfaces/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  cHJvamVjdChWaXNpYmxlTGlzdCxFbWJlZGRlZEszc3VyZmFjZSk=
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTU5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwcm9qZWN0LFZpc2libGVMaXN0LEVtYmVkZGVkSzNz
│ │ ├── ./usr/share/doc/Macaulay2/Kronecker/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  a3JvbmVja2VyTm9ybWFsRm9ybShNYXRyaXgp
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTA5Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoa3JvbmVja2VyTm9ybWFsRm9ybSxNYXRyaXgpLCJr
│ │ ├── ./usr/share/doc/Macaulay2/KustinMiller/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=37
│ │ │  a3VzdGluTWlsbGVyQ29tcGxleCguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=577
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHRvIHByaW50IGludGVybWVk
│ │ │  aWF0ZSBkYXRhIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VE97bmV3IERvY3VtZW50VGFn
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  UmVhbFhE
│ │ │  #:len=499
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidXNlIGV4dGVuZGVkIGV4cG9uZW50IHJl
│ │ │  YWwgbnVtYmVycyIsIERlc2NyaXB0aW9uID0+IChUVHsiUmVhbFhEIn0sIiAtLSBhIHN0cmF0ZWd5
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -7,55 +7,55 @@
│ │ │  
│ │ │  i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i3 : time LLL m;
│ │ │ - -- used 0.00617129s (cpu); 0.00927838s (thread); 0s (gc)
│ │ │ + -- used 0.00799891s (cpu); 0.00714432s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275576s (cpu); 0.0282043s (thread); 0s (gc)
│ │ │ + -- used 0.0229392s (cpu); 0.023325s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.0981684s (cpu); 0.0994356s (thread); 0s (gc)
│ │ │ + -- used 0.123509s (cpu); 0.123787s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0120219s (cpu); 0.0123051s (thread); 0s (gc)
│ │ │ + -- used 0.0476131s (cpu); 0.0508222s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0464998s (cpu); 0.049258s (thread); 0s (gc)
│ │ │ + -- used 0.0546132s (cpu); 0.0550711s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0584038s (cpu); 0.0599435s (thread); 0s (gc)
│ │ │ + -- used 0.085402s (cpu); 0.0853611s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.363093s (cpu); 0.36638s (thread); 0s (gc)
│ │ │ + -- used 0.399975s (cpu); 0.399851s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11236s (cpu); 0.113921s (thread); 0s (gc)
│ │ │ + -- used 0.191798s (cpu); 0.192027s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -162,64 +162,64 @@
│ │ │                
    i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time LLL m;
│ │ │ - -- used 0.00617129s (cpu); 0.00927838s (thread); 0s (gc)
│ │ │ + -- used 0.00799891s (cpu); 0.00714432s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275576s (cpu); 0.0282043s (thread); 0s (gc)
│ │ │ + -- used 0.0229392s (cpu); 0.023325s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.0981684s (cpu); 0.0994356s (thread); 0s (gc)
│ │ │ + -- used 0.123509s (cpu); 0.123787s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0120219s (cpu); 0.0123051s (thread); 0s (gc)
│ │ │ + -- used 0.0476131s (cpu); 0.0508222s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0464998s (cpu); 0.049258s (thread); 0s (gc)
│ │ │ + -- used 0.0546132s (cpu); 0.0550711s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0584038s (cpu); 0.0599435s (thread); 0s (gc)
│ │ │ + -- used 0.085402s (cpu); 0.0853611s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.363093s (cpu); 0.36638s (thread); 0s (gc)
│ │ │ + -- used 0.399975s (cpu); 0.399851s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │            
│ │ │                
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11236s (cpu); 0.113921s (thread); 0s (gc)
│ │ │ + -- used 0.191798s (cpu); 0.192027s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -116,50 +116,50 @@ │ │ │ │ 50 50 │ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ i2 : m = syz m1; │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ i3 : time LLL m; │ │ │ │ - -- used 0.00617129s (cpu); 0.00927838s (thread); 0s (gc) │ │ │ │ + -- used 0.00799891s (cpu); 0.00714432s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o3 : Matrix ZZ <-- ZZ │ │ │ │ i4 : time LLL(m, Strategy=>CohenEngine); │ │ │ │ - -- used 0.0275576s (cpu); 0.0282043s (thread); 0s (gc) │ │ │ │ + -- used 0.0229392s (cpu); 0.023325s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : time LLL(m, Strategy=>CohenTopLevel); │ │ │ │ - -- used 0.0981684s (cpu); 0.0994356s (thread); 0s (gc) │ │ │ │ + -- used 0.123509s (cpu); 0.123787s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o5 : Matrix ZZ <-- ZZ │ │ │ │ i6 : time LLL(m, Strategy=>{Givens,RealFP}); │ │ │ │ - -- used 0.0120219s (cpu); 0.0123051s (thread); 0s (gc) │ │ │ │ + -- used 0.0476131s (cpu); 0.0508222s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o6 : Matrix ZZ <-- ZZ │ │ │ │ i7 : time LLL(m, Strategy=>{Givens,RealQP}); │ │ │ │ - -- used 0.0464998s (cpu); 0.049258s (thread); 0s (gc) │ │ │ │ + -- used 0.0546132s (cpu); 0.0550711s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o7 : Matrix ZZ <-- ZZ │ │ │ │ i8 : time LLL(m, Strategy=>{Givens,RealXD}); │ │ │ │ - -- used 0.0584038s (cpu); 0.0599435s (thread); 0s (gc) │ │ │ │ + -- used 0.085402s (cpu); 0.0853611s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o8 : Matrix ZZ <-- ZZ │ │ │ │ i9 : time LLL(m, Strategy=>{Givens,RealRR}); │ │ │ │ - -- used 0.363093s (cpu); 0.36638s (thread); 0s (gc) │ │ │ │ + -- used 0.399975s (cpu); 0.399851s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o9 : Matrix ZZ <-- ZZ │ │ │ │ i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP}); │ │ │ │ - -- used 0.11236s (cpu); 0.113921s (thread); 0s (gc) │ │ │ │ + -- used 0.191798s (cpu); 0.192027s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o10 : Matrix ZZ <-- ZZ │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _N_T_L │ │ │ │ * Function: _L_L_L -- compute an LLL basis │ │ │ │ * Option key: _S_t_r_a_t_e_g_y -- an optional argument │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ YXJlSXNvbW9ycGhpYyhNYXRyaXgsTWF0cml4KQ== │ │ │ #:len=289 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhcmVJc29tb3JwaGljLE1hdHJpeCxNYXRyaXgpLCJh │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ @@ -16,14 +16,14 @@ │ │ │ │ │ │ 3 8 │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ - -- used 0.953075s (cpu); 0.458804s (thread); 0s (gc) │ │ │ + -- used 1.84567s (cpu); 0.453203s (thread); 0s (gc) │ │ │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ - -- used 0.694874s (cpu); 0.330095s (thread); 0s (gc) │ │ │ + -- used 1.22661s (cpu); 0.36468s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ @@ -114,19 +114,19 @@ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ 
    i5 : P = convexHull(M);
    │ │ │ │ │ │ │ │ │ 
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.953075s (cpu); 0.458804s (thread); 0s (gc)
    │ │ │ + -- used 1.84567s (cpu); 0.453203s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.694874s (cpu); 0.330095s (thread); 0s (gc)
    │ │ │ + -- used 1.22661s (cpu); 0.36468s (thread); 0s (gc) │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use areIsomorphic:

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ | 0 0 1 0 1 0 1 1 | │ │ │ │ | 0 0 0 1 0 1 1 1 | │ │ │ │ │ │ │ │ 3 8 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ │ - -- used 0.953075s (cpu); 0.458804s (thread); 0s (gc) │ │ │ │ + -- used 1.84567s (cpu); 0.453203s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.694874s (cpu); 0.330095s (thread); 0s (gc) │ │ │ │ + -- used 1.22661s (cpu); 0.36468s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee aarreeIIssoommoorrpphhiicc:: ********** │ │ │ │ * areIsomorphic(Matrix,Matrix) │ │ │ │ * areIsomorphic(Polyhedron,Polyhedron) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _a_r_e_I_s_o_m_o_r_p_h_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/LexIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ TGV4SWRlYWxz │ │ │ #:len=470 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciB3b3JraW5nIHdp │ │ │ dGggbGV4IGlkZWFscyIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsiIixFTXsiTGV4 │ │ ├── ./usr/share/doc/Macaulay2/Licenses/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ TGljZW5zZXM= │ │ │ #:len=349 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMg │ │ │ cGFja2FnZSBleGFtaW5lcyB0aGUgdmVyc2lvbiBudW1iZXIgb2YgdGhlIHZhcmlvdXMgcGFja2Fn │ │ ├── ./usr/share/doc/Macaulay2/LieTypes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ TGllQWxnZWJyYU1vZHVsZSA9PSBaWg== │ │ │ #:len=205 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU2MywgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9s │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ bGluZWFyVHJ1bmNhdGlvbnNCb3VuZA== │ │ │ #:len=2355 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYm91bmRzIHRoZSByZWdpb24gd2hlcmUg │ │ │ dHJ1bmNhdGlvbnMgb2YgYSBtb2R1bGUgaGF2ZSBsaW5lYXIgcmVzb2x1dGlvbnMiLCAibGluZW51 │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ @@ -29,21 +29,21 @@ │ │ │ i5 : findRegion({{0,0},{4,4}},M,f) │ │ │ │ │ │ o5 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ - -- .182944s elapsed │ │ │ + -- .206109s elapsed │ │ │ │ │ │ o6 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}}) │ │ │ - -- .0430672s elapsed │ │ │ + -- .0116023s elapsed │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ @@ -30,21 +30,21 @@ │ │ │ i5 : apply(L, d -> isLinearComplex res prune truncate(d,M)) │ │ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ - -- 3.82481s elapsed │ │ │ + -- 14.5753s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .0225246s elapsed │ │ │ + -- .0193842s elapsed │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ @@ -117,23 +117,23 @@ │ │ │ │ │ │
      │ │ │

      If some degrees d are known to satisfy f(d,M), then they can be specified using the option Inner in order to expedite the computation. Similarly, degrees not above those given in Outer will be assumed not to satisfy f(d,M). If f takes options these can also be given to findRegion.

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .182944s elapsed
│ │ │ + -- .206109s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
      │ │ │ 
      i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0430672s elapsed
│ │ │ + -- .0116023s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
      │ │ │
      │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -49,22 +49,22 @@ │ │ │ │ │ │ │ │ o5 : List │ │ │ │ If some degrees d are known to satisfy f(d,M), then they can be specified using │ │ │ │ the option Inner in order to expedite the computation. Similarly, degrees not │ │ │ │ above those given in Outer will be assumed not to satisfy f(d,M). If f takes │ │ │ │ options these can also be given to findRegion. │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ │ - -- .182944s elapsed │ │ │ │ + -- .206109s elapsed │ │ │ │ │ │ │ │ o6 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{ │ │ │ │ {1,1}}) │ │ │ │ - -- .0430672s elapsed │ │ │ │ + -- .0116023s elapsed │ │ │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCoonnttrriibbuuttoorrss ********** │ │ │ │ Mahrud Sayrafi contributed to the code for this function. │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ @@ -112,23 +112,23 @@ │ │ │ │ │ │
    │ │ │

    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 i-th step of the resolution of $M$.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 3.82481s elapsed
│ │ │ + -- 14.5753s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0225246s elapsed
│ │ │ + -- .0193842s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -49,21 +49,21 @@ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ 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 i-th step of the resolution of $M$. │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ │ - -- 3.82481s elapsed │ │ │ │ + -- 14.5753s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .0225246s elapsed │ │ │ │ + -- .0193842s elapsed │ │ │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ In general linearTruncationsBound will not find the minimal degrees where $M$ │ │ │ │ has a linear resolution but will be faster than repeatedly truncating $M$. │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ TG9jYWxSaW5ncw== │ │ │ #:len=5276 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTG9jYWxpemF0aW9ucyBvZiBwb2x5bm9t │ │ │ aWFsIHJpbmdzIGF0IHByaW1lIGlkZWFscyIsICJsaW5lbnVtIiA9PiA2NCwgImZpbGVuYW1lIiA9 │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ - -- .418238s elapsed │ │ │ + -- .294097s elapsed │ │ │ │ │ │ o5 = {1, 3, 6, 7, 6, 3, 1} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : oo//sum │ │ │ │ │ │ @@ -44,21 +44,21 @@ │ │ │ │ │ │ 2 3 │ │ │ o10 = ideal (x , y ) │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds │ │ │ - -- .0114708s elapsed │ │ │ + -- .0102323s elapsed │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ - -- .341941s elapsed │ │ │ + -- .374546s elapsed │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ @@ -107,15 +107,15 @@ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP     │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .418238s elapsed
│ │ │ + -- .294097s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ 
    i6 : oo//sum
│ │ │ @@ -147,23 +147,23 @@
│ │ │                2   3
│ │ │  o10 = ideal (x , y )
│ │ │  
│ │ │  o10 : Ideal of RP
    │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0114708s elapsed
│ │ │ + -- .0102323s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .341941s elapsed
│ │ │ + -- .374546s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -42,15 +42,15 @@ │ │ │ │ i4 : M = RP^1/I │ │ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ │ │ 1 │ │ │ │ o4 : RP-module, quotient of RP │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ │ - -- .418238s elapsed │ │ │ │ + -- .294097s elapsed │ │ │ │ │ │ │ │ o5 = {1, 3, 6, 7, 6, 3, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : oo//sum │ │ │ │ │ │ │ │ o6 = 27 │ │ │ │ @@ -66,21 +66,21 @@ │ │ │ │ i10 : q = ideal"x2,y3" │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o10 = ideal (x , y ) │ │ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds │ │ │ │ - -- .0114708s elapsed │ │ │ │ + -- .0102323s elapsed │ │ │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ │ - -- .341941s elapsed │ │ │ │ + -- .374546s elapsed │ │ │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Hilbert-Samuel function with respect to a parameter ideal other than the │ │ │ │ maximal ideal can be slower. │ │ ├── ./usr/share/doc/Macaulay2/M0nbar/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ dGV4KEN1cnZlQ2xhc3NSZXByZXNlbnRhdGl2ZU0wbmJhcik= │ │ │ #:len=923 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydCB0byBUZVggZm9ybWF0Iiwg │ │ │ ImxpbmVudW0iID0+IDE1ODQsIElucHV0cyA9PiB7U1BBTntUVHsiQyJ9LCIsICIsU1BBTnsiYW4g │ │ ├── ./usr/share/doc/Macaulay2/MCMApproximations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ Y29BcHByb3hpbWF0aW9uKC4uLixUb3RhbD0+Li4uKQ== │ │ │ #:len=293 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tjb0FwcHJveGltYXRpb24sVG90YWxdLCJjb0FwcHJv │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=24 │ │ │ dW5pbnN0YWxsUGFja2FnZShTdHJpbmcp │ │ │ #:len=297 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjkxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh1bmluc3RhbGxQYWNrYWdlLFN0cmluZyksInVuaW5z │ │ │ @@ -12973,17 +12973,17 @@ │ │ │ YXMgdGhlIG5hbWUgb3IgdmFsdWUgb2YgYW4gb3B0aW9uYWwgYXJndW1lbnQiLCJzeW1ib2xzIHVz │ │ │ ZWQgYXMgdGhlIG5hbWUgb3IgdmFsdWUgb2YgYW4gb3B0aW9uYWwgYXJndW1lbnQiLCJNYWNhdWxh │ │ │ eTJEb2MifSwgS2V5ID0+ICJzeW1ib2xzIHVzZWQgYXMgdGhlIG5hbWUgb3IgdmFsdWUgb2YgYW4g │ │ │ b3B0aW9uYWwgYXJndW1lbnQiLCBTdWJub2RlcyA9PiBNRU5VeyJTeW1ib2xzIHVzZWQgYXMgYW4g │ │ │ b3B0aW9uIG5hbWUgb3IgdmFsdWUiLFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiU3l6eWdpZXMi │ │ │ LCJTeXp5Z2llcyIsIk1hY2F1bGF5MkRvYyJ9fSwiU3ltYm9scyB1c2VkIGFzIGFuIG9wdGlvbiBu │ │ │ YW1lIixUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IlRocmVhZHMiLCJUaHJlYWRzIiwiTWFjYXVs │ │ │ -YXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiU3RyYXRlZ3kiLCJTdHJhdGVneSIs │ │ │ -Ik1hY2F1bGF5MkRvYyJ9fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkdlbmVyaWMiLCJHZW5l │ │ │ -cmljIiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiRGVncmVlR3Jv │ │ │ +YXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiR2VuZXJpYyIsIkdlbmVyaWMiLCJN │ │ │ +YWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFnIGZyb20geyJTdHJhdGVneSIsIlN0cmF0 │ │ │ +ZWd5IiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiRGVncmVlR3Jv │ │ │ dXAiLCJEZWdyZWVHcm91cCIsIk1hY2F1bGF5MkRvYyJ9fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJv │ │ │ bSB7IlZhcmlhYmxlcyIsIlZhcmlhYmxlcyIsIk1hY2F1bGF5MkRvYyJ9fSxUT3tuZXcgRG9jdW1l │ │ │ bnRUYWcgZnJvbSB7IkRlZ3JlZUxpZnQiLCJEZWdyZWVMaWZ0IiwiTWFjYXVsYXkyRG9jIn19LFRP │ │ │ e25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVmFyaWFibGVCYXNlTmFtZSIsIlZhcmlhYmxlQmFzZU5h │ │ │ bWUiLCJNYWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFnIGZyb20geyJEZWdyZWVSYW5r │ │ │ IiwiRGVncmVlUmFuayIsIk1hY2F1bGF5MkRvYyJ9fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7 │ │ │ IkludmVyc2VzIiwiSW52ZXJzZXMiLCJNYWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFn │ │ │ @@ -12992,21 +12992,21 @@ │ │ │ dFRhZyBmcm9tIHsiTW9ub21pYWxTaXplIiwiTW9ub21pYWxTaXplIiwiTWFjYXVsYXkyRG9jIn19 │ │ │ LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiTG9jYWwiLCJMb2NhbCIsIk1hY2F1bGF5MkRvYyJ9 │ │ │ fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkhlZnQiLCJIZWZ0IiwiTWFjYXVsYXkyRG9jIn19 │ │ │ LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiQ29uc3RhbnRzIiwiQ29uc3RhbnRzIiwiTWFjYXVs │ │ │ YXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiU2tld0NvbW11dGF0aXZlIiwiU2tl │ │ │ d0NvbW11dGF0aXZlIiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsi │ │ │ RGVncmVlTWFwIiwiRGVncmVlTWFwIiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRh │ │ │ -ZyBmcm9tIHsiVmVyYm9zaXR5IiwiVmVyYm9zaXR5IiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBE │ │ │ -b2N1bWVudFRhZyBmcm9tIHsiQ29kaW1lbnNpb25MaW1pdCIsIkNvZGltZW5zaW9uTGltaXQiLCJN │ │ │ -YWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFnIGZyb20geyJEZWdyZWVMaW1pdCIsIkRl │ │ │ -Z3JlZUxpbWl0IiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVmVy │ │ │ -aWZ5IiwiVmVyaWZ5IiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsi │ │ │ -QmFzaXNFbGVtZW50TGltaXQiLCJCYXNpc0VsZW1lbnRMaW1pdCIsIk1hY2F1bGF5MkRvYyJ9fSxU │ │ │ -T3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IlBhaXJMaW1pdCIsIlBhaXJMaW1pdCIsIk1hY2F1bGF5 │ │ │ +ZyBmcm9tIHsiRGVncmVlTGltaXQiLCJEZWdyZWVMaW1pdCIsIk1hY2F1bGF5MkRvYyJ9fSxUT3tu │ │ │ +ZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkJhc2lzRWxlbWVudExpbWl0IiwiQmFzaXNFbGVtZW50TGlt │ │ │ +aXQiLCJNYWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFnIGZyb20geyJQYWlyTGltaXQi │ │ │ +LCJQYWlyTGltaXQiLCJNYWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFnIGZyb20geyJW │ │ │ +ZXJib3NpdHkiLCJWZXJib3NpdHkiLCJNYWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50VGFn │ │ │ +IGZyb20geyJDb2RpbWVuc2lvbkxpbWl0IiwiQ29kaW1lbnNpb25MaW1pdCIsIk1hY2F1bGF5MkRv │ │ │ +YyJ9fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IlZlcmlmeSIsIlZlcmlmeSIsIk1hY2F1bGF5 │ │ │ MkRvYyJ9fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkNvZWZmaWNpZW50UmluZyIsIkNvZWZm │ │ │ aWNpZW50UmluZyIsIk1hY2F1bGF5MkRvYyJ9fSxUT3tuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkZv │ │ │ bGxvd0xpbmtzIiwiRm9sbG93TGlua3MiLCJNYWNhdWxheTJEb2MifX0sVE97bmV3IERvY3VtZW50 │ │ │ VGFnIGZyb20geyJFeGNsdWRlIiwiRXhjbHVkZSIsIk1hY2F1bGF5MkRvYyJ9fSxUT3tuZXcgRG9j │ │ │ dW1lbnRUYWcgZnJvbSB7Ikluc3RhbGxQcmVmaXgiLCJJbnN0YWxsUHJlZml4IiwiTWFjYXVsYXky │ │ │ RG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiU3l6eWd5TWF0cml4IiwiU3l6eWd5TWF0 │ │ │ cml4IiwiTWFjYXVsYXkyRG9jIn19LFRPe25ldyBEb2N1bWVudFRhZyBmcm9tIHsiQ2hhbmdlTWF0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out │ │ │ @@ -5,12 +5,12 @@ │ │ │ i2 : f │ │ │ │ │ │ o2 = 1073741824 │ │ │ │ │ │ i3 : (c = Command "date";) │ │ │ │ │ │ i4 : c │ │ │ -Sun Feb 9 23:54:36 UTC 2025 │ │ │ +Sat Jul 19 03:40:32 UTC 2025 │ │ │ │ │ │ o4 = 0 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 9579076464446459296 │ │ │ │ │ │ i1 : filename = temporaryFileName () | ".dbm" │ │ │ │ │ │ -o1 = /tmp/M2-11758-0/0.dbm │ │ │ +o1 = /tmp/M2-13205-0/0.dbm │ │ │ │ │ │ i2 : x = openDatabaseOut filename │ │ │ │ │ │ -o2 = /tmp/M2-11758-0/0.dbm │ │ │ +o2 = /tmp/M2-13205-0/0.dbm │ │ │ │ │ │ o2 : Database │ │ │ │ │ │ i3 : x#"first" = "hi there" │ │ │ │ │ │ o3 = hi there │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Fast__Nonminimal.out │ │ │ @@ -9,25 +9,25 @@ │ │ │ i2 : S = ring I │ │ │ │ │ │ o2 = S │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ │ │ i3 : elapsedTime C = res(I, FastNonminimal => true) │ │ │ - -- 1.80778s elapsed │ │ │ + -- 1.86095s elapsed │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 576 135 14 │ │ │ o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 11 │ │ │ │ │ │ o3 : ChainComplex │ │ │ │ │ │ i4 : elapsedTime C1 = res ideal(I_*) │ │ │ - -- .973592s elapsed │ │ │ + -- .845975s elapsed │ │ │ │ │ │ 1 35 140 385 819 1080 819 385 140 35 1 │ │ │ o4 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 11 │ │ │ │ │ │ o4 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___File_sp_lt_lt_sp__Thing.out │ │ │ @@ -12,19 +12,19 @@ │ │ │ │ │ │ o2 = stdio │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : fn = temporaryFileName() │ │ │ │ │ │ -o3 = /tmp/M2-11872-0/0 │ │ │ +o3 = /tmp/M2-13439-0/0 │ │ │ │ │ │ i4 : fn << "hi there" << endl << close │ │ │ │ │ │ -o4 = /tmp/M2-11872-0/0 │ │ │ +o4 = /tmp/M2-13439-0/0 │ │ │ │ │ │ o4 : File │ │ │ │ │ │ i5 : get fn │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ @@ -49,27 +49,27 @@ │ │ │ x + 10x + 45x + 120x + 210x + 252x + 210x + 120x + 45x + 10x + 1 │ │ │ o8 = stdio │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : fn << f << close │ │ │ │ │ │ -o9 = /tmp/M2-11872-0/0 │ │ │ +o9 = /tmp/M2-13439-0/0 │ │ │ │ │ │ o9 : File │ │ │ │ │ │ i10 : get fn │ │ │ │ │ │ o10 = 10 9 8 7 6 5 4 3 2 │ │ │ x + 10x + 45x + 120x + 210x + 252x + 210x + 120x + 45x + 10x │ │ │ + 1 │ │ │ │ │ │ i11 : fn << toExternalString f << close │ │ │ │ │ │ -o11 = /tmp/M2-11872-0/0 │ │ │ +o11 = /tmp/M2-13439-0/0 │ │ │ │ │ │ o11 : File │ │ │ │ │ │ i12 : get fn │ │ │ │ │ │ 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+ │ │ │ 1 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ @@ -1,19 +1,19 @@ │ │ │ -- -*- M2-comint -*- hash: 1731899428494721487 │ │ │ │ │ │ i1 : s = GCstats() │ │ │ │ │ │ -o1 = HashTable{"bytesAlloc" => 15334137450 } │ │ │ +o1 = HashTable{"bytesAlloc" => 15449393626 } │ │ │ "GC_free_space_divisor" => 3 │ │ │ "GC_LARGE_ALLOC_WARN_INTERVAL" => 1 │ │ │ "gcCpuTimeSecs" => 0 │ │ │ - "heapSize" => 194183168 │ │ │ - "numGCs" => 836 │ │ │ - "numGCThreads" => 6 │ │ │ + "heapSize" => 226713600 │ │ │ + "numGCs" => 824 │ │ │ + "numGCThreads" => 16 │ │ │ │ │ │ o1 : HashTable │ │ │ │ │ │ i2 : s#"heapSize" │ │ │ │ │ │ -o2 = 194183168 │ │ │ +o2 = 226713600 │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ @@ -40,26 +40,26 @@ │ │ │ o6 : PolynomialRing │ │ │ │ │ │ i7 : I = truncate(8, monomialCurveIdeal(R,{1,4,5,9})); │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : time gens gb I; │ │ │ - -- used 0.0628534s (cpu); 0.0628529s (thread); 0s (gc) │ │ │ + -- used 0.0217891s (cpu); 0.0217888s (thread); 0s (gc) │ │ │ │ │ │ 1 428 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ i9 : time J1 = saturate(I); │ │ │ - -- used 0.605726s (cpu); 0.339625s (thread); 0s (gc) │ │ │ + -- used 0.842929s (cpu); 0.221206s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : time J = saturate(I, MinimalGenerators => false); │ │ │ - -- used 0.000202009s (cpu); 0.000202159s (thread); 0s (gc) │ │ │ + -- used 0.000106109s (cpu); 0.00010645s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : numgens J │ │ │ │ │ │ o11 = 7 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ @@ -3,13 +3,13 @@ │ │ │ i1 : M = random(RR^200, RR^200); │ │ │ │ │ │ 200 200 │ │ │ o1 : Matrix RR <-- RR │ │ │ 53 53 │ │ │ │ │ │ i2 : time SVD(M); │ │ │ - -- used 0.0510253s (cpu); 0.0510246s (thread); 0s (gc) │ │ │ + -- used 0.0263697s (cpu); 0.0263675s (thread); 0s (gc) │ │ │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ - -- used 0.0514493s (cpu); 0.0514556s (thread); 0s (gc) │ │ │ + -- used 0.0263246s (cpu); 0.026337s (thread); 0s (gc) │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out │ │ │ @@ -351,15 +351,15 @@ │ │ │ | b e h k n q | │ │ │ | c f i l o r | │ │ │ │ │ │ 3 │ │ │ o58 : R-module, quotient of R │ │ │ │ │ │ i59 : time C = resolution M │ │ │ - -- used 0.00187636s (cpu); 0.00186805s (thread); 0s (gc) │ │ │ + -- used 0.00145835s (cpu); 0.0014456s (thread); 0s (gc) │ │ │ │ │ │ 3 6 15 18 6 │ │ │ o59 = R <-- R <-- R <-- R <-- R <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ o59 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ @@ -14,10 +14,10 @@ │ │ │ │ │ │ i4 : peek read f │ │ │ │ │ │ o4 = "hi there" │ │ │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ -o5 = false │ │ │ +o5 = true │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1330379359420 │ │ │ │ │ │ i1 : benchmark "sqrt 2p100000" │ │ │ │ │ │ -o1 = .0002909794917276939 │ │ │ +o1 = .0003412512839561764 │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i2 : S = ring I │ │ │ │ │ │ o2 = S │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ │ │ i3 : elapsedTime C = res(I, FastNonminimal => true) │ │ │ - -- 2.45141s elapsed │ │ │ + -- 2.12843s elapsed │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 576 135 14 │ │ │ o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 11 │ │ │ │ │ │ o3 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ @@ -2,15 +2,15 @@ │ │ │ │ │ │ i1 : n = 0 │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : t = schedule(() -> while true do n = n+1) │ │ │ │ │ │ -o2 = <> │ │ │ +o2 = <> │ │ │ │ │ │ o2 : Task │ │ │ │ │ │ i3 : sleep 1 │ │ │ │ │ │ o3 = 0 │ │ │ │ │ │ @@ -18,29 +18,29 @@ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ o4 : Task │ │ │ │ │ │ i5 : n │ │ │ │ │ │ -o5 = 709345 │ │ │ +o5 = 1093202 │ │ │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ o6 = 0 │ │ │ │ │ │ i7 : t │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ o7 : Task │ │ │ │ │ │ i8 : n │ │ │ │ │ │ -o8 = 1451545 │ │ │ +o8 = 2219186 │ │ │ │ │ │ i9 : isReady t │ │ │ │ │ │ o9 = false │ │ │ │ │ │ i10 : cancelTask t │ │ │ │ │ │ @@ -53,22 +53,22 @@ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ o12 : Task │ │ │ │ │ │ i13 : n │ │ │ │ │ │ -o13 = 1451960 │ │ │ +o13 = 2219384 │ │ │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ o14 = 0 │ │ │ │ │ │ i15 : n │ │ │ │ │ │ -o15 = 1451960 │ │ │ +o15 = 2219384 │ │ │ │ │ │ i16 : isReady t │ │ │ │ │ │ o16 = false │ │ │ │ │ │ i17 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ @@ -1,19 +1,19 @@ │ │ │ -- -*- M2-comint -*- hash: 8535510246140175278 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10626-0/0 │ │ │ +o1 = /tmp/M2-10905-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10626-0/0 │ │ │ +o2 = /tmp/M2-10905-0/0 │ │ │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ -o3 = /tmp/M2-10626-0/0/ │ │ │ +o3 = /tmp/M2-10905-0/0/ │ │ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ -o4 = /tmp/M2-10626-0/0/ │ │ │ +o4 = /tmp/M2-10905-0/0/ │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ @@ -1,27 +1,27 @@ │ │ │ -- -*- M2-comint -*- hash: 10986518019608335719 │ │ │ │ │ │ i1 : run "uname -a" │ │ │ -Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 (2025-02-07) x86_64 GNU/Linux │ │ │ +Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.35-1 (2025-07-03) x86_64 GNU/Linux │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close │ │ │ ba │ │ │ ad │ │ │ │ │ │ o2 = !grep a │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : peek get "!uname -a" │ │ │ │ │ │ -o3 = "Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 │ │ │ +o3 = "Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ " │ │ │ - (2025-02-07) x86_64 GNU/Linux │ │ │ + 6.12.35-1 (2025-07-03) x86_64 GNU/Linux │ │ │ │ │ │ i4 : f = openInOut "!egrep '^in'" │ │ │ │ │ │ o4 = !egrep '^in' │ │ │ │ │ │ o4 : File │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ @@ -126,15 +126,15 @@ │ │ │ │ │ │ ZZ │ │ │ o23 : Ideal of ----[x..z, w] │ │ │ 1277 │ │ │ │ │ │ i24 : gb I │ │ │ │ │ │ - -- registering gb 5 at 0x7fe96c5bd700 │ │ │ + -- registering gb 5 at 0x7fdd33824700 │ │ │ │ │ │ -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4 │ │ │ -- number of monomials = 8 │ │ │ -- #reduction steps = 2 │ │ │ -- #spairs done = 6 │ │ │ -- ncalls = 0 │ │ │ -- nloop = 0 │ │ │ @@ -177,15 +177,15 @@ │ │ │ │ │ │ i32 : f = random(R^1,R^{-3,-3,-5,-6}); │ │ │ │ │ │ 1 4 │ │ │ o32 : Matrix R <-- R │ │ │ │ │ │ i33 : time betti gb f │ │ │ - -- used 0.307492s (cpu); 0.307469s (thread); 0s (gc) │ │ │ + -- used 0.152258s (cpu); 0.151986s (thread); 0s (gc) │ │ │ │ │ │ 0 1 │ │ │ o33 = total: 1 53 │ │ │ 0: 1 . │ │ │ 1: . . │ │ │ 2: . 2 │ │ │ 3: . 1 │ │ │ @@ -208,15 +208,15 @@ │ │ │ │ │ │ 3 5 8 9 12 14 17 │ │ │ o35 = 1 - 2T - T + 2T + 2T - T - 2T + T │ │ │ │ │ │ o35 : ZZ[T] │ │ │ │ │ │ i36 : time betti gb f │ │ │ - -- used 0.00800087s (cpu); 0.00553291s (thread); 0s (gc) │ │ │ + -- used 0.00179524s (cpu); 0.00217529s (thread); 0s (gc) │ │ │ │ │ │ 0 1 │ │ │ o36 = total: 1 53 │ │ │ 0: 1 . │ │ │ 1: . . │ │ │ 2: . 2 │ │ │ 3: . 1 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spresolutions.out │ │ │ @@ -36,16 +36,16 @@ │ │ │ << res M << endl << endl; │ │ │ break; │ │ │ ) else ( │ │ │ << "-- computation interrupted" << endl; │ │ │ status M.cache.resolution; │ │ │ << "-- continuing the computation" << endl; │ │ │ )) │ │ │ - -- used 1.30377s (cpu); 0.973789s (thread); 0s (gc) │ │ │ - -- used 0.508987s (cpu); 0.396067s (thread); 0s (gc) │ │ │ + -- used 1.13974s (cpu); 0.98458s (thread); 0s (gc) │ │ │ + -- used 0.232854s (cpu); 0.148313s (thread); 0s (gc) │ │ │ -- computation started: │ │ │ -- computation interrupted │ │ │ -- continuing the computation │ │ │ -- computation complete │ │ │ 4 11 89 122 40 │ │ │ R <-- R <-- R <-- R <-- R <-- 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__Directory_lp__String_cm__String_rp.out │ │ │ @@ -1,76 +1,76 @@ │ │ │ -- -*- M2-comint -*- hash: 11422793294564310273 │ │ │ │ │ │ i1 : src = temporaryFileName() | "/" │ │ │ │ │ │ -o1 = /tmp/M2-11326-0/0/ │ │ │ +o1 = /tmp/M2-12313-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11326-0/1/ │ │ │ +o2 = /tmp/M2-12313-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11326-0/0/a/ │ │ │ +o3 = /tmp/M2-12313-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11326-0/0/b/ │ │ │ +o4 = /tmp/M2-12313-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11326-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12313-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11326-0/0/a/f │ │ │ +o6 = /tmp/M2-12313-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11326-0/0/a/g │ │ │ +o7 = /tmp/M2-12313-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11326-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12313-0/0/b/c/g │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : stack findFiles src │ │ │ │ │ │ -o9 = /tmp/M2-11326-0/0/ │ │ │ - /tmp/M2-11326-0/0/b/ │ │ │ - /tmp/M2-11326-0/0/b/c/ │ │ │ - /tmp/M2-11326-0/0/b/c/g │ │ │ - /tmp/M2-11326-0/0/a/ │ │ │ - /tmp/M2-11326-0/0/a/g │ │ │ - /tmp/M2-11326-0/0/a/f │ │ │ +o9 = /tmp/M2-12313-0/0/ │ │ │ + /tmp/M2-12313-0/0/b/ │ │ │ + /tmp/M2-12313-0/0/b/c/ │ │ │ + /tmp/M2-12313-0/0/b/c/g │ │ │ + /tmp/M2-12313-0/0/a/ │ │ │ + /tmp/M2-12313-0/0/a/f │ │ │ + /tmp/M2-12313-0/0/a/g │ │ │ │ │ │ i10 : copyDirectory(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-11326-0/0/b/c/g -> /tmp/M2-11326-0/1/b/c/g │ │ │ - -- copying: /tmp/M2-11326-0/0/a/g -> /tmp/M2-11326-0/1/a/g │ │ │ - -- copying: /tmp/M2-11326-0/0/a/f -> /tmp/M2-11326-0/1/a/f │ │ │ + -- copying: /tmp/M2-12313-0/0/b/c/g -> /tmp/M2-12313-0/1/b/c/g │ │ │ + -- copying: /tmp/M2-12313-0/0/a/f -> /tmp/M2-12313-0/1/a/f │ │ │ + -- copying: /tmp/M2-12313-0/0/a/g -> /tmp/M2-12313-0/1/a/g │ │ │ │ │ │ i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11326-0/0/b/c/g not newer than /tmp/M2-11326-0/1/b/c/g │ │ │ - -- skipping: /tmp/M2-11326-0/0/a/g not newer than /tmp/M2-11326-0/1/a/g │ │ │ - -- skipping: /tmp/M2-11326-0/0/a/f not newer than /tmp/M2-11326-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12313-0/0/b/c/g not newer than /tmp/M2-12313-0/1/b/c/g │ │ │ + -- skipping: /tmp/M2-12313-0/0/a/f not newer than /tmp/M2-12313-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12313-0/0/a/g not newer than /tmp/M2-12313-0/1/a/g │ │ │ │ │ │ i12 : stack findFiles dst │ │ │ │ │ │ -o12 = /tmp/M2-11326-0/1/ │ │ │ - /tmp/M2-11326-0/1/a/ │ │ │ - /tmp/M2-11326-0/1/a/f │ │ │ - /tmp/M2-11326-0/1/a/g │ │ │ - /tmp/M2-11326-0/1/b/ │ │ │ - /tmp/M2-11326-0/1/b/c/ │ │ │ - /tmp/M2-11326-0/1/b/c/g │ │ │ +o12 = /tmp/M2-12313-0/1/ │ │ │ + /tmp/M2-12313-0/1/b/ │ │ │ + /tmp/M2-12313-0/1/b/c/ │ │ │ + /tmp/M2-12313-0/1/b/c/g │ │ │ + /tmp/M2-12313-0/1/a/ │ │ │ + /tmp/M2-12313-0/1/a/f │ │ │ + /tmp/M2-12313-0/1/a/g │ │ │ │ │ │ i13 : get (dst|"b/c/g") │ │ │ │ │ │ o13 = ho there │ │ │ │ │ │ i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out │ │ │ @@ -1,41 +1,41 @@ │ │ │ -- -*- M2-comint -*- hash: 11539475420155775110 │ │ │ │ │ │ i1 : src = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11095-0/0 │ │ │ +o1 = /tmp/M2-11862-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-11095-0/1 │ │ │ +o2 = /tmp/M2-11862-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11095-0/0 │ │ │ +o3 = /tmp/M2-11862-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : copyFile(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-11095-0/0 -> /tmp/M2-11095-0/1 │ │ │ + -- copying: /tmp/M2-11862-0/0 -> /tmp/M2-11862-0/1 │ │ │ │ │ │ i5 : get dst │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11095-0/0 not newer than /tmp/M2-11095-0/1 │ │ │ + -- skipping: /tmp/M2-11862-0/0 not newer than /tmp/M2-11862-0/1 │ │ │ │ │ │ i7 : src << "ho there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11095-0/0 │ │ │ +o7 = /tmp/M2-11862-0/0 │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11095-0/0 not newer than /tmp/M2-11095-0/1 │ │ │ + -- skipping: /tmp/M2-11862-0/0 not newer than /tmp/M2-11862-0/1 │ │ │ │ │ │ i9 : get dst │ │ │ │ │ │ o9 = hi there │ │ │ │ │ │ i10 : removeFile src │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ @@ -1,23 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 15508153783232232453 │ │ │ │ │ │ i1 : t1 = cpuTime() │ │ │ │ │ │ -o1 = 258.991604772 │ │ │ +o1 = 244.703091515 │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ │ │ i2 : for i from 0 to 1000000 do 223131321321*324234324324; │ │ │ │ │ │ i3 : t2 = cpuTime() │ │ │ │ │ │ -o3 = 261.1503754559999 │ │ │ +o3 = 245.69947021 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : t2-t1 │ │ │ │ │ │ -o4 = 2.15877068399999 │ │ │ +o4 = .9963786950000042 │ │ │ │ │ │ o4 : RR (of precision 53) │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out │ │ │ @@ -1,24 +1,24 @@ │ │ │ -- -*- M2-comint -*- hash: 3660839476107967259 │ │ │ │ │ │ i1 : currentTime() │ │ │ │ │ │ -o1 = 1739145345 │ │ │ +o1 = 1752896489 │ │ │ │ │ │ i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 ) │ │ │ │ │ │ -o2 = 55.11132206304336 │ │ │ +o2 = 55.54707846942887 │ │ │ │ │ │ o2 : RR (of precision 53) │ │ │ │ │ │ i3 : 12 * (oo - floor oo) │ │ │ │ │ │ -o3 = 1.335864756520323 │ │ │ +o3 = 6.564941633146418 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : run "date" │ │ │ -Sun Feb 9 23:55:45 UTC 2025 │ │ │ +Sat Jul 19 03:41:29 UTC 2025 │ │ │ │ │ │ o4 = 0 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ @@ -1,8 +1,8 @@ │ │ │ -- -*- M2-comint -*- hash: 1330565958025 │ │ │ │ │ │ i1 : elapsedTime sleep 1 │ │ │ - -- 1.00014s elapsed │ │ │ + -- 1.00013s elapsed │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ @@ -1,14 +1,14 @@ │ │ │ -- -*- M2-comint -*- hash: 1731106803207298715 │ │ │ │ │ │ i1 : elapsedTiming sleep 1 │ │ │ │ │ │ o1 = 0 │ │ │ - -- 1.00014 seconds │ │ │ + -- 1.00009 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{1.00014, 0} │ │ │ +o2 = Time{1.00009, 0} │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ @@ -6,15 +6,15 @@ │ │ │ │ │ │ 3 3 2 3 │ │ │ o2 = ideal (- s - s*t + x - 1, - t - 3t - t + y, - s*t + z) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : time leadTerm gens gb I │ │ │ - -- used 0.254683s (cpu); 0.254684s (thread); 0s (gc) │ │ │ + -- used 0.117515s (cpu); 0.117267s (thread); 0s (gc) │ │ │ │ │ │ o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4 │ │ │ ------------------------------------------------------------------------ │ │ │ 563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2 │ │ │ ------------------------------------------------------------------------ │ │ │ 267076255345488270sy3z4 5256861933965245618410txyz6 │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ 3 3 2 3 │ │ │ o7 = ideal (- s - s*t + x - 1, - t - 3t + y - t, - s*t + z) │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : time G = eliminate(I,{s,t}) │ │ │ - -- used 0.4515s (cpu); 0.275672s (thread); 0s (gc) │ │ │ + -- used 0.121517s (cpu); 0.121522s (thread); 0s (gc) │ │ │ │ │ │ 3 9 2 9 2 8 2 6 3 9 2 7 8 │ │ │ o8 = ideal(x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z + │ │ │ ------------------------------------------------------------------------ │ │ │ 7 2 2 5 3 6 3 7 3 5 4 3 6 9 7 │ │ │ 7x*y z - 324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -154,15 +154,15 @@ │ │ │ i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}]; │ │ │ │ │ │ i11 : I1 = substitute(I,R1); │ │ │ │ │ │ o11 : Ideal of R1 │ │ │ │ │ │ i12 : time G = eliminate(I1,{s,t}) │ │ │ - -- used 0.0600603s (cpu); 0.060069s (thread); 0s (gc) │ │ │ + -- used 0.0460749s (cpu); 0.0460784s (thread); 0s (gc) │ │ │ │ │ │ 3 9 2 6 3 3 6 9 2 8 5 4 2 7 │ │ │ o12 = ideal(x y - 3x y z + 3x*y z - z - 6x y z - 15x*y z + 21y z - │ │ │ ----------------------------------------------------------------------- │ │ │ 2 9 2 5 3 6 3 7 3 2 6 3 6 7 2 │ │ │ 3x y - 324x y z + 6x*y z - y z - 405x*y z - 3y z + 7x*y z - │ │ │ ----------------------------------------------------------------------- │ │ │ @@ -228,15 +228,15 @@ │ │ │ │ │ │ 3 3 2 3 │ │ │ o16 = map (A, B, {s + s*t + 1, t + 3t + t, s*t }) │ │ │ │ │ │ o16 : RingMap A <-- B │ │ │ │ │ │ i17 : time G = kernel F │ │ │ - -- used 0.404083s (cpu); 0.234457s (thread); 0s (gc) │ │ │ + -- used 0.320628s (cpu); 0.139295s (thread); 0s (gc) │ │ │ │ │ │ 3 9 2 9 2 8 2 6 3 9 2 7 8 │ │ │ o17 = ideal(x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z + │ │ │ ----------------------------------------------------------------------- │ │ │ 7 2 2 5 3 6 3 7 3 5 4 3 6 9 7 │ │ │ 7x*y z - 324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z │ │ │ ----------------------------------------------------------------------- │ │ │ @@ -297,23 +297,23 @@ │ │ │ i19 : use ring I │ │ │ │ │ │ o19 = R │ │ │ │ │ │ o19 : PolynomialRing │ │ │ │ │ │ i20 : time f1 = resultant(I_0,I_2,s) │ │ │ - -- used 0.00172022s (cpu); 0.00171993s (thread); 0s (gc) │ │ │ + -- used 0.00135128s (cpu); 0.00134931s (thread); 0s (gc) │ │ │ │ │ │ 9 9 7 3 │ │ │ o20 = x*t - t - z*t - z │ │ │ │ │ │ o20 : R │ │ │ │ │ │ i21 : time f2 = resultant(I_1,f1,t) │ │ │ - -- used 0.0562473s (cpu); 0.0562272s (thread); 0s (gc) │ │ │ + -- used 0.033369s (cpu); 0.033381s (thread); 0s (gc) │ │ │ │ │ │ 3 9 2 9 2 8 2 6 3 9 2 7 8 7 2 │ │ │ o21 = - x y + 3x y + 6x y z + 3x y z - 3x*y + x y z - 12x*y z - 7x*y z + │ │ │ ----------------------------------------------------------------------- │ │ │ 2 5 3 6 3 7 3 5 4 3 6 9 7 8 │ │ │ 324x y z - 6x*y z + y z + 15x*y z - 3x*y z + y - 2x*y z + 6y z + │ │ │ ----------------------------------------------------------------------- │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ @@ -59,15 +59,15 @@ │ │ │ Version => 0.0 │ │ │ package prefix => /usr/ │ │ │ PackageIsLoaded => true │ │ │ pkgname => Foo │ │ │ private dictionary => Foo#"private dictionary" │ │ │ processed documentation => MutableHashTable{} │ │ │ raw documentation => MutableHashTable{} │ │ │ - source directory => /tmp/M2-10387-0/88-rundir/ │ │ │ + source directory => /tmp/M2-10448-0/88-rundir/ │ │ │ source file => stdio │ │ │ test inputs => MutableList{} │ │ │ │ │ │ i7 : dictionaryPath │ │ │ │ │ │ o7 = {Foo.Dictionary, PackageCitations.Dictionary, Varieties.Dictionary, │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 7475038936570224899 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10721-0/0 │ │ │ +o1 = /tmp/M2-11100-0/0 │ │ │ │ │ │ i2 : fileExists fn │ │ │ │ │ │ o2 = false │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10721-0/0 │ │ │ +o3 = /tmp/M2-11100-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : fileExists fn │ │ │ │ │ │ o4 = true │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out │ │ │ @@ -1,28 +1,28 @@ │ │ │ -- -*- M2-comint -*- hash: 1216695447195237994 │ │ │ │ │ │ i1 : f = temporaryFileName() << "hi there" │ │ │ │ │ │ -o1 = /tmp/M2-12302-0/0 │ │ │ +o1 = /tmp/M2-14347-0/0 │ │ │ │ │ │ o1 : File │ │ │ │ │ │ i2 : fileLength f │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : close f │ │ │ │ │ │ -o3 = /tmp/M2-12302-0/0 │ │ │ +o3 = /tmp/M2-14347-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : filename = toString f │ │ │ │ │ │ -o4 = /tmp/M2-12302-0/0 │ │ │ +o4 = /tmp/M2-14347-0/0 │ │ │ │ │ │ i5 : fileLength filename │ │ │ │ │ │ o5 = 8 │ │ │ │ │ │ i6 : get filename │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out │ │ │ @@ -1,25 +1,25 @@ │ │ │ -- -*- M2-comint -*- hash: 11202140621123993633 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11497-0/0 │ │ │ +o1 = /tmp/M2-12664-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-11497-0/0 │ │ │ +o2 = /tmp/M2-12664-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : fileMode f │ │ │ │ │ │ o3 = 420 │ │ │ │ │ │ i4 : close f │ │ │ │ │ │ -o4 = /tmp/M2-11497-0/0 │ │ │ +o4 = /tmp/M2-12664-0/0 │ │ │ │ │ │ o4 : File │ │ │ │ │ │ i5 : removeFile fn │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 4782570202197464532 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11114-0/0 │ │ │ +o1 = /tmp/M2-11901-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-11114-0/0 │ │ │ +o2 = /tmp/M2-11901-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : fileMode fn │ │ │ │ │ │ o3 = 420 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 17473878267845575442 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10998-0/0 │ │ │ +o1 = /tmp/M2-11665-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-10998-0/0 │ │ │ +o2 = /tmp/M2-11665-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : m = 7 + 7*8 + 7*64 │ │ │ │ │ │ o3 = 511 │ │ │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ │ │ i5 : fileMode f │ │ │ │ │ │ o5 = 511 │ │ │ │ │ │ i6 : close f │ │ │ │ │ │ -o6 = /tmp/M2-10998-0/0 │ │ │ +o6 = /tmp/M2-11665-0/0 │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : fileMode fn │ │ │ │ │ │ o7 = 511 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 16772784390799334723 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-12132-0/0 │ │ │ +o1 = /tmp/M2-14017-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12132-0/0 │ │ │ +o2 = /tmp/M2-14017-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : m = fileMode fn │ │ │ │ │ │ o3 = 420 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out │ │ │ @@ -1,7 +1,7 @@ │ │ │ -- -*- M2-comint -*- hash: 1331310711075 │ │ │ │ │ │ i1 : currentTime() - fileTime "." │ │ │ │ │ │ -o1 = 48 │ │ │ +o1 = 39 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.out │ │ │ @@ -29,15 +29,15 @@ │ │ │ {4} | 0 x2-3 y3-1 | │ │ │ │ │ │ 3 3 │ │ │ o6 : Matrix R <-- R │ │ │ │ │ │ i7 : syz f │ │ │ │ │ │ - -- registering gb 0 at 0x7f53c265d000 │ │ │ + -- registering gb 0 at 0x7f598969b000 │ │ │ │ │ │ -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3 │ │ │ -- number of monomials = 9 │ │ │ -- #reduction steps = 6 │ │ │ -- #spairs done = 6 │ │ │ -- ncalls = 0 │ │ │ -- nloop = 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out │ │ │ @@ -10,11 +10,11 @@ │ │ │ │ │ │ o2 = hi there │ │ │ │ │ │ i3 : removeFile "test-file" │ │ │ │ │ │ i4 : get "!date" │ │ │ │ │ │ -o4 = Sun Feb 9 23:55:06 UTC 2025 │ │ │ +o4 = Sat Jul 19 03:40:58 UTC 2025 │ │ │ │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances_lp__Type_rp.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ defaultPrecision => 53 │ │ │ engineDebugLevel => 0 │ │ │ errorDepth => 0 │ │ │ gbTrace => 0 │ │ │ interpreterDepth => 1 │ │ │ lineNumber => 2 │ │ │ loadDepth => 3 │ │ │ - maxAllowableThreads => 7 │ │ │ + maxAllowableThreads => 17 │ │ │ maxExponent => 1073741823 │ │ │ minExponent => -1073741824 │ │ │ numTBBThreads => 0 │ │ │ o1 => 2432902008176640000 │ │ │ oo => 2432902008176640000 │ │ │ printingAccuracy => -1 │ │ │ printingLeadLimit => 5 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Canceled_lp__Task_rp.out │ │ │ @@ -2,15 +2,15 @@ │ │ │ │ │ │ i1 : n = 0 │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : t = schedule(() -> while true do n = n + 1) │ │ │ │ │ │ -o2 = <> │ │ │ +o2 = <> │ │ │ │ │ │ o2 : Task │ │ │ │ │ │ i3 : sleep 1 │ │ │ │ │ │ o3 = 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Directory.out │ │ │ @@ -2,19 +2,19 @@ │ │ │ │ │ │ i1 : isDirectory "." │ │ │ │ │ │ o1 = true │ │ │ │ │ │ i2 : fn = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10543-0/0 │ │ │ +o2 = /tmp/M2-10742-0/0 │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10543-0/0 │ │ │ +o3 = /tmp/M2-10742-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : isDirectory fn │ │ │ │ │ │ o4 = false │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ @@ -75,15 +75,15 @@ │ │ │ o17 = false │ │ │ │ │ │ i18 : isPrime(m*m*m1*m1*m2^6) │ │ │ │ │ │ o18 = false │ │ │ │ │ │ i19 : elapsedTime facs = factor(m*m1) │ │ │ - -- 4.34633s elapsed │ │ │ + -- 4.34639s elapsed │ │ │ │ │ │ o19 = 1000000000000000000000000000057*1000000000000000000010000000083 │ │ │ │ │ │ o19 : Expression of class Product │ │ │ │ │ │ i20 : facs = facs//toList/toList │ │ │ │ │ │ @@ -97,17 +97,17 @@ │ │ │ │ │ │ i22 : m3 = nextPrime (m^3) │ │ │ │ │ │ o22 = 10000000000000000000000000001710000000000000000000000000097470000000000 │ │ │ 00000000000000185613 │ │ │ │ │ │ i23 : elapsedTime isPrime m3 │ │ │ - -- .0563776s elapsed │ │ │ + -- .0467009s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isPseudoprime m3 │ │ │ - -- .000134241s elapsed │ │ │ + -- .000125735s elapsed │ │ │ │ │ │ o24 = true │ │ │ │ │ │ i25 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 4782205245758053629 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-12340-0/0 │ │ │ +o1 = /tmp/M2-14425-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12340-0/0 │ │ │ +o2 = /tmp/M2-14425-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : isRegularFile fn │ │ │ │ │ │ o3 = true │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 5113372159204571746 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10866-0/0 │ │ │ +o1 = /tmp/M2-11393-0/0 │ │ │ │ │ │ i2 : makeDirectory (dir|"/a/b/c") │ │ │ │ │ │ -o2 = /tmp/M2-10866-0/0/a/b/c │ │ │ +o2 = /tmp/M2-11393-0/0/a/b/c │ │ │ │ │ │ i3 : removeDirectory (dir|"/a/b/c") │ │ │ │ │ │ i4 : removeDirectory (dir|"/a/b") │ │ │ │ │ │ i5 : removeDirectory (dir|"/a") │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ @@ -1,7 +1,7 @@ │ │ │ -- -*- M2-comint -*- hash: 1331887830690 │ │ │ │ │ │ i1 : maxAllowableThreads │ │ │ │ │ │ -o1 = 7 │ │ │ +o1 = 17 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ @@ -3,31 +3,31 @@ │ │ │ i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2) │ │ │ │ │ │ o1 = fib │ │ │ │ │ │ o1 : FunctionClosure │ │ │ │ │ │ i2 : time fib 28 │ │ │ - -- used 1.31977s (cpu); 0.95198s (thread); 0s (gc) │ │ │ + -- used 0.894368s (cpu); 0.618538s (thread); 0s (gc) │ │ │ │ │ │ o2 = 514229 │ │ │ │ │ │ i3 : fib = memoize fib │ │ │ │ │ │ o3 = fib │ │ │ │ │ │ o3 : FunctionClosure │ │ │ │ │ │ i4 : time fib 28 │ │ │ - -- used 6.6595e-05s (cpu); 6.6164e-05s (thread); 0s (gc) │ │ │ + -- used 4.8801e-05s (cpu); 4.8141e-05s (thread); 0s (gc) │ │ │ │ │ │ o4 = 514229 │ │ │ │ │ │ i5 : time fib 28 │ │ │ - -- used 4.138e-06s (cpu); 3.807e-06s (thread); 0s (gc) │ │ │ + -- used 2.916e-06s (cpu); 2.365e-06s (thread); 0s (gc) │ │ │ │ │ │ o5 = 514229 │ │ │ │ │ │ i6 : fib = memoize( n -> fib(n-1) + fib(n-2) ) │ │ │ │ │ │ o6 = fib │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ @@ -18,20 +18,20 @@ │ │ │ {13 => (poincare, BettiTally) } │ │ │ {14 => (hilbertPolynomial, ZZ, BettiTally) } │ │ │ {15 => (degree, BettiTally) } │ │ │ {16 => (hilbertSeries, ZZ, BettiTally) } │ │ │ {17 => (^, Ring, BettiTally) } │ │ │ {18 => (regularity, BettiTally) } │ │ │ {19 => (mathML, BettiTally) } │ │ │ - {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)} │ │ │ - {21 => (truncate, BettiTally, ZZ, ZZ) } │ │ │ - {22 => (codim, BettiTally) } │ │ │ - {23 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ - {24 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ - {25 => (dual, BettiTally) } │ │ │ + {20 => (dual, BettiTally) } │ │ │ + {21 => (codim, BettiTally) } │ │ │ + {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)} │ │ │ + {23 => (truncate, BettiTally, ZZ, ZZ) } │ │ │ + {24 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ + {25 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ │ │ │ o1 : NumberedVerticalList │ │ │ │ │ │ i2 : methods resolution │ │ │ │ │ │ o2 = {0 => (resolution, Ideal) } │ │ │ {1 => (resolution, Module)} │ │ │ @@ -60,20 +60,20 @@ │ │ │ {1 => (++, Module, GradedModule)} │ │ │ {2 => (++, Module, Module) } │ │ │ │ │ │ o4 : NumberedVerticalList │ │ │ │ │ │ i5 : methods( Matrix, Matrix ) │ │ │ │ │ │ -o5 = {0 => (+, Matrix, Matrix) } │ │ │ - {1 => (-, Matrix, Matrix) } │ │ │ - {2 => (contract', Matrix, Matrix) } │ │ │ - {3 => (contract, Matrix, Matrix) } │ │ │ - {4 => (diff, Matrix, Matrix) } │ │ │ - {5 => (diff', Matrix, Matrix) } │ │ │ +o5 = {0 => (diff, Matrix, Matrix) } │ │ │ + {1 => (contract, Matrix, Matrix) } │ │ │ + {2 => (diff', Matrix, Matrix) } │ │ │ + {3 => (-, Matrix, Matrix) } │ │ │ + {4 => (contract', Matrix, Matrix) } │ │ │ + {5 => (+, Matrix, Matrix) } │ │ │ {6 => (markedGB, Matrix, Matrix) } │ │ │ {7 => (Hom, Matrix, Matrix) } │ │ │ {8 => (==, Matrix, Matrix) } │ │ │ {9 => (*, Matrix, Matrix) } │ │ │ {10 => (|, Matrix, Matrix) } │ │ │ {11 => (||, Matrix, Matrix) } │ │ │ {12 => (subquotient, Matrix, Matrix) } │ │ │ @@ -88,18 +88,18 @@ │ │ │ {21 => (quotient', Matrix, Matrix) } │ │ │ {22 => (quotient, Matrix, Matrix) } │ │ │ {23 => (remainder', Matrix, Matrix) } │ │ │ {24 => (remainder, Matrix, Matrix) } │ │ │ {25 => (%, Matrix, Matrix) } │ │ │ {26 => (pushout, Matrix, Matrix) } │ │ │ {27 => (solve, Matrix, Matrix) } │ │ │ - {28 => (intersection, Matrix, Matrix, Matrix, Matrix) } │ │ │ - {29 => (pullback, Matrix, Matrix) } │ │ │ - {30 => (tensor, Matrix, Matrix) } │ │ │ - {31 => (intersection, Matrix, Matrix) } │ │ │ + {28 => (pullback, Matrix, Matrix) } │ │ │ + {29 => (intersection, Matrix, Matrix, Matrix, Matrix) } │ │ │ + {30 => (intersection, Matrix, Matrix) } │ │ │ + {31 => (tensor, Matrix, Matrix) } │ │ │ {32 => (substitute, Matrix, Matrix) } │ │ │ {33 => (checkDegrees, Matrix, Matrix) } │ │ │ {34 => (isIsomorphic, Matrix, Matrix) } │ │ │ {35 => (coneFromVData, Matrix, Matrix) } │ │ │ {36 => (coneFromHData, Matrix, Matrix) } │ │ │ {37 => (fan, Matrix, Matrix, List) } │ │ │ {38 => (fan, Matrix, Matrix, Sequence) } │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i2 : S = ring I │ │ │ │ │ │ o2 = S │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ │ │ i3 : elapsedTime C = minimalBetti I │ │ │ - -- 2.37905s elapsed │ │ │ + -- 4.50648s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ o3 = total: 1 35 140 385 819 1080 819 385 140 35 1 │ │ │ 0: 1 . . . . . . . . . . │ │ │ 1: . 35 140 189 84 . . . . . . │ │ │ 2: . . . 196 735 1080 735 196 . . . │ │ │ 3: . . . . . . 84 189 140 35 . │ │ │ @@ -26,44 +26,44 @@ │ │ │ o3 : BettiTally │ │ │ │ │ │ i4 : I = ideal I_*; │ │ │ │ │ │ o4 : Ideal of S │ │ │ │ │ │ i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2) │ │ │ - -- .748075s elapsed │ │ │ + -- 1.64364s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ o5 = total: 1 35 140 385 819 1080 735 196 │ │ │ 0: 1 . . . . . . . │ │ │ 1: . 35 140 189 84 . . . │ │ │ 2: . . . 196 735 1080 735 196 │ │ │ │ │ │ o5 : BettiTally │ │ │ │ │ │ i6 : I = ideal I_*; │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5) │ │ │ - -- .0319761s elapsed │ │ │ + -- .0578398s elapsed │ │ │ │ │ │ 0 1 2 3 4 │ │ │ o7 = total: 1 35 140 189 84 │ │ │ 0: 1 . . . . │ │ │ 1: . 35 140 189 84 │ │ │ │ │ │ o7 : BettiTally │ │ │ │ │ │ i8 : I = ideal I_*; │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5) │ │ │ - -- 1.22946s elapsed │ │ │ + -- 1.38309s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ o9 = total: 1 35 140 385 819 1080 │ │ │ 0: 1 . . . . . │ │ │ 1: . 35 140 189 84 . │ │ │ 2: . . . 196 735 1080 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -1,22 +1,22 @@ │ │ │ -- -*- M2-comint -*- hash: 15555226809509933135 │ │ │ │ │ │ i1 : p = temporaryFileName() | "/" │ │ │ │ │ │ -o1 = /tmp/M2-10885-0/0/ │ │ │ +o1 = /tmp/M2-11432-0/0/ │ │ │ │ │ │ i2 : mkdir p │ │ │ │ │ │ i3 : isDirectory p │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : (fn = p | "foo") << "hi there" << close │ │ │ │ │ │ -o4 = /tmp/M2-10885-0/0/foo │ │ │ +o4 = /tmp/M2-11432-0/0/foo │ │ │ │ │ │ o4 : File │ │ │ │ │ │ i5 : get fn │ │ │ │ │ │ o5 = hi there │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_move__File_lp__String_cm__String_rp.out │ │ │ @@ -1,31 +1,31 @@ │ │ │ -- -*- M2-comint -*- hash: 4857944042471093218 │ │ │ │ │ │ i1 : src = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10759-0/0 │ │ │ +o1 = /tmp/M2-11178-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10759-0/1 │ │ │ +o2 = /tmp/M2-11178-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10759-0/0 │ │ │ +o3 = /tmp/M2-11178-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : moveFile(src,dst,Verbose=>true) │ │ │ ---moving: /tmp/M2-10759-0/0 -> /tmp/M2-10759-0/1 │ │ │ +--moving: /tmp/M2-11178-0/0 -> /tmp/M2-11178-0/1 │ │ │ │ │ │ i5 : get dst │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ i6 : bak = moveFile(dst,Verbose=>true) │ │ │ ---backup file created: /tmp/M2-10759-0/1.bak │ │ │ +--backup file created: /tmp/M2-11178-0/1.bak │ │ │ │ │ │ -o6 = /tmp/M2-10759-0/1.bak │ │ │ +o6 = /tmp/M2-11178-0/1.bak │ │ │ │ │ │ i7 : removeFile bak │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_nanosleep.out │ │ │ @@ -1,8 +1,8 @@ │ │ │ -- -*- M2-comint -*- hash: 1331114612441 │ │ │ │ │ │ i1 : elapsedTime nanosleep 500000000 │ │ │ - -- .500227s elapsed │ │ │ + -- .500097s elapsed │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ @@ -5,26 +5,26 @@ │ │ │ o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : L = random toList (1..10000); │ │ │ │ │ │ i3 : elapsedTime apply(1..100, n -> sort L); │ │ │ - -- .60735s elapsed │ │ │ + -- .740176s elapsed │ │ │ │ │ │ i4 : elapsedTime parallelApply(1..100, n -> sort L); │ │ │ - -- .295654s elapsed │ │ │ + -- .163264s elapsed │ │ │ │ │ │ i5 : allowableThreads │ │ │ │ │ │ o5 = 5 │ │ │ │ │ │ i6 : allowableThreads = maxAllowableThreads │ │ │ │ │ │ -o6 = 7 │ │ │ +o6 = 17 │ │ │ │ │ │ i7 : R = ZZ/101[x,y,z]; │ │ │ │ │ │ i8 : I = (ideal vars R)^2 │ │ │ │ │ │ 2 2 2 │ │ │ o8 = ideal (x , x*y, x*z, y , y*z, z ) │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ │ │ o17 : Task │ │ │ │ │ │ i18 : schedule t'; │ │ │ │ │ │ i19 : t' │ │ │ │ │ │ -o19 = <> │ │ │ +o19 = <> │ │ │ │ │ │ o19 : Task │ │ │ │ │ │ i20 : taskResult t' │ │ │ │ │ │ 1 6 8 3 │ │ │ o20 = R <-- R <-- R <-- R <-- 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ @@ -67,15 +67,15 @@ │ │ │ i3 : S = ring I │ │ │ │ │ │ o3 = S │ │ │ │ │ │ o3 : PolynomialRing │ │ │ │ │ │ i4 : elapsedTime minimalBetti I │ │ │ - -- 3.11318s elapsed │ │ │ + -- 2.56513s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ o4 = total: 1 35 140 385 819 1080 819 385 140 35 1 │ │ │ 0: 1 . . . . . . . . . . │ │ │ 1: . 35 140 189 84 . . . . . . │ │ │ 2: . . . 196 735 1080 735 196 . . . │ │ │ 3: . . . . . . 84 189 140 35 . │ │ │ @@ -84,15 +84,15 @@ │ │ │ o4 : BettiTally │ │ │ │ │ │ i5 : I = ideal I_*; │ │ │ │ │ │ o5 : Ideal of S │ │ │ │ │ │ i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true) │ │ │ - -- 2.33669s elapsed │ │ │ + -- 2.33464s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ o6 = total: 1 35 140 385 819 1080 819 385 140 35 1 │ │ │ 0: 1 . . . . . . . . . . │ │ │ 1: . 35 140 189 84 . . . . . . │ │ │ 2: . . . 196 735 1080 735 196 . . . │ │ │ 3: . . . . . . 84 189 140 35 . │ │ │ @@ -105,15 +105,15 @@ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : numTBBThreads = 1 │ │ │ │ │ │ o8 = 1 │ │ │ │ │ │ i9 : elapsedTime minimalBetti(I) │ │ │ - -- 2.03997s elapsed │ │ │ + -- 2.11254s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ o9 = total: 1 35 140 385 819 1080 819 385 140 35 1 │ │ │ 0: 1 . . . . . . . . . . │ │ │ 1: . 35 140 189 84 . . . . . . │ │ │ 2: . . . 196 735 1080 735 196 . . . │ │ │ 3: . . . . . . 84 189 140 35 . │ │ │ @@ -132,15 +132,15 @@ │ │ │ o11 = 0 │ │ │ │ │ │ i12 : I = ideal I_*; │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : elapsedTime freeResolution(I, Strategy => Nonminimal) │ │ │ - -- 2.05922s elapsed │ │ │ + -- 4.14556s elapsed │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 576 135 14 │ │ │ o13 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ o13 : Complex │ │ │ @@ -150,15 +150,15 @@ │ │ │ o14 = 1 │ │ │ │ │ │ i15 : I = ideal I_*; │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ │ │ i16 : elapsedTime freeResolution(I, Strategy => Nonminimal) │ │ │ - -- 1.99304s elapsed │ │ │ + -- 2.07918s elapsed │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 576 135 14 │ │ │ o16 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ o16 : Complex │ │ │ @@ -174,43 +174,43 @@ │ │ │ o18 : PolynomialRing │ │ │ │ │ │ i19 : I = ideal random(S^1, S^{4:-5}); │ │ │ │ │ │ o19 : Ideal of S │ │ │ │ │ │ i20 : elapsedTime groebnerBasis(I, Strategy => "F4"); │ │ │ - -- 4.49471s elapsed │ │ │ + -- 5.22478s elapsed │ │ │ │ │ │ 1 108 │ │ │ o20 : Matrix S <-- S │ │ │ │ │ │ i21 : numTBBThreads = 1 │ │ │ │ │ │ o21 = 1 │ │ │ │ │ │ i22 : I = ideal I_*; │ │ │ │ │ │ o22 : Ideal of S │ │ │ │ │ │ i23 : elapsedTime groebnerBasis(I, Strategy => "F4"); │ │ │ - -- 7.78738s elapsed │ │ │ + -- 6.40932s elapsed │ │ │ │ │ │ 1 108 │ │ │ o23 : Matrix S <-- S │ │ │ │ │ │ i24 : numTBBThreads = 10 │ │ │ │ │ │ o24 = 10 │ │ │ │ │ │ i25 : I = ideal I_*; │ │ │ │ │ │ o25 : Ideal of S │ │ │ │ │ │ i26 : elapsedTime groebnerBasis(I, Strategy => "F4"); │ │ │ - -- 4.79613s elapsed │ │ │ + -- 3.59757s elapsed │ │ │ │ │ │ 1 108 │ │ │ o26 : Matrix S <-- S │ │ │ │ │ │ i27 : needsPackage "AssociativeAlgebras" │ │ │ │ │ │ o27 = AssociativeAlgebras │ │ │ @@ -233,15 +233,15 @@ │ │ │ o30 = ideal (5a + 2b*c + 3c*b, 3a*c + 5b + 2c*a, 2a*b + 3b*a + 5c ) │ │ │ │ │ │ ZZ │ │ │ o30 : Ideal of ---<|a, b, c|> │ │ │ 101 │ │ │ │ │ │ i31 : elapsedTime NCGB(I, 22); │ │ │ - -- 1.26729s elapsed │ │ │ + -- 1.10176s elapsed │ │ │ │ │ │ ZZ 1 ZZ 148 │ │ │ o31 : Matrix (---<|a, b, c|>) <-- (---<|a, b, c|>) │ │ │ 101 101 │ │ │ │ │ │ i32 : I = ideal I_* │ │ │ │ │ │ @@ -253,14 +253,14 @@ │ │ │ 101 │ │ │ │ │ │ i33 : numTBBThreads = 1 │ │ │ │ │ │ o33 = 1 │ │ │ │ │ │ i34 : elapsedTime NCGB(I, 22); │ │ │ - -- 1.1978s elapsed │ │ │ + -- 1.24741s elapsed │ │ │ │ │ │ ZZ 1 ZZ 148 │ │ │ o34 : Matrix (---<|a, b, c|>) <-- (---<|a, b, c|>) │ │ │ 101 101 │ │ │ │ │ │ i35 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ @@ -146,65 +146,65 @@ │ │ │ o26 : ZZ[T] │ │ │ │ │ │ i27 : gbTrace = 3 │ │ │ │ │ │ o27 = 3 │ │ │ │ │ │ i28 : time poincare I │ │ │ - -- used 0.00336927s (cpu); 1.6521e-05s (thread); 0s (gc) │ │ │ + -- used 0.00192958s (cpu); 7.865e-06s (thread); 0s (gc) │ │ │ │ │ │ 3 6 9 │ │ │ o28 = 1 - 3T + 3T - T │ │ │ │ │ │ o28 : ZZ[T] │ │ │ │ │ │ i29 : time gens gb I; │ │ │ │ │ │ - -- registering gb 19 at 0x7fec4bc63e00 │ │ │ + -- registering gb 19 at 0x7f188450ee00 │ │ │ │ │ │ -- [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 │ │ │ -- number of monomials = 4186 │ │ │ -- #reduction steps = 38 │ │ │ -- #spairs done = 11 │ │ │ -- ncalls = 10 │ │ │ -- nloop = 29 │ │ │ -- nsaved = 0 │ │ │ - -- -- used 0.0206106s (cpu); 0.0205024s (thread); 0s (gc) │ │ │ + -- -- used 0.0100611s (cpu); 0.0099988s (thread); 0s (gc) │ │ │ │ │ │ 1 11 │ │ │ o29 : Matrix R <-- R │ │ │ │ │ │ i30 : R = QQ[a..d]; │ │ │ │ │ │ i31 : I = ideal random(R^1, R^{3:-3}); │ │ │ │ │ │ - -- registering gb 20 at 0x7fec4bc63c40 │ │ │ + -- registering gb 20 at 0x7f188450ec40 │ │ │ │ │ │ -- [gb]number of (nonminimal) gb elements = 0 │ │ │ -- number of monomials = 0 │ │ │ -- #reduction steps = 0 │ │ │ -- #spairs done = 0 │ │ │ -- ncalls = 0 │ │ │ -- nloop = 0 │ │ │ -- nsaved = 0 │ │ │ -- │ │ │ o31 : Ideal of R │ │ │ │ │ │ i32 : time p = poincare I │ │ │ │ │ │ - -- registering gb 21 at 0x7fec4bc638c0 │ │ │ + -- registering gb 21 at 0x7f188450e8c0 │ │ │ │ │ │ -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11 │ │ │ -- number of monomials = 267 │ │ │ -- #reduction steps = 236 │ │ │ -- #spairs done = 30 │ │ │ -- ncalls = 10 │ │ │ -- nloop = 20 │ │ │ -- nsaved = 0 │ │ │ - -- -- used 0.00799675s (cpu); 0.00881508s (thread); 0s (gc) │ │ │ + -- -- used 0.00388814s (cpu); 0.00363116s (thread); 0s (gc) │ │ │ │ │ │ 3 6 9 │ │ │ o32 = 1 - 3T + 3T - T │ │ │ │ │ │ o32 : ZZ[T] │ │ │ │ │ │ i33 : S = QQ[a..d, MonomialOrder => Eliminate 2] │ │ │ @@ -254,27 +254,27 @@ │ │ │ │ │ │ i36 : gbTrace = 3 │ │ │ │ │ │ o36 = 3 │ │ │ │ │ │ i37 : time gens gb J; │ │ │ │ │ │ - -- registering gb 22 at 0x7fec4bc63700 │ │ │ + -- registering gb 22 at 0x7f188450e700 │ │ │ │ │ │ -- [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 │ │ │ -- 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 │ │ │ -- {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 │ │ │ -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39 │ │ │ -- number of monomials = 1051 │ │ │ -- #reduction steps = 284 │ │ │ -- #spairs done = 53 │ │ │ -- ncalls = 46 │ │ │ -- nloop = 54 │ │ │ -- nsaved = 0 │ │ │ - -- -- used 0.0679993s (cpu); 0.0696023s (thread); 0s (gc) │ │ │ + -- -- used 0.0560149s (cpu); 0.0553724s (thread); 0s (gc) │ │ │ │ │ │ 1 39 │ │ │ o37 : Matrix S <-- S │ │ │ │ │ │ i38 : selectInSubring(1, gens gb J) │ │ │ │ │ │ o38 = | 243873059890414515367459726418219472801881021280016638460434780718278 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_process__I__D.out │ │ │ @@ -1,7 +1,7 @@ │ │ │ -- -*- M2-comint -*- hash: 1330513630563 │ │ │ │ │ │ i1 : processID() │ │ │ │ │ │ -o1 = 10387 │ │ │ +o1 = 10448 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ 123 110 13 │ │ │ o2 = x + x + x + 1 │ │ │ │ │ │ o2 : R │ │ │ │ │ │ i3 : time factor f │ │ │ - -- used 0.00278858s (cpu); 0.0027806s (thread); 0s (gc) │ │ │ + -- used 0.00239599s (cpu); 0.00238081s (thread); 0s (gc) │ │ │ │ │ │ 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 │ │ │ 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) │ │ │ │ │ │ o3 : Expression of class Product │ │ │ │ │ │ i4 : g = () -> factor f │ │ │ @@ -38,11 +38,11 @@ │ │ │ o6 = h │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ i7 : for i to 10 do (g();h();h()) │ │ │ │ │ │ i8 : profileSummary │ │ │ -g: 11 times, used .0266591 seconds │ │ │ -h: 22 times, used .0532723 seconds │ │ │ +g: 11 times, used .0234191 seconds │ │ │ +h: 22 times, used .0666404 seconds │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ @@ -13,15 +13,15 @@ │ │ │ i5 : codim I, degree I │ │ │ │ │ │ o5 = (2, 10) │ │ │ │ │ │ o5 : Sequence │ │ │ │ │ │ i6 : time randomKRationalPoint(I) │ │ │ - -- used 0.186982s (cpu); 0.129358s (thread); 0s (gc) │ │ │ + -- used 0.224136s (cpu); 0.0635276s (thread); 0s (gc) │ │ │ │ │ │ o6 = ideal (x - 53x , x + 8x , x - 4x ) │ │ │ 2 3 1 3 0 3 │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ i7 : R=kk[x_0..x_5]; │ │ │ @@ -33,15 +33,15 @@ │ │ │ i9 : codim I, degree I │ │ │ │ │ │ o9 = (3, 10) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : time randomKRationalPoint(I) │ │ │ - -- used 0.403357s (cpu); 0.338406s (thread); 0s (gc) │ │ │ + -- used 0.390978s (cpu); 0.207144s (thread); 0s (gc) │ │ │ │ │ │ o10 = ideal (x - 27x , x - 16x , x - 9x , x + 44x , x - 52x ) │ │ │ 4 5 3 5 2 5 1 5 0 5 │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2] │ │ │ @@ -58,12 +58,12 @@ │ │ │ │ │ │ i14 : I=ideal random(n,R); │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ │ │ i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c); │ │ │ #select(degs,d->d==1))),f->f>0)) │ │ │ - -- used 3.98035s (cpu); 2.47525s (thread); 0s (gc) │ │ │ + -- used 3.84275s (cpu); 2.01402s (thread); 0s (gc) │ │ │ │ │ │ o15 = 58 │ │ │ │ │ │ i16 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 20910736704070514 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11649-0/0 │ │ │ +o1 = /tmp/M2-12976-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-11649-0/0 │ │ │ +o2 = /tmp/M2-12976-0/0 │ │ │ │ │ │ i3 : (fn = dir | "/" | "foo") << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11649-0/0/foo │ │ │ +o3 = /tmp/M2-12976-0/0/foo │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : readDirectory dir │ │ │ │ │ │ o4 = {., .., foo} │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reading_spfiles.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 13513555104200944796 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11232-0/0 │ │ │ +o1 = /tmp/M2-12139-0/0 │ │ │ │ │ │ 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 │ │ │ │ │ │ -o2 = /tmp/M2-11232-0/0 │ │ │ +o2 = /tmp/M2-12139-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : get fn │ │ │ │ │ │ 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 │ │ │ +8*y^3 │ │ │ @@ -38,15 +38,15 @@ │ │ │ │ │ │ o6 : Expression of class Product │ │ │ │ │ │ i7 : fn << "sample = 2^100 │ │ │ print sample │ │ │ " << close │ │ │ │ │ │ -o7 = /tmp/M2-11232-0/0 │ │ │ +o7 = /tmp/M2-12139-0/0 │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : get fn │ │ │ │ │ │ o8 = sample = 2^100 │ │ │ print sample │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_readlink.out │ │ │ @@ -1,12 +1,12 @@ │ │ │ -- -*- M2-comint -*- hash: 4408639611478781130 │ │ │ │ │ │ i1 : p = temporaryFileName () │ │ │ │ │ │ -o1 = /tmp/M2-11961-0/0 │ │ │ +o1 = /tmp/M2-13608-0/0 │ │ │ │ │ │ i2 : symlinkFile ("foo", p) │ │ │ │ │ │ i3 : readlink p │ │ │ │ │ │ o3 = foo │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out │ │ │ @@ -1,39 +1,39 @@ │ │ │ -- -*- M2-comint -*- hash: 324072347213224656 │ │ │ │ │ │ i1 : realpath "." │ │ │ │ │ │ -o1 = /tmp/M2-10387-0/83-rundir/ │ │ │ +o1 = /tmp/M2-10448-0/83-rundir/ │ │ │ │ │ │ i2 : p = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-11980-0/0 │ │ │ +o2 = /tmp/M2-13647-0/0 │ │ │ │ │ │ i3 : q = temporaryFileName() │ │ │ │ │ │ -o3 = /tmp/M2-11980-0/1 │ │ │ +o3 = /tmp/M2-13647-0/1 │ │ │ │ │ │ i4 : symlinkFile(p,q) │ │ │ │ │ │ i5 : p << close │ │ │ │ │ │ -o5 = /tmp/M2-11980-0/0 │ │ │ +o5 = /tmp/M2-13647-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : readlink q │ │ │ │ │ │ -o6 = /tmp/M2-11980-0/0 │ │ │ +o6 = /tmp/M2-13647-0/0 │ │ │ │ │ │ i7 : realpath q │ │ │ │ │ │ -o7 = /tmp/M2-11980-0/0 │ │ │ +o7 = /tmp/M2-13647-0/0 │ │ │ │ │ │ i8 : removeFile p │ │ │ │ │ │ i9 : removeFile q │ │ │ │ │ │ i10 : realpath "" │ │ │ │ │ │ -o10 = /tmp/M2-10387-0/83-rundir/ │ │ │ +o10 = /tmp/M2-10448-0/83-rundir/ │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ @@ -1,15 +1,15 @@ │ │ │ -- -*- M2-comint -*- hash: 1729384374372662693 │ │ │ │ │ │ i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --")) │ │ │ │ │ │ i2 : collectGarbage() │ │ │ --finalization: (1)[7]: -- finalizing sequence #8 -- │ │ │ ---finalization: (2)[4]: -- finalizing sequence #5 -- │ │ │ ---finalization: (3)[1]: -- finalizing sequence #2 -- │ │ │ ---finalization: (4)[2]: -- finalizing sequence #3 -- │ │ │ ---finalization: (5)[3]: -- finalizing sequence #4 -- │ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 -- │ │ │ +--finalization: (2)[3]: -- finalizing sequence #4 -- │ │ │ +--finalization: (3)[0]: -- finalizing sequence #1 -- │ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 -- │ │ │ +--finalization: (5)[4]: -- finalizing sequence #5 -- │ │ │ +--finalization: (6)[1]: -- finalizing sequence #2 -- │ │ │ --finalization: (7)[6]: -- finalizing sequence #7 -- │ │ │ ---finalization: (8)[0]: -- finalizing sequence #1 -- │ │ │ +--finalization: (8)[2]: -- finalizing sequence #3 -- │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 8532980310097060089 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10923-0/0 │ │ │ +o1 = /tmp/M2-11510-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10923-0/0 │ │ │ +o2 = /tmp/M2-11510-0/0 │ │ │ │ │ │ i3 : readDirectory dir │ │ │ │ │ │ o3 = {., ..} │ │ │ │ │ │ o3 : List │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__Path.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 1731420232148149387 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10465-0/0 │ │ │ +o1 = /tmp/M2-10584-0/0 │ │ │ │ │ │ i2 : rootPath | fn │ │ │ │ │ │ -o2 = /tmp/M2-10465-0/0 │ │ │ +o2 = /tmp/M2-10584-0/0 │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__U__R__I.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 1731420231525572968 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11592-0/0 │ │ │ +o1 = /tmp/M2-12859-0/0 │ │ │ │ │ │ i2 : rootURI | fn │ │ │ │ │ │ -o2 = file:///tmp/M2-11592-0/0 │ │ │ +o2 = file:///tmp/M2-12859-0/0 │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out │ │ │ @@ -25,19 +25,19 @@ │ │ │ o4 = image | x2 x2-y2 xyz7 | │ │ │ │ │ │ 1 │ │ │ o4 : R-module, submodule of R │ │ │ │ │ │ i5 : f = temporaryFileName() │ │ │ │ │ │ -o5 = /tmp/M2-11478-0/0 │ │ │ +o5 = /tmp/M2-12625-0/0 │ │ │ │ │ │ i6 : f << toString (p,m,M) << close │ │ │ │ │ │ -o6 = /tmp/M2-11478-0/0 │ │ │ +o6 = /tmp/M2-12625-0/0 │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : get f │ │ │ │ │ │ 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, │ │ │ x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}}) │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_schedule.out │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ │ │ i4 : taskResult t │ │ │ │ │ │ o4 = 8 │ │ │ │ │ │ i5 : u = schedule(f,4) │ │ │ │ │ │ -o5 = <> │ │ │ +o5 = <> │ │ │ │ │ │ o5 : Task │ │ │ │ │ │ i6 : taskResult u │ │ │ │ │ │ o6 = 16 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_serial__Number.out │ │ │ @@ -1,15 +1,15 @@ │ │ │ -- -*- M2-comint -*- hash: 5271760183816554957 │ │ │ │ │ │ i1 : serialNumber asdf │ │ │ │ │ │ -o1 = 1528251 │ │ │ +o1 = 1628251 │ │ │ │ │ │ i2 : serialNumber foo │ │ │ │ │ │ -o2 = 1528253 │ │ │ +o2 = 1628253 │ │ │ │ │ │ i3 : serialNumber ZZ │ │ │ │ │ │ o3 = 1000050 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ @@ -191,18 +191,18 @@ │ │ │ o25 = 40 │ │ │ │ │ │ i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A; │ │ │ │ │ │ i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B; │ │ │ │ │ │ i30 : time X = solve(A,B); │ │ │ - -- used 0.000236533s (cpu); 0.000229119s (thread); 0s (gc) │ │ │ + -- used 0.000140143s (cpu); 0.000128931s (thread); 0s (gc) │ │ │ │ │ │ i31 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.000168847s (cpu); 0.000168906s (thread); 0s (gc) │ │ │ + -- used 9.3294e-05s (cpu); 9.3475e-05s (thread); 0s (gc) │ │ │ │ │ │ i32 : norm(A*X-B) │ │ │ │ │ │ o32 = 5.111850690840453e-15 │ │ │ │ │ │ o32 : RR (of precision 53) │ │ │ │ │ │ @@ -211,18 +211,18 @@ │ │ │ o33 = 100 │ │ │ │ │ │ i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A; │ │ │ │ │ │ i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B; │ │ │ │ │ │ i38 : time X = solve(A,B); │ │ │ - -- used 0.484036s (cpu); 0.306285s (thread); 0s (gc) │ │ │ + -- used 0.145216s (cpu); 0.145221s (thread); 0s (gc) │ │ │ │ │ │ i39 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.238738s (cpu); 0.23868s (thread); 0s (gc) │ │ │ + -- used 0.12495s (cpu); 0.12496s (thread); 0s (gc) │ │ │ │ │ │ i40 : norm(A*X-B) │ │ │ │ │ │ o40 = 1.491578274689709814082355885932e-28 │ │ │ │ │ │ o40 : RR (of precision 100) │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__Directory_lp__String_cm__String_rp.out │ │ │ @@ -1,60 +1,60 @@ │ │ │ -- -*- M2-comint -*- hash: 2989513528213557691 │ │ │ │ │ │ i1 : src = temporaryFileName() | "/" │ │ │ │ │ │ -o1 = /tmp/M2-11272-0/0/ │ │ │ +o1 = /tmp/M2-12219-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11272-0/1/ │ │ │ +o2 = /tmp/M2-12219-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11272-0/0/a/ │ │ │ +o3 = /tmp/M2-12219-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11272-0/0/b/ │ │ │ +o4 = /tmp/M2-12219-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11272-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12219-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11272-0/0/a/f │ │ │ +o6 = /tmp/M2-12219-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11272-0/0/a/g │ │ │ +o7 = /tmp/M2-12219-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11272-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12219-0/0/b/c/g │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : symlinkDirectory(src,dst,Verbose=>true) │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11272-0/1/b/c/g │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11272-0/1/a/g │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11272-0/1/a/f │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12219-0/1/b/c/g │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12219-0/1/a/f │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12219-0/1/a/g │ │ │ │ │ │ i10 : get (dst|"b/c/g") │ │ │ │ │ │ o10 = ho there │ │ │ │ │ │ i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true) │ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11272-0/1/b/c/g │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11272-0/1/a/g │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11272-0/1/a/f │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12219-0/1/b/c/g │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12219-0/1/a/f │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12219-0/1/a/g │ │ │ │ │ │ i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d │ │ │ │ │ │ o12 = rm │ │ │ │ │ │ o12 : FunctionClosure │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__File.out │ │ │ @@ -1,12 +1,12 @@ │ │ │ -- -*- M2-comint -*- hash: 9343844672940306595 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11345-0/0 │ │ │ +o1 = /tmp/M2-12352-0/0 │ │ │ │ │ │ i2 : symlinkFile("qwert", fn) │ │ │ │ │ │ i3 : fileExists fn │ │ │ │ │ │ o3 = false │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_temporary__File__Name.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 1731926531291302106 │ │ │ │ │ │ i1 : temporaryFileName () | ".tex" │ │ │ │ │ │ -o1 = /tmp/M2-12321-0/0.tex │ │ │ +o1 = /tmp/M2-14386-0/0.tex │ │ │ │ │ │ i2 : temporaryFileName () | ".html" │ │ │ │ │ │ -o2 = /tmp/M2-12321-0/1.html │ │ │ +o2 = /tmp/M2-14386-0/1.html │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ @@ -1,8 +1,8 @@ │ │ │ -- -*- M2-comint -*- hash: 1332435500723 │ │ │ │ │ │ i1 : time 3^30 │ │ │ - -- used 1.3776e-05s (cpu); 5.69e-06s (thread); 0s (gc) │ │ │ + -- used 2.0499e-05s (cpu); 5.37e-06s (thread); 0s (gc) │ │ │ │ │ │ o1 = 205891132094649 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out │ │ │ @@ -1,14 +1,14 @@ │ │ │ -- -*- M2-comint -*- hash: 1730988300469098603 │ │ │ │ │ │ i1 : timing 3^30 │ │ │ │ │ │ o1 = 205891132094649 │ │ │ - -- .000014217 seconds │ │ │ + -- .000015719 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{.000014217, 205891132094649} │ │ │ +o2 = Time{.000015719, 205891132094649} │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out │ │ │ @@ -33,15 +33,15 @@ │ │ │ "memtailor version" => 1.0 │ │ │ "mpfi version" => 1.5.4 │ │ │ "mpfr version" => 4.2.1 │ │ │ "mpsolve version" => 3.2.1 │ │ │ "mysql version" => not present │ │ │ "normaliz version" => 3.10.4 │ │ │ "ntl version" => 11.5.1 │ │ │ - "operating system release" => 6.1.0-31-amd64 │ │ │ + "operating system release" => 6.12.35+deb13-cloud-amd64 │ │ │ "operating system" => Linux │ │ │ "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings PruneComplex BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples Divisor EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieTypes ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties Licenses TestIdeals FrobeniusThresholds Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization NonminimalComplexes CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups │ │ │ "pointer size" => 8 │ │ │ "python version" => 3.13.2 │ │ │ "readline version" => 8.2 │ │ │ "scscp version" => not present │ │ │ "tbb version" => 2022.0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Basis__Element__Limit.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │ │ │ │
    │ │ │

    BasisElementLimit -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Codimension__Limit.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    CodimensionLimit -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Coefficient__Ring.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    CoefficientRing -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ o2 = 1073741824 │ │ │ │ │ │ │ │ │ 
    i3 : (c = Command "date";)
    │ │ │ │ │ │ │ │ │ 
    i4 : c
│ │ │ -Sun Feb  9 23:54:36 UTC 2025
│ │ │ +Sat Jul 19 03:40:32 UTC 2025
│ │ │  
│ │ │  o4 = 0
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ in a file), then it gets executed with empty argument list. │ │ │ │ i1 : (f = Command ( () -> 2^30 );) │ │ │ │ i2 : f │ │ │ │ │ │ │ │ o2 = 1073741824 │ │ │ │ i3 : (c = Command "date";) │ │ │ │ i4 : c │ │ │ │ -Sun Feb 9 23:54:36 UTC 2025 │ │ │ │ +Sat Jul 19 03:40:32 UTC 2025 │ │ │ │ │ │ │ │ o4 = 0 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_u_n -- run an external command │ │ │ │ * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: ********** │ │ │ │ * ? Command (missing documentation) │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html │ │ │ @@ -44,20 +44,20 @@ │ │ │

    Database -- the class of all database files

    │ │ │
    │ │ │

    Description

    │ │ │ A database file is just like a hash table, except both the keys and values have to be strings. In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-11758-0/0.dbm
    │ │ │ +o1 = /tmp/M2-13205-0/0.dbm │ │ │ 
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11758-0/0.dbm
│ │ │ +o2 = /tmp/M2-13205-0/0.dbm
│ │ │  
│ │ │  o2 : Database
    │ │ │ 
    i3 : x#"first" = "hi there"
│ │ │  
│ │ │  o3 = hi there
    │ │ │ ├── html2text {} │ │ │ │ @@ -6,18 +6,18 @@ │ │ │ │ ************ DDaattaabbaassee ---- tthhee ccllaassss ooff aallll ddaattaabbaassee ffiilleess ************ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ A database file is just like a hash table, except both the keys and values have │ │ │ │ to be strings. In this example we create a database file, store a few entries, │ │ │ │ remove an entry with _r_e_m_o_v_e, close the file, and then remove the file. │ │ │ │ i1 : filename = temporaryFileName () | ".dbm" │ │ │ │ │ │ │ │ -o1 = /tmp/M2-11758-0/0.dbm │ │ │ │ +o1 = /tmp/M2-13205-0/0.dbm │ │ │ │ i2 : x = openDatabaseOut filename │ │ │ │ │ │ │ │ -o2 = /tmp/M2-11758-0/0.dbm │ │ │ │ +o2 = /tmp/M2-13205-0/0.dbm │ │ │ │ │ │ │ │ o2 : Database │ │ │ │ i3 : x#"first" = "hi there" │ │ │ │ │ │ │ │ o3 = hi there │ │ │ │ i4 : x#"first" │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Degree__Group.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │ │ │ │
    │ │ │

    DegreeGroup -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Degree__Limit.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    DegreeLimit -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Degree__Map.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    DegreeMap -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Fast__Nonminimal.html │ │ │ @@ -84,26 +84,26 @@ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ o2 : PolynomialRing │ │ │ 
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.80778s elapsed
│ │ │ + -- 1.86095s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    │ │ │ 
    i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- .973592s elapsed
│ │ │ + -- .845975s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -30,28 +30,28 @@ │ │ │ │ 0,5 1,5 2,5 3,5 4,5 0,6 1,6 2,6 3,6 4,6 5,6 │ │ │ │ i2 : S = ring I │ │ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ i3 : elapsedTime C = res(I, FastNonminimal => true) │ │ │ │ - -- 1.80778s elapsed │ │ │ │ + -- 1.86095s elapsed │ │ │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 │ │ │ │ 576 135 14 │ │ │ │ o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S │ │ │ │ <-- S <-- S <-- 0 │ │ │ │ │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 │ │ │ │ 9 10 11 │ │ │ │ │ │ │ │ o3 : ChainComplex │ │ │ │ i4 : elapsedTime C1 = res ideal(I_*) │ │ │ │ - -- .973592s elapsed │ │ │ │ + -- .845975s elapsed │ │ │ │ │ │ │ │ 1 35 140 385 819 1080 819 385 140 │ │ │ │ 35 1 │ │ │ │ o4 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S │ │ │ │ <-- S <-- S <-- 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___File_sp_lt_lt_sp__Thing.html │ │ │ @@ -97,20 +97,20 @@ │ │ │ o2 = stdio │ │ │ │ │ │ o2 : File │ │ │ 
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11872-0/0
    │ │ │ +o3 = /tmp/M2-13439-0/0 │ │ │ 
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11872-0/0
│ │ │ +o4 = /tmp/M2-13439-0/0
│ │ │  
│ │ │  o4 : File
    │ │ │ 
    i5 : get fn
│ │ │  
│ │ │  o5 = hi there
    │ │ │ @@ -139,29 +139,29 @@ │ │ │ o8 = stdio │ │ │ │ │ │ o8 : File │ │ │ 
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11872-0/0
│ │ │ +o9 = /tmp/M2-13439-0/0
│ │ │  
│ │ │  o9 : File
    │ │ │ 
    i10 : get fn
│ │ │  
│ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │        + 1
    │ │ │ 
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11872-0/0
│ │ │ +o11 = /tmp/M2-13439-0/0
│ │ │  
│ │ │  o11 : File
    │ │ │ 
    i12 : get fn
│ │ │  
│ │ │  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+
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,18 +38,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11872-0/0
│ │ │ │ +o3 = /tmp/M2-13439-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11872-0/0
│ │ │ │ +o4 = /tmp/M2-13439-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : R = QQ[x]
│ │ │ │  
│ │ │ │ @@ -68,25 +68,25 @@
│ │ │ │   10      9      8       7       6       5       4       3      2
│ │ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │ │  o8 = stdio
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : fn << f << close
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11872-0/0
│ │ │ │ +o9 = /tmp/M2-13439-0/0
│ │ │ │  
│ │ │ │  o9 : File
│ │ │ │  i10 : get fn
│ │ │ │  
│ │ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │ │        + 1
│ │ │ │  i11 : fn << toExternalString f << close
│ │ │ │  
│ │ │ │ -o11 = /tmp/M2-11872-0/0
│ │ │ │ +o11 = /tmp/M2-13439-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  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+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -45,31 +45,31 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats 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.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 15334137450        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 15449393626        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 194183168
│ │ │ -               "numGCs" => 836
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 226713600
│ │ │ +               "numGCs" => 824
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
    
│ │ │      		          
    
│ │ │          
    The value returned is a hash table, from which individual bits of information can be easily extracted, as follows.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 194183168
    
│ │ │ +o2 = 226713600
│ │ │      		          
    
│ │ │          
    Any entries whose keys are all upper case give the values of environment variables affecting the operation of the garbage collector that have been specified by the user.
    
│ │ │          
    For further information about the individual items in the table, we refer the user to the source code and documentation of the garbage collector.
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,28 +8,28 @@
│ │ │ │  Macaulay2 uses the Hans Boehm _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r to reclaim unused memory. The
│ │ │ │  function GCstats 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.
│ │ │ │  i1 : s = GCstats()
│ │ │ │  
│ │ │ │ -o1 = HashTable{"bytesAlloc" => 15334137450        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 15449393626        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 194183168
│ │ │ │ -               "numGCs" => 836
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 226713600
│ │ │ │ +               "numGCs" => 824
│ │ │ │ +               "numGCThreads" => 16
│ │ │ │  
│ │ │ │  o1 : HashTable
│ │ │ │  The value returned is a hash table, from which individual bits of information
│ │ │ │  can be easily extracted, as follows.
│ │ │ │  i2 : s#"heapSize"
│ │ │ │  
│ │ │ │ -o2 = 194183168
│ │ │ │ +o2 = 226713600
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Generic.html
│ │ │ @@ -34,15 +34,15 @@
│ │ │      
    
│ │ │        
│ │ │        
    
│ │ │  
    
│ │ │    
│ │ │    
│ │ │ -
    next | previous | forward | backward | up | index | toc      
    
│ │ │ +next | previous | forward | backward | up | index | toc      
    
│ │ │      
    
│ │ │  
        
    
│ │ │        
    Generic -- an optional argument
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  A symbol used as the name of an optional argument.      
    
│ │ │        
    
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -106,28 +106,28 @@
│ │ │            
    i7 : I = truncate(8, monomialCurveIdeal(R,{1,4,5,9}));
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ 
    i8 : time gens gb I;
│ │ │ - -- used 0.0628534s (cpu); 0.0628529s (thread); 0s (gc)
│ │ │ + -- used 0.0217891s (cpu); 0.0217888s (thread); 0s (gc)
│ │ │  
│ │ │               1      428
│ │ │  o8 : Matrix R  <-- R
    │ │ │ 
    i9 : time J1 = saturate(I);
│ │ │ - -- used 0.605726s (cpu); 0.339625s (thread); 0s (gc)
│ │ │ + -- used 0.842929s (cpu); 0.221206s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    │ │ │ 
    i10 : time J = saturate(I, MinimalGenerators => false);
│ │ │ - -- used 0.000202009s (cpu); 0.000202159s (thread); 0s (gc)
│ │ │ + -- used 0.000106109s (cpu); 0.00010645s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of R
    │ │ │ 
    i11 : numgens J
│ │ │  
│ │ │  o11 = 7
    │ │ │ ├── html2text {} │ │ │ │ @@ -44,24 +44,24 @@ │ │ │ │ o6 = R │ │ │ │ │ │ │ │ o6 : PolynomialRing │ │ │ │ i7 : I = truncate(8, monomialCurveIdeal(R,{1,4,5,9})); │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ i8 : time gens gb I; │ │ │ │ - -- used 0.0628534s (cpu); 0.0628529s (thread); 0s (gc) │ │ │ │ + -- used 0.0217891s (cpu); 0.0217888s (thread); 0s (gc) │ │ │ │ │ │ │ │ 1 428 │ │ │ │ o8 : Matrix R <-- R │ │ │ │ i9 : time J1 = saturate(I); │ │ │ │ - -- used 0.605726s (cpu); 0.339625s (thread); 0s (gc) │ │ │ │ + -- used 0.842929s (cpu); 0.221206s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : time J = saturate(I, MinimalGenerators => false); │ │ │ │ - -- used 0.000202009s (cpu); 0.000202159s (thread); 0s (gc) │ │ │ │ + -- used 0.000106109s (cpu); 0.00010645s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : numgens J │ │ │ │ │ │ │ │ o11 = 7 │ │ │ │ i12 : numgens J1 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Pair__Limit.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │ │ │ │
    │ │ │

    PairLimit -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html │ │ │ @@ -61,19 +61,19 @@ │ │ │ │ │ │ 200 200 │ │ │ o1 : Matrix RR <-- RR │ │ │ 53 53 │ │ │ 
    i2 : time SVD(M);
│ │ │ - -- used 0.0510253s (cpu); 0.0510246s (thread); 0s (gc)
    │ │ │ + -- used 0.0263697s (cpu); 0.0263675s (thread); 0s (gc) │ │ │ 
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0514493s (cpu); 0.0514556s (thread); 0s (gc)
    │ │ │ + -- used 0.0263246s (cpu); 0.026337s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │
    │ │ │

    Further information

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -12,17 +12,17 @@ │ │ │ │ For large matrices, this algorithm is often much faster. │ │ │ │ i1 : M = random(RR^200, RR^200); │ │ │ │ │ │ │ │ 200 200 │ │ │ │ o1 : Matrix RR <-- RR │ │ │ │ 53 53 │ │ │ │ i2 : time SVD(M); │ │ │ │ - -- used 0.0510253s (cpu); 0.0510246s (thread); 0s (gc) │ │ │ │ + -- used 0.0263697s (cpu); 0.0263675s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0514493s (cpu); 0.0514556s (thread); 0s (gc) │ │ │ │ + -- used 0.0263246s (cpu); 0.026337s (thread); 0s (gc) │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _S_V_D -- singular value decomposition of a matrix │ │ │ │ * Option key: _D_i_v_i_d_e_C_o_n_q_u_e_r -- an optional argument │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: ********** │ │ │ │ * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- Use the lapack divide and conquer SVD │ │ │ │ algorithm │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Strategy.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
      │ │ │ │ │ │
      │ │ │
      │ │ │ │ │ │ │ │ │ -
      next | previous | forward | backward | up | index | toc
      │ │ │ +next | previous | forward | backward | up | index | toc
      │ │ │
    │ │ │
    │ │ │

    Strategy -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Threads.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    Threads -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Verbosity.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    Verbosity -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Verify.html │ │ │ @@ -34,15 +34,15 @@ │ │ │
    │ │ │ │ │ │
    │ │ │
    │ │ │ │ │ │ │ │ │ -
    next | previous | forward | backward | up | index | toc
    │ │ │ +next | previous | forward | backward | up | index | toc
    │ │ │
    │ │ │
    │ │ │

    Verify -- an optional argument

    │ │ │
    │ │ │

    Description

    │ │ │ A symbol used as the name of an optional argument.
    │ │ │
    │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html │ │ │ @@ -554,15 +554,15 @@ │ │ │ 3 │ │ │ o58 : R-module, quotient of R │ │ │ │ │ │ │ │ │ We may use resolution to produce a projective resolution of it, and time to report the time required. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i59 : time C = resolution M
│ │ │ - -- used 0.00187636s (cpu); 0.00186805s (thread); 0s (gc)
│ │ │ + -- used 0.00145835s (cpu); 0.0014456s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -386,15 +386,15 @@ │ │ │ │ | c f i l o r | │ │ │ │ │ │ │ │ 3 │ │ │ │ o58 : R-module, quotient of R │ │ │ │ We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to │ │ │ │ report the time required. │ │ │ │ i59 : time C = resolution M │ │ │ │ - -- used 0.00187636s (cpu); 0.00186805s (thread); 0s (gc) │ │ │ │ + -- used 0.00145835s (cpu); 0.0014456s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 6 15 18 6 │ │ │ │ o59 = R <-- R <-- R <-- R <-- R <-- 0 │ │ │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o59 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ 
    i4 : peek read f
│ │ │  
│ │ │  o4 = "hi there"
    │ │ │ 
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    │ │ │ +o5 = true │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -23,10 +23,10 @@ │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : peek read f │ │ │ │ │ │ │ │ o4 = "hi there" │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ │ │ -o5 = false │ │ │ │ +o5 = true │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html │ │ │ @@ -67,15 +67,15 @@ │ │ │
    │ │ │
    │ │ │

    Description

    │ │ │ Produces an accurate timing for the code contained in the string s. The value returned is the number of seconds. │ │ │ │ │ │ │ │ │ 
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0002909794917276939
│ │ │ +o1 = .0003412512839561764
│ │ │  
│ │ │  o1 : RR (of precision 53)
    │ │ │
    │ │ │ The snippet of code provided will be run enough times to register meaningfully on the clock, and the garbage collector will be called beforehand.
    │ │ │
    │ │ │

    For the programmer

    │ │ │ ├── html2text {} │ │ │ │ @@ -13,14 +13,14 @@ │ │ │ │ o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds it takes to evaluate the code │ │ │ │ in s │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Produces an accurate timing for the code contained in the string s. The value │ │ │ │ returned is the number of seconds. │ │ │ │ i1 : benchmark "sqrt 2p100000" │ │ │ │ │ │ │ │ -o1 = .0002909794917276939 │ │ │ │ +o1 = .0003412512839561764 │ │ │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ The snippet of code provided will be run enough times to register meaningfully │ │ │ │ on the clock, and the garbage collector will be called beforehand. │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.45141s elapsed
│ │ │ + -- 2.12843s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,15 +27,15 @@ │ │ │ │ 0,5 1,5 2,5 3,5 4,5 0,6 1,6 2,6 3,6 4,6 5,6 │ │ │ │ i2 : S = ring I │ │ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ i3 : elapsedTime C = res(I, FastNonminimal => true) │ │ │ │ - -- 2.45141s elapsed │ │ │ │ + -- 2.12843s elapsed │ │ │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 │ │ │ │ 576 135 14 │ │ │ │ o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S │ │ │ │ <-- S <-- S <-- 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ @@ -76,15 +76,15 @@ │ │ │ 
    i1 : n = 0
│ │ │  
│ │ │  o1 = 0
    │ │ │ │ │ │ │ │ │ 
    i2 : t = schedule(() -> while true do n = n+1)
│ │ │  
│ │ │ -o2 = <<task, running>>
│ │ │ +o2 = <<task, created>>
│ │ │  
│ │ │  o2 : Task
    │ │ │ │ │ │ │ │ │ 
    i3 : sleep 1
│ │ │  
│ │ │  o3 = 0
    │ │ │ @@ -95,15 +95,15 @@ │ │ │ o4 = <<task, running>> │ │ │ │ │ │ o4 : Task │ │ │ │ │ │ │ │ │ 
    i5 : n
│ │ │  
│ │ │ -o5 = 709345
    │ │ │ +o5 = 1093202 │ │ │ │ │ │ │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ │ │ │ │ │ │ @@ -112,15 +112,15 @@ │ │ │ o7 = <<task, running>> │ │ │ │ │ │ o7 : Task │ │ │ │ │ │ │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 1451545
    │ │ │ +o8 = 2219186 │ │ │ │ │ │ │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ │ │ │ │ │ │ @@ -138,25 +138,25 @@ │ │ │ o12 = <<task, canceled>> │ │ │ │ │ │ o12 : Task │ │ │ │ │ │ │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 1451960
    │ │ │ +o13 = 2219384 │ │ │ │ │ │ │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │ │ │ │ │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 1451960
    │ │ │ +o15 = 2219384 │ │ │ │ │ │ │ │ │ 
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -16,39 +16,39 @@ │ │ │ │ stop. │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : n = 0 │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ i2 : t = schedule(() -> while true do n = n+1) │ │ │ │ │ │ │ │ -o2 = <> │ │ │ │ +o2 = <> │ │ │ │ │ │ │ │ o2 : Task │ │ │ │ i3 : sleep 1 │ │ │ │ │ │ │ │ o3 = 0 │ │ │ │ i4 : t │ │ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ │ │ o4 : Task │ │ │ │ i5 : n │ │ │ │ │ │ │ │ -o5 = 709345 │ │ │ │ +o5 = 1093202 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 1451545 │ │ │ │ +o8 = 2219186 │ │ │ │ i9 : isReady t │ │ │ │ │ │ │ │ o9 = false │ │ │ │ i10 : cancelTask t │ │ │ │ i11 : sleep 2 │ │ │ │ stdio:2:25:(3):[1]: error: interrupted │ │ │ │ │ │ │ │ @@ -56,19 +56,19 @@ │ │ │ │ i12 : t │ │ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ │ │ o12 : Task │ │ │ │ i13 : n │ │ │ │ │ │ │ │ -o13 = 1451960 │ │ │ │ +o13 = 2219384 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 1451960 │ │ │ │ +o15 = 2219384 │ │ │ │ i16 : isReady t │ │ │ │ │ │ │ │ o16 = false │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html │ │ │ @@ -70,30 +70,30 @@ │ │ │
    │ │ │

    Change the current working directory to dir.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10626-0/0
    │ │ │ +o1 = /tmp/M2-10905-0/0 │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10626-0/0
    │ │ │ +o2 = /tmp/M2-10905-0/0 │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10626-0/0/
    │ │ │ +o3 = /tmp/M2-10905-0/0/ │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10626-0/0/
    │ │ │ +o4 = /tmp/M2-10905-0/0/ │ │ │
    │ │ │
    │ │ │

    If dir is omitted, then the current working directory is changed to the user's home directory.

    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -11,23 +11,23 @@ │ │ │ │ o dir, a _s_t_r_i_n_g, │ │ │ │ * Outputs: │ │ │ │ o a _s_t_r_i_n_g, the new working directory; │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Change the current working directory to dir. │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ │ │ -o1 = /tmp/M2-10626-0/0 │ │ │ │ +o1 = /tmp/M2-10905-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-10626-0/0 │ │ │ │ +o2 = /tmp/M2-10905-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-10626-0/0/ │ │ │ │ +o3 = /tmp/M2-10905-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-10626-0/0/ │ │ │ │ +o4 = /tmp/M2-10905-0/0/ │ │ │ │ If dir is omitted, then the current working directory is changed to the user's │ │ │ │ home directory. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ @@ -42,15 +42,15 @@ │ │ │
    │ │ │
    │ │ │

    communicating with programs

    │ │ │
    │ │ │ 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 run command. │ │ │ │ │ │ │ │ │ 
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 (2025-02-07) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.35-1 (2025-07-03) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
    │ │ │
    │ │ │ To run a program and provide it with input, one way is use the operator <<, 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. │ │ │ │ │ │ │ │ │ 
    i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │ @@ -62,17 +62,17 @@
│ │ │  o2 : File
    │ │ │
    │ │ │ 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 get 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. │ │ │ │ │ │ │ │ │ 
    i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1
│ │ │ +o3 = "Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       "
│ │ │ -     (2025-02-07) x86_64 GNU/Linux
    │ │ │ + 6.12.35-1 (2025-07-03) x86_64 GNU/Linux │ │ │
    │ │ │ Bidirectional communication with a program is also possible. We use openInOut 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 egrep and use it to locate the symbol names in Macaulay2 that begin with in. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : f = openInOut "!egrep '^in'"
│ │ │  
│ │ │  o4 = !egrep '^in'
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,16 +4,16 @@
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ ccoommmmuunniiccaattiinngg wwiitthh pprrooggrraammss ************
│ │ │ │  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 _r_u_n command.
│ │ │ │  i1 : run "uname -a"
│ │ │ │ -Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1 (2025-02-
│ │ │ │ -07) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.35-1
│ │ │ │ +(2025-07-03) x86_64 GNU/Linux
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  To run a program and provide it with input, one way is use the operator _<_<,
│ │ │ │  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.
│ │ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │ │   ba
│ │ │ │ @@ -25,17 +25,17 @@
│ │ │ │  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 _g_e_t 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.
│ │ │ │  i3 : peek get "!uname -a"
│ │ │ │  
│ │ │ │ -o3 = "Linux sbuild 6.1.0-31-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.128-1
│ │ │ │ +o3 = "Linux sbuild 6.12.35+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │       "
│ │ │ │ -     (2025-02-07) x86_64 GNU/Linux
│ │ │ │ +     6.12.35-1 (2025-07-03) x86_64 GNU/Linux
│ │ │ │  Bidirectional communication with a program is also possible. We use _o_p_e_n_I_n_O_u_t
│ │ │ │  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 egrep and use it to locate the symbol names in Macaulay2 that
│ │ │ │  begin with in.
│ │ │ │  i4 : f = openInOut "!egrep '^in'"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html
│ │ │ @@ -215,15 +215,15 @@
│ │ │                  ZZ
│ │ │  o23 : Ideal of ----[x..z, w]
│ │ │                 1277
    │ │ │ 
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7fe96c5bd700
│ │ │ +   -- registering gb 5 at 0x7fdd33824700
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -301,15 +301,15 @@
│ │ │  
    i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │  
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    │ │ │ 
    i33 : time betti gb f
│ │ │ - -- used 0.307492s (cpu); 0.307469s (thread); 0s (gc)
│ │ │ + -- used 0.152258s (cpu); 0.151986s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -339,15 +339,15 @@
│ │ │              3    5     8     9    12     14    17
│ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │  
│ │ │  o35 : ZZ[T]
    │ │ │ 
    i36 : time betti gb f
│ │ │ - -- used 0.00800087s (cpu); 0.00553291s (thread); 0s (gc)
│ │ │ + -- used 0.00179524s (cpu); 0.00217529s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -139,15 +139,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7fe96c5bd700
│ │ │ │ +   -- registering gb 5 at 0x7fdd33824700
│ │ │ │  
│ │ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │ │     -- number of monomials                = 8
│ │ │ │     -- #reduction steps = 2
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │ @@ -212,15 +212,15 @@
│ │ │ │  
│ │ │ │  o31 : ZZ[T]
│ │ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │ │  
│ │ │ │                1      4
│ │ │ │  o32 : Matrix R  <-- R
│ │ │ │  i33 : time betti gb f
│ │ │ │ - -- used 0.307492s (cpu); 0.307469s (thread); 0s (gc)
│ │ │ │ + -- used 0.152258s (cpu); 0.151986s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -244,15 +244,15 @@
│ │ │ │  i35 : poincare cokernel f = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6) -- cache poincare
│ │ │ │  
│ │ │ │              3    5     8     9    12     14    17
│ │ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : time betti gb f
│ │ │ │ - -- used 0.00800087s (cpu); 0.00553291s (thread); 0s (gc)
│ │ │ │ + -- used 0.00179524s (cpu); 0.00217529s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_spresolutions.html
│ │ │ @@ -92,16 +92,16 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 1.30377s (cpu); 0.973789s (thread); 0s (gc)
│ │ │ - -- used 0.508987s (cpu); 0.396067s (thread); 0s (gc)
│ │ │ + -- used 1.13974s (cpu); 0.98458s (thread); 0s (gc)
│ │ │ + -- used 0.232854s (cpu); 0.148313s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,16 +50,16 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 1.30377s (cpu); 0.973789s (thread); 0s (gc)
│ │ │ │ - -- used 0.508987s (cpu); 0.396067s (thread); 0s (gc)
│ │ │ │ + -- used 1.13974s (cpu); 0.98458s (thread); 0s (gc)
│ │ │ │ + -- used 0.232854s (cpu); 0.148313s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,90 +85,90 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11326-0/0/
    
│ │ │ +o1 = /tmp/M2-12313-0/0/
│ │ │      		          
                  
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11326-0/1/
    
│ │ │ +o2 = /tmp/M2-12313-0/1/
│ │ │      		          
                  
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11326-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12313-0/0/a/
│ │ │      		          
                  
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11326-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12313-0/0/b/
│ │ │      		          
                  
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11326-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12313-0/0/b/c/
│ │ │      		          
                  
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11326-0/0/a/f
│ │ │ +o6 = /tmp/M2-12313-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │      		          
                  
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11326-0/0/a/g
│ │ │ +o7 = /tmp/M2-12313-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │      		          
                  
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11326-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12313-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │      		          
                  
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11326-0/0/
│ │ │ -     /tmp/M2-11326-0/0/b/
│ │ │ -     /tmp/M2-11326-0/0/b/c/
│ │ │ -     /tmp/M2-11326-0/0/b/c/g
│ │ │ -     /tmp/M2-11326-0/0/a/
│ │ │ -     /tmp/M2-11326-0/0/a/g
│ │ │ -     /tmp/M2-11326-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-12313-0/0/
│ │ │ +     /tmp/M2-12313-0/0/b/
│ │ │ +     /tmp/M2-12313-0/0/b/c/
│ │ │ +     /tmp/M2-12313-0/0/b/c/g
│ │ │ +     /tmp/M2-12313-0/0/a/
│ │ │ +     /tmp/M2-12313-0/0/a/f
│ │ │ +     /tmp/M2-12313-0/0/a/g
│ │ │      		          
                  
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11326-0/0/b/c/g -> /tmp/M2-11326-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11326-0/0/a/g -> /tmp/M2-11326-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11326-0/0/a/f -> /tmp/M2-11326-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-12313-0/0/b/c/g -> /tmp/M2-12313-0/1/b/c/g
│ │ │ + -- copying: /tmp/M2-12313-0/0/a/f -> /tmp/M2-12313-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12313-0/0/a/g -> /tmp/M2-12313-0/1/a/g
│ │ │      		          
                  
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11326-0/0/b/c/g not newer than /tmp/M2-11326-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11326-0/0/a/g not newer than /tmp/M2-11326-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11326-0/0/a/f not newer than /tmp/M2-11326-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-12313-0/0/b/c/g not newer than /tmp/M2-12313-0/1/b/c/g
│ │ │ + -- skipping: /tmp/M2-12313-0/0/a/f not newer than /tmp/M2-12313-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12313-0/0/a/g not newer than /tmp/M2-12313-0/1/a/g
│ │ │      		          
                  
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11326-0/1/
│ │ │ -      /tmp/M2-11326-0/1/a/
│ │ │ -      /tmp/M2-11326-0/1/a/f
│ │ │ -      /tmp/M2-11326-0/1/a/g
│ │ │ -      /tmp/M2-11326-0/1/b/
│ │ │ -      /tmp/M2-11326-0/1/b/c/
│ │ │ -      /tmp/M2-11326-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-12313-0/1/
│ │ │ +      /tmp/M2-12313-0/1/b/
│ │ │ +      /tmp/M2-12313-0/1/b/c/
│ │ │ +      /tmp/M2-12313-0/1/b/c/g
│ │ │ +      /tmp/M2-12313-0/1/a/
│ │ │ +      /tmp/M2-12313-0/1/a/f
│ │ │ +      /tmp/M2-12313-0/1/a/g
│ │ │      		          
                  
    i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,68 +26,68 @@
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o a copy of the directory tree rooted at src is created, rooted at
│ │ │ │              dst
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11326-0/0/
│ │ │ │ +o1 = /tmp/M2-12313-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11326-0/1/
│ │ │ │ +o2 = /tmp/M2-12313-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11326-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12313-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11326-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12313-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11326-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12313-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11326-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12313-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11326-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12313-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11326-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12313-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11326-0/0/
│ │ │ │ -     /tmp/M2-11326-0/0/b/
│ │ │ │ -     /tmp/M2-11326-0/0/b/c/
│ │ │ │ -     /tmp/M2-11326-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11326-0/0/a/
│ │ │ │ -     /tmp/M2-11326-0/0/a/g
│ │ │ │ -     /tmp/M2-11326-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-12313-0/0/
│ │ │ │ +     /tmp/M2-12313-0/0/b/
│ │ │ │ +     /tmp/M2-12313-0/0/b/c/
│ │ │ │ +     /tmp/M2-12313-0/0/b/c/g
│ │ │ │ +     /tmp/M2-12313-0/0/a/
│ │ │ │ +     /tmp/M2-12313-0/0/a/f
│ │ │ │ +     /tmp/M2-12313-0/0/a/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11326-0/0/b/c/g -> /tmp/M2-11326-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11326-0/0/a/g -> /tmp/M2-11326-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11326-0/0/a/f -> /tmp/M2-11326-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12313-0/0/b/c/g -> /tmp/M2-12313-0/1/b/c/g
│ │ │ │ + -- copying: /tmp/M2-12313-0/0/a/f -> /tmp/M2-12313-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12313-0/0/a/g -> /tmp/M2-12313-0/1/a/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11326-0/0/b/c/g not newer than /tmp/M2-11326-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11326-0/0/a/g not newer than /tmp/M2-11326-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11326-0/0/a/f not newer than /tmp/M2-11326-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12313-0/0/b/c/g not newer than /tmp/M2-12313-0/1/b/c/g
│ │ │ │ + -- skipping: /tmp/M2-12313-0/0/a/f not newer than /tmp/M2-12313-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12313-0/0/a/g not newer than /tmp/M2-12313-0/1/a/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11326-0/1/
│ │ │ │ -      /tmp/M2-11326-0/1/a/
│ │ │ │ -      /tmp/M2-11326-0/1/a/f
│ │ │ │ -      /tmp/M2-11326-0/1/a/g
│ │ │ │ -      /tmp/M2-11326-0/1/b/
│ │ │ │ -      /tmp/M2-11326-0/1/b/c/
│ │ │ │ -      /tmp/M2-11326-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-12313-0/1/
│ │ │ │ +      /tmp/M2-12313-0/1/b/
│ │ │ │ +      /tmp/M2-12313-0/1/b/c/
│ │ │ │ +      /tmp/M2-12313-0/1/b/c/g
│ │ │ │ +      /tmp/M2-12313-0/1/a/
│ │ │ │ +      /tmp/M2-12313-0/1/a/f
│ │ │ │ +      /tmp/M2-12313-0/1/a/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -81,51 +81,51 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,37 +19,37 @@
│ │ │ │            o Verbose => a _B_o_o_l_e_a_n_ _v_a_l_u_e, default value false, whether to report
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o the file may be copied
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11095-0/0
│ │ │ │ +o1 = /tmp/M2-11862-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11095-0/1
│ │ │ │ +o2 = /tmp/M2-11862-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11095-0/0
│ │ │ │ +o3 = /tmp/M2-11862-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11095-0/0 -> /tmp/M2-11095-0/1
│ │ │ │ + -- copying: /tmp/M2-11862-0/0 -> /tmp/M2-11862-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11095-0/0 not newer than /tmp/M2-11095-0/1
│ │ │ │ + -- skipping: /tmp/M2-11862-0/0 not newer than /tmp/M2-11862-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11095-0/0
│ │ │ │ +o7 = /tmp/M2-11862-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11095-0/0 not newer than /tmp/M2-11095-0/1
│ │ │ │ + -- skipping: /tmp/M2-11862-0/0 not newer than /tmp/M2-11862-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -61,32 +61,32 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
                  
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11095-0/0
    
│ │ │ +o1 = /tmp/M2-11862-0/0
│ │ │      		          
                  
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11095-0/1
    
│ │ │ +o2 = /tmp/M2-11862-0/1
│ │ │      		          
                  
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11095-0/0
│ │ │ +o3 = /tmp/M2-11862-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11095-0/0 -> /tmp/M2-11095-0/1
    
│ │ │ + -- copying: /tmp/M2-11862-0/0 -> /tmp/M2-11862-0/1
│ │ │      		          
                  
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │      		          
                  
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11095-0/0 not newer than /tmp/M2-11095-0/1
    
│ │ │ + -- skipping: /tmp/M2-11862-0/0 not newer than /tmp/M2-11862-0/1
│ │ │      		          
                  
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11095-0/0
│ │ │ +o7 = /tmp/M2-11862-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │      		          
                  
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11095-0/0 not newer than /tmp/M2-11095-0/1
    
│ │ │ + -- skipping: /tmp/M2-11862-0/0 not newer than /tmp/M2-11862-0/1
│ │ │      		          
                  
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 258.991604772
│ │ │ +o1 = 244.703091515
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │      		          
                  
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │      		          
                  
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 261.1503754559999
│ │ │ +o3 = 245.69947021
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │      		          
                  
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 2.15877068399999
│ │ │ +o4 = .9963786950000042
│ │ │  
│ │ │  o4 : RR (of precision 53)
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,26 +9,26 @@
│ │ │ │              cpuTime()
│ │ │ │      * Outputs:
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds of cpu time used since the
│ │ │ │              program was started
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : t1 = cpuTime()
│ │ │ │  
│ │ │ │ -o1 = 258.991604772
│ │ │ │ +o1 = 244.703091515
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 261.1503754559999
│ │ │ │ +o3 = 245.69947021
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 2.15877068399999
│ │ │ │ +o4 = .9963786950000042
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -61,42 +61,42 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1739145345
    
│ │ │ +o1 = 1752896489
│ │ │      		          
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.11132206304336
│ │ │ +o2 = 55.54707846942887
│ │ │  
│ │ │  o2 : RR (of precision 53)
    
│ │ │      		          
    
│ │ │          
    We can also compute how many months account for the fractional part of that number.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 1.335864756520323
│ │ │ +o3 = 6.564941633146418
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │      		          
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : run "date"
│ │ │ -Sun Feb  9 23:55:45 UTC 2025
│ │ │ +Sat Jul 19 03:41:29 UTC 2025
│ │ │  
│ │ │  o4 = 0
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    For the programmer
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,28 @@
│ │ │ │              currentTime()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the current time, in seconds since 00:00:00 1970-01-01
│ │ │ │              UTC, the beginning of the epoch
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : currentTime()
│ │ │ │  
│ │ │ │ -o1 = 1739145345
│ │ │ │ +o1 = 1752896489
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 55.11132206304336
│ │ │ │ +o2 = 55.54707846942887
│ │ │ │  
│ │ │ │  o2 : RR (of precision 53)
│ │ │ │  We can also compute how many months account for the fractional part of that
│ │ │ │  number.
│ │ │ │  i3 : 12 * (oo - floor oo)
│ │ │ │  
│ │ │ │ -o3 = 1.335864756520323
│ │ │ │ +o3 = 6.564941633146418
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Sun Feb  9 23:55:45 UTC 2025
│ │ │ │ +Sat Jul 19 03:41:29 UTC 2025
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -54,15 +54,15 @@
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00014s elapsed
│ │ │ + -- 1.00013s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │  ********** SSyynnooppssiiss **********
│ │ │ │      *   Usage:
│ │ │ │              elapsedTime e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the
│ │ │ │  value of e.
│ │ │ │  i1 : elapsedTime sleep 1
│ │ │ │ - -- 1.00014s elapsed
│ │ │ │ + -- 1.00013s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _G_C_s_t_a_t_s -- information about the status of the garbage collector
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html
│ │ │ @@ -46,22 +46,22 @@
│ │ │          
    Description
    
│ │ │  elapsedTiming e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of time elapsed, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00014 seconds
│ │ │ +     -- 1.00009 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │      		          
                  
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00014, 0}
    
│ │ │ +o2 = Time{1.00009, 0}
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  where t is the number of seconds of time elapsed, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : elapsedTiming sleep 1
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │ -     -- 1.00014 seconds
│ │ │ │ +     -- 1.00009 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00014, 0}
│ │ │ │ +o2 = Time{1.00009, 0}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html
│ │ │ @@ -53,15 +53,15 @@
│ │ │                 3                   3     2               3
│ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ 
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.254683s (cpu); 0.254684s (thread); 0s (gc)
│ │ │ + -- used 0.117515s (cpu); 0.117267s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -140,15 +140,15 @@
│ │ │                 3                   3     2               3
│ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ 
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.4515s (cpu); 0.275672s (thread); 0s (gc)
│ │ │ + -- used 0.121517s (cpu); 0.121522s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -215,15 +215,15 @@
│ │ │            
    i11 : I1 = substitute(I,R1);
│ │ │  
│ │ │  o11 : Ideal of R1
    │ │ │ 
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0600603s (cpu); 0.060069s (thread); 0s (gc)
│ │ │ + -- used 0.0460749s (cpu); 0.0460784s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -297,15 +297,15 @@
│ │ │                     3             3     2         3
│ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │  
│ │ │  o16 : RingMap A <-- B
    │ │ │ 
    i17 : time G = kernel F
│ │ │ - -- used 0.404083s (cpu); 0.234457s (thread); 0s (gc)
│ │ │ + -- used 0.320628s (cpu); 0.139295s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -372,24 +372,24 @@
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : PolynomialRing
    │ │ │ 
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00172022s (cpu); 0.00171993s (thread); 0s (gc)
│ │ │ + -- used 0.00135128s (cpu); 0.00134931s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
    │ │ │ 
    i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0562473s (cpu); 0.0562272s (thread); 0s (gc)
│ │ │ + -- used 0.033369s (cpu); 0.033381s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time leadTerm gens gb I
│ │ │ │ - -- used 0.254683s (cpu); 0.254684s (thread); 0s (gc)
│ │ │ │ + -- used 0.117515s (cpu); 0.117267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ │ - -- used 0.4515s (cpu); 0.275672s (thread); 0s (gc)
│ │ │ │ + -- used 0.121517s (cpu); 0.121522s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -156,15 +156,15 @@
│ │ │ │  Sometimes giving the variables different degrees will speed up the
│ │ │ │  computations. Here, we set the degrees of x, y, and z to be the total degrees.
│ │ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │ │  i11 : I1 = substitute(I,R1);
│ │ │ │  
│ │ │ │  o11 : Ideal of R1
│ │ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ │ - -- used 0.0600603s (cpu); 0.060069s (thread); 0s (gc)
│ │ │ │ + -- used 0.0460749s (cpu); 0.0460784s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7
│ │ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2
│ │ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -227,15 +227,15 @@
│ │ │ │  i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
│ │ │ │  
│ │ │ │                     3             3     2         3
│ │ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │ │  
│ │ │ │  o16 : RingMap A <-- B
│ │ │ │  i17 : time G = kernel F
│ │ │ │ - -- used 0.404083s (cpu); 0.234457s (thread); 0s (gc)
│ │ │ │ + -- used 0.320628s (cpu); 0.139295s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -296,22 +296,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00172022s (cpu); 0.00171993s (thread); 0s (gc)
│ │ │ │ + -- used 0.00135128s (cpu); 0.00134931s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9    9      7    3
│ │ │ │  o20 = x*t  - t  - z*t  - z
│ │ │ │  
│ │ │ │  o20 : R
│ │ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ │ - -- used 0.0562473s (cpu); 0.0562272s (thread); 0s (gc)
│ │ │ │ + -- used 0.033369s (cpu); 0.033381s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10387-0/88-rundir/
│ │ │ +             source directory => /tmp/M2-10448-0/88-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    │ │ │ 
    i7 : dictionaryPath
│ │ │  
│ │ │  o7 = {Foo.Dictionary, PackageCitations.Dictionary, Varieties.Dictionary,
│ │ │ ├── html2text {}
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-10387-0/88-rundir/
│ │ │ │ +             source directory => /tmp/M2-10448-0/88-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, PackageCitations.Dictionary, Varieties.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -67,25 +67,25 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10721-0/0
    
│ │ │ +o1 = /tmp/M2-11100-0/0
│ │ │      		          
                  
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │      		          
                  
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10721-0/0
│ │ │ +o3 = /tmp/M2-11100-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : fileExists fn
│ │ │  
│ │ │  o4 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,21 +11,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether a file with the filename or path fn exists
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10721-0/0
│ │ │ │ +o1 = /tmp/M2-11100-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10721-0/0
│ │ │ │ +o3 = /tmp/M2-11100-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -68,34 +68,34 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,28 +12,28 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the length of the file f or the file whose name is f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The length of an open output file is determined from the internal count of the
│ │ │ │  number of bytes written so far.
│ │ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12302-0/0
│ │ │ │ +o1 = /tmp/M2-14347-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12302-0/0
│ │ │ │ +o3 = /tmp/M2-14347-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12302-0/0
│ │ │ │ +o4 = /tmp/M2-14347-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -69,32 +69,32 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
                  
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12302-0/0
│ │ │ +o1 = /tmp/M2-14347-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │      		          
                  
    i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
    
│ │ │      		          
                  
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12302-0/0
│ │ │ +o3 = /tmp/M2-14347-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12302-0/0
    
│ │ │ +o4 = /tmp/M2-14347-0/0
│ │ │      		          
                  
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11497-0/0
    
│ │ │ +o1 = /tmp/M2-12664-0/0
│ │ │      		          
                  
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11497-0/0
│ │ │ +o2 = /tmp/M2-12664-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
                  
    i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
    
│ │ │      		          
                  
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11497-0/0
│ │ │ +o4 = /tmp/M2-12664-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │      		          
                  
    i5 : removeFile fn
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,24 @@
│ │ │ │      * Inputs:
│ │ │ │            o f
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the open file f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11497-0/0
│ │ │ │ +o1 = /tmp/M2-12664-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11497-0/0
│ │ │ │ +o2 = /tmp/M2-12664-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11497-0/0
│ │ │ │ +o4 = /tmp/M2-12664-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -69,20 +69,20 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o9 = (3, 10)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11114-0/0
    
│ │ │ +o1 = /tmp/M2-11901-0/0
│ │ │      		          
                  
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11114-0/0
│ │ │ +o2 = /tmp/M2-11901-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
                  
    i3 : fileMode fn
│ │ │  
│ │ │  o3 = 420
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the file located at the filename or path fn
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11114-0/0
│ │ │ │ +o1 = /tmp/M2-11901-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11114-0/0
│ │ │ │ +o2 = /tmp/M2-11901-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -73,20 +73,20 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10998-0/0
    
│ │ │ +o1 = /tmp/M2-11665-0/0
│ │ │      		          
                  
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10998-0/0
│ │ │ +o2 = /tmp/M2-11665-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
                  
    i3 : m = 7 + 7*8 + 7*64
│ │ │  
│ │ │  o3 = 511
    
│ │ │ @@ -98,15 +98,15 @@
│ │ │      		              
    i5 : fileMode f
│ │ │  
│ │ │  o5 = 511
    
│ │ │      		          
                  
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-10998-0/0
│ │ │ +o6 = /tmp/M2-11665-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │      		          
                  
    i7 : fileMode fn
│ │ │  
│ │ │  o7 = 511
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,30 +12,30 @@
│ │ │ │            o mo
│ │ │ │            o f
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the open file f is set to mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10998-0/0
│ │ │ │ +o1 = /tmp/M2-11665-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10998-0/0
│ │ │ │ +o2 = /tmp/M2-11665-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │ │  
│ │ │ │  o3 = 511
│ │ │ │  i4 : fileMode(m,f)
│ │ │ │  i5 : fileMode f
│ │ │ │  
│ │ │ │  o5 = 511
│ │ │ │  i6 : close f
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10998-0/0
│ │ │ │ +o6 = /tmp/M2-11665-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -73,20 +73,20 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12132-0/0
    
│ │ │ +o1 = /tmp/M2-14017-0/0
│ │ │      		          
                  
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12132-0/0
│ │ │ +o2 = /tmp/M2-14017-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
                  
    i3 : m = fileMode fn
│ │ │  
│ │ │  o3 = 420
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o fn
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the file located at the filename or path fn is set to
│ │ │ │              mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12132-0/0
│ │ │ │ +o1 = /tmp/M2-14017-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12132-0/0
│ │ │ │ +o2 = /tmp/M2-14017-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  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.        
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 48
    
│ │ │ +o1 = 39
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,13 +19,13 @@
│ │ │ │              returns null if no error occurs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  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.
│ │ │ │  i1 : currentTime() - fileTime "."
│ │ │ │  
│ │ │ │ -o1 = 48
│ │ │ │ +o1 = 39
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │               3      3
│ │ │  o6 : Matrix R  <-- R
│ │ │  
    		          
                  
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f53c265d000
│ │ │ +   -- registering gb 0 at 0x7f598969b000
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       {3} | x2-3  0     -z4+2 |
│ │ │ │       {4} | 0     x2-3  y3-1  |
│ │ │ │  
│ │ │ │               3      3
│ │ │ │  o6 : Matrix R  <-- R
│ │ │ │  i7 : syz f
│ │ │ │  
│ │ │ │ -   -- registering gb 0 at 0x7f53c265d000
│ │ │ │ +   -- registering gb 0 at 0x7f598969b000
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  
    		          
                  
    i3 : removeFile "test-file"
    
│ │ │      		          
                  
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sun Feb  9 23:55:06 UTC 2025
    
│ │ │ +o4 = Sat Jul 19 03:40:58 UTC 2025
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Sun Feb  9 23:55:06 UTC 2025
│ │ │ │ +o4 = Sat Jul 19 03:40:58 UTC 2025
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances_lp__Type_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Canceled_lp__Task_rp.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │  
    		              
    i1 : n = 0
│ │ │  
│ │ │  o1 = 0
    
│ │ │      		          
                  
    i2 : t = schedule(() -> while true do n = n + 1)
│ │ │  
│ │ │ -o2 = <<task, created>>
│ │ │ +o2 = <<task, running>>
│ │ │  
│ │ │  o2 : Task
    
│ │ │      		          
                  
    i3 : sleep 1
│ │ │  
│ │ │  o3 = 0
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o whether the task t has been canceled
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : n = 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : t = schedule(() -> while true do n = n + 1)
│ │ │ │  
│ │ │ │ -o2 = <>
│ │ │ │ +o2 = <>
│ │ │ │  
│ │ │ │  o2 : Task
│ │ │ │  i3 : sleep 1
│ │ │ │  
│ │ │ │  o3 = 0
│ │ │ │  i4 : isCanceled t
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -72,20 +72,20 @@
│ │ │      		              
    i1 : isDirectory "."
│ │ │  
│ │ │  o1 = true
    
│ │ │      		          
                  
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10543-0/0
    
│ │ │ +o2 = /tmp/M2-10742-0/0
│ │ │      		          
                  
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10543-0/0
│ │ │ +o3 = /tmp/M2-10742-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : isDirectory fn
│ │ │  
│ │ │  o4 = false
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,18 +14,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : isDirectory "."
│ │ │ │  
│ │ │ │  o1 = true
│ │ │ │  i2 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10543-0/0
│ │ │ │ +o2 = /tmp/M2-10742-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10543-0/0
│ │ │ │ +o3 = /tmp/M2-10742-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -175,15 +175,15 @@
│ │ │            
                  
    i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │  
│ │ │  o18 = false
    
│ │ │      		          
                  
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 4.34633s elapsed
│ │ │ + -- 4.34639s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │      		          
                  
    i20 : facs = facs//toList/toList
│ │ │ @@ -201,21 +201,21 @@
│ │ │  
    		              
    i22 : m3 = nextPrime (m^3)
│ │ │  
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
    
│ │ │      		          
                  
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0563776s elapsed
│ │ │ + -- .0467009s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │      		          
                  
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000134241s elapsed
│ │ │ + -- .000125735s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 4.34633s elapsed
│ │ │ │ + -- 4.34639s elapsed
│ │ │ │  
│ │ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │ │  
│ │ │ │  o19 : Expression of class Product
│ │ │ │  i20 : facs = facs//toList/toList
│ │ │ │  
│ │ │ │  o20 = {{1000000000000000000000000000057, 1},
│ │ │ │ @@ -98,19 +98,19 @@
│ │ │ │  o20 : List
│ │ │ │  i21 : assert(set facs === set {{m,1}, {m1,1}})
│ │ │ │  i22 : m3 = nextPrime (m^3)
│ │ │ │  
│ │ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │ │        00000000000000185613
│ │ │ │  i23 : elapsedTime isPrime m3
│ │ │ │ - -- .0563776s elapsed
│ │ │ │ + -- .0467009s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000134241s elapsed
│ │ │ │ + -- .000125735s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -67,20 +67,20 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  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.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12340-0/0
    
│ │ │ +o1 = /tmp/M2-14425-0/0
│ │ │      		          
                  
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12340-0/0
│ │ │ +o2 = /tmp/M2-14425-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
                  
    i3 : isRegularFile fn
│ │ │  
│ │ │  o3 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,18 +14,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a regular file
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  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.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12340-0/0
│ │ │ │ +o1 = /tmp/M2-14425-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12340-0/0
│ │ │ │ +o2 = /tmp/M2-14425-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -77,20 +77,20 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,18 +14,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10866-0/0
│ │ │ │ +o1 = /tmp/M2-11393-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10866-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11393-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -61,15 +61,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
                  
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10866-0/0
    
│ │ │ +o1 = /tmp/M2-11393-0/0
│ │ │      		          
                  
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10866-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11393-0/0/a/b/c
│ │ │      		          
                  
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │      		          
                  
    i4 : removeDirectory (dir|"/a/b")
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,12 +10,12 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -51,34 +51,34 @@
│ │ │  
│ │ │  o1 = fib
│ │ │  
│ │ │  o1 : FunctionClosure
│ │ │  
    		          
                  
    i2 : time fib 28
│ │ │ - -- used 1.31977s (cpu); 0.95198s (thread); 0s (gc)
│ │ │ + -- used 0.894368s (cpu); 0.618538s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │      		          
                  
    i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │      		          
                  
    i4 : time fib 28
│ │ │ - -- used 6.6595e-05s (cpu); 6.6164e-05s (thread); 0s (gc)
│ │ │ + -- used 4.8801e-05s (cpu); 4.8141e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │      		          
                  
    i5 : time fib 28
│ │ │ - -- used 4.138e-06s (cpu); 3.807e-06s (thread); 0s (gc)
│ │ │ + -- used 2.916e-06s (cpu); 2.365e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
    
│ │ │      		          
    
│ │ │          
    An optional second argument to memoize provides a list of initial values, each of the form x => v, where v is the value to be provided for the argument x.
    
│ │ │          
    Alternatively, values can be provided after defining the memoized function using the syntax f x = v. A slightly more efficient implementation of the above would be
    
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,28 +10,28 @@
│ │ │ │  arguments are presented.
│ │ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │ │  
│ │ │ │  o1 = fib
│ │ │ │  
│ │ │ │  o1 : FunctionClosure
│ │ │ │  i2 : time fib 28
│ │ │ │ - -- used 1.31977s (cpu); 0.95198s (thread); 0s (gc)
│ │ │ │ + -- used 0.894368s (cpu); 0.618538s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 6.6595e-05s (cpu); 6.6164e-05s (thread); 0s (gc)
│ │ │ │ + -- used 4.8801e-05s (cpu); 4.8141e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.138e-06s (cpu); 3.807e-06s (thread); 0s (gc)
│ │ │ │ + -- used 2.916e-06s (cpu); 2.365e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 514229
│ │ │ │  An optional second argument to memoize provides a list of initial values, each
│ │ │ │  of the form x => v, where v is the value to be provided for the argument x.
│ │ │ │  Alternatively, values can be provided after defining the memoized function
│ │ │ │  using the syntax f x = v. A slightly more efficient implementation of the above
│ │ │ │  would be
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_methods.html
│ │ │ @@ -86,20 +86,20 @@
│ │ │       {13 => (poincare, BettiTally)                                }
│ │ │       {14 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {15 => (degree, BettiTally)                                  }
│ │ │       {16 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {17 => (^, Ring, BettiTally)                                 }
│ │ │       {18 => (regularity, BettiTally)                              }
│ │ │       {19 => (mathML, BettiTally)                                  }
│ │ │ -     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {21 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {25 => (dual, BettiTally)                                    }
│ │ │ +     {20 => (dual, BettiTally)                                    }
│ │ │ +     {21 => (codim, BettiTally)                                   }
│ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {25 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │              
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                
                    
    i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ @@ -186,20 +186,20 @@
│ │ │                
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ -     {2 => (contract', Matrix, Matrix)                           }
│ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (diff', Matrix, Matrix)                               }
│ │ │ +o5 = {0 => (diff, Matrix, Matrix)                                }
│ │ │ +     {1 => (contract, Matrix, Matrix)                            }
│ │ │ +     {2 => (diff', Matrix, Matrix)                               }
│ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │ +     {4 => (contract', Matrix, Matrix)                           }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -214,18 +214,18 @@
│ │ │       {21 => (quotient', Matrix, Matrix)                          }
│ │ │       {22 => (quotient, Matrix, Matrix)                           }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (remainder, Matrix, Matrix)                          }
│ │ │       {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (intersection, Matrix, Matrix, Matrix, Matrix)       }
│ │ │ -     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersection, Matrix, Matrix)                       }
│ │ │ +     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {29 => (intersection, Matrix, Matrix, Matrix, Matrix)       }
│ │ │ +     {30 => (intersection, Matrix, Matrix)                       }
│ │ │ +     {31 => (tensor, Matrix, Matrix)                             }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (checkDegrees, Matrix, Matrix)                       }
│ │ │       {34 => (isIsomorphic, Matrix, Matrix)                       }
│ │ │       {35 => (coneFromVData, Matrix, Matrix)                      }
│ │ │       {36 => (coneFromHData, Matrix, Matrix)                      }
│ │ │       {37 => (fan, Matrix, Matrix, List)                          }
│ │ │       {38 => (fan, Matrix, Matrix, Sequence)                      }
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,20 +30,20 @@
│ │ │ │       {13 => (poincare, BettiTally)                                }
│ │ │ │       {14 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {15 => (degree, BettiTally)                                  }
│ │ │ │       {16 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {17 => (^, Ring, BettiTally)                                 }
│ │ │ │       {18 => (regularity, BettiTally)                              }
│ │ │ │       {19 => (mathML, BettiTally)                                  }
│ │ │ │ -     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ -     {21 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {25 => (dual, BettiTally)                                    }
│ │ │ │ +     {20 => (dual, BettiTally)                                    }
│ │ │ │ +     {21 => (codim, BettiTally)                                   }
│ │ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ +     {25 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ │       {1 => (resolution, Module)}
│ │ │ │       {2 => (resolution, Matrix)}
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {2 => (contract', Matrix, Matrix)                           }
│ │ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {5 => (diff', Matrix, Matrix)                               }
│ │ │ │ +o5 = {0 => (diff, Matrix, Matrix)                                }
│ │ │ │ +     {1 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {2 => (diff', Matrix, Matrix)                               }
│ │ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {4 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -113,18 +113,18 @@
│ │ │ │       {21 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {22 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │       {24 => (remainder, Matrix, Matrix)                          }
│ │ │ │       {25 => (%, Matrix, Matrix)                                  }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (intersection, Matrix, Matrix, Matrix, Matrix)       }
│ │ │ │ -     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │ -     {31 => (intersection, Matrix, Matrix)                       }
│ │ │ │ +     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {29 => (intersection, Matrix, Matrix, Matrix, Matrix)       }
│ │ │ │ +     {30 => (intersection, Matrix, Matrix)                       }
│ │ │ │ +     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (checkDegrees, Matrix, Matrix)                       }
│ │ │ │       {34 => (isIsomorphic, Matrix, Matrix)                       }
│ │ │ │       {35 => (coneFromVData, Matrix, Matrix)                      }
│ │ │ │       {36 => (coneFromHData, Matrix, Matrix)                      }
│ │ │ │       {37 => (fan, Matrix, Matrix, List)                          }
│ │ │ │       {38 => (fan, Matrix, Matrix, Sequence)                      }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │      		          
                  
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 2.37905s elapsed
│ │ │ + -- 4.50648s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -121,15 +121,15 @@
│ │ │            
                  
    i4 : I = ideal I_*;
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │      		          
                  
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .748075s elapsed
│ │ │ + -- 1.64364s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -138,15 +138,15 @@
│ │ │            
                  
    i6 : I = ideal I_*;
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │      		          
                  
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0319761s elapsed
│ │ │ + -- .0578398s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -154,15 +154,15 @@
│ │ │            
                  
    i8 : I = ideal I_*;
│ │ │  
│ │ │  o8 : Ideal of S
    
│ │ │      		          
                  
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.22946s elapsed
│ │ │ + -- 1.38309s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,15 +44,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ │ - -- 2.37905s elapsed
│ │ │ │ + -- 4.50648s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -61,40 +61,40 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  One can compute smaller parts of the Betti table, by using _D_e_g_r_e_e_L_i_m_i_t and/or
│ │ │ │  _L_e_n_g_t_h_L_i_m_i_t.
│ │ │ │  i4 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ │ - -- .748075s elapsed
│ │ │ │ + -- 1.64364s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7
│ │ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │ │           0: 1  .   .   .   .    .   .   .
│ │ │ │           1: . 35 140 189  84    .   .   .
│ │ │ │           2: .  .   . 196 735 1080 735 196
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ │ - -- .0319761s elapsed
│ │ │ │ + -- .0578398s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3  4
│ │ │ │  o7 = total: 1 35 140 189 84
│ │ │ │           0: 1  .   .   .  .
│ │ │ │           1: . 35 140 189 84
│ │ │ │  
│ │ │ │  o7 : BettiTally
│ │ │ │  i8 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ │ - -- 1.22946s elapsed
│ │ │ │ + -- 1.38309s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -70,28 +70,28 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-10885-0/0/
    
│ │ │ +o1 = /tmp/M2-11432-0/0/
│ │ │      		          
                  
    i2 : mkdir p
    
│ │ │      		          
                  
    i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
    
│ │ │      		          
                  
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10885-0/0/foo
│ │ │ +o4 = /tmp/M2-11432-0/0/foo
│ │ │  
│ │ │  o4 : File
    
│ │ │      		          
                  
    i5 : get fn
│ │ │  
│ │ │  o5 = hi there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,22 +12,22 @@
│ │ │ │      * Consequences:
│ │ │ │            o a directory will be created at the path p
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Only one directory will be made, so the components of the path p other than the
│ │ │ │  last must already exist.
│ │ │ │  i1 : p = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10885-0/0/
│ │ │ │ +o1 = /tmp/M2-11432-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10885-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11432-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -85,42 +85,42 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10759-0/0
    
│ │ │ +o1 = /tmp/M2-11178-0/0
│ │ │      		          
                  
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10759-0/1
    
│ │ │ +o2 = /tmp/M2-11178-0/1
│ │ │      		          
                  
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10759-0/0
│ │ │ +o3 = /tmp/M2-11178-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10759-0/0 -> /tmp/M2-10759-0/1
    
│ │ │ +--moving: /tmp/M2-11178-0/0 -> /tmp/M2-11178-0/1
│ │ │      		          
                  
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │      		          
                  
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10759-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11178-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10759-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11178-0/1.bak
│ │ │      		          
                  
    i7 : removeFile bak
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,31 +21,31 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10759-0/0
│ │ │ │ +o1 = /tmp/M2-11178-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10759-0/1
│ │ │ │ +o2 = /tmp/M2-11178-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10759-0/0
│ │ │ │ +o3 = /tmp/M2-11178-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10759-0/0 -> /tmp/M2-10759-0/1
│ │ │ │ +--moving: /tmp/M2-11178-0/0 -> /tmp/M2-11178-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-10759-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11178-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10759-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11178-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -43,15 +43,15 @@
│ │ │  
        
    
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500227s elapsed
│ │ │ + -- .500097s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,14 +4,14 @@
│ │ │ │  [q                   ]
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ nnaannoosslleeeepp ---- sslleeeepp ffoorr aa ggiivveenn nnuummbbeerr ooff nnaannoosseeccoonnddss ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  nanosleep n -- sleeps for n nanoseconds.
│ │ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ │ - -- .500227s elapsed
│ │ │ │ + -- .500097s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_l_e_e_p -- sleep for a while
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _n_a_n_o_s_l_e_e_p is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -60,19 +60,19 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i2 : L = random toList (1..10000);
    
│ │ │      		          
                  
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .60735s elapsed
    
│ │ │ + -- .740176s elapsed
│ │ │      		          
                  
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .295654s elapsed
    
│ │ │ + -- .163264s elapsed
│ │ │      		          
    
│ │ │          
    
│ │ │            
    You will have to try it on your examples to see how much they speed up.
    
│ │ │            
    Warning: Threads computing in parallel can give wrong answers if their code is not "thread safe", 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.
    
│ │ │            
    The rest of this document describes how to control parallel tasks more directly.
    
│ │ │            
    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 allowableThreads, and may be examined and changed as follows. (allowableThreads is temporarily increased if necessary inside parallelApply.)
    
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
    		              
    i5 : allowableThreads
│ │ │  
│ │ │  o5 = 5
    
│ │ │      		          
                  
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │      		          
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │ @@ -191,15 +191,15 @@
│ │ │          
    
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i18 : schedule t';
    
│ │ │      		          
                  
    i19 : t'
│ │ │  
│ │ │ -o19 = <<task, running>>
│ │ │ +o19 = <<task, created>>
│ │ │  
│ │ │  o19 : Task
    
│ │ │      		          
                  
    i20 : taskResult t'
│ │ │  
│ │ │         1      6      8      3
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = random toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .60735s elapsed
│ │ │ │ + -- .740176s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .295654s elapsed
│ │ │ │ + -- .163264s elapsed
│ │ │ │  You will have to try it on your examples to see how much they speed up.
│ │ │ │  Warning: Threads computing in parallel can give wrong answers if their code is
│ │ │ │  not "thread safe", 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
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s, and may be examined and changed as follows. (_a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s
│ │ │ │  is temporarily increased if necessary inside _p_a_r_a_l_l_e_l_A_p_p_l_y.)
│ │ │ │  i5 : allowableThreads
│ │ │ │  
│ │ │ │  o5 = 5
│ │ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │ │  
│ │ │ │ -o6 = 7
│ │ │ │ +o6 = 17
│ │ │ │  To run a function in another thread use _s_c_h_e_d_u_l_e, as in the following example.
│ │ │ │  i7 : R = ZZ/101[x,y,z];
│ │ │ │  i8 : I = (ideal vars R)^2
│ │ │ │  
│ │ │ │               2             2        2
│ │ │ │  o8 = ideal (x , x*y, x*z, y , y*z, z )
│ │ │ │  
│ │ │ │ @@ -99,15 +99,15 @@
│ │ │ │  o17 = <>
│ │ │ │  
│ │ │ │  o17 : Task
│ │ │ │  Start it running with _s_c_h_e_d_u_l_e.
│ │ │ │  i18 : schedule t';
│ │ │ │  i19 : t'
│ │ │ │  
│ │ │ │ -o19 = <>
│ │ │ │ +o19 = <>
│ │ │ │  
│ │ │ │  o19 : Task
│ │ │ │  i20 : taskResult t'
│ │ │ │  
│ │ │ │         1      6      8      3
│ │ │ │  o20 = R  <-- R  <-- R  <-- R  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │      		          
                  
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 3.11318s elapsed
│ │ │ + -- 2.56513s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -142,15 +142,15 @@
│ │ │            
                  
    i5 : I = ideal I_*;
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │      		          
                  
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 2.33669s elapsed
│ │ │ + -- 2.33464s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -166,15 +166,15 @@
│ │ │            
                  
    i8 : numTBBThreads = 1
│ │ │  
│ │ │  o8 = 1
    
│ │ │      		          
                  
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 2.03997s elapsed
│ │ │ + -- 2.11254s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -199,15 +199,15 @@
│ │ │            
                  
    i12 : I = ideal I_*;
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │      		          
                  
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.05922s elapsed
│ │ │ + -- 4.14556s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -220,15 +220,15 @@
│ │ │            
                  
    i15 : I = ideal I_*;
│ │ │  
│ │ │  o15 : Ideal of S
    
│ │ │      		          
                  
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.99304s elapsed
│ │ │ + -- 2.07918s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -253,15 +253,15 @@
│ │ │            
                  
    i19 : I = ideal random(S^1, S^{4:-5});
│ │ │  
│ │ │  o19 : Ideal of S
    
│ │ │      		          
                  
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.49471s elapsed
│ │ │ + -- 5.22478s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │      		          
                  
    i21 : numTBBThreads = 1
│ │ │  
│ │ │ @@ -270,15 +270,15 @@
│ │ │            
                  
    i22 : I = ideal I_*;
│ │ │  
│ │ │  o22 : Ideal of S
    
│ │ │      		          
                  
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.78738s elapsed
│ │ │ + -- 6.40932s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │      		          
                  
    i24 : numTBBThreads = 10
│ │ │  
│ │ │ @@ -287,15 +287,15 @@
│ │ │            
                  
    i25 : I = ideal I_*;
│ │ │  
│ │ │  o25 : Ideal of S
    
│ │ │      		          
                  
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.79613s elapsed
│ │ │ + -- 3.59757s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    For Groebner basis computation in associative algebras, ParallelizeByDegree is not relevant.  In this case, use numTBBThreads to control the amount of parallelism.
    
│ │ │ @@ -328,15 +328,15 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
    
│ │ │      		          
                  
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.26729s elapsed
│ │ │ + -- 1.10176s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │      		          
                  
    i32 : I = ideal I_*
│ │ │ @@ -351,15 +351,15 @@
│ │ │            
                  
    i33 : numTBBThreads = 1
│ │ │  
│ │ │  o33 = 1
    
│ │ │      		          
                  
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.1978s elapsed
│ │ │ + -- 1.24741s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │      		          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -92,30 +92,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 3.11318s elapsed
│ │ │ │ + -- 2.56513s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │           4: .  .   .   .   .    .   .   .   .  .  1
│ │ │ │  
│ │ │ │  o4 : BettiTally
│ │ │ │  i5 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ │ - -- 2.33669s elapsed
│ │ │ │ + -- 2.33464s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -125,15 +125,15 @@
│ │ │ │  i7 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o8 = 1
│ │ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ │ - -- 2.03997s elapsed
│ │ │ │ + -- 2.11254s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -148,15 +148,15 @@
│ │ │ │  i11 : numTBBThreads = 0
│ │ │ │  
│ │ │ │  o11 = 0
│ │ │ │  i12 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.05922s elapsed
│ │ │ │ + -- 4.14556s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -167,15 +167,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 1.99304s elapsed
│ │ │ │ + -- 2.07918s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -194,37 +194,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.49471s elapsed
│ │ │ │ + -- 5.22478s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o20 : Matrix S  <-- S
│ │ │ │  i21 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o21 = 1
│ │ │ │  i22 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o22 : Ideal of S
│ │ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 7.78738s elapsed
│ │ │ │ + -- 6.40932s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o23 : Matrix S  <-- S
│ │ │ │  i24 : numTBBThreads = 10
│ │ │ │  
│ │ │ │  o24 = 10
│ │ │ │  i25 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.79613s elapsed
│ │ │ │ + -- 3.59757s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Groebner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -245,15 +245,15 @@
│ │ │ │                 2                         2                         2
│ │ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.26729s elapsed
│ │ │ │ + -- 1.10176s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  i32 : I = ideal I_*
│ │ │ │  
│ │ │ │                 2                         2                         2
│ │ │ │ @@ -262,15 +262,15 @@
│ │ │ │                  ZZ
│ │ │ │  o32 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i33 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o33 = 1
│ │ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.1978s elapsed
│ │ │ │ + -- 1.24741s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -314,34 +314,34 @@
│ │ │            
                  
    i27 : gbTrace = 3
│ │ │  
│ │ │  o27 = 3
    
│ │ │      		          
                  
    i28 : time poincare I
│ │ │ - -- used 0.00336927s (cpu); 1.6521e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00192958s (cpu); 7.865e-06s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │      		          
                  
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fec4bc63e00
│ │ │ +   -- registering gb 19 at 0x7f188450ee00
│ │ │  
│ │ │     -- [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
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0206106s (cpu); 0.0205024s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0100611s (cpu); 0.0099988s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    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.
    
│ │ │ @@ -349,39 +349,39 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ @@ -436,27 +436,27 @@
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ @@ -92,16 +92,16 @@
│ │ │  o6 : FunctionClosure
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i30 : R = QQ[a..d];
    
│ │ │      		          
                  
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 20 at 0x7fec4bc63c40
│ │ │ +   -- registering gb 20 at 0x7f188450ec40
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │     -- nsaved = 0
│ │ │     -- 
│ │ │  o31 : Ideal of R
    
│ │ │      		          
                  
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 21 at 0x7fec4bc638c0
│ │ │ +   -- registering gb 21 at 0x7f188450e8c0
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.00799675s (cpu); 0.00881508s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00388814s (cpu); 0.00363116s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │      		          
                  
    i36 : gbTrace = 3
│ │ │  
│ │ │  o36 = 3
    
│ │ │      		          
                  
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7fec4bc63700
│ │ │ +   -- registering gb 22 at 0x7f188450e700
│ │ │  
│ │ │     -- [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
│ │ │     -- 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
│ │ │     -- {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
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0679993s (cpu); 0.0696023s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0560149s (cpu); 0.0553724s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
    
│ │ │      		          
                  
    i38 : selectInSubring(1, gens gb J)
│ │ │ ├── html2text {}
│ │ │ │ @@ -179,66 +179,66 @@
│ │ │ │  o26 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o26 : ZZ[T]
│ │ │ │  i27 : gbTrace = 3
│ │ │ │  
│ │ │ │  o27 = 3
│ │ │ │  i28 : time poincare I
│ │ │ │ - -- used 0.00336927s (cpu); 1.6521e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00192958s (cpu); 7.865e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7fec4bc63e00
│ │ │ │ +   -- registering gb 19 at 0x7f188450ee00
│ │ │ │  
│ │ │ │     -- [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
│ │ │ │     -- number of monomials                = 4186
│ │ │ │     -- #reduction steps = 38
│ │ │ │     -- #spairs done = 11
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 29
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0206106s (cpu); 0.0205024s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0100611s (cpu); 0.0099988s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  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.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 20 at 0x7fec4bc63c40
│ │ │ │ +   -- registering gb 20 at 0x7f188450ec40
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 21 at 0x7fec4bc638c0
│ │ │ │ +   -- registering gb 21 at 0x7f188450e8c0
│ │ │ │  
│ │ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 267
│ │ │ │     -- #reduction steps = 236
│ │ │ │     -- #spairs done = 30
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 20
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.00799675s (cpu); 0.00881508s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00388814s (cpu); 0.00363116s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -283,30 +283,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 22 at 0x7fec4bc63700
│ │ │ │ +   -- registering gb 22 at 0x7f188450e700
│ │ │ │  
│ │ │ │     -- [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
│ │ │ │     -- 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
│ │ │ │     -- {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
│ │ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements
│ │ │ │  = 39
│ │ │ │     -- number of monomials                = 1051
│ │ │ │     -- #reduction steps = 284
│ │ │ │     -- #spairs done = 53
│ │ │ │     -- ncalls = 46
│ │ │ │     -- nloop = 54
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0679993s (cpu); 0.0696023s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0560149s (cpu); 0.0553724s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 243873059890414515367459726418219472801881021280016638460434780718278
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -61,15 +61,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : processID()
│ │ │  
│ │ │ -o1 = 10387
    
│ │ │ +o1 = 10448
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,13 +9,13 @@
│ │ │ │      *   Usage:
│ │ │ │              processID()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the process identifier of the current Macaulay2 process
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : processID()
│ │ │ │  
│ │ │ │ -o1 = 10387
│ │ │ │ +o1 = 10448
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_r_o_u_p_I_D -- the process group identifier
│ │ │ │      * _s_e_t_G_r_o_u_p_I_D -- set the process group identifier
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_c_e_s_s_I_D is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        123    110    13
│ │ │  o2 = x    + x    + x   + 1
│ │ │  
│ │ │  o2 : R
    
│ │ │      		          
                  
    i3 : time factor f
│ │ │ - -- used 0.00278858s (cpu); 0.0027806s (thread); 0s (gc)
│ │ │ + -- used 0.00239599s (cpu); 0.00238081s (thread); 0s (gc)
│ │ │  
│ │ │                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
│ │ │  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)
│ │ │  
│ │ │  o3 : Expression of class Product
    
│ │ │      		          
              
                  
    i7 : for i to 10 do (g();h();h())
    
│ │ │      		          
                  
    i8 : profileSummary
│ │ │ -g: 11 times, used .0266591 seconds
│ │ │ -h: 22 times, used .0532723 seconds
    
│ │ │ +g: 11 times, used .0234191 seconds
│ │ │ +h: 22 times, used .0666404 seconds
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Ways to use profile:
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i2 : f = (x^110+1)*(x^13+1)
│ │ │ │  
│ │ │ │        123    110    13
│ │ │ │  o2 = x    + x    + x   + 1
│ │ │ │  
│ │ │ │  o2 : R
│ │ │ │  i3 : time factor f
│ │ │ │ - -- used 0.00278858s (cpu); 0.0027806s (thread); 0s (gc)
│ │ │ │ + -- used 0.00239599s (cpu); 0.00238081s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                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
│ │ │ │ @@ -50,14 +50,14 @@
│ │ │ │  i6 : h = profile("h", () -> factor f)
│ │ │ │  
│ │ │ │  o6 = h
│ │ │ │  
│ │ │ │  o6 : FunctionClosure
│ │ │ │  i7 : for i to 10 do (g();h();h())
│ │ │ │  i8 : profileSummary
│ │ │ │ -g: 11 times, used .0266591 seconds
│ │ │ │ -h: 22 times, used .0532723 seconds
│ │ │ │ +g: 11 times, used .0234191 seconds
│ │ │ │ +h: 22 times, used .0666404 seconds
│ │ │ │  ********** WWaayyss ttoo uussee pprrooffiillee:: **********
│ │ │ │      * profile(Function)
│ │ │ │      * profile(String,Function)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_f_i_l_e is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  
│ │ │  o5 = (2, 10)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
    		          
                  
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.186982s (cpu); 0.129358s (thread); 0s (gc)
│ │ │ + -- used 0.224136s (cpu); 0.0635276s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │      		          
              
                  
    i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.403357s (cpu); 0.338406s (thread); 0s (gc)
│ │ │ + -- used 0.390978s (cpu); 0.207144s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │      		          
    
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
    		              
    i14 : I=ideal random(n,R);
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │      		          
                  
    i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 3.98035s (cpu); 2.47525s (thread); 0s (gc)
│ │ │ + -- used 3.84275s (cpu); 2.01402s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Ways to use randomKRationalPoint:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : codim I, degree I
│ │ │ │  
│ │ │ │  o5 = (2, 10)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.186982s (cpu); 0.129358s (thread); 0s (gc)
│ │ │ │ + -- used 0.224136s (cpu); 0.0635276s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │ │               2      3   1     3   0     3
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : R=kk[x_0..x_5];
│ │ │ │  i8 : I=minors(3,random(R^5,R^{3:-1}));
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : codim I, degree I
│ │ │ │  
│ │ │ │  o9 = (3, 10)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.403357s (cpu); 0.338406s (thread); 0s (gc)
│ │ │ │ + -- used 0.390978s (cpu); 0.207144s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  The claim that $63 \%$ of the intersections contain a K-rational point can be
│ │ │ │  experimentally tested:
│ │ │ │ @@ -70,14 +70,14 @@
│ │ │ │  o13 : RR (of precision 53)
│ │ │ │  i14 : I=ideal random(n,R);
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c-
│ │ │ │  >degree c);
│ │ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ │ - -- used 3.98035s (cpu); 2.47525s (thread); 0s (gc)
│ │ │ │ + -- used 3.84275s (cpu); 2.01402s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -67,25 +67,25 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11649-0/0
    
│ │ │ +o1 = /tmp/M2-12976-0/0
│ │ │      		          
                  
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11649-0/0
    
│ │ │ +o2 = /tmp/M2-12976-0/0
│ │ │      		          
                  
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11649-0/0/foo
│ │ │ +o3 = /tmp/M2-12976-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : readDirectory dir
│ │ │  
│ │ │  o4 = {., .., foo}
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,21 +11,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Outputs:
│ │ │ │            o a _l_i_s_t, the list of filenames stored in the directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11649-0/0
│ │ │ │ +o1 = /tmp/M2-12976-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11649-0/0
│ │ │ │ +o2 = /tmp/M2-12976-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11649-0/0/foo
│ │ │ │ +o3 = /tmp/M2-12976-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │  o4 = {., .., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -44,20 +44,20 @@
│ │ │        
    reading files
    
│ │ │        
    
│ │ │  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 get 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 expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11232-0/0
    
│ │ │ +o1 = /tmp/M2-12139-0/0
│ │ │      		          
                  
    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
│ │ │  
│ │ │ -o2 = /tmp/M2-11232-0/0
│ │ │ +o2 = /tmp/M2-12139-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │  
                  
    i3 : get fn
│ │ │ @@ -96,15 +96,15 @@
│ │ │          
    
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
│ │ │            
│ │ │  
│ │ │          
                  
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11232-0/0
│ │ │ +o7 = /tmp/M2-12139-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │      		          
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i8 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,20 +7,20 @@
│ │ │ │  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 _g_e_t 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 expression.
│ │ │ │  First we create the file by writing the desired text to it.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11232-0/0
│ │ │ │ +o1 = /tmp/M2-12139-0/0
│ │ │ │  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
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11232-0/0
│ │ │ │ +o2 = /tmp/M2-12139-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  Now we get the contents of the file, as a single string.
│ │ │ │  i3 : get fn
│ │ │ │  
│ │ │ │  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
│ │ │ │       +8*y^3
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  o6 : Expression of class Product
│ │ │ │  Often a file will contain code written in the Macaulay2 language. Let's create
│ │ │ │  such a file.
│ │ │ │  i7 : fn << "sample = 2^100
│ │ │ │       print sample
│ │ │ │       " << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11232-0/0
│ │ │ │ +o7 = /tmp/M2-12139-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -67,15 +67,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-11961-0/0
    
│ │ │ +o1 = /tmp/M2-13608-0/0
│ │ │      		          
                  
    i2 : symlinkFile ("foo", p)
    
│ │ │      		          
                  
    i3 : readlink p
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a symbolic link
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11961-0/0
│ │ │ │ +o1 = /tmp/M2-13608-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_e_a_d_l_i_n_k is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -67,59 +67,59 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10387-0/83-rundir/
    
│ │ │ +o1 = /tmp/M2-10448-0/83-rundir/
│ │ │      		          
                  
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11980-0/0
    
│ │ │ +o2 = /tmp/M2-13647-0/0
│ │ │      		          
                  
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11980-0/1
    
│ │ │ +o3 = /tmp/M2-13647-0/1
│ │ │      		          
                  
    i4 : symlinkFile(p,q)
    
│ │ │      		          
                  
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11980-0/0
│ │ │ +o5 = /tmp/M2-13647-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │      		          
                  
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11980-0/0
    
│ │ │ +o6 = /tmp/M2-13647-0/0
│ │ │      		          
                  
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11980-0/0
    
│ │ │ +o7 = /tmp/M2-13647-0/0
│ │ │      		          
                  
    i8 : removeFile p
    
│ │ │      		          
                  
    i9 : removeFile q
    
│ │ │      		          
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10387-0/83-rundir/
    
│ │ │ +o10 = /tmp/M2-10448-0/83-rundir/
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  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.      
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,39 +13,39 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename, or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, a pathname to fn passing through no symbolic links, and
│ │ │ │              ending with a slash if fn refers to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : realpath "."
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10387-0/83-rundir/
│ │ │ │ +o1 = /tmp/M2-10448-0/83-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11980-0/0
│ │ │ │ +o2 = /tmp/M2-13647-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11980-0/1
│ │ │ │ +o3 = /tmp/M2-13647-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11980-0/0
│ │ │ │ +o5 = /tmp/M2-13647-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11980-0/0
│ │ │ │ +o6 = /tmp/M2-13647-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11980-0/0
│ │ │ │ +o7 = /tmp/M2-13647-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10387-0/83-rundir/
│ │ │ │ +o10 = /tmp/M2-10448-0/83-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  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.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -74,21 +74,21 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │      		          
                  
    i2 : collectGarbage() 
│ │ │  --finalization: (1)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (2)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (4)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (2)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (3)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (6)[1]: -- finalizing sequence #2 --
│ │ │  --finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (8)[0]: -- finalizing sequence #1 --
    
│ │ │ +--finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  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.      
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,21 +16,21 @@
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │  --finalization: (1)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (2)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (4)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (2)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (3)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (6)[1]: -- finalizing sequence #2 --
│ │ │ │  --finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  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.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -69,20 +69,20 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  o1 : Matroid
│ │ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │ │  
│ │ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │ │  
│ │ │ │  o2 : Matroid
│ │ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ │ - -- .203206s elapsed
│ │ │ │ + -- .113405s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │  
│ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │  
│ │ │  o6 : Matroid
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10923-0/0
    
│ │ │ +o1 = /tmp/M2-11510-0/0
│ │ │      		          
                  
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10923-0/0
    
│ │ │ +o2 = /tmp/M2-11510-0/0
│ │ │      		          
                  
    i3 : readDirectory dir
│ │ │  
│ │ │  o3 = {., ..}
│ │ │  
│ │ │  o3 : List
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is removed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10923-0/0
│ │ │ │ +o1 = /tmp/M2-11510-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10923-0/0
│ │ │ │ +o2 = /tmp/M2-11510-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │  o3 = {., ..}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -62,20 +62,20 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    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 C:/, so it is safe always to concatenate with the value of rootPath, even when it is unknown whether the external program has been compiled under Cygwin.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10465-0/0
    
│ │ │ +o1 = /tmp/M2-10584-0/0
│ │ │      		          
                  
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10465-0/0
    
│ │ │ +o2 = /tmp/M2-10584-0/0
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │            
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  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 C:/, so it is safe
│ │ │ │  always to concatenate with the value of _r_o_o_t_P_a_t_h, even when it is unknown
│ │ │ │  whether the external program has been compiled under Cygwin.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10465-0/0
│ │ │ │ +o1 = /tmp/M2-10584-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10465-0/0
│ │ │ │ +o2 = /tmp/M2-10584-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -62,20 +62,20 @@
│ │ │        
      
│ │ │          
      Description
      
│ │ │          
      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 C:/, so it is safe always to concatenate with the value of rootPath, even when it is unknown whether the external program has been compiled under Cygwin.
      
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11592-0/0
      
│ │ │ +o1 = /tmp/M2-12859-0/0
│ │ │        		          
                    
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11592-0/0
      
│ │ │ +o2 = file:///tmp/M2-12859-0/0
│ │ │        		          
      
│ │ │        
      
│ │ │        
      
│ │ │          
      See also
      
│ │ │          
        
│ │ │            
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  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 C:/, so it is safe
│ │ │ │  always to concatenate with the value of _r_o_o_t_P_a_t_h, even when it is unknown
│ │ │ │  whether the external program has been compiled under Cygwin.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11592-0/0
│ │ │ │ +o1 = /tmp/M2-12859-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11592-0/0
│ │ │ │ +o2 = file:///tmp/M2-12859-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -74,20 +74,20 @@
│ │ │  
│ │ │                               1
│ │ │  o4 : R-module, submodule of R
│ │ │  
    		          
                  
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-11478-0/0
    
│ │ │ +o5 = /tmp/M2-12625-0/0
│ │ │      		          
                  
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11478-0/0
│ │ │ +o6 = /tmp/M2-12625-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │      		          
                  
    i7 : get f
│ │ │  
│ │ │  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,
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,18 +28,18 @@
│ │ │ │  
│ │ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o4 : R-module, submodule of R
│ │ │ │  i5 : f = temporaryFileName()
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11478-0/0
│ │ │ │ +o5 = /tmp/M2-12625-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11478-0/0
│ │ │ │ +o6 = /tmp/M2-12625-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : get f
│ │ │ │  
│ │ │ │  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,
│ │ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ │ │  i8 : (p',m',M') = value get f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_schedule.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
    		              
    i4 : taskResult t
│ │ │  
│ │ │  o4 = 8
    
│ │ │      		          
                  
    i5 : u = schedule(f,4)
│ │ │  
│ │ │ -o5 = <<task, result available, task done>>
│ │ │ +o5 = <<task, created>>
│ │ │  
│ │ │  o5 : Task
    
│ │ │      		          
                  
    i6 : taskResult u
│ │ │  
│ │ │  o6 = 16
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  
│ │ │ │  o3 : Task
│ │ │ │  i4 : taskResult t
│ │ │ │  
│ │ │ │  o4 = 8
│ │ │ │  i5 : u = schedule(f,4)
│ │ │ │  
│ │ │ │ -o5 = <>
│ │ │ │ +o5 = <>
│ │ │ │  
│ │ │ │  o5 : Task
│ │ │ │  i6 : taskResult u
│ │ │ │  
│ │ │ │  o6 = 16
│ │ │ │  ********** WWaayyss ttoo uussee sscchheedduullee:: **********
│ │ │ │      * schedule(Function)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_serial__Number.html
│ │ │ @@ -67,20 +67,20 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1528251
    
│ │ │ +o1 = 1628251
│ │ │      		          
                  
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1528253
    
│ │ │ +o2 = 1628253
│ │ │      		          
                  
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,16 +11,16 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1528251
│ │ │ │ +o1 = 1628251
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1528253
│ │ │ │ +o2 = 1628253
│ │ │ │  i3 : serialNumber ZZ
│ │ │ │  
│ │ │ │  o3 = 1000050
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _s_e_r_i_a_l_N_u_m_b_e_r is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -320,19 +320,19 @@
│ │ │  
    		              
    i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
    
│ │ │      		          
                  
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │      		          
                  
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000236533s (cpu); 0.000229119s (thread); 0s (gc)
    
│ │ │ + -- used 0.000140143s (cpu); 0.000128931s (thread); 0s (gc)
│ │ │      		          
                  
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000168847s (cpu); 0.000168906s (thread); 0s (gc)
    
│ │ │ + -- used 9.3294e-05s (cpu); 9.3475e-05s (thread); 0s (gc)
│ │ │      		          
                  
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │  
│ │ │  o32 : RR (of precision 53)
    
│ │ │ @@ -353,19 +353,19 @@
│ │ │      		              
    i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
    
│ │ │      		          
                  
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │      		          
                  
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.484036s (cpu); 0.306285s (thread); 0s (gc)
    
│ │ │ + -- used 0.145216s (cpu); 0.145221s (thread); 0s (gc)
│ │ │      		          
                  
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.238738s (cpu); 0.23868s (thread); 0s (gc)
    
│ │ │ + -- used 0.12495s (cpu); 0.12496s (thread); 0s (gc)
│ │ │      		          
                  
    i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │  
│ │ │  o40 : RR (of precision 100)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -195,33 +195,33 @@
│ │ │ │  i24 : printingPrecision = 4;
│ │ │ │  i25 : N = 40
│ │ │ │  
│ │ │ │  o25 = 40
│ │ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │ │  i30 : time X = solve(A,B);
│ │ │ │ - -- used 0.000236533s (cpu); 0.000229119s (thread); 0s (gc)
│ │ │ │ + -- used 0.000140143s (cpu); 0.000128931s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000168847s (cpu); 0.000168906s (thread); 0s (gc)
│ │ │ │ + -- used 9.3294e-05s (cpu); 9.3475e-05s (thread); 0s (gc)
│ │ │ │  i32 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o32 = 5.111850690840453e-15
│ │ │ │  
│ │ │ │  o32 : RR (of precision 53)
│ │ │ │  Over higher precision RR or CC, these routines will be much slower than the
│ │ │ │  lower precision lapack routines.
│ │ │ │  i33 : N = 100
│ │ │ │  
│ │ │ │  o33 = 100
│ │ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │ │  i38 : time X = solve(A,B);
│ │ │ │ - -- used 0.484036s (cpu); 0.306285s (thread); 0s (gc)
│ │ │ │ + -- used 0.145216s (cpu); 0.145221s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.238738s (cpu); 0.23868s (thread); 0s (gc)
│ │ │ │ + -- used 0.12495s (cpu); 0.12496s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html
│ │ │┄ Ordering differences only
│ │ │ @@ -53,18 +53,18 @@
│ │ │  Syzygies -- an optional argument          
│ │ │          
│ │ │          
    Symbols used as an option name
    
│ │ │          
      
│ │ │            
    • 
│ │ │  Threads -- an optional argument          
      • 
│ │ │            
    • 
│ │ │ -Strategy -- an optional argument          
      • 
│ │ │ -          
    • 
│ │ │  Generic -- an optional argument          
      • 
│ │ │            
    • 
│ │ │ +Strategy -- an optional argument          
      • 
│ │ │ +          
    • 
│ │ │  DegreeGroup -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  Variables -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  DegreeLift -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  VariableBaseName -- an optional argument          
      • 
│ │ │ @@ -85,26 +85,26 @@
│ │ │            
    • 
│ │ │  Constants -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  SkewCommutative -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  DegreeMap -- an optional argument          
      • 
│ │ │            
    • 
│ │ │ -Verbosity -- an optional argument          
      • 
│ │ │ -          
    • 
│ │ │ -CodimensionLimit -- an optional argument          
      • 
│ │ │ -          
    • 
│ │ │  DegreeLimit -- an optional argument          
      • 
│ │ │            
    • 
│ │ │ -Verify -- an optional argument          
      • 
│ │ │ -          
    • 
│ │ │  BasisElementLimit -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  PairLimit -- an optional argument          
      • 
│ │ │            
    • 
│ │ │ +Verbosity -- an optional argument          
      • 
│ │ │ +          
    • 
│ │ │ +CodimensionLimit -- an optional argument          
      • 
│ │ │ +          
    • 
│ │ │ +Verify -- an optional argument          
      • 
│ │ │ +          
    • 
│ │ │  CoefficientRing -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  FollowLinks -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  Exclude -- an optional argument          
      • 
│ │ │            
    • 
│ │ │  InstallPrefix -- an optional argument          
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -6,36 +6,36 @@
│ │ │ │  ===============================================================================
│ │ │ │  ************ ssyymmbboollss uusseedd aass tthhee nnaammee oorr vvaalluuee ooff aann ooppttiioonnaall aarrgguummeenntt ************
│ │ │ │  ******** MMeennuu ********
│ │ │ │  ****** SSyymmbboollss uusseedd aass aann ooppttiioonn nnaammee oorr vvaalluuee ******
│ │ │ │      * _S_y_z_y_g_i_e_s -- an optional argument
│ │ │ │  ****** SSyymmbboollss uusseedd aass aann ooppttiioonn nnaammee ******
│ │ │ │      * _T_h_r_e_a_d_s -- an optional argument
│ │ │ │ -    * _S_t_r_a_t_e_g_y -- an optional argument
│ │ │ │      * _G_e_n_e_r_i_c -- an optional argument
│ │ │ │ +    * _S_t_r_a_t_e_g_y -- an optional argument
│ │ │ │      * _D_e_g_r_e_e_G_r_o_u_p -- an optional argument
│ │ │ │      * _V_a_r_i_a_b_l_e_s -- an optional argument
│ │ │ │      * _D_e_g_r_e_e_L_i_f_t -- an optional argument
│ │ │ │      * _V_a_r_i_a_b_l_e_B_a_s_e_N_a_m_e -- an optional argument
│ │ │ │      * _D_e_g_r_e_e_R_a_n_k -- an optional argument
│ │ │ │      * _I_n_v_e_r_s_e_s -- an optional argument
│ │ │ │      * _D_e_g_r_e_e_s -- an optional argument
│ │ │ │      * _J_o_i_n -- an optional argument
│ │ │ │      * _M_o_n_o_m_i_a_l_S_i_z_e -- an optional argument
│ │ │ │      * _L_o_c_a_l -- an optional argument
│ │ │ │      * _H_e_f_t -- an optional argument
│ │ │ │      * _C_o_n_s_t_a_n_t_s -- an optional argument
│ │ │ │      * _S_k_e_w_C_o_m_m_u_t_a_t_i_v_e -- an optional argument
│ │ │ │      * _D_e_g_r_e_e_M_a_p -- an optional argument
│ │ │ │ -    * _V_e_r_b_o_s_i_t_y -- an optional argument
│ │ │ │ -    * _C_o_d_i_m_e_n_s_i_o_n_L_i_m_i_t -- an optional argument
│ │ │ │      * _D_e_g_r_e_e_L_i_m_i_t -- an optional argument
│ │ │ │ -    * _V_e_r_i_f_y -- an optional argument
│ │ │ │      * _B_a_s_i_s_E_l_e_m_e_n_t_L_i_m_i_t -- an optional argument
│ │ │ │      * _P_a_i_r_L_i_m_i_t -- an optional argument
│ │ │ │ +    * _V_e_r_b_o_s_i_t_y -- an optional argument
│ │ │ │ +    * _C_o_d_i_m_e_n_s_i_o_n_L_i_m_i_t -- an optional argument
│ │ │ │ +    * _V_e_r_i_f_y -- an optional argument
│ │ │ │      * _C_o_e_f_f_i_c_i_e_n_t_R_i_n_g -- an optional argument
│ │ │ │      * _F_o_l_l_o_w_L_i_n_k_s -- an optional argument
│ │ │ │      * _E_x_c_l_u_d_e -- an optional argument
│ │ │ │      * _I_n_s_t_a_l_l_P_r_e_f_i_x -- an optional argument
│ │ │ │      * _S_y_z_y_g_y_M_a_t_r_i_x -- an optional argument
│ │ │ │      * _C_h_a_n_g_e_M_a_t_r_i_x -- an optional argument
│ │ │ │      * _M_i_n_i_m_a_l_M_a_t_r_i_x -- an optional argument
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,73 +85,73 @@
│ │ │        
│ │ │        
      
│ │ │          
      Description
      
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11272-0/0/
      
│ │ │ +o1 = /tmp/M2-12219-0/0/
│ │ │        		          
                    
      i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11272-0/1/
      
│ │ │ +o2 = /tmp/M2-12219-0/1/
│ │ │        		          
                    
      i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11272-0/0/a/
      
│ │ │ +o3 = /tmp/M2-12219-0/0/a/
│ │ │        		          
                    
      i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11272-0/0/b/
      
│ │ │ +o4 = /tmp/M2-12219-0/0/b/
│ │ │        		          
                    
      i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11272-0/0/b/c/
      
│ │ │ +o5 = /tmp/M2-12219-0/0/b/c/
│ │ │        		          
                    
      i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11272-0/0/a/f
│ │ │ +o6 = /tmp/M2-12219-0/0/a/f
│ │ │  
│ │ │  o6 : File
      
│ │ │        		          
                    
      i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11272-0/0/a/g
│ │ │ +o7 = /tmp/M2-12219-0/0/a/g
│ │ │  
│ │ │  o7 : File
      
│ │ │        		          
                    
      i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11272-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12219-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
      
│ │ │        		          
                    
      i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11272-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11272-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11272-0/1/a/f
      
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12219-0/1/b/c/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12219-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12219-0/1/a/g
│ │ │        		          
                    
      i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
      
│ │ │        		          
                    
      i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11272-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11272-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11272-0/1/a/f
      
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12219-0/1/b/c/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12219-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12219-0/1/a/g
│ │ │        		          
      
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │  
                    
      i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │  
│ │ │  o12 = rm
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,53 +31,53 @@
│ │ │ │            o The directory tree rooted at src is duplicated by a directory tree
│ │ │ │              rooted at dst. The files in the source tree are represented by
│ │ │ │              relative symbolic links in the destination tree to the original
│ │ │ │              files in the source tree.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11272-0/0/
│ │ │ │ +o1 = /tmp/M2-12219-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11272-0/1/
│ │ │ │ +o2 = /tmp/M2-12219-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11272-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12219-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11272-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12219-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11272-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12219-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11272-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12219-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11272-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12219-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11272-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12219-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11272-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11272-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11272-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12219-0/1/b/c/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12219-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12219-0/1/a/g
│ │ │ │  i10 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o10 = ho there
│ │ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11272-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11272-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11272-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12219-0/1/b/c/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12219-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12219-0/1/a/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │        
│ │ │        
      
│ │ │          
      Description
      
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
                    
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11345-0/0
      
│ │ │ +o1 = /tmp/M2-12352-0/0
│ │ │        		          
                    
      i2 : symlinkFile("qwert", fn)
      
│ │ │        		          
                    
      i3 : fileExists fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11345-0/0
│ │ │ │ +o1 = /tmp/M2-12352-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -61,20 +61,20 @@
│ │ │        
│ │ │        
      
│ │ │          
      Description
      
│ │ │  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.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12321-0/0.tex
      
│ │ │ +o1 = /tmp/M2-14386-0/0.tex
│ │ │        		          
                    
      i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12321-0/1.html
      
│ │ │ +o2 = /tmp/M2-14386-0/1.html
│ │ │        		          
      
│ │ │          
      This function will work under Unix, and also under Windows if you have a directory on the same drive called /tmp.
      
│ │ │          
      If the name of the temporary file will be given to an external program, it may be necessary to concatenate it with rootPath or rootURI to enable the external program to find the file.
      
│ │ │          
      The temporary file name is derived from the value of the environment variable TMPDIR, if it has one.
      
│ │ │          
      If fork 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 fork was called.
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,18 +12,18 @@
│ │ │ │            o a unique temporary file name.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  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.
│ │ │ │  i1 : temporaryFileName () | ".tex"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12321-0/0.tex
│ │ │ │ +o1 = /tmp/M2-14386-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12321-0/1.html
│ │ │ │ +o2 = /tmp/M2-14386-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -54,15 +54,15 @@
│ │ │          
│ │ │        
│ │ │        
      
│ │ │          
      Description
      
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │  
│ │ │          
                    
      i1 : time 3^30
│ │ │ - -- used 1.3776e-05s (cpu); 5.69e-06s (thread); 0s (gc)
│ │ │ + -- used 2.0499e-05s (cpu); 5.37e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
      
│ │ │        		          
      
│ │ │        
      
│ │ │        
      
│ │ │          
      See also
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │      *   Usage:
│ │ │ │              time e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value
│ │ │ │  of e. The time used by the the current thread and garbage collection during the
│ │ │ │  evaluation of e is also shown.
│ │ │ │  i1 : time 3^30
│ │ │ │ - -- used 1.3776e-05s (cpu); 5.69e-06s (thread); 0s (gc)
│ │ │ │ + -- used 2.0499e-05s (cpu); 5.37e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -46,22 +46,22 @@
│ │ │          
      Description
      
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
      
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000014217 seconds
│ │ │ +     -- .000015719 seconds
│ │ │  
│ │ │  o1 : Time
      
│ │ │        		          
                    
      i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000014217, 205891132094649}
      
│ │ │ +o2 = Time{.000015719, 205891132094649}
│ │ │        		          
      
│ │ │        
      
│ │ │        
      
│ │ │          
      See also
      
│ │ │          
        
│ │ │            
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,20 +9,20 @@
│ │ │ │  is the number of seconds of cpu timing used, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : timing 3^30
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │ -     -- .000014217 seconds
│ │ │ │ +     -- .000015719 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000014217, 205891132094649}
│ │ │ │ +o2 = Time{.000015719, 205891132094649}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │                 "memtailor version" => 1.0
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.1
│ │ │                 "mpsolve version" => 3.2.1
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.10.4
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.1.0-31-amd64
│ │ │ +               "operating system release" => 6.12.35+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings PruneComplex BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples Divisor EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieTypes ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties Licenses TestIdeals FrobeniusThresholds Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization NonminimalComplexes CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.2
│ │ │                 "readline version" => 8.2
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.0
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │                 "memtailor version" => 1.0
│ │ │ │                 "mpfi version" => 1.5.4
│ │ │ │                 "mpfr version" => 4.2.1
│ │ │ │                 "mpsolve version" => 3.2.1
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.10.4
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.1.0-31-amd64
│ │ │ │ +               "operating system release" => 6.12.35+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato Depth
│ │ │ │  Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements
│ │ │ │  LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings
│ │ │ │  SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/toc.html
│ │ │┄ Ordering differences only
│ │ │ @@ -7399,19 +7399,19 @@
│ │ │              
        • 
│ │ │              
      • 
│ │ │                
        
│ │ │  Threads -- an optional argument              
        
│ │ │              
        • 
│ │ │              
      • 
│ │ │                
        
│ │ │ -Strategy -- an optional argument              
        
│ │ │ +Generic -- an optional argument              
      
│ │ │              
│ │ │              
    • 
│ │ │                
      
│ │ │ -Generic -- an optional argument              
      
│ │ │ +Strategy -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │  DegreeGroup -- an optional argument              
      
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ @@ -7463,35 +7463,35 @@
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │  DegreeMap -- an optional argument              
      
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ -Verbosity -- an optional argument              
      
│ │ │ +DegreeLimit -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ -CodimensionLimit -- an optional argument              
      
│ │ │ +BasisElementLimit -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ -DegreeLimit -- an optional argument              
      
│ │ │ +PairLimit -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ -Verify -- an optional argument              
      
│ │ │ +Verbosity -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ -BasisElementLimit -- an optional argument              
      
│ │ │ +CodimensionLimit -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ -PairLimit -- an optional argument              
      
│ │ │ +Verify -- an optional argument              
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │  CoefficientRing -- an optional argument              
      
│ │ │              
      • 
│ │ │              
    • 
│ │ │                
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -2047,36 +2047,36 @@
│ │ │ │      * _S_y_m_b_o_l_ ___ _R_i_n_g -- get a ring variable by name
│ │ │ │      * _S_y_m_b_o_l_ ___ _T_h_i_n_g -- index variable
│ │ │ │      * _S_y_m_b_o_l_ ___ _T_h_i_n_g_ _=_ _T_h_i_n_g -- assignment to an indexed variable
│ │ │ │      * _s_y_m_b_o_l_B_o_d_y -- symbol bodies
│ │ │ │      * _s_y_m_b_o_l_s_ _u_s_e_d_ _a_s_ _t_h_e_ _n_a_m_e_ _o_r_ _v_a_l_u_e_ _o_f_ _a_n_ _o_p_t_i_o_n_a_l_ _a_r_g_u_m_e_n_t
│ │ │ │            o _S_y_z_y_g_i_e_s -- an optional argument
│ │ │ │            o _T_h_r_e_a_d_s -- an optional argument
│ │ │ │ -          o _S_t_r_a_t_e_g_y -- an optional argument
│ │ │ │            o _G_e_n_e_r_i_c -- an optional argument
│ │ │ │ +          o _S_t_r_a_t_e_g_y -- an optional argument
│ │ │ │            o _D_e_g_r_e_e_G_r_o_u_p -- an optional argument
│ │ │ │            o _V_a_r_i_a_b_l_e_s -- an optional argument
│ │ │ │            o _D_e_g_r_e_e_L_i_f_t -- an optional argument
│ │ │ │            o _V_a_r_i_a_b_l_e_B_a_s_e_N_a_m_e -- an optional argument
│ │ │ │            o _D_e_g_r_e_e_R_a_n_k -- an optional argument
│ │ │ │            o _I_n_v_e_r_s_e_s -- an optional argument
│ │ │ │            o _D_e_g_r_e_e_s -- an optional argument
│ │ │ │            o _J_o_i_n -- an optional argument
│ │ │ │            o _M_o_n_o_m_i_a_l_S_i_z_e -- an optional argument
│ │ │ │            o _L_o_c_a_l -- an optional argument
│ │ │ │            o _H_e_f_t -- an optional argument
│ │ │ │            o _C_o_n_s_t_a_n_t_s -- an optional argument
│ │ │ │            o _S_k_e_w_C_o_m_m_u_t_a_t_i_v_e -- an optional argument
│ │ │ │            o _D_e_g_r_e_e_M_a_p -- an optional argument
│ │ │ │ -          o _V_e_r_b_o_s_i_t_y -- an optional argument
│ │ │ │ -          o _C_o_d_i_m_e_n_s_i_o_n_L_i_m_i_t -- an optional argument
│ │ │ │            o _D_e_g_r_e_e_L_i_m_i_t -- an optional argument
│ │ │ │ -          o _V_e_r_i_f_y -- an optional argument
│ │ │ │            o _B_a_s_i_s_E_l_e_m_e_n_t_L_i_m_i_t -- an optional argument
│ │ │ │            o _P_a_i_r_L_i_m_i_t -- an optional argument
│ │ │ │ +          o _V_e_r_b_o_s_i_t_y -- an optional argument
│ │ │ │ +          o _C_o_d_i_m_e_n_s_i_o_n_L_i_m_i_t -- an optional argument
│ │ │ │ +          o _V_e_r_i_f_y -- an optional argument
│ │ │ │            o _C_o_e_f_f_i_c_i_e_n_t_R_i_n_g -- an optional argument
│ │ │ │            o _F_o_l_l_o_w_L_i_n_k_s -- an optional argument
│ │ │ │            o _E_x_c_l_u_d_e -- an optional argument
│ │ │ │            o _I_n_s_t_a_l_l_P_r_e_f_i_x -- an optional argument
│ │ │ │            o _S_y_z_y_g_y_M_a_t_r_i_x -- an optional argument
│ │ │ │            o _C_h_a_n_g_e_M_a_t_r_i_x -- an optional argument
│ │ │ │            o _M_i_n_i_m_a_l_M_a_t_r_i_x -- an optional argument
│ │ ├── ./usr/share/doc/Macaulay2/MapleInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  c3RvcmU=
│ │ │  #:len=651
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RvcmUgcmVzdWx0IG9mIGEgTWFwbGUg
│ │ │  Y29tcHV0YXRpb24gaW4gYSBmaWxlLiIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsi
│ │ ├── ./usr/share/doc/Macaulay2/Markov/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  Z2F1c3NJZGVhbChSaW5nLExpc3Qp
│ │ │  #:len=239
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjE4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnYXVzc0lkZWFsLFJpbmcsTGlzdCksImdhdXNzSWRl
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   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|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   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|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 3.13287s (cpu); 1.71368s (thread); 0s (gc)
│ │ │ + -- used 3.79413s (cpu); 2.17435s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -139,15 +139,15 @@
│ │ │          
      • 
│ │ │          
      
│ │ │            
      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.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │  
                    
      i8 : time netList primaryDecomposition J
│ │ │ - -- used 3.13287s (cpu); 1.71368s (thread); 0s (gc)
│ │ │ + -- used 3.79413s (cpu); 2.17435s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  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.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 3.13287s (cpu); 1.71368s (thread); 0s (gc)
│ │ │ │ + -- used 3.79413s (cpu); 2.17435s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  YWxnZWJyYWljTWF0cm9pZENpcmN1aXRz
│ │ │  #:len=1461
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIGJhc2VzIG9mIHRoZSBhbGdlYnJh
│ │ │  aWMgbWF0cm9pZCIsICJsaW5lbnVtIiA9PiAyMTIxLCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aXNBU01VbmlvbihMaXN0KQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTA2MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNBU01VbmlvbixMaXN0KSwiaXNBU01VbmlvbihM
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out
│ │ │ @@ -212,17 +212,17 @@
│ │ │        | 1 -1 1 |
│ │ │        | 0 1  0 |
│ │ │  
│ │ │                 3       3
│ │ │  o22 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i23 : time schubertRegularity B
│ │ │ - -- used 0.0762177s (cpu); 0.0314497s (thread); 0s (gc)
│ │ │ + -- used 0.0847111s (cpu); 0.0254131s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.010001s (cpu); 0.0137971s (thread); 0s (gc)
│ │ │ + -- used 0.0100116s (cpu); 0.0120553s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out
│ │ │ @@ -166,17 +166,17 @@
│ │ │        z   z   z   , z   z   z    - z   z   , z   z   z    - z   z   )
│ │ │         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
│ │ │  
│ │ │  o14 : Ideal of QQ[z   ..z   ]
│ │ │                     1,1   5,5
│ │ │  
│ │ │  i15 : time schubertRegularity p
│ │ │ - -- used 0.000811421s (cpu); 0.000282139s (thread); 0s (gc)
│ │ │ + -- used 0.00265319s (cpu); 0.000216806s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 5
│ │ │  
│ │ │  i16 : time regularity comodule I
│ │ │ - -- used 0.0148788s (cpu); 0.0151641s (thread); 0s (gc)
│ │ │ + -- used 0.00928066s (cpu); 0.012135s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : w = {2,1,4,3}
│ │ │  
│ │ │  o1 = {2, 1, 4, 3}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00400002s (cpu); 0.00427234s (thread); 0s (gc)
│ │ │ + -- used 0.00400087s (cpu); 0.00359277s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
│ │ │  
│ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00234252s (cpu); 0.00182869s (thread); 0s (gc)
│ │ │ + -- used 0.00162331s (cpu); 0.00159932s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -333,21 +333,21 @@
│ │ │          
      
│ │ │          
      
│ │ │            
      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].
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i23 : time schubertRegularity B
│ │ │ - -- used 0.0762177s (cpu); 0.0314497s (thread); 0s (gc)
│ │ │ + -- used 0.0847111s (cpu); 0.0254131s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │        		          
                    
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.010001s (cpu); 0.0137971s (thread); 0s (gc)
│ │ │ + -- used 0.0100116s (cpu); 0.0120553s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
      
│ │ │        		          
      
│ │ │          
      
│ │ │            
      Functions for investigating ASM varieties
      
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -241,19 +241,19 @@
│ │ │ │  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].
│ │ │ │  i23 : time schubertRegularity B
│ │ │ │ - -- used 0.0762177s (cpu); 0.0314497s (thread); 0s (gc)
│ │ │ │ + -- used 0.0847111s (cpu); 0.0254131s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.010001s (cpu); 0.0137971s (thread); 0s (gc)
│ │ │ │ + -- used 0.0100116s (cpu); 0.0120553s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -268,21 +268,21 @@
│ │ │          
      
│ │ │          
      
│ │ │            
      Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW21].
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      i15 : time schubertRegularity p
│ │ │ - -- used 0.000811421s (cpu); 0.000282139s (thread); 0s (gc)
│ │ │ + -- used 0.00265319s (cpu); 0.000216806s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 5
      
│ │ │        		          
                    
      i16 : time regularity comodule I
│ │ │ - -- used 0.0148788s (cpu); 0.0151641s (thread); 0s (gc)
│ │ │ + -- used 0.00928066s (cpu); 0.012135s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │        		          
      
│ │ │          
      
│ │ │            
      Functions for investigating matrix Schubert varieties
      
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -532,19 +532,19 @@
│ │ │ │  
│ │ │ │  o14 : Ideal of QQ[z   ..z   ]
│ │ │ │                     1,1   5,5
│ │ │ │  Finally, this package contains functions for investigating homological
│ │ │ │  invariants of matrix Schubert varieties efficiently through combinatorial
│ │ │ │  algorithms produced in [PSW21].
│ │ │ │  i15 : time schubertRegularity p
│ │ │ │ - -- used 0.000811421s (cpu); 0.000282139s (thread); 0s (gc)
│ │ │ │ + -- used 0.00265319s (cpu); 0.000216806s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 5
│ │ │ │  i16 : time regularity comodule I
│ │ │ │ - -- used 0.0148788s (cpu); 0.0151641s (thread); 0s (gc)
│ │ │ │ + -- used 0.00928066s (cpu); 0.012135s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines a
│ │ │ │        Schubert determinantal ideal or, more generally, an alternating sign
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -77,26 +77,26 @@
│ │ │  
│ │ │  o1 = {2, 1, 4, 3}
│ │ │  
│ │ │  o1 : List
│ │ │  
    		          
                  
    i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00400002s (cpu); 0.00427234s (thread); 0s (gc)
│ │ │ + -- used 0.00400087s (cpu); 0.00359277s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
    
│ │ │      		          
                  
    i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00234252s (cpu); 0.00182869s (thread); 0s (gc)
│ │ │ + -- used 0.00162331s (cpu); 0.00159932s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,24 +19,24 @@
│ │ │ │  PipeDream.
│ │ │ │  i1 : w = {2,1,4,3}
│ │ │ │  
│ │ │ │  o1 = {2, 1, 4, 3}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time grothendieckPolynomial w
│ │ │ │ - -- used 0.00400002s (cpu); 0.00427234s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400087s (cpu); 0.00359277s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o2 : QQ[x ..x ]
│ │ │ │           1   4
│ │ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ │ - -- used 0.00234252s (cpu); 0.00182869s (thread); 0s (gc)
│ │ │ │ + -- used 0.00162331s (cpu); 0.00159932s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxhdHMsTWF0cm9pZCxaWixTdHJpbmcpLCJmbGF0
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.1578s (cpu); 0.0717953s (thread); 0s (gc)
│ │ │ + -- used 0.154694s (cpu); 0.085799s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.145769s (cpu); 0.0633631s (thread); 0s (gc)
│ │ │ + -- used 0.137435s (cpu); 0.0758691s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -76,15 +76,15 @@
│ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │        -----------------------------------------------------------------------
│ │ │        groundSet, dual, storedRepresentation}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : time fVector R10
│ │ │ - -- used 0.000363171s (cpu); 0.00029826s (thread); 0s (gc)
│ │ │ + -- used 0.000307677s (cpu); 0.000157315s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_all__Minors.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │  
│ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │  
│ │ │  o2 : Matroid
│ │ │  
│ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .203206s elapsed
│ │ │ + -- .113405s elapsed
│ │ │  
│ │ │  i4 : #L
│ │ │  
│ │ │  o4 = 64
│ │ │  
│ │ │  i5 : netList L_{0..4}
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out
│ │ │ @@ -33,14 +33,14 @@
│ │ │  i6 : F7 = specificMatroid "fano"
│ │ │  
│ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │  
│ │ │  o6 : Matroid
│ │ │  
│ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.228153s (cpu); 0.0609033s (thread); 0s (gc)
│ │ │ + -- used 0.230943s (cpu); 0.100864s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.62715s (cpu); 1.07556s (thread); 0s (gc)
│ │ │ + -- used 1.76526s (cpu); 1.2321s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │  
│ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │  
│ │ │  o4 : Matroid
│ │ │  
│ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0160021s (cpu); 0.0130043s (thread); 0s (gc)
│ │ │ + -- used 0.0120031s (cpu); 0.0109728s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.93772s (cpu); 2.56293s (thread); 0s (gc)
│ │ │ + -- used 6.14969s (cpu); 3.30257s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_quick__Isomorphism__Test.out
│ │ │ @@ -37,15 +37,15 @@
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.00245681s (cpu); 0.000493465s (thread); 0s (gc)
│ │ │ + -- used 0.00133375s (cpu); 0.00053877s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : value oo === false
│ │ │  
│ │ │  o11 = true
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_set__Representation.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i5 : keys M.cache
│ │ │  
│ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime fVector M
│ │ │ - -- .0916028s elapsed
│ │ │ + -- .0815457s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -122,21 +122,21 @@
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │      		          
                  
    i10 : time isWellDefined R10
│ │ │ - -- used 0.1578s (cpu); 0.0717953s (thread); 0s (gc)
│ │ │ + -- used 0.154694s (cpu); 0.085799s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │      		          
                  
    i11 : time fVector R10
│ │ │ - -- used 0.145769s (cpu); 0.0633631s (thread); 0s (gc)
│ │ │ + -- used 0.137435s (cpu); 0.0758691s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -150,15 +150,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        groundSet, dual, storedRepresentation}
│ │ │  
│ │ │  o12 : List
    
│ │ │      		          
                  
    i13 : time fVector R10
│ │ │ - -- used 0.000363171s (cpu); 0.00029826s (thread); 0s (gc)
│ │ │ + -- used 0.000307677s (cpu); 0.000157315s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -71,19 +71,19 @@
│ │ │ │  o8 : Matroid
│ │ │ │  i9 : keys R10.cache
│ │ │ │  
│ │ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : time isWellDefined R10
│ │ │ │ - -- used 0.1578s (cpu); 0.0717953s (thread); 0s (gc)
│ │ │ │ + -- used 0.154694s (cpu); 0.085799s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.145769s (cpu); 0.0633631s (thread); 0s (gc)
│ │ │ │ + -- used 0.137435s (cpu); 0.0758691s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ │ │ @@ -93,15 +93,15 @@
│ │ │ │  
│ │ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        groundSet, dual, storedRepresentation}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : time fVector R10
│ │ │ │ - -- used 0.000363171s (cpu); 0.00029826s (thread); 0s (gc)
│ │ │ │ + -- used 0.000307677s (cpu); 0.000157315s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │  
│ │ │  o2 : Matroid
    
│ │ │      		          
                  
    i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .203206s elapsed
    
│ │ │ + -- .113405s elapsed
│ │ │      		          
                  
    i4 : #L
│ │ │  
│ │ │  o4 = 64
    
│ │ │      		          
              
                  
    i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.228153s (cpu); 0.0609033s (thread); 0s (gc)
    
│ │ │ + -- used 0.230943s (cpu); 0.100864s (thread); 0s (gc)
│ │ │      		          
                  
    i8 : #autF7
│ │ │  
│ │ │  o8 = 168
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  symmetric group S_7:
│ │ │ │  i6 : F7 = specificMatroid "fano"
│ │ │ │  
│ │ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │ │  
│ │ │ │  o6 : Matroid
│ │ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ │ - -- used 0.228153s (cpu); 0.0609033s (thread); 0s (gc)
│ │ │ │ + -- used 0.230943s (cpu); 0.100864s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │            
                  
    i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
    
│ │ │      		          
                  
    i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.62715s (cpu); 1.07556s (thread); 0s (gc)
│ │ │ + -- used 1.76526s (cpu); 1.2321s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 1.62715s (cpu); 1.07556s (thread); 0s (gc)
│ │ │ │ + -- used 1.76526s (cpu); 1.2321s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │      		          
                  
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0160021s (cpu); 0.0130043s (thread); 0s (gc)
│ │ │ + -- used 0.0120031s (cpu); 0.0109728s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │      		          
                  
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.93772s (cpu); 2.56293s (thread); 0s (gc)
│ │ │ + -- used 6.14969s (cpu); 3.30257s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │ │  
│ │ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │ │  
│ │ │ │  o4 : Matroid
│ │ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ │ - -- used 0.0160021s (cpu); 0.0130043s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120031s (cpu); 0.0109728s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 4.93772s (cpu); 2.56293s (thread); 0s (gc)
│ │ │ │ + -- used 6.14969s (cpu); 3.30257s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │            
                  
    i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │  
│ │ │  o9 = true
    
│ │ │      		          
                  
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.00245681s (cpu); 0.000493465s (thread); 0s (gc)
│ │ │ + -- used 0.00133375s (cpu); 0.00053877s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
    
│ │ │      		          
                  
    i11 : value oo === false
│ │ │  
│ │ │  o11 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  o7 = a "matroid" of rank 7 on 11 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ │ - -- used 0.00245681s (cpu); 0.000493465s (thread); 0s (gc)
│ │ │ │ + -- used 0.00133375s (cpu); 0.00053877s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o5 : List
│ │ │      		          
                  
    i6 : elapsedTime fVector M
│ │ │ - -- .0916028s elapsed
│ │ │ + -- .0815457s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o4 : Matrix QQ  <-- QQ
│ │ │ │  i5 : keys M.cache
│ │ │ │  
│ │ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime fVector M
│ │ │ │ - -- .0916028s elapsed
│ │ │ │ + -- .0815457s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MergeTeX/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  bWVyZ2VUZVgoLi4uLFBhdGg9Pi4uLik=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttZXJnZVRlWCxQYXRoXSwibWVyZ2VUZVgoLi4uLFBh
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  ZGVjb21wb3NlKElkZWFsLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzEwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1soZGVjb21wb3NlLElkZWFsKSxTdHJhdGVneV0sImRl
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
│ │ │ @@ -5,16 +5,16 @@
│ │ │  i2 : R = ZZ/101[w..z];
│ │ │  
│ │ │  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);
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0119314)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .0526113)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0453312)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .00252051)  #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : 0
│ │ │ - -- .0346098s elapsed
│ │ │ + --  Time taken : .0000215
│ │ │ + -- .0776985s elapsed
│ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000400291s elapsed
│ │ │ + -- .000515263s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0108096s elapsed
│ │ │ + -- .0111118s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out
│ │ │ @@ -29,22 +29,22 @@
│ │ │  o5 = 840
│ │ │  
│ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .0803977s elapsed
│ │ │ + -- .106302s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00150213s elapsed
│ │ │ + -- .00131765s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00113311s elapsed
│ │ │ + -- .00102472s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -58,20 +58,20 @@
│ │ │            
                  
    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);
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │      		          
                  
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0119314)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .0526113)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0453312)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .00252051)  #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : 0
│ │ │ - -- .0346098s elapsed
│ │ │ + --  Time taken : .0000215
│ │ │ + -- .0776985s elapsed
│ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,20 +11,20 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  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);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Birational        (time .0119314)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Linear            (time .0526113)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0453312)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .00252051)  #primes = 0 #prunedViaCodim = 0
│ │ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and
│ │ │ │  selecting minimal primes...
│ │ │ │ - --  Time taken : 0
│ │ │ │ - -- .0346098s elapsed
│ │ │ │ + --  Time taken : .0000215
│ │ │ │ + -- .0776985s elapsed
│ │ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -123,23 +123,23 @@
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │      		          
                  
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000400291s elapsed
│ │ │ + -- .000515263s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │      		          
                  
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0108096s elapsed
│ │ │ + -- .0111118s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │      		          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,21 +63,21 @@
│ │ │ │  i5 : I = intersect(ideal(a^2,b^2,c), ideal(a,b^3,c^2))
│ │ │ │  
│ │ │ │               2        2   3     2
│ │ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ │ - -- .000400291s elapsed
│ │ │ │ + -- .000515263s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0108096s elapsed
│ │ │ │ + -- .0111118s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -115,27 +115,27 @@
│ │ │            
    i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │  
│ │ │  o6 = true
    │ │ │ 
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .0803977s elapsed
│ │ │ + -- .106302s elapsed
│ │ │  
│ │ │  o7 = true
    │ │ │ 
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00150213s elapsed
│ │ │ + -- .00131765s elapsed
│ │ │  
│ │ │  o8 = true
    │ │ │ 
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00113311s elapsed
│ │ │ + -- .00102472s elapsed
│ │ │  
│ │ │  o9 = false
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -51,23 +51,23 @@ │ │ │ │ i5 : D = product(I_*/degree/sum) │ │ │ │ │ │ │ │ o5 = 840 │ │ │ │ i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0 │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime radicalContainment(x_0, I) │ │ │ │ - -- .0803977s elapsed │ │ │ │ + -- .106302s elapsed │ │ │ │ │ │ │ │ o7 = true │ │ │ │ i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar") │ │ │ │ - -- .00150213s elapsed │ │ │ │ + -- .00131765s elapsed │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar") │ │ │ │ - -- .00113311s elapsed │ │ │ │ + -- .00102472s elapsed │ │ │ │ │ │ │ │ o9 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_d_i_c_a_l -- the radical of an ideal │ │ │ │ ********** WWaayyss ttoo uussee rraaddiiccaallCCoonnttaaiinnmmeenntt:: ********** │ │ │ │ * radicalContainment(Ideal,Ideal) │ │ │ │ * radicalContainment(RingElement,Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Miura/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ c2NhbGFyTXVsdGlwbGljYXRpb24= │ │ │ #:len=1333 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQWRkIFJlZHVjZWQgSWRlYWwgTXVsdGlw │ │ │ bGUgVGltZXMiLCAibGluZW51bSIgPT4gMjAwLCBJbnB1dHMgPT4ge1NQQU57VFR7IkoifSwiLCAi │ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ bXVsdGlSZWVzSWRlYWwoSWRlYWwp │ │ │ #:len=280 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0aVJlZXNJZGVhbCxJZGVhbCksIm11bHRpUmVl │ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ @@ -57,29 +57,29 @@ │ │ │ i9 : J = ideal vars U │ │ │ │ │ │ o9 = ideal (a, b, c) │ │ │ │ │ │ o9 : Ideal of U │ │ │ │ │ │ i10 : time multiReesIdeal J │ │ │ - -- used 0.133142s (cpu); 0.0729098s (thread); 0s (gc) │ │ │ + -- used 0.654047s (cpu); 0.199659s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o10 = ideal (c*X - b*X , b*X - a*X , a*X - c*X , c*X - a*X , b*X - c*X , │ │ │ 1 2 1 2 1 2 0 2 0 2 │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 │ │ │ a*X - b*X , X - X X , X X - X , X - X X ) │ │ │ 0 2 1 0 2 0 1 2 0 1 2 │ │ │ │ │ │ o10 : Ideal of U[X ..X ] │ │ │ 0 2 │ │ │ │ │ │ i11 : time multiReesIdeal (J, a) │ │ │ - -- used 0.0112772s (cpu); 0.0102657s (thread); 0s (gc) │ │ │ + -- used 0.464114s (cpu); 0.0791983s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o11 = ideal (c*X - b*X , b*X - a*X , a*X - c*X , c*X - a*X , b*X - c*X , │ │ │ 1 2 1 2 1 2 0 2 0 2 │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 │ │ │ a*X - b*X , X - X X , X X - X , X - X X ) │ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ @@ -157,30 +157,30 @@ │ │ │ │ │ │ o9 = ideal (a, b, c) │ │ │ │ │ │ o9 : Ideal of U │ │ │ │ │ │ │ │ │ 
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.133142s (cpu); 0.0729098s (thread); 0s (gc)
│ │ │ + -- used 0.654047s (cpu); 0.199659s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │  
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    │ │ │ │ │ │ │ │ │ 
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0112772s (cpu); 0.0102657s (thread); 0s (gc)
│ │ │ + -- used 0.464114s (cpu); 0.0791983s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,28 +80,28 @@
│ │ │ │  i8 : U = T/minors(2,m);
│ │ │ │  i9 : J = ideal vars U
│ │ │ │  
│ │ │ │  o9 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o9 : Ideal of U
│ │ │ │  i10 : time multiReesIdeal J
│ │ │ │ - -- used 0.133142s (cpu); 0.0729098s (thread); 0s (gc)
│ │ │ │ + -- used 0.654047s (cpu); 0.199659s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │ │  
│ │ │ │  o10 : Ideal of U[X ..X ]
│ │ │ │                    0   2
│ │ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ │ - -- used 0.0112772s (cpu); 0.0102657s (thread); 0s (gc)
│ │ │ │ + -- used 0.464114s (cpu); 0.0791983s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  ZGVmb3JtTUNNTW9kdWxl
│ │ │  #:len=2237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmVyc2FsIGRlZm9ybWF0aW9uIG9mIE1D
│ │ │  TS1tb2R1bGUgb24gaHlwZXJzdXJmYWNlIiwgRGVzY3JpcHRpb24gPT4gKCJUaGlzIGlzIHRoZSBt
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
│ │ │  
│ │ │  i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.448688s (cpu); 0.330463s (thread); 0s (gc)
│ │ │ + -- used 0.484335s (cpu); 0.3451s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -70,15 +70,15 @@
│ │ │  o10 = cokernel | x2 y2  |
│ │ │                 | -y -x2 |
│ │ │  
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
│ │ │  
│ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.699233s (cpu); 0.550851s (thread); 0s (gc)
│ │ │ + -- used 0.775042s (cpu); 0.503718s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -134,15 +134,15 @@
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    │ │ │ │ │ │ │ │ │ 
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.448688s (cpu); 0.330463s (thread); 0s (gc)
│ │ │ + -- used 0.484335s (cpu); 0.3451s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │                 | -y -x2 |
│ │ │  
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    │ │ │ │ │ │ │ │ │ 
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.699233s (cpu); 0.550851s (thread); 0s (gc)
│ │ │ + -- used 0.775042s (cpu); 0.503718s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ │ ├── html2text {}
│ │ │ │ @@ -71,15 +71,15 @@
│ │ │ │  i7 : N0 = module ideal (x^2,y^2)
│ │ │ │  
│ │ │ │  o7 = image | x2 y2 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o7 : R-module, submodule of R
│ │ │ │  i8 : (S,N) = time deformMCMModule N0
│ │ │ │ - -- used 0.448688s (cpu); 0.330463s (thread); 0s (gc)
│ │ │ │ + -- used 0.484335s (cpu); 0.3451s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │ │                    {8} | xxi_4-y+xi_3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │ │       -x2-xxi_2-xi_1                                           |
│ │ │ │  
│ │ │ │ @@ -104,15 +104,15 @@
│ │ │ │  
│ │ │ │  o10 = cokernel | x2 y2  |
│ │ │ │                 | -y -x2 |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o10 : R-module, quotient of R
│ │ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ │ - -- used 0.699233s (cpu); 0.550851s (thread); 0s (gc)
│ │ │ │ + -- used 0.775042s (cpu); 0.503718s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │ │                      | xxi_4-y+xi_3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c3BhcnNlTW9ub2Ryb215U29sdmUoLi4uLE51bWJlck9mUmVwZWF0cz0+Li4uKQ==
│ │ │  #:len=345
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzcGFyc2VNb25vZHJvbXlTb2x2ZSxOdW1iZXJPZlJl
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,27 +3,27 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00293529s elapsed
│ │ │ - -- .00270725s elapsed
│ │ │ - -- .000331491s elapsed
│ │ │ - -- .00264599s elapsed
│ │ │ - -- .0619881s elapsed
│ │ │ - -- .00035814s elapsed
│ │ │ - -- .0029127s elapsed
│ │ │ - -- .00246511s elapsed
│ │ │ - -- .000212939s elapsed
│ │ │ - -- .00238499s elapsed
│ │ │ - -- .0024502s elapsed
│ │ │ - -- .0119058s elapsed
│ │ │ ---backup directory created: /tmp/M2-86321-0/1
│ │ │ + -- .00397159s elapsed
│ │ │ + -- .003989s elapsed
│ │ │ + -- .000379439s elapsed
│ │ │ + -- .00420825s elapsed
│ │ │ + -- .0179321s elapsed
│ │ │ + -- .000300622s elapsed
│ │ │ + -- .00252447s elapsed
│ │ │ + -- .00236236s elapsed
│ │ │ + -- .000238896s elapsed
│ │ │ + -- .0428276s elapsed
│ │ │ + -- .00257593s elapsed
│ │ │ + -- .000249075s elapsed
│ │ │ +--backup directory created: /tmp/M2-152514-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out
│ │ │ @@ -14,128 +14,128 @@
│ │ │  
│ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │  
│ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │  
│ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20,
│ │ │ +o9 = {{0, 9, 15, 20, 10, 8, 14, 12, 17, 1, 4, 2, 3, 13, 6, 18, 16, 19, 5, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18}, {15, 1, 0, 7, 4, 8, 6, 12, 13, 9, 10, 2, 16, 17, 14, 11, 3, 5, 19,
│ │ │ +     11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 18}, {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3,
│ │ │ +     7, 11}, {0, 1, 6, 3, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16,
│ │ │ +     18, 19, 20}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
│ │ │ +     1, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14,
│ │ │ +     18, 1, 19, 5, 7, 11}, {3, 1, 6, 5, 16, 2, 0, 8, 9, 13, 10, 12, 4, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 16, 19, 15, 17, 3}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13,
│ │ │ +     14, 7, 17, 11, 18, 19, 20}, {0, 1, 6, 20, 7, 15, 11, 17, 9, 5, 10, 3, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 18, 16, 19, 15, 17, 3}, {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8,
│ │ │ +     13, 14, 18, 16, 19, 2, 8, 12}, {0, 1, 15, 20, 4, 8, 6, 12, 17, 9, 10, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 16, 1, 17, 0, 3, 18, 19, 20}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9,
│ │ │ +     3, 13, 14, 18, 16, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │ +     4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 15, 16, 17, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8,
│ │ │ +     10, 12, 4, 13, 14, 18, 16, 19, 15, 17, 3}, {13, 10, 5, 15, 4, 12, 6, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 1, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {13, 10, 8, 15, 4, 7, 6,
│ │ │ +     7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {15, 1, 0, 7, 4, 8, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 9, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7,
│ │ │ +     12, 13, 9, 10, 2, 16, 17, 14, 11, 3, 5, 19, 20, 18}, {13, 10, 5, 15, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {15, 9, 0, 7,
│ │ │ +     12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 8, 14, 12, 13, 1, 4, 2, 16, 17, 6, 11, 3, 5, 19, 20, 18}, {0, 1, 2,
│ │ │ +     4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6,
│ │ │ +     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7},
│ │ │ +     2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {16, 1, 3, 19, 9, 2, 4, 8, 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5,
│ │ │ +     1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7}, {3, 6, 1, 19, 16, 2, 0, 8, 10, 13, 9, 12, 14, 15, 4, 20, 17, 18, 11,
│ │ │ +     {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 7}, {3, 1, 2, 19, 4, 16, 6, 0, 8, 9, 10, 13, 12, 15, 14, 20, 17, 18,
│ │ │ +     20}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, 20, 13, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 5, 7}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14,
│ │ │ +     11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 11, 5, 7}, {16, 6, 3, 19, 1, 2, 10, 8, 15, 14, 9, 12, 17, 0, 4, 20,
│ │ │ +     17, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1,
│ │ │ +     20, 13, 18, 11, 5, 7}, {13, 10, 8, 15, 4, 7, 6, 11, 12, 9, 14, 5, 2, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 0, 3, 19, 20, 18}, {19, 14, 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20,
│ │ │ +     1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 0, 18, 13, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2,
│ │ │ +     13, 14, 18, 16, 19, 15, 17, 3}, {15, 9, 0, 7, 10, 8, 14, 12, 13, 1, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 1, 17, 0, 3, 19, 20, 18}, {16, 14, 4, 19, 2, 3, 8, 15, 6, 12, 1, 17,
│ │ │ +     2, 16, 17, 6, 11, 3, 5, 19, 20, 18}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 10, 20, 13, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13,
│ │ │ +     13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {16, 1, 3, 19, 9, 2, 4, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {7, 1, 9, 0, 12, 17, 2, 3, 4,
│ │ │ +     15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, 7}, {3, 6, 1, 19, 16, 2, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 15, 6, 11, 14, 13, 5, 16, 19, 20, 18}, {3, 1, 6, 19, 0, 2, 13, 8,
│ │ │ +     8, 10, 13, 9, 12, 14, 15, 4, 20, 17, 18, 11, 5, 7}, {3, 1, 2, 19, 4, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 16, 10, 12, 4, 15, 14, 20, 17, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7,
│ │ │ +     6, 0, 8, 9, 10, 13, 12, 15, 14, 20, 17, 18, 11, 5, 7}, {1, 6, 3, 19, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16,
│ │ │ +     2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {16, 6, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {0, 1, 2, 3,
│ │ │ +     1, 2, 10, 8, 15, 14, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {13, 10, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {13, 10,
│ │ │ +     15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {19, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {13,
│ │ │ +     4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, 10, 0, 18, 13, 11, 5, 7}, {13,
│ │ │       ------------------------------------------------------------------------
│ │ │       10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {10, 9, 7, 0, 12, 17, 2, 3, 11, 8, 4, 15, 5, 14, 6, 13, 1, 16, 19, 20,
│ │ │ +     {16, 14, 4, 19, 2, 3, 8, 15, 6, 12, 1, 17, 9, 0, 10, 20, 13, 18, 11, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18}, {11, 10, 12, 13, 6, 20, 9, 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3,
│ │ │ +     7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12,
│ │ │ +     19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4,
│ │ │ +     18, 11, 5, 7}, {7, 1, 9, 0, 12, 17, 2, 3, 4, 8, 10, 15, 6, 11, 14, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     0, 3, 15, 17}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8,
│ │ │ +     5, 16, 19, 20, 18}, {3, 1, 6, 19, 0, 2, 13, 8, 9, 16, 10, 12, 4, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     0, 12, 17, 3, 15}, {13, 1, 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14,
│ │ │ +     20, 17, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 0, 12, 17, 3, 15}, {6, 10, 7, 13, 2, 20, 8, 18, 11, 12, 14, 19, 5, 9,
│ │ │ +     1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 16, 4, 0, 3, 15, 17}, {11, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2,
│ │ │ +     10, 4, 20, 14, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 1, 16, 7, 0, 3, 15, 17}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 14, 16, 8, 1, 12, 17, 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7,
│ │ │ +     5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 19, 8, 9, 1, 16, 4, 0, 3, 15, 17}, {7, 10, 8, 13, 6, 20, 9, 18, 12,
│ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {10, 9, 7, 0, 12, 17, 2, 3, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 14, 19, 2, 11, 1, 16, 5, 0, 3, 15, 17}, {0, 9, 15, 20, 10, 8, 14, 12,
│ │ │ +     8, 4, 15, 5, 14, 6, 13, 1, 16, 19, 20, 18}, {11, 10, 12, 13, 6, 20, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 1, 4, 2, 3, 13, 6, 18, 16, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13,
│ │ │ +     18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, 15, 17}, {13, 10, 19, 2, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0, 1, 6, 3, 7, 2, 11,
│ │ │ +     9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {6, 10, 12, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 9, 5, 10, 12, 4, 13, 14, 15, 16, 17, 18, 19, 20}, {10, 9, 15, 20, 0,
│ │ │ +     11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, 3, 15, 17}, {13, 10, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {10, 9, 15, 20,
│ │ │ +     2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {13, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {3, 1, 6, 5,
│ │ │ +     19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, 8, 0, 12, 17, 3, 15}, {6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 2, 0, 8, 9, 13, 10, 12, 4, 15, 14, 7, 17, 11, 18, 19, 20}, {0, 1, 6,
│ │ │ +     10, 7, 13, 2, 20, 8, 18, 11, 12, 14, 19, 5, 9, 1, 16, 4, 0, 3, 15, 17},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 7, 15, 11, 17, 9, 5, 10, 3, 4, 13, 14, 18, 16, 19, 2, 8, 12}, {0, 1,
│ │ │ +     {11, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 5, 1, 16, 7, 0, 3, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 20, 4, 8, 6, 12, 17, 9, 10, 2, 3, 13, 14, 18, 16, 19, 5, 7, 11},
│ │ │ +     17}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18, 11, 14, 16, 8, 1, 12, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7,
│ │ │ +     3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 18, 16, 19, 15,
│ │ │ +     3, 15, 17}, {7, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 11, 1, 16, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 3}}
│ │ │ +     0, 3, 15, 17}}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -96,27 +96,27 @@
│ │ │                
    i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
    
│ │ │            
│ │ │            
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │            
│ │ │            
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00293529s elapsed
│ │ │ - -- .00270725s elapsed
│ │ │ - -- .000331491s elapsed
│ │ │ - -- .00264599s elapsed
│ │ │ - -- .0619881s elapsed
│ │ │ - -- .00035814s elapsed
│ │ │ - -- .0029127s elapsed
│ │ │ - -- .00246511s elapsed
│ │ │ - -- .000212939s elapsed
│ │ │ - -- .00238499s elapsed
│ │ │ - -- .0024502s elapsed
│ │ │ - -- .0119058s elapsed
│ │ │ ---backup directory created: /tmp/M2-86321-0/1
│ │ │ + -- .00397159s elapsed
│ │ │ + -- .003989s elapsed
│ │ │ + -- .000379439s elapsed
│ │ │ + -- .00420825s elapsed
│ │ │ + -- .0179321s elapsed
│ │ │ + -- .000300622s elapsed
│ │ │ + -- .00252447s elapsed
│ │ │ + -- .00236236s elapsed
│ │ │ + -- .000238896s elapsed
│ │ │ + -- .0428276s elapsed
│ │ │ + -- .00257593s elapsed
│ │ │ + -- .000249075s elapsed
│ │ │ +--backup directory created: /tmp/M2-152514-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,27 +23,27 @@
│ │ │ │            o npaths, an _i_n_t_e_g_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s.
│ │ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ │ - -- .00293529s elapsed
│ │ │ │ - -- .00270725s elapsed
│ │ │ │ - -- .000331491s elapsed
│ │ │ │ - -- .00264599s elapsed
│ │ │ │ - -- .0619881s elapsed
│ │ │ │ - -- .00035814s elapsed
│ │ │ │ - -- .0029127s elapsed
│ │ │ │ - -- .00246511s elapsed
│ │ │ │ - -- .000212939s elapsed
│ │ │ │ - -- .00238499s elapsed
│ │ │ │ - -- .0024502s elapsed
│ │ │ │ - -- .0119058s elapsed
│ │ │ │ ---backup directory created: /tmp/M2-86321-0/1
│ │ │ │ + -- .00397159s elapsed
│ │ │ │ + -- .003989s elapsed
│ │ │ │ + -- .000379439s elapsed
│ │ │ │ + -- .00420825s elapsed
│ │ │ │ + -- .0179321s elapsed
│ │ │ │ + -- .000300622s elapsed
│ │ │ │ + -- .00252447s elapsed
│ │ │ │ + -- .00236236s elapsed
│ │ │ │ + -- .000238896s elapsed
│ │ │ │ + -- .0428276s elapsed
│ │ │ │ + -- .00257593s elapsed
│ │ │ │ + -- .000249075s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-152514-0/1
│ │ │ │    H01: 1
│ │ │ │    H10: 1
│ │ │ │  number of paths tracked: 2
│ │ │ │  found 1 points in the fiber so far
│ │ │ │    H01: 1
│ │ │ │    H10: 1
│ │ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Group.html
│ │ │ @@ -104,131 +104,131 @@
│ │ │            
│ │ │            
│ │ │                
    i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
    
│ │ │            
│ │ │            
│ │ │                
    i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20,
│ │ │ +o9 = {{0, 9, 15, 20, 10, 8, 14, 12, 17, 1, 4, 2, 3, 13, 6, 18, 16, 19, 5, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18}, {15, 1, 0, 7, 4, 8, 6, 12, 13, 9, 10, 2, 16, 17, 14, 11, 3, 5, 19,
│ │ │ +     11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 18}, {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3,
│ │ │ +     7, 11}, {0, 1, 6, 3, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16,
│ │ │ +     18, 19, 20}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
│ │ │ +     1, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14,
│ │ │ +     18, 1, 19, 5, 7, 11}, {3, 1, 6, 5, 16, 2, 0, 8, 9, 13, 10, 12, 4, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 16, 19, 15, 17, 3}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13,
│ │ │ +     14, 7, 17, 11, 18, 19, 20}, {0, 1, 6, 20, 7, 15, 11, 17, 9, 5, 10, 3, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 18, 16, 19, 15, 17, 3}, {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8,
│ │ │ +     13, 14, 18, 16, 19, 2, 8, 12}, {0, 1, 15, 20, 4, 8, 6, 12, 17, 9, 10, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 16, 1, 17, 0, 3, 18, 19, 20}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9,
│ │ │ +     3, 13, 14, 18, 16, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │ +     4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 15, 16, 17, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8,
│ │ │ +     10, 12, 4, 13, 14, 18, 16, 19, 15, 17, 3}, {13, 10, 5, 15, 4, 12, 6, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 1, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {13, 10, 8, 15, 4, 7, 6,
│ │ │ +     7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {15, 1, 0, 7, 4, 8, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 9, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7,
│ │ │ +     12, 13, 9, 10, 2, 16, 17, 14, 11, 3, 5, 19, 20, 18}, {13, 10, 5, 15, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {15, 9, 0, 7,
│ │ │ +     12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 8, 14, 12, 13, 1, 4, 2, 16, 17, 6, 11, 3, 5, 19, 20, 18}, {0, 1, 2,
│ │ │ +     4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6,
│ │ │ +     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7},
│ │ │ +     2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {16, 1, 3, 19, 9, 2, 4, 8, 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5,
│ │ │ +     1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7}, {3, 6, 1, 19, 16, 2, 0, 8, 10, 13, 9, 12, 14, 15, 4, 20, 17, 18, 11,
│ │ │ +     {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 7}, {3, 1, 2, 19, 4, 16, 6, 0, 8, 9, 10, 13, 12, 15, 14, 20, 17, 18,
│ │ │ +     20}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, 20, 13, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 5, 7}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14,
│ │ │ +     11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 11, 5, 7}, {16, 6, 3, 19, 1, 2, 10, 8, 15, 14, 9, 12, 17, 0, 4, 20,
│ │ │ +     17, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1,
│ │ │ +     20, 13, 18, 11, 5, 7}, {13, 10, 8, 15, 4, 7, 6, 11, 12, 9, 14, 5, 2, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 0, 3, 19, 20, 18}, {19, 14, 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20,
│ │ │ +     1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 0, 18, 13, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2,
│ │ │ +     13, 14, 18, 16, 19, 15, 17, 3}, {15, 9, 0, 7, 10, 8, 14, 12, 13, 1, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 1, 17, 0, 3, 19, 20, 18}, {16, 14, 4, 19, 2, 3, 8, 15, 6, 12, 1, 17,
│ │ │ +     2, 16, 17, 6, 11, 3, 5, 19, 20, 18}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 10, 20, 13, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13,
│ │ │ +     13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {16, 1, 3, 19, 9, 2, 4, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {7, 1, 9, 0, 12, 17, 2, 3, 4,
│ │ │ +     15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, 7}, {3, 6, 1, 19, 16, 2, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 15, 6, 11, 14, 13, 5, 16, 19, 20, 18}, {3, 1, 6, 19, 0, 2, 13, 8,
│ │ │ +     8, 10, 13, 9, 12, 14, 15, 4, 20, 17, 18, 11, 5, 7}, {3, 1, 2, 19, 4, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 16, 10, 12, 4, 15, 14, 20, 17, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7,
│ │ │ +     6, 0, 8, 9, 10, 13, 12, 15, 14, 20, 17, 18, 11, 5, 7}, {1, 6, 3, 19, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16,
│ │ │ +     2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {16, 6, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {0, 1, 2, 3,
│ │ │ +     1, 2, 10, 8, 15, 14, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {13, 10, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {13, 10,
│ │ │ +     15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {19, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {13,
│ │ │ +     4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, 10, 0, 18, 13, 11, 5, 7}, {13,
│ │ │       ------------------------------------------------------------------------
│ │ │       10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {10, 9, 7, 0, 12, 17, 2, 3, 11, 8, 4, 15, 5, 14, 6, 13, 1, 16, 19, 20,
│ │ │ +     {16, 14, 4, 19, 2, 3, 8, 15, 6, 12, 1, 17, 9, 0, 10, 20, 13, 18, 11, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18}, {11, 10, 12, 13, 6, 20, 9, 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3,
│ │ │ +     7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12,
│ │ │ +     19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4,
│ │ │ +     18, 11, 5, 7}, {7, 1, 9, 0, 12, 17, 2, 3, 4, 8, 10, 15, 6, 11, 14, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     0, 3, 15, 17}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8,
│ │ │ +     5, 16, 19, 20, 18}, {3, 1, 6, 19, 0, 2, 13, 8, 9, 16, 10, 12, 4, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     0, 12, 17, 3, 15}, {13, 1, 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14,
│ │ │ +     20, 17, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 0, 12, 17, 3, 15}, {6, 10, 7, 13, 2, 20, 8, 18, 11, 12, 14, 19, 5, 9,
│ │ │ +     1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 16, 4, 0, 3, 15, 17}, {11, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2,
│ │ │ +     10, 4, 20, 14, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 1, 16, 7, 0, 3, 15, 17}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 14, 16, 8, 1, 12, 17, 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7,
│ │ │ +     5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 19, 8, 9, 1, 16, 4, 0, 3, 15, 17}, {7, 10, 8, 13, 6, 20, 9, 18, 12,
│ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {10, 9, 7, 0, 12, 17, 2, 3, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 14, 19, 2, 11, 1, 16, 5, 0, 3, 15, 17}, {0, 9, 15, 20, 10, 8, 14, 12,
│ │ │ +     8, 4, 15, 5, 14, 6, 13, 1, 16, 19, 20, 18}, {11, 10, 12, 13, 6, 20, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 1, 4, 2, 3, 13, 6, 18, 16, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13,
│ │ │ +     18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, 15, 17}, {13, 10, 19, 2, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0, 1, 6, 3, 7, 2, 11,
│ │ │ +     9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {6, 10, 12, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 9, 5, 10, 12, 4, 13, 14, 15, 16, 17, 18, 19, 20}, {10, 9, 15, 20, 0,
│ │ │ +     11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, 3, 15, 17}, {13, 10, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {10, 9, 15, 20,
│ │ │ +     2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {13, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {3, 1, 6, 5,
│ │ │ +     19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, 8, 0, 12, 17, 3, 15}, {6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 2, 0, 8, 9, 13, 10, 12, 4, 15, 14, 7, 17, 11, 18, 19, 20}, {0, 1, 6,
│ │ │ +     10, 7, 13, 2, 20, 8, 18, 11, 12, 14, 19, 5, 9, 1, 16, 4, 0, 3, 15, 17},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 7, 15, 11, 17, 9, 5, 10, 3, 4, 13, 14, 18, 16, 19, 2, 8, 12}, {0, 1,
│ │ │ +     {11, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 5, 1, 16, 7, 0, 3, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 20, 4, 8, 6, 12, 17, 9, 10, 2, 3, 13, 14, 18, 16, 19, 5, 7, 11},
│ │ │ +     17}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18, 11, 14, 16, 8, 1, 12, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7,
│ │ │ +     3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 18, 16, 19, 15,
│ │ │ +     3, 15, 17}, {7, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 11, 1, 16, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 3}}
│ │ │ +     0, 3, 15, 17}}
│ │ │  
│ │ │  o9 : List
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    Caveat

    │ │ │ ├── html2text {} │ │ │ │ @@ -32,131 +32,131 @@ │ │ │ │ i4 : varMatrix = gateMatrix{{t_1,t_2}}; │ │ │ │ i5 : phi = transpose gateMatrix{{t_1^3, t_1^2*t_2, t_1*t_2^2, t_2^3}}; │ │ │ │ i6 : loss = sum for i from 0 to 3 list (u_i - phi_(i,0))^2; │ │ │ │ i7 : dLoss = diff(varMatrix, gateMatrix{{loss}}); │ │ │ │ i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss); │ │ │ │ i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10}) │ │ │ │ │ │ │ │ -o9 = {{13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, │ │ │ │ +o9 = {{0, 9, 15, 20, 10, 8, 14, 12, 17, 1, 4, 2, 3, 13, 6, 18, 16, 19, 5, 7, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 18}, {15, 1, 0, 7, 4, 8, 6, 12, 13, 9, 10, 2, 16, 17, 14, 11, 3, 5, 19, │ │ │ │ + 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 20, 18}, {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, │ │ │ │ + 7, 11}, {0, 1, 6, 3, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 15, 16, 17, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, │ │ │ │ + 18, 19, 20}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, │ │ │ │ + 1, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 16, 17, 18, 19, 20}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, │ │ │ │ + 18, 1, 19, 5, 7, 11}, {3, 1, 6, 5, 16, 2, 0, 8, 9, 13, 10, 12, 4, 15, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 18, 16, 19, 15, 17, 3}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, │ │ │ │ + 14, 7, 17, 11, 18, 19, 20}, {0, 1, 6, 20, 7, 15, 11, 17, 9, 5, 10, 3, 4, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 14, 18, 16, 19, 15, 17, 3}, {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, │ │ │ │ + 13, 14, 18, 16, 19, 2, 8, 12}, {0, 1, 15, 20, 4, 8, 6, 12, 17, 9, 10, 2, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 11, 16, 1, 17, 0, 3, 18, 19, 20}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, │ │ │ │ + 3, 13, 14, 18, 16, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, │ │ │ │ + 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 11, 12, 13, 14, 15, 16, 17, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, │ │ │ │ + 10, 12, 4, 13, 14, 18, 16, 19, 15, 17, 3}, {13, 10, 5, 15, 4, 12, 6, 2, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 15, 1, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {13, 10, 8, 15, 4, 7, 6, │ │ │ │ + 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {15, 1, 0, 7, 4, 8, 6, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 11, 12, 9, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, │ │ │ │ + 12, 13, 9, 10, 2, 16, 17, 14, 11, 3, 5, 19, 20, 18}, {13, 10, 5, 15, 4, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {15, 9, 0, 7, │ │ │ │ + 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 10, 8, 14, 12, 13, 1, 4, 2, 16, 17, 6, 11, 3, 5, 19, 20, 18}, {0, 1, 2, │ │ │ │ + 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, 1, 2, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, │ │ │ │ + 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, │ │ │ │ + 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {16, 1, 3, 19, 9, 2, 4, 8, 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, │ │ │ │ + 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 7}, {3, 6, 1, 19, 16, 2, 0, 8, 10, 13, 9, 12, 14, 15, 4, 20, 17, 18, 11, │ │ │ │ + {13, 10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 18, 19, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 5, 7}, {3, 1, 2, 19, 4, 16, 6, 0, 8, 9, 10, 13, 12, 15, 14, 20, 17, 18, │ │ │ │ + 20}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, 20, 13, 18, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 11, 5, 7}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, │ │ │ │ + 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 18, 11, 5, 7}, {16, 6, 3, 19, 1, 2, 10, 8, 15, 14, 9, 12, 17, 0, 4, 20, │ │ │ │ + 17, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 13, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, │ │ │ │ + 20, 13, 18, 11, 5, 7}, {13, 10, 8, 15, 4, 7, 6, 11, 12, 9, 14, 5, 2, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 17, 0, 3, 19, 20, 18}, {19, 14, 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, │ │ │ │ + 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 10, 0, 18, 13, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, │ │ │ │ + 13, 14, 18, 16, 19, 15, 17, 3}, {15, 9, 0, 7, 10, 8, 14, 12, 13, 1, 4, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 16, 1, 17, 0, 3, 19, 20, 18}, {16, 14, 4, 19, 2, 3, 8, 15, 6, 12, 1, 17, │ │ │ │ + 2, 16, 17, 6, 11, 3, 5, 19, 20, 18}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 9, 0, 10, 20, 13, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, │ │ │ │ + 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 12, 13, 14, 15, 16, 17, 18, 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, │ │ │ │ + 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {16, 1, 3, 19, 9, 2, 4, 8, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {7, 1, 9, 0, 12, 17, 2, 3, 4, │ │ │ │ + 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, 7}, {3, 6, 1, 19, 16, 2, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 8, 10, 15, 6, 11, 14, 13, 5, 16, 19, 20, 18}, {3, 1, 6, 19, 0, 2, 13, 8, │ │ │ │ + 8, 10, 13, 9, 12, 14, 15, 4, 20, 17, 18, 11, 5, 7}, {3, 1, 2, 19, 4, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 9, 16, 10, 12, 4, 15, 14, 20, 17, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, │ │ │ │ + 6, 0, 8, 9, 10, 13, 12, 15, 14, 20, 17, 18, 11, 5, 7}, {1, 6, 3, 19, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, │ │ │ │ + 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {16, 6, 3, 19, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {0, 1, 2, 3, │ │ │ │ + 1, 2, 10, 8, 15, 14, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {13, 10, 8, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {13, 10, │ │ │ │ + 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {19, 14, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {13, │ │ │ │ + 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, 10, 0, 18, 13, 11, 5, 7}, {13, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {10, 9, 7, 0, 12, 17, 2, 3, 11, 8, 4, 15, 5, 14, 6, 13, 1, 16, 19, 20, │ │ │ │ + {16, 14, 4, 19, 2, 3, 8, 15, 6, 12, 1, 17, 9, 0, 10, 20, 13, 18, 11, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 18}, {11, 10, 12, 13, 6, 20, 9, 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, │ │ │ │ + 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 15, 17}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, │ │ │ │ + 19, 20}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 17, 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, │ │ │ │ + 18, 11, 5, 7}, {7, 1, 9, 0, 12, 17, 2, 3, 4, 8, 10, 15, 6, 11, 14, 13, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 0, 3, 15, 17}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, │ │ │ │ + 5, 16, 19, 20, 18}, {3, 1, 6, 19, 0, 2, 13, 8, 9, 16, 10, 12, 4, 15, 14, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 0, 12, 17, 3, 15}, {13, 1, 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, │ │ │ │ + 20, 17, 18, 11, 5, 7}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 8, 0, 12, 17, 3, 15}, {6, 10, 7, 13, 2, 20, 8, 18, 11, 12, 14, 19, 5, 9, │ │ │ │ + 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 1, 16, 4, 0, 3, 15, 17}, {11, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, │ │ │ │ + 10, 4, 20, 14, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 5, 1, 16, 7, 0, 3, 15, 17}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18, │ │ │ │ + 13, 14, 15, 16, 17, 18, 19, 20}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 11, 14, 16, 8, 1, 12, 17, 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, │ │ │ │ + 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 14, 19, 8, 9, 1, 16, 4, 0, 3, 15, 17}, {7, 10, 8, 13, 6, 20, 9, 18, 12, │ │ │ │ + 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {10, 9, 7, 0, 12, 17, 2, 3, 11, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 4, 14, 19, 2, 11, 1, 16, 5, 0, 3, 15, 17}, {0, 9, 15, 20, 10, 8, 14, 12, │ │ │ │ + 8, 4, 15, 5, 14, 6, 13, 1, 16, 19, 20, 18}, {11, 10, 12, 13, 6, 20, 9, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 17, 1, 4, 2, 3, 13, 6, 18, 16, 19, 5, 7, 11}, {10, 9, 15, 20, 0, 8, 13, │ │ │ │ + 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, 15, 17}, {13, 10, 19, 2, 6, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0, 1, 6, 3, 7, 2, 11, │ │ │ │ + 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {6, 10, 12, 13, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 8, 9, 5, 10, 12, 4, 13, 14, 15, 16, 17, 18, 19, 20}, {10, 9, 15, 20, 0, │ │ │ │ + 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, 3, 15, 17}, {13, 10, 19, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {10, 9, 15, 20, │ │ │ │ + 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {13, 1, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {3, 1, 6, 5, │ │ │ │ + 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, 8, 0, 12, 17, 3, 15}, {6, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 16, 2, 0, 8, 9, 13, 10, 12, 4, 15, 14, 7, 17, 11, 18, 19, 20}, {0, 1, 6, │ │ │ │ + 10, 7, 13, 2, 20, 8, 18, 11, 12, 14, 19, 5, 9, 1, 16, 4, 0, 3, 15, 17}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 20, 7, 15, 11, 17, 9, 5, 10, 3, 4, 13, 14, 18, 16, 19, 2, 8, 12}, {0, 1, │ │ │ │ + {11, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 5, 1, 16, 7, 0, 3, 15, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 15, 20, 4, 8, 6, 12, 17, 9, 10, 2, 3, 13, 14, 18, 16, 19, 5, 7, 11}, │ │ │ │ + 17}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18, 11, 14, 16, 8, 1, 12, 17, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, │ │ │ │ + 3, 15}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 11}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 18, 16, 19, 15, │ │ │ │ + 3, 15, 17}, {7, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 11, 1, 16, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 17, 3}} │ │ │ │ + 0, 3, 15, 17}} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This is still somewhat experimental. │ │ │ │ ********** WWaayyss ttoo uussee mmoonnooddrroommyyGGrroouupp:: ********** │ │ │ │ * monodromyGroup(System) │ │ │ │ * monodromyGroup(System,AbstractPoint,List) │ │ ├── ./usr/share/doc/Macaulay2/MonomialAlgebras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ ZGVjb21wb3NlSG9tb2dlbmVvdXNNQShMaXN0KQ== │ │ │ #:len=308 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTYxMSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZGVjb21wb3NlSG9tb2dlbmVvdXNNQSxMaXN0KSwi │ │ ├── ./usr/share/doc/Macaulay2/MonomialIntegerPrograms/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ R3JhZGVkQmV0dGlz │ │ │ #:len=431 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiB7fSwgImxpbmVudW0iID0+IDExNDYs │ │ │ IHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkdyYWRlZEJldHRp │ │ ├── ./usr/share/doc/Macaulay2/MonomialOrbits/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ aGlsYmVydFJlcHJlc2VudGF0aXZlcw== │ │ │ #:len=3397 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCByZXByZXNlbnRhdGl2ZXMgb2Yg │ │ │ bW9ub21pYWwgaWRlYWxzIHVuZGVyIHBlcm11dGF0aW9ucyBvZiB0aGUgdmFyaWFibGVzIiwgImxp │ │ ├── ./usr/share/doc/Macaulay2/Msolve/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ bXNvbHZlUlVSKElkZWFsKQ== │ │ │ #:len=227 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjM2LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtc29sdmVSVVIsSWRlYWwpLCJtc29sdmVSVVIoSWRl │ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i2 : I = ideal(x, y, z) │ │ │ │ │ │ o2 = ideal (x, y, z) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : msolveGB(I, Verbosity => 2, Threads => 6) │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-144574-0/0-in.ms -o /tmp/M2-144574-0/0-out.ms │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-266668-0/0-in.ms -o /tmp/M2-266668-0/0-out.ms │ │ │ │ │ │ --------------- INPUT DATA --------------- │ │ │ #variables 3 │ │ │ #equations 3 │ │ │ #invalid equations 0 │ │ │ field characteristic 0 │ │ │ homogeneous input? 1 │ │ │ @@ -45,25 +45,25 @@ │ │ │ time(rd) time of the current f4 round in seconds given │ │ │ for real and cpu time │ │ │ -------------------------------------------------------- │ │ │ │ │ │ deg sel pairs mat density new data time(rd) in sec (real|cpu) │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ -reduce final basis 3 x 3 33.33% 3 new 0 zero 0.02 | 0.08 │ │ │ +reduce final basis 3 x 3 33.33% 3 new 0 zero 0.00 | 0.00 │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ -overall(elapsed) 0.04 sec │ │ │ -overall(cpu) 0.11 sec │ │ │ +overall(elapsed) 0.00 sec │ │ │ +overall(cpu) 0.00 sec │ │ │ select 0.00 sec 0.0% │ │ │ -symbolic prep. 0.00 sec 0.0% │ │ │ -update 0.01 sec 35.5% │ │ │ -convert 0.02 sec 64.3% │ │ │ -linear algebra 0.00 sec 0.0% │ │ │ +symbolic prep. 0.00 sec 0.3% │ │ │ +update 0.00 sec 77.6% │ │ │ +convert 0.00 sec 3.1% │ │ │ +linear algebra 0.00 sec 1.5% │ │ │ reduce gb 0.00 sec 0.0% │ │ │ ----------------------------------------- │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ size of basis 3 │ │ │ #terms in basis 3 │ │ │ #pairs reduced 0 │ │ │ @@ -89,15 +89,15 @@ │ │ │ ----------------------------------------- │ │ │ │ │ │ multi-modular steps │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ {1}{2}<100.00%> │ │ │ │ │ │ ------------------------------------------------------------------------------------ │ │ │ -msolve overall time 0.31 sec (elapsed) / 0.91 sec (cpu) │ │ │ +msolve overall time 0.14 sec (elapsed) / 0.11 sec (cpu) │ │ │ ------------------------------------------------------------------------------------ │ │ │ │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ Max coeff. bitsize 1 │ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/_msolve__Real__Solutions.out │ │ │ @@ -11,87 +11,87 @@ │ │ │ 2 2 │ │ │ o2 = ideal (x - x, y - 5) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : rationalIntervalSols = msolveRealSolutions I │ │ │ │ │ │ - 4294967295 4294967297 4801919417 9603838835 1 │ │ │ -o3 = {{{----------, ----------}, {----------, ----------}}, {{- ----------, │ │ │ - 4294967296 4294967296 2147483648 4294967296 4294967296 │ │ │ - ------------------------------------------------------------------------ │ │ │ - 1 4801919417 9603838835 4294967295 4294967297 │ │ │ - ----------}, {----------, ----------}}, {{----------, ----------}, {- │ │ │ - 4294967296 2147483648 4294967296 4294967296 4294967296 │ │ │ - ------------------------------------------------------------------------ │ │ │ - 9603838835 4801919417 1 1 9603838835 │ │ │ - ----------, - ----------}}, {{- ----------, ----------}, {- ----------, │ │ │ - 4294967296 2147483648 4294967296 4294967296 4294967296 │ │ │ - ------------------------------------------------------------------------ │ │ │ - 4801919417 │ │ │ - - ----------}}} │ │ │ - 2147483648 │ │ │ + 4294967295 4294967297 4801919417 9603838835 4294967295 │ │ │ +o3 = {{{----------, ----------}, {----------, ----------}}, {{----------, │ │ │ + 4294967296 4294967296 2147483648 4294967296 4294967296 │ │ │ + ------------------------------------------------------------------------ │ │ │ + 4294967297 9603838835 4801919417 1 1 │ │ │ + ----------}, {- ----------, - ----------}}, {{- ----------, ----------}, │ │ │ + 4294967296 4294967296 2147483648 4294967296 4294967296 │ │ │ + ------------------------------------------------------------------------ │ │ │ + 4801919417 9603838835 1 1 9603838835 │ │ │ + {----------, ----------}}, {{- ----------, ----------}, {- ----------, - │ │ │ + 2147483648 4294967296 4294967296 4294967296 4294967296 │ │ │ + ------------------------------------------------------------------------ │ │ │ + 4801919417 │ │ │ + ----------}}} │ │ │ + 2147483648 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : rationalApproxSols = msolveRealSolutions(I, QQ) │ │ │ │ │ │ - 19207677669 19207677669 19207677669 │ │ │ -o4 = {{0, -----------}, {1, -----------}, {0, - -----------}, {1, - │ │ │ - 8589934592 8589934592 8589934592 │ │ │ + 19207677669 19207677669 19207677669 │ │ │ +o4 = {{1, -----------}, {1, - -----------}, {0, -----------}, {0, - │ │ │ + 8589934592 8589934592 8589934592 │ │ │ ------------------------------------------------------------------------ │ │ │ 19207677669 │ │ │ -----------}} │ │ │ 8589934592 │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : floatIntervalSols = msolveRealSolutions(I, RRi) │ │ │ │ │ │ -o5 = {{[1,1], [2.23607,2.23607]}, {[-2.32831e-10,2.32831e-10], │ │ │ +o5 = {{[1,1], [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]}, │ │ │ ------------------------------------------------------------------------ │ │ │ - [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]}, │ │ │ + {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ ------------------------------------------------------------------------ │ │ │ {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10) │ │ │ │ │ │ -o6 = {{[-.000976562,.000976562], [-2.23633,-2.23535]}, {[.999023,1.00098], │ │ │ +o6 = {{[.999023,1.00098], [2.23535,2.23633]}, {[.999023,1.00098], │ │ │ ------------------------------------------------------------------------ │ │ │ [-2.23633,-2.23535]}, {[-.000976562,.000976562], [2.23535,2.23633]}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {[.999023,1.00098], [2.23535,2.23633]}} │ │ │ + {[-.000976562,.000976562], [-2.23633,-2.23535]}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : floatApproxSols = msolveRealSolutions(I, RR) │ │ │ │ │ │ -o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}} │ │ │ +o7 = {{1, 2.23607}, {1, -2.23607}, {0, 2.23607}, {0, -2.23607}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : floatApproxSols = msolveRealSolutions(I, RR_10) │ │ │ │ │ │ -o8 = {{1, -2.23584}, {0, -2.23584}, {1, 2.23584}, {0, 2.23584}} │ │ │ +o8 = {{1, 2.23584}, {1, -2.23584}, {0, 2.23584}, {0, -2.23584}} │ │ │ │ │ │ o8 : List │ │ │ │ │ │ i9 : I = ideal {(x-1)*x^3, (y^2-5)^2} │ │ │ │ │ │ 4 3 4 2 │ │ │ o9 = ideal (x - x , y - 10y + 25) │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : floatApproxSols = msolveRealSolutions(I, RRi) │ │ │ │ │ │ -o10 = {{[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}, {[1,1], │ │ │ +o10 = {{[1,1], [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]}, │ │ │ ----------------------------------------------------------------------- │ │ │ - [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ + {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {[1,1], [2.23607,2.23607]}} │ │ │ + {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}} │ │ │ │ │ │ o10 : List │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html │ │ │ @@ -99,78 +99,78 @@ │ │ │ o2 = ideal (x - x, y - 5) │ │ │ │ │ │ o2 : Ideal of R     │ │ │ │ │ │ │ │ │ 
    i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │  
│ │ │ -        4294967295  4294967297    4801919417  9603838835             1     
│ │ │ -o3 = {{{----------, ----------}, {----------, ----------}}, {{- ----------,
│ │ │ -        4294967296  4294967296    2147483648  4294967296        4294967296 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -          1        4801919417  9603838835      4294967295  4294967297     
│ │ │ -     ----------}, {----------, ----------}}, {{----------, ----------}, {-
│ │ │ -     4294967296    2147483648  4294967296      4294967296  4294967296     
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     9603838835    4801919417             1           1          9603838835 
│ │ │ -     ----------, - ----------}}, {{- ----------, ----------}, {- ----------,
│ │ │ -     4294967296    2147483648        4294967296  4294967296      4294967296 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       4801919417
│ │ │ -     - ----------}}}
│ │ │ -       2147483648
│ │ │ +        4294967295  4294967297    4801919417  9603838835      4294967295 
│ │ │ +o3 = {{{----------, ----------}, {----------, ----------}}, {{----------,
│ │ │ +        4294967296  4294967296    2147483648  4294967296      4294967296 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     4294967297      9603838835    4801919417             1           1      
│ │ │ +     ----------}, {- ----------, - ----------}}, {{- ----------, ----------},
│ │ │ +     4294967296      4294967296    2147483648        4294967296  4294967296  
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +      4801919417  9603838835             1           1          9603838835   
│ │ │ +     {----------, ----------}}, {{- ----------, ----------}, {- ----------, -
│ │ │ +      2147483648  4294967296        4294967296  4294967296      4294967296   
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     4801919417
│ │ │ +     ----------}}}
│ │ │ +     2147483648
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ 
    i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │  
│ │ │ -          19207677669       19207677669         19207677669        
│ │ │ -o4 = {{0, -----------}, {1, -----------}, {0, - -----------}, {1, -
│ │ │ -           8589934592        8589934592          8589934592        
│ │ │ +          19207677669         19207677669       19207677669        
│ │ │ +o4 = {{1, -----------}, {1, - -----------}, {0, -----------}, {0, -
│ │ │ +           8589934592          8589934592        8589934592        
│ │ │       ------------------------------------------------------------------------
│ │ │       19207677669
│ │ │       -----------}}
│ │ │        8589934592
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ 
    i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o5 = {{[1,1], [2.23607,2.23607]}, {[-2.32831e-10,2.32831e-10],
│ │ │ +o5 = {{[1,1], [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │ +     {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │       {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ 
    i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │  
│ │ │ -o6 = {{[-.000976562,.000976562], [-2.23633,-2.23535]}, {[.999023,1.00098],
│ │ │ +o6 = {{[.999023,1.00098], [2.23535,2.23633]}, {[.999023,1.00098],
│ │ │       ------------------------------------------------------------------------
│ │ │       [-2.23633,-2.23535]}, {[-.000976562,.000976562], [2.23535,2.23633]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[.999023,1.00098], [2.23535,2.23633]}}
│ │ │ +     {[-.000976562,.000976562], [-2.23633,-2.23535]}}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ 
    i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │  
│ │ │ -o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}}
│ │ │ +o7 = {{1, 2.23607}, {1, -2.23607}, {0, 2.23607}, {0, -2.23607}}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ 
    i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │  
│ │ │ -o8 = {{1, -2.23584}, {0, -2.23584}, {1, 2.23584}, {0, 2.23584}}
│ │ │ +o8 = {{1, 2.23584}, {1, -2.23584}, {0, 2.23584}, {0, -2.23584}}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │
    │ │ │

    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.

    │ │ │
    │ │ │ @@ -182,19 +182,19 @@ │ │ │ o9 = ideal (x - x , y - 10y + 25) │ │ │ │ │ │ o9 : Ideal of R     │ │ │ │ │ │ │ │ │ 
    i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o10 = {{[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}, {[1,1],
│ │ │ +o10 = {{[1,1], [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]},
│ │ │ +      {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {[1,1], [2.23607,2.23607]}}
│ │ │ +      {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o10 : List
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use msolveRealSolutions:

    │ │ │ ├── html2text {} │ │ │ │ @@ -44,86 +44,86 @@ │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o2 = ideal (x - x, y - 5) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : rationalIntervalSols = msolveRealSolutions I │ │ │ │ │ │ │ │ - 4294967295 4294967297 4801919417 9603838835 1 │ │ │ │ -o3 = {{{----------, ----------}, {----------, ----------}}, {{- ----------, │ │ │ │ - 4294967296 4294967296 2147483648 4294967296 4294967296 │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 1 4801919417 9603838835 4294967295 4294967297 │ │ │ │ - ----------}, {----------, ----------}}, {{----------, ----------}, {- │ │ │ │ - 4294967296 2147483648 4294967296 4294967296 4294967296 │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 9603838835 4801919417 1 1 9603838835 │ │ │ │ - ----------, - ----------}}, {{- ----------, ----------}, {- ----------, │ │ │ │ - 4294967296 2147483648 4294967296 4294967296 4294967296 │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 4801919417 │ │ │ │ - - ----------}}} │ │ │ │ - 2147483648 │ │ │ │ + 4294967295 4294967297 4801919417 9603838835 4294967295 │ │ │ │ +o3 = {{{----------, ----------}, {----------, ----------}}, {{----------, │ │ │ │ + 4294967296 4294967296 2147483648 4294967296 4294967296 │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 4294967297 9603838835 4801919417 1 1 │ │ │ │ + ----------}, {- ----------, - ----------}}, {{- ----------, ----------}, │ │ │ │ + 4294967296 4294967296 2147483648 4294967296 4294967296 │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 4801919417 9603838835 1 1 9603838835 │ │ │ │ + {----------, ----------}}, {{- ----------, ----------}, {- ----------, - │ │ │ │ + 2147483648 4294967296 4294967296 4294967296 4294967296 │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 4801919417 │ │ │ │ + ----------}}} │ │ │ │ + 2147483648 │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : rationalApproxSols = msolveRealSolutions(I, QQ) │ │ │ │ │ │ │ │ - 19207677669 19207677669 19207677669 │ │ │ │ -o4 = {{0, -----------}, {1, -----------}, {0, - -----------}, {1, - │ │ │ │ - 8589934592 8589934592 8589934592 │ │ │ │ + 19207677669 19207677669 19207677669 │ │ │ │ +o4 = {{1, -----------}, {1, - -----------}, {0, -----------}, {0, - │ │ │ │ + 8589934592 8589934592 8589934592 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 19207677669 │ │ │ │ -----------}} │ │ │ │ 8589934592 │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : floatIntervalSols = msolveRealSolutions(I, RRi) │ │ │ │ │ │ │ │ -o5 = {{[1,1], [2.23607,2.23607]}, {[-2.32831e-10,2.32831e-10], │ │ │ │ +o5 = {{[1,1], [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]}, │ │ │ │ + {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10) │ │ │ │ │ │ │ │ -o6 = {{[-.000976562,.000976562], [-2.23633,-2.23535]}, {[.999023,1.00098], │ │ │ │ +o6 = {{[.999023,1.00098], [2.23535,2.23633]}, {[.999023,1.00098], │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ [-2.23633,-2.23535]}, {[-.000976562,.000976562], [2.23535,2.23633]}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {[.999023,1.00098], [2.23535,2.23633]}} │ │ │ │ + {[-.000976562,.000976562], [-2.23633,-2.23535]}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : floatApproxSols = msolveRealSolutions(I, RR) │ │ │ │ │ │ │ │ -o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}} │ │ │ │ +o7 = {{1, 2.23607}, {1, -2.23607}, {0, 2.23607}, {0, -2.23607}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : floatApproxSols = msolveRealSolutions(I, RR_10) │ │ │ │ │ │ │ │ -o8 = {{1, -2.23584}, {0, -2.23584}, {1, 2.23584}, {0, 2.23584}} │ │ │ │ +o8 = {{1, 2.23584}, {1, -2.23584}, {0, 2.23584}, {0, -2.23584}} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ 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. │ │ │ │ i9 : I = ideal {(x-1)*x^3, (y^2-5)^2} │ │ │ │ │ │ │ │ 4 3 4 2 │ │ │ │ o9 = ideal (x - x , y - 10y + 25) │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : floatApproxSols = msolveRealSolutions(I, RRi) │ │ │ │ │ │ │ │ -o10 = {{[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}, {[1,1], │ │ │ │ +o10 = {{[1,1], [2.23607,2.23607]}, {[1,1], [-2.23607,-2.23607]}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ │ + {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {[1,1], [2.23607,2.23607]}} │ │ │ │ + {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]}} │ │ │ │ │ │ │ │ o10 : List │ │ │ │ ********** WWaayyss ttoo uussee mmssoollvveeRReeaallSSoolluuttiioonnss:: ********** │ │ │ │ * msolveRealSolutions(Ideal) │ │ │ │ * msolveRealSolutions(Ideal,Ring) │ │ │ │ * msolveRealSolutions(Ideal,RingFamily) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html │ │ │ @@ -66,15 +66,15 @@ │ │ │ │ │ │ o2 = ideal (x, y, z) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ 
    i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-144574-0/0-in.ms -o /tmp/M2-144574-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-266668-0/0-in.ms -o /tmp/M2-266668-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -102,25 +102,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.02 | 0.08         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.04 sec
│ │ │ -overall(cpu)            0.11 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.01 sec  35.5%
│ │ │ -convert                 0.02 sec  64.3%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ +update                  0.00 sec  77.6%
│ │ │ +convert                 0.00 sec   3.1%
│ │ │ +linear algebra          0.00 sec   1.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -146,15 +146,15 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%>
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.31 sec (elapsed) /  0.91 sec (cpu)
│ │ │ +msolve overall time           0.14 sec (elapsed) /  0.11 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │   
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  Max coeff. bitsize                1
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,16 +31,16 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal(x, y, z)
│ │ │ │  
│ │ │ │  o2 = ideal (x, y, z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-144574-0/0-in.ms -o /
│ │ │ │ -tmp/M2-144574-0/0-out.ms
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-266668-0/0-in.ms -o /
│ │ │ │ +tmp/M2-266668-0/0-out.ms
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -72,26 +72,26 @@
│ │ │ │  deg     sel   pairs        mat          density            new data
│ │ │ │  time(rd) in sec (real|cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero
│ │ │ │ -0.02 | 0.08
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.04 sec
│ │ │ │ -overall(cpu)            0.11 sec
│ │ │ │ +overall(elapsed)        0.00 sec
│ │ │ │ +overall(cpu)            0.00 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.01 sec  35.5%
│ │ │ │ -convert                 0.02 sec  64.3%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ │ +update                  0.00 sec  77.6%
│ │ │ │ +convert                 0.00 sec   3.1%
│ │ │ │ +linear algebra          0.00 sec   1.5%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -119,15 +119,15 @@
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  {1}{2}<100.00%>
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.31 sec (elapsed) /  0.91 sec (cpu)
│ │ │ │ +msolve overall time           0.14 sec (elapsed) /  0.11 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ ├── ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZGVncmVlT2ZNYXAoSWRlYWwp
│ │ │  #:len=283
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjk3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/MultigradedBGG/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  ZGVncmVlKERpZmZlcmVudGlhbE1vZHVsZSk=
│ │ │  #:len=1348
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgZGVncmVlIG9mIHRo
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│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  VXNlTWF0cm9pZFNwZWVkdXA=
│ │ │  #:len=723
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgYXJndW1lbnQiLCBEZXNj
│ │ │  cmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoZSBvcHRpb24gVXNlTWF0cm9pZFNwZWVkdXAg
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  Z3JHcihJZGVhbCk=
│ │ │  #:len=249
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnckdyLElkZWFsKSwiZ3JHcihJZGVhbCkiLCJNdWx0
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │  
│ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime jMult I
│ │ │ - -- .0212025s elapsed
│ │ │ + -- .0204144s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .24679s elapsed
│ │ │ + -- .496935s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .155598s elapsed
│ │ │ + -- .803903s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │               2        3
│ │ │  o2 = ideal (x , x*y, y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .179396s elapsed
│ │ │ + -- .282645s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out
│ │ │ @@ -13,17 +13,17 @@
│ │ │       ------------------------------------------------------------------------
│ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monjMult I
│ │ │ - -- .215844s elapsed
│ │ │ + -- .599575s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.04785s elapsed
│ │ │ + -- 2.17659s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -83,27 +83,27 @@
│ │ │  
│ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime jMult I
│ │ │ - -- .0212025s elapsed
│ │ │ + -- .0204144s elapsed
│ │ │  
│ │ │  o3 = 2
    │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime monjMult I
│ │ │ - -- .24679s elapsed
│ │ │ + -- .496935s elapsed
│ │ │  
│ │ │  o4 = 2
    │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .155598s elapsed
│ │ │ + -- .803903s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -21,23 +21,23 @@ │ │ │ │ o1 : PolynomialRing │ │ │ │ i2 : I = ideal"xy,yz,zx" │ │ │ │ │ │ │ │ o2 = ideal (x*y, y*z, x*z) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime jMult I │ │ │ │ - -- .0212025s elapsed │ │ │ │ + -- .0204144s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ i4 : elapsedTime monjMult I │ │ │ │ - -- .24679s elapsed │ │ │ │ + -- .496935s elapsed │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : elapsedTime multiplicitySequence I │ │ │ │ - -- .155598s elapsed │ │ │ │ + -- .803903s elapsed │ │ │ │ │ │ │ │ o5 = HashTable{2 => 3} │ │ │ │ 3 => 2 │ │ │ │ │ │ │ │ o5 : HashTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal │ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ @@ -84,15 +84,15 @@ │ │ │ 2 3 │ │ │ o2 = ideal (x , x*y, y ) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .179396s elapsed
│ │ │ + -- .282645s elapsed
│ │ │  
│ │ │  o3 = 2
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,15 +23,15 @@ │ │ │ │ i2 : I = ideal"x2,xy,y3" │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o2 = ideal (x , x*y, y ) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime monAnalyticSpread I │ │ │ │ - -- .179396s elapsed │ │ │ │ + -- .282645s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_P -- the Newton polyhedron of a monomial ideal │ │ │ │ ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: ********** │ │ │ │ * monAnalyticSpread(Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ @@ -87,21 +87,21 @@ │ │ │ 10 11 8 12 9 11 10 10 11 9 12 8 │ │ │ x y , x y , x y , x y , x y , x y ) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime monjMult I
│ │ │ - -- .215844s elapsed
│ │ │ + -- .599575s elapsed
│ │ │  
│ │ │  o3 = 192
    │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime jMult I
│ │ │ - -- 1.04785s elapsed
│ │ │ + -- 2.17659s elapsed
│ │ │  
│ │ │  o4 = 192
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -25,19 +25,19 @@ │ │ │ │ o2 = ideal (x y , x y , x y , x y , x y , x y , x y , x y , x y , │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 10 11 8 12 9 11 10 10 11 9 12 8 │ │ │ │ x y , x y , x y , x y , x y , x y ) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime monjMult I │ │ │ │ - -- .215844s elapsed │ │ │ │ + -- .599575s elapsed │ │ │ │ │ │ │ │ o3 = 192 │ │ │ │ i4 : elapsedTime jMult I │ │ │ │ - -- 1.04785s elapsed │ │ │ │ + -- 2.17659s elapsed │ │ │ │ │ │ │ │ o4 = 192 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal │ │ │ │ * _j_M_u_l_t -- the j-multiplicity of an ideal │ │ │ │ * _m_o_n_R_e_d_u_c_t_i_o_n -- the minimal monomial reduction of a monomial ideal │ │ │ │ * _N_P -- the Newton polyhedron of a monomial ideal │ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=41 │ │ │ bG9nQ2Fub25pY2FsVGhyZXNob2xkKENlbnRyYWxBcnJhbmdlbWVudCk= │ │ │ #:len=361 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA2OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobG9nQ2Fub25pY2FsVGhyZXNob2xkLENlbnRyYWxB │ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ TXVsdElkZWFs │ │ │ #:len=1359 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIG11bHRpcGxpZXIg │ │ │ aWRlYWwgb2YgYSBnaXZlbiBudW1iZXIuIiwgImxpbmVudW0iID0+IDY0OCwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ dGV4TWF0aChSQVQp │ │ │ #:len=209 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDE5OSwgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4TWF0 │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ @@ -13,15 +13,15 @@ │ │ │ o4 : ProjectiveVariety, curve in PP^8 │ │ │ │ │ │ i5 : ? X │ │ │ │ │ │ o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 │ │ │ │ │ │ i6 : time f = X ===> Y; │ │ │ - -- used 3.61187s (cpu); 2.07472s (thread); 0s (gc) │ │ │ + -- used 8.10238s (cpu); 3.70567s (thread); 0s (gc) │ │ │ │ │ │ o6 : MultirationalMap (automorphism of PP^8) │ │ │ │ │ │ i7 : f X │ │ │ │ │ │ o7 = Y │ │ │ │ │ │ @@ -38,15 +38,15 @@ │ │ │ o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ │ │ i10 : W = random({{2},{1}},Y); │ │ │ │ │ │ o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ │ │ i11 : time g = V ===> W; │ │ │ - -- used 3.44985s (cpu); 1.81373s (thread); 0s (gc) │ │ │ + -- used 8.51391s (cpu); 3.89965s (thread); 0s (gc) │ │ │ │ │ │ o11 : MultirationalMap (automorphism of PP^8) │ │ │ │ │ │ i12 : g||W │ │ │ │ │ │ o12 = multi-rational map consisting of one single rational map │ │ │ source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1 │ │ │ @@ -129,15 +129,15 @@ │ │ │ o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9 │ │ │ │ │ │ i16 : ? Z │ │ │ │ │ │ o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2 │ │ │ │ │ │ i17 : time h = Z ===> GG_K(1,4) │ │ │ - -- used 7.98739s (cpu); 4.64055s (thread); 0s (gc) │ │ │ + -- used 12.2511s (cpu); 8.12525s (thread); 0s (gc) │ │ │ │ │ │ o17 = h │ │ │ │ │ │ o17 : MultirationalMap (isomorphism from PP^9 to PP^9) │ │ │ │ │ │ i18 : h || GG_K(1,4) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ @@ -7,15 +7,15 @@ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi)); │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 │ │ │ │ │ │ i4 : time X = Phi^* Y; │ │ │ - -- used 4.74547s (cpu); 3.30375s (thread); 0s (gc) │ │ │ + -- used 3.6681s (cpu); 2.93148s (thread); 0s (gc) │ │ │ │ │ │ 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 ) │ │ │ │ │ │ i5 : dim X, degree X, degrees X │ │ │ │ │ │ o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1), │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ @@ -11,26 +11,26 @@ │ │ │ o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7 x PP^7) │ │ │ │ │ │ i4 : Z = source Phi; │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7 │ │ │ │ │ │ i5 : time Phi Z; │ │ │ - -- used 0.0929525s (cpu); 0.0923825s (thread); 0s (gc) │ │ │ + -- used 0.796659s (cpu); 0.226534s (thread); 0s (gc) │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7 │ │ │ │ │ │ i6 : dim oo, degree oo, degrees oo │ │ │ │ │ │ o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)}) │ │ │ │ │ │ o6 : Sequence │ │ │ │ │ │ i7 : time Phi (point Z + point Z + point Z) │ │ │ - -- used 2.12072s (cpu); 1.27842s (thread); 0s (gc) │ │ │ + -- used 5.82399s (cpu); 2.79473s (thread); 0s (gc) │ │ │ │ │ │ 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 │ │ │ │ │ │ o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7 │ │ │ │ │ │ i8 : dim oo, degree oo, degrees oo │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ @@ -11,22 +11,22 @@ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of │ │ │ target variety: hypersurface in PP^4 defined by a form of degree 2 │ │ │ ------------------------------------------------------------------------ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ │ │ i4 : time degree(Phi,Strategy=>"random point") │ │ │ - -- used 4.21068s (cpu); 2.53763s (thread); 0s (gc) │ │ │ + -- used 8.60794s (cpu); 1.99728s (thread); 0s (gc) │ │ │ │ │ │ o4 = 2 │ │ │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ - -- used 0.312606s (cpu); 0.236089s (thread); 0s (gc) │ │ │ + -- used 0.309751s (cpu); 0.242231s (thread); 0s (gc) │ │ │ │ │ │ o5 = 2 │ │ │ │ │ │ i6 : time degree Phi │ │ │ - -- used 0.326294s (cpu); 0.24353s (thread); 0s (gc) │ │ │ + -- used 0.172914s (cpu); 0.173914s (thread); 0s (gc) │ │ │ │ │ │ o6 = 2 │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ @@ -3,12 +3,12 @@ │ │ │ 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}; │ │ │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ i3 : time degree Phi │ │ │ - -- used 0.571211s (cpu); 0.424891s (thread); 0s (gc) │ │ │ + -- used 0.510038s (cpu); 0.32469s (thread); 0s (gc) │ │ │ │ │ │ o3 = 1 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ @@ -1,52 +1,52 @@ │ │ │ -- -*- M2-comint -*- hash: 11533721324852072161 │ │ │ │ │ │ i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4); │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5) │ │ │ │ │ │ i2 : time ? Phi │ │ │ - -- used 0.0017476s (cpu); 0.000210294s (thread); 0s (gc) │ │ │ + -- used 0.00108592s (cpu); 0.000124764s (thread); 0s (gc) │ │ │ │ │ │ o2 = multi-rational map consisting of 2 rational maps │ │ │ source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 │ │ │ target variety: PP^4 x PP^5 │ │ │ ------------------------------------------------------------------------ │ │ │ hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ │ │ │ i3 : image Phi; │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5 │ │ │ │ │ │ i4 : time ? Phi │ │ │ - -- used 0.00400281s (cpu); 0.000328677s (thread); 0s (gc) │ │ │ + -- used 0.00226652s (cpu); 0.000256101s (thread); 0s (gc) │ │ │ │ │ │ o4 = multi-rational map consisting of 2 rational maps │ │ │ 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 │ │ │ target variety: PP^4 x PP^5 │ │ │ dominance: false │ │ │ image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ │ │ │ i5 : time describe Phi │ │ │ - -- used 1.5069s (cpu); 1.10379s (thread); 0s (gc) │ │ │ + -- used 2.96765s (cpu); 2.59941s (thread); 0s (gc) │ │ │ │ │ │ o5 = multi-rational map consisting of 2 rational maps │ │ │ 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 │ │ │ target variety: PP^4 x PP^5 │ │ │ base locus: empty subscheme of PP^4 x PP^5 │ │ │ dominance: false │ │ │ image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ multidegree: {51, 51, 51, 51, 51} │ │ │ degree: 1 │ │ │ degree sequence (map 1/2): [(1,0), (0,2)] │ │ │ degree sequence (map 2/2): [(0,1), (2,0)] │ │ │ coefficient ring: ZZ/65521 │ │ │ │ │ │ i6 : time ? Phi │ │ │ - -- used 0.000149972s (cpu); 0.000376616s (thread); 0s (gc) │ │ │ + -- used 0.000105859s (cpu); 0.000315862s (thread); 0s (gc) │ │ │ │ │ │ o6 = multi-rational map consisting of 2 rational maps │ │ │ 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 │ │ │ target variety: PP^4 x PP^5 │ │ │ base locus: empty subscheme of PP^4 x PP^5 │ │ │ dominance: false │ │ │ image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ @@ -3,45 +3,45 @@ │ │ │ i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true) │ │ │ │ │ │ o1 = Phi │ │ │ │ │ │ o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ i2 : time (Phi1,Phi2) = graph Phi │ │ │ - -- used 0.0199683s (cpu); 0.0203147s (thread); 0s (gc) │ │ │ + -- used 0.195567s (cpu); 0.058218s (thread); 0s (gc) │ │ │ │ │ │ o2 = (Phi1, Phi2) │ │ │ │ │ │ o2 : Sequence │ │ │ │ │ │ i3 : Phi1; │ │ │ │ │ │ o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4) │ │ │ │ │ │ i4 : Phi2; │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ i5 : time (Phi21,Phi22) = graph Phi2 │ │ │ - -- used 0.0308626s (cpu); 0.0318301s (thread); 0s (gc) │ │ │ + -- used 0.0835289s (cpu); 0.0292452s (thread); 0s (gc) │ │ │ │ │ │ o5 = (Phi21, Phi22) │ │ │ │ │ │ o5 : Sequence │ │ │ │ │ │ i6 : Phi21; │ │ │ │ │ │ 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) │ │ │ │ │ │ i7 : Phi22; │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ i8 : time (Phi211,Phi212) = graph Phi21 │ │ │ - -- used 0.188571s (cpu); 0.120925s (thread); 0s (gc) │ │ │ + -- used 0.598917s (cpu); 0.172583s (thread); 0s (gc) │ │ │ │ │ │ o8 = (Phi211, Phi212) │ │ │ │ │ │ o8 : Sequence │ │ │ │ │ │ i9 : Phi211; │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ @@ -11,25 +11,25 @@ │ │ │ o3 : RationalMap (quadratic rational map from PP^4 to PP^4) │ │ │ │ │ │ i4 : Phi = rationalMap {f,g}; │ │ │ │ │ │ o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4) │ │ │ │ │ │ i5 : time Z = image Phi; │ │ │ - -- used 0.193919s (cpu); 0.194375s (thread); 0s (gc) │ │ │ + -- used 0.326902s (cpu); 0.260371s (thread); 0s (gc) │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4 │ │ │ │ │ │ i6 : dim Z, degree Z, degrees Z │ │ │ │ │ │ o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)}) │ │ │ │ │ │ o6 : Sequence │ │ │ │ │ │ i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4"); │ │ │ - -- used 8.21524s (cpu); 3.17664s (thread); 0s (gc) │ │ │ + -- used 21.9591s (cpu); 4.80259s (thread); 0s (gc) │ │ │ │ │ │ o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4 │ │ │ │ │ │ i8 : assert(Z == Z') │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ @@ -4,25 +4,25 @@ │ │ │ │ │ │ i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6 │ │ │ Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2)); │ │ │ │ │ │ o2 : MultirationalMap (rational map from PP^6 to GG(2,4)) │ │ │ │ │ │ i3 : time Psi = inverse2 Phi; │ │ │ - -- used 0.380852s (cpu); 0.296207s (thread); 0s (gc) │ │ │ + -- used 1.37433s (cpu); 0.614694s (thread); 0s (gc) │ │ │ │ │ │ o3 : MultirationalMap (birational map from GG(2,4) to PP^6) │ │ │ │ │ │ i4 : assert(Phi * Psi == 1) │ │ │ │ │ │ i5 : Phi' = Phi || Phi; │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4)) │ │ │ │ │ │ i6 : time Psi' = inverse2 Phi'; │ │ │ - -- used 1.86808s (cpu); 1.29771s (thread); 0s (gc) │ │ │ + -- used 2.85058s (cpu); 2.01873s (thread); 0s (gc) │ │ │ │ │ │ o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6) │ │ │ │ │ │ i7 : assert(Phi' * Psi' == 1) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ @@ -7,33 +7,33 @@ │ │ │ │ │ │ i2 : -- we see Phi as a dominant map │ │ │ Phi = rationalMap(Phi,image Phi); │ │ │ │ │ │ o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ i3 : time inverse Phi; │ │ │ - -- used 0.0442329s (cpu); 0.0417169s (thread); 0s (gc) │ │ │ + -- used 0.807672s (cpu); 0.131399s (thread); 0s (gc) │ │ │ │ │ │ o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4) │ │ │ │ │ │ i4 : Psi = last graph Phi; │ │ │ │ │ │ o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ i5 : time inverse Psi; │ │ │ - -- used 0.177065s (cpu); 0.0983666s (thread); 0s (gc) │ │ │ + -- used 0.64434s (cpu); 0.281716s (thread); 0s (gc) │ │ │ │ │ │ o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5) │ │ │ │ │ │ i6 : Eta = first graph Psi; │ │ │ │ │ │ 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) │ │ │ │ │ │ i7 : time inverse Eta; │ │ │ - -- used 0.453153s (cpu); 0.286087s (thread); 0s (gc) │ │ │ + -- used 2.39753s (cpu); 0.700055s (thread); 0s (gc) │ │ │ │ │ │ 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) │ │ │ │ │ │ i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1) │ │ │ │ │ │ i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ @@ -6,32 +6,32 @@ │ │ │ o2 : RationalMap (quadratic rational map from PP^3 to PP^2) │ │ │ │ │ │ i3 : Phi = rationalMap {f,f}; │ │ │ │ │ │ o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2) │ │ │ │ │ │ i4 : time isIsomorphism Phi │ │ │ - -- used 0.00369049s (cpu); 1.033e-05s (thread); 0s (gc) │ │ │ + -- used 0.00389415s (cpu); 5.791e-06s (thread); 0s (gc) │ │ │ │ │ │ o4 = false │ │ │ │ │ │ i5 : Psi = first graph Phi; │ │ │ │ │ │ o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3) │ │ │ │ │ │ i6 : time isIsomorphism Psi │ │ │ - -- used 0.323078s (cpu); 0.152253s (thread); 0s (gc) │ │ │ + -- used 1.31022s (cpu); 0.409418s (thread); 0s (gc) │ │ │ │ │ │ o6 = false │ │ │ │ │ │ i7 : Eta = first graph Psi; │ │ │ │ │ │ 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) │ │ │ │ │ │ i8 : time isIsomorphism Eta │ │ │ - -- used 1.39409s (cpu); 0.718132s (thread); 0s (gc) │ │ │ + -- used 3.7902s (cpu); 1.51658s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : assert(o8 and (not o6) and (not o4)) │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ @@ -3,24 +3,24 @@ │ │ │ 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}); │ │ │ │ │ │ i2 : Phi = rationalMap {f,g}; │ │ │ │ │ │ o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2) │ │ │ │ │ │ i3 : time isMorphism Phi │ │ │ - -- used 0.230726s (cpu); 0.159482s (thread); 0s (gc) │ │ │ + -- used 0.29478s (cpu); 0.230752s (thread); 0s (gc) │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : time Psi = first graph Phi; │ │ │ - -- used 0.158078s (cpu); 0.0838548s (thread); 0s (gc) │ │ │ + -- used 0.955989s (cpu); 0.247508s (thread); 0s (gc) │ │ │ │ │ │ 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) │ │ │ │ │ │ i5 : time isMorphism Psi │ │ │ - -- used 4.10881s (cpu); 2.89424s (thread); 0s (gc) │ │ │ + -- used 6.74807s (cpu); 5.82189s (thread); 0s (gc) │ │ │ │ │ │ o5 = true │ │ │ │ │ │ i6 : assert((not o3) and o5) │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out │ │ │ @@ -3,24 +3,24 @@ │ │ │ i1 : K = ZZ/333331; │ │ │ │ │ │ i2 : X = PP_K^(1,7); -- rational normal curve of degree 7 │ │ │ │ │ │ o2 : ProjectiveVariety, curve in PP^7 │ │ │ │ │ │ i3 : time f = linearlyNormalEmbedding X; │ │ │ - -- used 0.00799341s (cpu); 0.00882943s (thread); 0s (gc) │ │ │ + -- used 0.00803099s (cpu); 0.00781773s (thread); 0s (gc) │ │ │ │ │ │ o3 : MultirationalMap (automorphism of X) │ │ │ │ │ │ i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3 │ │ │ │ │ │ o4 : ProjectiveVariety, curve in PP^3 │ │ │ │ │ │ i5 : time g = linearlyNormalEmbedding Y; │ │ │ - -- used 0.560105s (cpu); 0.400122s (thread); 0s (gc) │ │ │ + -- used 1.43896s (cpu); 1.23947s (thread); 0s (gc) │ │ │ │ │ │ o5 : MultirationalMap (birational map from Y to curve in PP^7) │ │ │ │ │ │ i6 : assert(isIsomorphism g) │ │ │ │ │ │ i7 : describe g │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ 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}; │ │ │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ i3 : time multidegree Phi │ │ │ - -- used 0.549954s (cpu); 0.410961s (thread); 0s (gc) │ │ │ + -- used 1.24188s (cpu); 0.632383s (thread); 0s (gc) │ │ │ │ │ │ o3 = {66, 46, 31, 20} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : (degree source Phi,degree image Phi) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ @@ -1,21 +1,21 @@ │ │ │ -- -*- M2-comint -*- hash: 16199733219210081214 │ │ │ │ │ │ i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4); │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5) │ │ │ │ │ │ i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi) │ │ │ - -- used 0.00402065s (cpu); 0.00152952s (thread); 0s (gc) │ │ │ - -- used 0.156121s (cpu); 0.106571s (thread); 0s (gc) │ │ │ - -- used 0.216164s (cpu); 0.135241s (thread); 0s (gc) │ │ │ - -- used 0.180607s (cpu); 0.107667s (thread); 0s (gc) │ │ │ - -- used 0.127909s (cpu); 0.0773013s (thread); 0s (gc) │ │ │ + -- used 0.00215027s (cpu); 0.0011356s (thread); 0s (gc) │ │ │ + -- used 0.178981s (cpu); 0.116603s (thread); 0s (gc) │ │ │ + -- used 0.174219s (cpu); 0.109074s (thread); 0s (gc) │ │ │ + -- used 0.186167s (cpu); 0.113206s (thread); 0s (gc) │ │ │ + -- used 0.154433s (cpu); 0.0805549s (thread); 0s (gc) │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ - -- used 0.0486544s (cpu); 0.0500349s (thread); 0s (gc) │ │ │ + -- used 0.197938s (cpu); 0.0508185s (thread); 0s (gc) │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ @@ -3,26 +3,26 @@ │ │ │ i1 : K = ZZ/1000003; │ │ │ │ │ │ i2 : X = PP_K^({1,1,2},{3,2,3}); │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i3 : time p := point X │ │ │ - -- used 0.0160848s (cpu); 0.0148379s (thread); 0s (gc) │ │ │ + -- used 0.516235s (cpu); 0.0818124s (thread); 0s (gc) │ │ │ │ │ │ o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1]) │ │ │ │ │ │ o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i4 : Y = random({2,1,2},X); │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i5 : time q = point Y │ │ │ - -- used 2.0638s (cpu); 1.06252s (thread); 0s (gc) │ │ │ + -- used 2.70174s (cpu); 1.51899s (thread); 0s (gc) │ │ │ │ │ │ o5 = q │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ o4 : RationalMap (quadratic rational map from PP^4 to PP^4) │ │ │ │ │ │ i5 : Phi = rationalMap {f,g,h}; │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4) │ │ │ │ │ │ i6 : time segre Phi; │ │ │ - -- used 0.84109s (cpu); 0.510468s (thread); 0s (gc) │ │ │ + -- used 7.36297s (cpu); 1.6609s (thread); 0s (gc) │ │ │ │ │ │ o6 : RationalMap (rational map from PP^4 to PP^149) │ │ │ │ │ │ i7 : describe segre Phi │ │ │ │ │ │ o7 = rational map defined by forms of degree 6 │ │ │ source variety: PP^4 │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3) │ │ │ │ │ │ o1 = Phi │ │ │ │ │ │ o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2) │ │ │ │ │ │ i2 : time describe Phi │ │ │ - -- used 0.255978s (cpu); 0.169365s (thread); 0s (gc) │ │ │ + -- used 0.177758s (cpu); 0.106991s (thread); 0s (gc) │ │ │ │ │ │ o2 = multi-rational map consisting of 3 rational maps │ │ │ source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1) │ │ │ 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 │ │ │ base locus: empty subscheme of PP^3 x PP^2 │ │ │ dominance: true │ │ │ multidegree: {10, 14, 19, 25} │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ 
    i5 : ? X
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15
    │ │ │ │ │ │ │ │ │ 
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.61187s (cpu); 2.07472s (thread); 0s (gc)
│ │ │ + -- used 8.10238s (cpu); 3.70567s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    │ │ │ │ │ │ │ │ │ 
    i7 : f X
│ │ │  
│ │ │  o7 = Y
│ │ │ @@ -124,15 +124,15 @@
│ │ │            
│ │ │                
    i10 : W = random({{2},{1}},Y);
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    
│ │ │            
│ │ │            
│ │ │                
    i11 : time g = V ===> W;
│ │ │ - -- used 3.44985s (cpu); 1.81373s (thread); 0s (gc)
│ │ │ + -- used 8.51391s (cpu); 3.89965s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    
│ │ │            
│ │ │            
│ │ │                
    i12 : g||W
│ │ │  
│ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │ @@ -223,15 +223,15 @@
│ │ │            
│ │ │                
    i16 : ? Z
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    
│ │ │            
│ │ │            
│ │ │                
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 7.98739s (cpu); 4.64055s (thread); 0s (gc)
│ │ │ + -- used 12.2511s (cpu); 8.12525s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    
│ │ │            
│ │ │            
│ │ │                
    i18 : h || GG_K(1,4)
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  take(N,-2));
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │ │  i5 : ? X
│ │ │ │  
│ │ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15
│ │ │ │  i6 : time f = X ===> Y;
│ │ │ │ - -- used 3.61187s (cpu); 2.07472s (thread); 0s (gc)
│ │ │ │ + -- used 8.10238s (cpu); 3.70567s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i7 : f X
│ │ │ │  
│ │ │ │  o7 = Y
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, curve in PP^8
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │  i9 : V = random({{2},{1}},X);
│ │ │ │  
│ │ │ │  o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │ │  i10 : W = random({{2},{1}},Y);
│ │ │ │  
│ │ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │ │  i11 : time g = V ===> W;
│ │ │ │ - -- used 3.44985s (cpu); 1.81373s (thread); 0s (gc)
│ │ │ │ + -- used 8.51391s (cpu); 3.89965s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i12 : g||W
│ │ │ │  
│ │ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2
│ │ │ │  hypersurfaces of degrees 1^1 2^1
│ │ │ │ @@ -145,15 +145,15 @@
│ │ │ │  i15 : Z = projectiveVariety pfaffians(4,A);
│ │ │ │  
│ │ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │ │  i16 : ? Z
│ │ │ │  
│ │ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ │ - -- used 7.98739s (cpu); 4.64055s (thread); 0s (gc)
│ │ │ │ + -- used 12.2511s (cpu); 8.12525s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = h
│ │ │ │  
│ │ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
│ │ │ │  i18 : h || GG_K(1,4)
│ │ │ │  
│ │ │ │  o18 = multi-rational map consisting of one single rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │            
│ │ │                
    i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi));
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time X = Phi^* Y;
│ │ │ - -- used 4.74547s (cpu); 3.30375s (thread); 0s (gc)
│ │ │ + -- used 3.6681s (cpu); 2.93148s (thread); 0s (gc)
│ │ │  
│ │ │  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 )
    
│ │ │            
│ │ │            
│ │ │                
    i5 : dim X, degree X, degrees X
│ │ │  
│ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random(
│ │ │ │  {1,1},ring target Phi));
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │ │  i4 : time X = Phi^* Y;
│ │ │ │ - -- used 4.74547s (cpu); 3.30375s (thread); 0s (gc)
│ │ │ │ + -- used 3.6681s (cpu); 2.93148s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -88,28 +88,28 @@
│ │ │            
│ │ │                
    i4 : Z = source Phi;
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time Phi Z;
│ │ │ - -- used 0.0929525s (cpu); 0.0923825s (thread); 0s (gc)
│ │ │ + -- used 0.796659s (cpu); 0.226534s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    
│ │ │            
│ │ │            
│ │ │                
    i6 : dim oo, degree oo, degrees oo
│ │ │  
│ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │  
│ │ │  o6 : Sequence
    
│ │ │            
│ │ │            
│ │ │                
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.12072s (cpu); 1.27842s (thread); 0s (gc)
│ │ │ + -- used 5.82399s (cpu); 2.79473s (thread); 0s (gc)
│ │ │  
│ │ │  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 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
    
│ │ │            
│ │ │            
│ │ │                
    i8 : dim oo, degree oo, degrees oo
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,24 +26,24 @@
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^7 x PP^7)
│ │ │ │  i4 : Z = source Phi;
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
│ │ │ │  i5 : time Phi Z;
│ │ │ │ - -- used 0.0929525s (cpu); 0.0923825s (thread); 0s (gc)
│ │ │ │ + -- used 0.796659s (cpu); 0.226534s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │ │  
│ │ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ │ - -- used 2.12072s (cpu); 1.27842s (thread); 0s (gc)
│ │ │ │ + -- used 5.82399s (cpu); 2.79473s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
│ │ │ │  i8 : dim oo, degree oo, degrees oo
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html
│ │ │ @@ -88,27 +88,27 @@
│ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 4.21068s (cpu); 2.53763s (thread); 0s (gc)
│ │ │ + -- used 8.60794s (cpu); 1.99728s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.312606s (cpu); 0.236089s (thread); 0s (gc)
│ │ │ + -- used 0.309751s (cpu); 0.242231s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │            
│ │ │            
│ │ │                
    i6 : time degree Phi
│ │ │ - -- used 0.326294s (cpu); 0.24353s (thread); 0s (gc)
│ │ │ + -- used 0.172914s (cpu); 0.173914s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    
│ │ │            
│ │ │          
│ │ │          
    Note, as in the example above, that calculation times may vary depending on the strategy used.
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,23 +28,23 @@ │ │ │ │ │ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ │ source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of │ │ │ │ target variety: hypersurface in PP^4 defined by a form of degree 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ i4 : time degree(Phi,Strategy=>"random point") │ │ │ │ - -- used 4.21068s (cpu); 2.53763s (thread); 0s (gc) │ │ │ │ + -- used 8.60794s (cpu); 1.99728s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.312606s (cpu); 0.236089s (thread); 0s (gc) │ │ │ │ + -- used 0.309751s (cpu); 0.242231s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.326294s (cpu); 0.24353s (thread); 0s (gc) │ │ │ │ + -- used 0.172914s (cpu); 0.173914s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = 2 │ │ │ │ Note, as in the example above, that calculation times may vary depending on the │ │ │ │ strategy used. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map │ │ │ │ * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ │ │ │ 
    i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
    │ │ │ │ │ │ │ │ │ 
    i3 : time degree Phi
│ │ │ - -- used 0.571211s (cpu); 0.424891s (thread); 0s (gc)
│ │ │ + -- used 0.510038s (cpu); 0.32469s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ 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}; │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to │ │ │ │ PP^2 x PP^4) │ │ │ │ i3 : time degree Phi │ │ │ │ - -- used 0.571211s (cpu); 0.424891s (thread); 0s (gc) │ │ │ │ + -- used 0.510038s (cpu); 0.32469s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = 1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a │ │ │ │ probabilistic approach │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ @@ -75,40 +75,40 @@ │ │ │ │ │ │ 
    i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
    │ │ │ │ │ │ │ │ │ 
    i2 : time ? Phi
│ │ │ - -- used 0.0017476s (cpu); 0.000210294s (thread); 0s (gc)
│ │ │ + -- used 0.00108592s (cpu); 0.000124764s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ │ │ │ 
    i3 : image Phi;
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
    │ │ │ │ │ │ │ │ │ 
    i4 : time ? Phi
│ │ │ - -- used 0.00400281s (cpu); 0.000328677s (thread); 0s (gc)
│ │ │ + -- used 0.00226652s (cpu); 0.000256101s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       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 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ │ │ │ 
    i5 : time describe Phi
│ │ │ - -- used 1.5069s (cpu); 1.10379s (thread); 0s (gc)
│ │ │ + -- used 2.96765s (cpu); 2.59941s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       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 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -116,15 +116,15 @@
│ │ │       degree: 1
│ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    │ │ │ │ │ │ │ │ │ 
    i6 : time ? Phi
│ │ │ - -- used 0.000149972s (cpu); 0.000376616s (thread); 0s (gc)
│ │ │ + -- used 0.000105859s (cpu); 0.000315862s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       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 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,36 +17,36 @@
│ │ │ │  ? Phi is a lite version of describe Phi. The latter has a different behavior
│ │ │ │  than _d_e_s_c_r_i_b_e_(_R_a_t_i_o_n_a_l_M_a_p_), since it performs computations.
│ │ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4 x PP^5)
│ │ │ │  i2 : time ? Phi
│ │ │ │ - -- used 0.0017476s (cpu); 0.000210294s (thread); 0s (gc)
│ │ │ │ + -- used 0.00108592s (cpu); 0.000124764s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i3 : image Phi;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │ │  i4 : time ? Phi
│ │ │ │ - -- used 0.00400281s (cpu); 0.000328677s (thread); 0s (gc)
│ │ │ │ + -- used 0.00226652s (cpu); 0.000256101s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │ │       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
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i5 : time describe Phi
│ │ │ │ - -- used 1.5069s (cpu); 1.10379s (thread); 0s (gc)
│ │ │ │ + -- used 2.96765s (cpu); 2.59941s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │ │       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
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │ │       degree: 1
│ │ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │ │       coefficient ring: ZZ/65521
│ │ │ │  i6 : time ? Phi
│ │ │ │ - -- used 0.000149972s (cpu); 0.000376616s (thread); 0s (gc)
│ │ │ │ + -- used 0.000105859s (cpu); 0.000315862s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       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
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    │ │ │ │ │ │ │ │ │ 
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0199683s (cpu); 0.0203147s (thread); 0s (gc)
│ │ │ + -- used 0.195567s (cpu); 0.058218s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ 
    i3 : Phi1;
│ │ │ @@ -102,15 +102,15 @@
│ │ │            
│ │ │                
    i4 : Phi2;
│ │ │  
│ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.0308626s (cpu); 0.0318301s (thread); 0s (gc)
│ │ │ + -- used 0.0835289s (cpu); 0.0292452s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │            
│ │ │            
│ │ │                
    i6 : Phi21;
│ │ │ @@ -120,15 +120,15 @@
│ │ │            
│ │ │                
    i7 : Phi22;
│ │ │  
│ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
    
│ │ │            
│ │ │            
│ │ │                
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.188571s (cpu); 0.120925s (thread); 0s (gc)
│ │ │ + -- used 0.598917s (cpu); 0.172583s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │            
│ │ │            
│ │ │                
    i9 : Phi211;
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,43 +20,43 @@
│ │ │ │  Phi)^-1 * (last graph Phi) == Phi are always satisfied.
│ │ │ │  i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i2 : time (Phi1,Phi2) = graph Phi
│ │ │ │ - -- used 0.0199683s (cpu); 0.0203147s (thread); 0s (gc)
│ │ │ │ + -- used 0.195567s (cpu); 0.058218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (Phi1, Phi2)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : Phi1;
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4)
│ │ │ │  i4 : Phi2;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of
│ │ │ │  PP^4 x PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ │ - -- used 0.0308626s (cpu); 0.0318301s (thread); 0s (gc)
│ │ │ │ + -- used 0.0835289s (cpu); 0.0292452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = (Phi21, Phi22)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : Phi21;
│ │ │ │  
│ │ │ │  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)
│ │ │ │  i7 : Phi22;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of
│ │ │ │  PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │ │  i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ │ - -- used 0.188571s (cpu); 0.120925s (thread); 0s (gc)
│ │ │ │ + -- used 0.598917s (cpu); 0.172583s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (Phi211, Phi212)
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : Phi211;
│ │ │ │  
│ │ │ │  o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │            
│ │ │                
    i4 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time Z = image Phi;
│ │ │ - -- used 0.193919s (cpu); 0.194375s (thread); 0s (gc)
│ │ │ + -- used 0.326902s (cpu); 0.260371s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │            
│ │ │            
│ │ │                
    i6 : dim Z, degree Z, degrees Z
│ │ │  
│ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ @@ -103,15 +103,15 @@
│ │ │  o6 : Sequence
    
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 8.21524s (cpu); 3.17664s (thread); 0s (gc)
│ │ │ + -- used 21.9591s (cpu); 4.80259s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │      		          
                  
    i8 : assert(Z == Z')
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,26 +24,26 @@
│ │ │ │  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};
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i4 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │ │  i5 : time Z = image Phi;
│ │ │ │ - -- used 0.193919s (cpu); 0.194375s (thread); 0s (gc)
│ │ │ │ + -- used 0.326902s (cpu); 0.260371s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │ │  
│ │ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  Alternatively, the calculation can be performed using the Segre embedding as
│ │ │ │  follows:
│ │ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ │ - -- used 8.21524s (cpu); 3.17664s (thread); 0s (gc)
│ │ │ │ + -- used 21.9591s (cpu); 4.80259s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -78,29 +78,29 @@
│ │ │                
    i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6
│ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2));
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.380852s (cpu); 0.296207s (thread); 0s (gc)
│ │ │ + -- used 1.37433s (cpu); 0.614694s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │            
│ │ │            
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │            
│ │ │            
│ │ │                
    i5 : Phi' = Phi || Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
    
│ │ │            
│ │ │            
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.86808s (cpu); 1.29771s (thread); 0s (gc)
│ │ │ + -- used 2.85058s (cpu); 2.01873s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
    
│ │ │            
│ │ │            
│ │ │                
    i7 : assert(Phi' * Psi' == 1)
    
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,23 +25,23 @@
│ │ │ │  i2 : -- map defined by the cubics through the secant variety to the rational
│ │ │ │  normal curve of degree 6
│ │ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal
│ │ │ │  PP_K([6],2));
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ │ - -- used 0.380852s (cpu); 0.296207s (thread); 0s (gc)
│ │ │ │ + -- used 1.37433s (cpu); 0.614694s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │ │  i4 : assert(Phi * Psi == 1)
│ │ │ │  i5 : Phi' = Phi || Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ │ - -- used 1.86808s (cpu); 1.29771s (thread); 0s (gc)
│ │ │ │ + -- used 2.85058s (cpu); 2.01873s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -84,37 +84,37 @@
│ │ │                
    i2 : -- we see Phi as a dominant map
│ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.0442329s (cpu); 0.0417169s (thread); 0s (gc)
│ │ │ + -- used 0.807672s (cpu); 0.131399s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
    
│ │ │            
│ │ │            
│ │ │                
    i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time inverse Psi;
│ │ │ - -- used 0.177065s (cpu); 0.0983666s (thread); 0s (gc)
│ │ │ + -- used 0.64434s (cpu); 0.281716s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │            
│ │ │            
│ │ │                
    i6 : Eta = first graph Psi;
│ │ │  
│ │ │  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)
    
│ │ │            
│ │ │            
│ │ │                
    i7 : time inverse Eta;
│ │ │ - -- used 0.453153s (cpu); 0.286087s (thread); 0s (gc)
│ │ │ + -- used 2.39753s (cpu); 0.700055s (thread); 0s (gc)
│ │ │  
│ │ │  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)
    
│ │ │            
│ │ │            
│ │ │                
    i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
    
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,32 +25,32 @@
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from PP^4 to PP^5)
│ │ │ │  i2 : -- we see Phi as a dominant map
│ │ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i3 : time inverse Phi;
│ │ │ │ - -- used 0.0442329s (cpu); 0.0417169s (thread); 0s (gc)
│ │ │ │ + -- used 0.807672s (cpu); 0.131399s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i4 : Psi = last graph Phi;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time inverse Psi;
│ │ │ │ - -- used 0.177065s (cpu); 0.0983666s (thread); 0s (gc)
│ │ │ │ + -- used 0.64434s (cpu); 0.281716s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-
│ │ │ │  dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i6 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  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)
│ │ │ │  i7 : time inverse Eta;
│ │ │ │ - -- used 0.453153s (cpu); 0.286087s (thread); 0s (gc)
│ │ │ │ + -- used 2.39753s (cpu); 0.700055s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -79,37 +79,37 @@
│ │ │            
│ │ │                
    i3 : Phi = rationalMap {f,f};
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00369049s (cpu); 1.033e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00389415s (cpu); 5.791e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │            
│ │ │            
│ │ │                
    i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
    
│ │ │            
│ │ │            
│ │ │                
    i6 : time isIsomorphism Psi
│ │ │ - -- used 0.323078s (cpu); 0.152253s (thread); 0s (gc)
│ │ │ + -- used 1.31022s (cpu); 0.409418s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
    
│ │ │            
│ │ │            
│ │ │                
    i7 : Eta = first graph Psi;
│ │ │  
│ │ │  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)
    
│ │ │            
│ │ │            
│ │ │                
    i8 : time isIsomorphism Eta
│ │ │ - -- used 1.39409s (cpu); 0.718132s (thread); 0s (gc)
│ │ │ + -- used 3.7902s (cpu); 1.51658s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │            
│ │ │            
│ │ │                
    i9 : assert(o8 and (not o6) and (not o4))
    
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,31 +18,31 @@
│ │ │ │       ZZ/33331[a..d]; f = rationalMap {c^2-b*d,b*c-a*d,b^2-a*c};
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │ │  i3 : Phi = rationalMap {f,f};
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │ │  i4 : time isIsomorphism Phi
│ │ │ │ - -- used 0.00369049s (cpu); 1.033e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00389415s (cpu); 5.791e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : Psi = first graph Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to
│ │ │ │  PP^3)
│ │ │ │  i6 : time isIsomorphism Psi
│ │ │ │ - -- used 0.323078s (cpu); 0.152253s (thread); 0s (gc)
│ │ │ │ + -- used 1.31022s (cpu); 0.409418s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  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)
│ │ │ │  i8 : time isIsomorphism Eta
│ │ │ │ - -- used 1.39409s (cpu); 0.718132s (thread); 0s (gc)
│ │ │ │ + -- used 3.7902s (cpu); 1.51658s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -76,27 +76,27 @@
│ │ │            
│ │ │                
    i2 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.230726s (cpu); 0.159482s (thread); 0s (gc)
│ │ │ + -- used 0.29478s (cpu); 0.230752s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.158078s (cpu); 0.0838548s (thread); 0s (gc)
│ │ │ + -- used 0.955989s (cpu); 0.247508s (thread); 0s (gc)
│ │ │  
│ │ │  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)
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time isMorphism Psi
│ │ │ - -- used 4.10881s (cpu); 2.89424s (thread); 0s (gc)
│ │ │ + -- used 6.74807s (cpu); 5.82189s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │            
│ │ │            
│ │ │                
    i6 : assert((not o3) and o5)
    
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,24 +18,24 @@
│ │ │ │  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});
│ │ │ │  i2 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^4 x PP^2)
│ │ │ │  i3 : time isMorphism Phi
│ │ │ │ - -- used 0.230726s (cpu); 0.159482s (thread); 0s (gc)
│ │ │ │ + -- used 0.29478s (cpu); 0.230752s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.158078s (cpu); 0.0838548s (thread); 0s (gc)
│ │ │ │ + -- used 0.955989s (cpu); 0.247508s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  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)
│ │ │ │  i5 : time isMorphism Psi
│ │ │ │ - -- used 4.10881s (cpu); 2.89424s (thread); 0s (gc)
│ │ │ │ + -- used 6.74807s (cpu); 5.82189s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -74,26 +74,26 @@
│ │ │            
│ │ │                
    i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00799341s (cpu); 0.00882943s (thread); 0s (gc)
│ │ │ + -- used 0.00803099s (cpu); 0.00781773s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
    
│ │ │            
│ │ │            
│ │ │                
    i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.560105s (cpu); 0.400122s (thread); 0s (gc)
│ │ │ + -- used 1.43896s (cpu); 1.23947s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
    
│ │ │            
│ │ │            
│ │ │                
    i6 : assert(isIsomorphism g)
    
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,23 +14,23 @@
│ │ │ │              is a linear projection
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/333331;
│ │ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ │ - -- used 0.00799341s (cpu); 0.00882943s (thread); 0s (gc)
│ │ │ │ + -- used 0.00803099s (cpu); 0.00781773s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an
│ │ │ │  isomorphic projection of X in PP^3
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ │ - -- used 0.560105s (cpu); 0.400122s (thread); 0s (gc)
│ │ │ │ + -- used 1.43896s (cpu); 1.23947s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │            
│ │ │                
    i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.549954s (cpu); 0.410961s (thread); 0s (gc)
│ │ │ + -- used 1.24188s (cpu); 0.632383s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
    
│ │ │            
│ │ │            
│ │ │                
    i4 : (degree source Phi,degree image Phi)
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  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};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time multidegree Phi
│ │ │ │ - -- used 0.549954s (cpu); 0.410961s (thread); 0s (gc)
│ │ │ │ + -- used 1.24188s (cpu); 0.632383s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -76,27 +76,27 @@
│ │ │            
│ │ │                
    i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
    
│ │ │            
│ │ │            
│ │ │                
    i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00402065s (cpu); 0.00152952s (thread); 0s (gc)
│ │ │ - -- used 0.156121s (cpu); 0.106571s (thread); 0s (gc)
│ │ │ - -- used 0.216164s (cpu); 0.135241s (thread); 0s (gc)
│ │ │ - -- used 0.180607s (cpu); 0.107667s (thread); 0s (gc)
│ │ │ - -- used 0.127909s (cpu); 0.0773013s (thread); 0s (gc)
│ │ │ + -- used 0.00215027s (cpu); 0.0011356s (thread); 0s (gc)
│ │ │ + -- used 0.178981s (cpu); 0.116603s (thread); 0s (gc)
│ │ │ + -- used 0.174219s (cpu); 0.109074s (thread); 0s (gc)
│ │ │ + -- used 0.186167s (cpu); 0.113206s (thread); 0s (gc)
│ │ │ + -- used 0.154433s (cpu); 0.0805549s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.0486544s (cpu); 0.0500349s (thread); 0s (gc)
    
│ │ │ + -- used 0.197938s (cpu); 0.0508185s (thread); 0s (gc)
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    References

    │ │ │ ArXiv preprint: Computations with rational maps between multi-projective varieties.
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,25 +18,25 @@ │ │ │ │ This is calculated by means of the inverse image of an appropriate random │ │ │ │ subvariety of the target. │ │ │ │ i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4); │ │ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 to PP^5) │ │ │ │ i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi) │ │ │ │ - -- used 0.00402065s (cpu); 0.00152952s (thread); 0s (gc) │ │ │ │ - -- used 0.156121s (cpu); 0.106571s (thread); 0s (gc) │ │ │ │ - -- used 0.216164s (cpu); 0.135241s (thread); 0s (gc) │ │ │ │ - -- used 0.180607s (cpu); 0.107667s (thread); 0s (gc) │ │ │ │ - -- used 0.127909s (cpu); 0.0773013s (thread); 0s (gc) │ │ │ │ + -- used 0.00215027s (cpu); 0.0011356s (thread); 0s (gc) │ │ │ │ + -- used 0.178981s (cpu); 0.116603s (thread); 0s (gc) │ │ │ │ + -- used 0.174219s (cpu); 0.109074s (thread); 0s (gc) │ │ │ │ + -- used 0.186167s (cpu); 0.113206s (thread); 0s (gc) │ │ │ │ + -- used 0.154433s (cpu); 0.0805549s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ │ - -- used 0.0486544s (cpu); 0.0500349s (thread); 0s (gc) │ │ │ │ + -- used 0.197938s (cpu); 0.0508185s (thread); 0s (gc) │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e │ │ │ │ _v_a_r_i_e_t_i_e_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ │ │ map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html │ │ │ @@ -76,28 +76,28 @@ │ │ │ │ │ │ 
    i2 : X = PP_K^({1,1,2},{3,2,3});
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ 
    i3 : time p := point X
│ │ │ - -- used 0.0160848s (cpu); 0.0148379s (thread); 0s (gc)
│ │ │ + -- used 0.516235s (cpu); 0.0818124s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ 
    i4 : Y = random({2,1,2},X);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ 
    i5 : time q = point Y
│ │ │ - -- used 2.0638s (cpu); 1.06252s (thread); 0s (gc)
│ │ │ + -- used 2.70174s (cpu); 1.51899s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ 
    i6 : assert(isSubset(p,X) and isSubset(q,Y))
    │ │ │ ├── html2text {} │ │ │ │ @@ -15,25 +15,25 @@ │ │ │ │ o a _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e_ _v_a_r_i_e_t_y, a random rational point on $X$ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : K = ZZ/1000003; │ │ │ │ i2 : X = PP_K^({1,1,2},{3,2,3}); │ │ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9 │ │ │ │ i3 : time p := point X │ │ │ │ - -- used 0.0160848s (cpu); 0.0148379s (thread); 0s (gc) │ │ │ │ + -- used 0.516235s (cpu); 0.0818124s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1], │ │ │ │ [3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1]) │ │ │ │ │ │ │ │ o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i4 : Y = random({2,1,2},X); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ i5 : time q = point Y │ │ │ │ - -- used 2.0638s (cpu); 1.06252s (thread); 0s (gc) │ │ │ │ + -- used 2.70174s (cpu); 1.51899s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = q │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ │ │ The list of homogeneous coordinates can be obtained with the operator |-. │ │ │ │ i7 : |- p │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ │ │ │ 
    i5 : Phi = rationalMap {f,g,h};
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
    │ │ │ │ │ │ │ │ │ 
    i6 : time segre Phi;
│ │ │ - -- used 0.84109s (cpu); 0.510468s (thread); 0s (gc)
│ │ │ + -- used 7.36297s (cpu); 1.6609s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
    │ │ │ │ │ │ │ │ │ 
    i7 : describe segre Phi
│ │ │  
│ │ │  o7 = rational map defined by forms of degree 6
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  
│ │ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x
│ │ │ │  PP^4)
│ │ │ │  i6 : time segre Phi;
│ │ │ │ - -- used 0.84109s (cpu); 0.510468s (thread); 0s (gc)
│ │ │ │ + -- used 7.36297s (cpu); 1.6609s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    │ │ │ │ │ │ │ │ │ 
    i2 : time describe Phi
│ │ │ - -- used 0.255978s (cpu); 0.169365s (thread); 0s (gc)
│ │ │ + -- used 0.177758s (cpu); 0.106991s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       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 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to
│ │ │ │  threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i2 : time describe Phi
│ │ │ │ - -- used 0.255978s (cpu); 0.169365s (thread); 0s (gc)
│ │ │ │ + -- used 0.177758s (cpu); 0.106991s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of
│ │ │ │  multi-degree (1,1)
│ │ │ │       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
│ │ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ ├── ./usr/share/doc/Macaulay2/NAGtypes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  UG9seVNwYWNl
│ │ │  #:len=827
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwb2x5bm9taWFsIHZlY3RvciBzdWJz
│ │ │  cGFjZSIsICJsaW5lbnVtIiA9PiA4NTcsIFNlZUFsc28gPT4gRElWe0hFQURFUjJ7IlNlZSBhbHNv
│ │ ├── ./usr/share/doc/Macaulay2/NCAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  UVEgJSBOQ0dyb2VibmVyQmFzaXM=
│ │ │  #:len=317
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY3MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sICUsUVEsTkNHcm9lYm5lckJhc2lzKSwi
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  Z2VuZXJhdGVSYW5kb21HcmFwaHMoWlosWlosWlop
│ │ │  #:len=275
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExMiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2VuZXJhdGVSYW5kb21HcmFwaHMsWlosWlosWlop
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (69, 80, 81, 89, 94, 95, 96, 94, 100, 97, 96, 97, 96, 98, 99, 97, 99,
│ │ │ +o9 = (70, 81, 88, 94, 92, 98, 96, 93, 94, 97, 99, 97, 97, 99, 92, 96, 98, 96,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 97, 99, 99, 98, 98, 99, 98, 99, 99, 98)
│ │ │ +     99, 98, 96, 97, 100, 100, 99, 98, 99, 96, 98)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (21, 11, 5, 4, 0, 2, 6, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1,
│ │ │ +o10 = (17, 8, 5, 2, 1, 3, 2, 2, 1, 0, 3, 1, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 1, 1, 1, 0)
│ │ │ +      0, 0, 0, 0, 2, 0)
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DGg, D}W, D[{, Du[, Dww}
│ │ │ +o2 = {Dmg, DYS, DUC, DnW, Dhk}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 1729831171060067675
│ │ │  
│ │ │  i1 : R = QQ[a..e];
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, c}, {a, d}, {b, d}, {b, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {a, d}, {c, d}, {b, e}, {c, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │  i3 : graphComplement "Dhc"
│ │ │  
│ │ │  o3 = DUW
│ │ │  
│ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i5 : time graphComplement G;
│ │ │ - -- used 0.000482564s (cpu); 0.000422561s (thread); 0s (gc)
│ │ │ + -- used 0.000509146s (cpu); 0.000475473s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0467584s (cpu); 0.0451558s (thread); 0s (gc)
│ │ │ + -- used 0.0548717s (cpu); 0.0527698s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -93,26 +93,26 @@
│ │ │            
│ │ │            
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │            
│ │ │            
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (69, 80, 81, 89, 94, 95, 96, 94, 100, 97, 96, 97, 96, 98, 99, 97, 99,
│ │ │ +o9 = (70, 81, 88, 94, 92, 98, 96, 93, 94, 97, 99, 97, 97, 99, 92, 96, 98, 96,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 97, 99, 99, 98, 98, 99, 98, 99, 99, 98)
│ │ │ +     99, 98, 96, 97, 100, 100, 99, 98, 99, 96, 98)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │            
│ │ │            
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (21, 11, 5, 4, 0, 2, 6, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1,
│ │ │ +o10 = (17, 8, 5, 2, 1, 3, 2, 2, 1, 0, 3, 1, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 1, 1, 1, 0)
│ │ │ +      0, 0, 0, 0, 2, 0)
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -38,25 +38,25 @@ │ │ │ │ connected, at least as $n$ tends to infinity. │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => │ │ │ │ true}; │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o9 = (69, 80, 81, 89, 94, 95, 96, 94, 100, 97, 96, 97, 96, 98, 99, 97, 99, │ │ │ │ +o9 = (70, 81, 88, 94, 92, 98, 96, 93, 94, 97, 99, 97, 97, 99, 92, 96, 98, 96, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 94, 99, 97, 99, 99, 98, 98, 99, 98, 99, 99, 98) │ │ │ │ + 99, 98, 96, 97, 100, 100, 99, 98, 99, 96, 98) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (21, 11, 5, 4, 0, 2, 6, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, │ │ │ │ +o10 = (17, 8, 5, 2, 1, 3, 2, 2, 1, 0, 3, 1, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 0, 1, 1, 1, 0) │ │ │ │ + 0, 0, 0, 0, 2, 0) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ @@ -104,15 +104,15 @@ │ │ │ o1 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ │ │ │ │ o1 : List     │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DGg, D}W, D[{, Du[, Dww}
│ │ │ +o2 = {Dmg, DYS, DUC, DnW, Dhk}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ 
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,15 +38,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DGg, D}W, D[{, Du[, Dww}
│ │ │ │ +o2 = {Dmg, DYS, DUC, DnW, Dhk}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -89,23 +89,23 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : R = QQ[a..e];
    
│ │ │      		          
                  
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, c}, {a, d}, {b, d}, {b, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {a, d}, {c, d}, {b, e}, {c, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
    
│ │ │      		          
    
│ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -25,23 +25,23 @@ │ │ │ │ vertices with a given regularity. Note that some graphs may be isomorphic. │ │ │ │ If a _P_o_l_y_n_o_m_i_a_l_R_i_n_g $R$ is supplied instead, then the number of vertices is the │ │ │ │ number of generators. Moreover, the nauty-based strings are automatically │ │ │ │ converted to instances of the class _G_r_a_p_h in $R$. │ │ │ │ i1 : R = QQ[a..e]; │ │ │ │ i2 : generateRandomRegularGraphs(R, 3, 2) │ │ │ │ │ │ │ │ -o2 = {Graph{"edges" => {{a, c}, {a, d}, {b, d}, {b, e}, {c, e}}}, │ │ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}}, │ │ │ │ "ring" => R │ │ │ │ "vertices" => {a, b, c, d, e} │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}, │ │ │ │ + Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}}, │ │ │ │ "ring" => R │ │ │ │ "vertices" => {a, b, c, d, e} │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}} │ │ │ │ + Graph{"edges" => {{a, b}, {a, d}, {c, d}, {b, e}, {c, e}}}} │ │ │ │ "ring" => R │ │ │ │ "vertices" => {a, b, c, d, e} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The number of vertices $n$ must be positive as nauty cannot handle graphs with │ │ │ │ zero vertices. │ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ @@ -113,19 +113,19 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : G = generateBipartiteGraphs 7;
    │ │ │ 
    i5 : time graphComplement G;
│ │ │ - -- used 0.000482564s (cpu); 0.000422561s (thread); 0s (gc)
    │ │ │ + -- used 0.000509146s (cpu); 0.000475473s (thread); 0s (gc) │ │ │ 
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0467584s (cpu); 0.0451558s (thread); 0s (gc)
    │ │ │ + -- used 0.0548717s (cpu); 0.0527698s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -42,17 +42,17 @@ │ │ │ │ │ │ │ │ o3 = DUW │ │ │ │ Batch calls can be performed considerably faster when using the List input │ │ │ │ format. However, care should be taken as the returned list is entirely in │ │ │ │ Graph6 or Sparse6 format. │ │ │ │ i4 : G = generateBipartiteGraphs 7; │ │ │ │ i5 : time graphComplement G; │ │ │ │ - -- used 0.000482564s (cpu); 0.000422561s (thread); 0s (gc) │ │ │ │ + -- used 0.000509146s (cpu); 0.000475473s (thread); 0s (gc) │ │ │ │ i6 : time (graphComplement \ G); │ │ │ │ - -- used 0.0467584s (cpu); 0.0451558s (thread); 0s (gc) │ │ │ │ + -- used 0.0548717s (cpu); 0.0527698s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_o_m_p_l_e_m_e_n_t_G_r_a_p_h -- returns the complement of a graph or hypergraph │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ YWRkRWRnZXMoLi4uLE5vTmV3NUN5Y2xlcz0+Li4uKQ== │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTkxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGRFZGdlcyxOb05ldzVDeWNsZXNdLCJhZGRFZGdl │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ @@ -26,22 +26,22 @@ │ │ │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true}; │ │ │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected)) │ │ │ │ │ │ -o9 = (63, 83, 88, 90, 98, 97, 96, 95, 95, 98, 95, 95, 94, 99, 96, 96, 100, │ │ │ +o9 = (65, 83, 81, 97, 96, 93, 96, 98, 95, 94, 94, 99, 96, 96, 98, 98, 97, 97, │ │ │ ------------------------------------------------------------------------ │ │ │ - 97, 98, 97, 99, 100, 99, 98, 100, 99, 97, 97, 99) │ │ │ + 99, 99, 95, 98, 96, 96, 97, 99, 99, 99, 99) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected)) │ │ │ │ │ │ -o10 = (26, 9, 6, 2, 2, 1, 4, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, │ │ │ +o10 = (19, 9, 3, 2, 6, 5, 3, 3, 1, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, │ │ │ ----------------------------------------------------------------------- │ │ │ - 0, 0, 0, 0, 0, 1) │ │ │ + 0, 1, 1, 0, 0, 0) │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : generateRandomGraphs(5, 5) │ │ │ │ │ │ -o2 = {DXg, D?_, DlW, D}[, DJs} │ │ │ +o2 = {DnO, DU{, D_S, DJc, DH{} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : generateRandomGraphs(5, 5, RandomSeed => 314159) │ │ │ │ │ │ o3 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1331287392268 │ │ │ │ │ │ i1 : generateRandomRegularGraphs(5, 3, 2) │ │ │ │ │ │ -o1 = {DYc, DpS, DMg} │ │ │ +o1 = {DkK, DpS, DUW} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ @@ -13,13 +13,13 @@ │ │ │ 4 => {2, 1} │ │ │ │ │ │ o2 : Graph │ │ │ │ │ │ i3 : G = generateBipartiteGraphs 7; │ │ │ │ │ │ i4 : time graphComplement G; │ │ │ - -- used 0.000473788s (cpu); 0.000427112s (thread); 0s (gc) │ │ │ + -- used 0.000747031s (cpu); 0.000629351s (thread); 0s (gc) │ │ │ │ │ │ i5 : time (graphComplement \ G); │ │ │ - -- used 0.132423s (cpu); 0.0571132s (thread); 0s (gc) │ │ │ + -- used 0.143133s (cpu); 0.069226s (thread); 0s (gc) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ @@ -93,26 +93,26 @@ │ │ │ │ │ │ │ │ │ 
      i8 : prob = n -> log(n)/n;
      │ │ │ │ │ │ │ │ │ 
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (63, 83, 88, 90, 98, 97, 96, 95, 95, 98, 95, 95, 94, 99, 96, 96, 100,
│ │ │ +o9 = (65, 83, 81, 97, 96, 93, 96, 98, 95, 94, 94, 99, 96, 96, 98, 98, 97, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 97, 99, 100, 99, 98, 100, 99, 97, 97, 99)
│ │ │ +     99, 99, 95, 98, 96, 96, 97, 99, 99, 99, 99)
│ │ │  
│ │ │  o9 : Sequence
      │ │ │ │ │ │ │ │ │ 
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (26, 9, 6, 2, 2, 1, 4, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1,
│ │ │ +o10 = (19, 9, 3, 2, 6, 5, 3, 3, 1, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 1)
│ │ │ +      0, 1, 1, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
      │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -38,25 +38,25 @@ │ │ │ │ connected, at least as $n$ tends to infinity. │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => │ │ │ │ true}; │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o9 = (63, 83, 88, 90, 98, 97, 96, 95, 95, 98, 95, 95, 94, 99, 96, 96, 100, │ │ │ │ +o9 = (65, 83, 81, 97, 96, 93, 96, 98, 95, 94, 94, 99, 96, 96, 98, 98, 97, 97, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 97, 98, 97, 99, 100, 99, 98, 100, 99, 97, 97, 99) │ │ │ │ + 99, 99, 95, 98, 96, 96, 97, 99, 99, 99, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (26, 9, 6, 2, 2, 1, 4, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, │ │ │ │ +o10 = (19, 9, 3, 2, 6, 5, 3, 3, 1, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 0, 0, 0, 0, 1) │ │ │ │ + 0, 1, 1, 0, 0, 0) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ @@ -96,15 +96,15 @@ │ │ │ o1 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ │ │ │ │ o1 : List     │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DXg, D?_, DlW, D}[, DJs}
│ │ │ +o2 = {DnO, DU{, D_S, DJc, DH{}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ 
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,15 +31,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DXg, D?_, DlW, D}[, DJs}
│ │ │ │ +o2 = {DnO, DU{, D_S, DJc, DH{}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DYc, DpS, DMg}
│ │ │ +o1 = {DkK, DpS, DUW}
│ │ │  
│ │ │  o1 : List
    
│ │ │      		          
    
│ │ │
    │ │ │
    │ │ │

    Caveat

    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ * Outputs: │ │ │ │ o G, a _l_i_s_t, the randomly generated regular graphs │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ 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. │ │ │ │ i1 : generateRandomRegularGraphs(5, 3, 2) │ │ │ │ │ │ │ │ -o1 = {DYc, DpS, DMg} │ │ │ │ +o1 = {DkK, DpS, DUW} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The number of vertices $n$ must be positive as nauty cannot handle graphs with │ │ │ │ zero vertices. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ @@ -109,19 +109,19 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : G = generateBipartiteGraphs 7;
    │ │ │ 
    i4 : time graphComplement G;
│ │ │ - -- used 0.000473788s (cpu); 0.000427112s (thread); 0s (gc)
    │ │ │ + -- used 0.000747031s (cpu); 0.000629351s (thread); 0s (gc) │ │ │ 
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.132423s (cpu); 0.0571132s (thread); 0s (gc)
    │ │ │ + -- used 0.143133s (cpu); 0.069226s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use graphComplement:

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -39,16 +39,16 @@ │ │ │ │ │ │ │ │ o2 : Graph │ │ │ │ Batch calls can be performed considerably faster when using the List input │ │ │ │ format. However, care should be taken as the returned list is entirely in │ │ │ │ Graph6 or Sparse6 format. │ │ │ │ i3 : G = generateBipartiteGraphs 7; │ │ │ │ i4 : time graphComplement G; │ │ │ │ - -- used 0.000473788s (cpu); 0.000427112s (thread); 0s (gc) │ │ │ │ + -- used 0.000747031s (cpu); 0.000629351s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.132423s (cpu); 0.0571132s (thread); 0s (gc) │ │ │ │ + -- used 0.143133s (cpu); 0.069226s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _g_r_a_p_h_C_o_m_p_l_e_m_e_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/NoetherNormalization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=38 │ │ │ bm9ldGhlck5vcm1hbGl6YXRpb24oLi4uLFZlcmJvc2U9Pi4uLik= │ │ │ #:len=326 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzEzLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tub2V0aGVyTm9ybWFsaXphdGlvbixWZXJib3NlXSwi │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ bm9ldGhlcmlhbk9wZXJhdG9ycyhJZGVhbCxJZGVhbCk= │ │ │ #:len=2583 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTm9ldGhlcmlhbiBvcGVyYXRvcnMgb2Yg │ │ │ YSBwcmltYXJ5IGNvbXBvbmVudCIsICJsaW5lbnVtIiA9PiAyNzQ5LCBJbnB1dHMgPT4ge1NQQU57 │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ @@ -47,15 +47,15 @@ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ o5 = true │ │ │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ - -- .0926449s elapsed │ │ │ + -- .187677s elapsed │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ ------------------------------------------------------------------------ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ o6 : List │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ 
      i5 : isPrimary Q
│ │ │  
│ │ │  o5 = true
      │ │ │ │ │ │ │ │ │ 
      i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .0926449s elapsed
│ │ │ + -- .187677s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
      │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -51,15 +51,15 @@ │ │ │ │ 1 2 3 2 3 │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ │ - -- .0926449s elapsed │ │ │ │ + -- .187677s elapsed │ │ │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/NonminimalComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ Tm9ubWluaW1hbENvbXBsZXhlcw== │ │ │ #:len=668 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3VwcG9ydCBmb3IgY29tcHV0aW5nIGhv │ │ │ bW9sb2d5LCByYW5rcyBhbmQgU1ZEIGNvbXBsZXhlcywgZnJvbSBhIGNoYWluIGNvbXBsZXggb3Zl │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ VG9yaWNEaXZpc29yID09IFpa │ │ │ #:len=349 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQ0MSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sID09LFRvcmljRGl2aXNvcixaWiksIlRv │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ @@ -78,15 +78,15 @@ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ - -- used 0.298531s (cpu); 0.299707s (thread); 0s (gc) │ │ │ + -- used 0.36222s (cpu); 0.363444s (thread); 0s (gc) │ │ │ │ │ │ i15 : A2 = intersectionRing Y; │ │ │ │ │ │ i16 : assert (# rays Y === numgens A2) │ │ │ │ │ │ i17 : ideal A2 │ │ │ │ │ │ @@ -110,19 +110,19 @@ │ │ │ 2 2 2 2 2 2 2 2 3 2 │ │ │ (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 ) │ │ │ 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 │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ i19 : for i to dim Y list time hilbertFunction (i, A2) │ │ │ - -- used 0.00144935s (cpu); 0.00144823s (thread); 0s (gc) │ │ │ - -- used 2.2452e-05s (cpu); 8.0521e-05s (thread); 0s (gc) │ │ │ - -- used 8.756e-06s (cpu); 6.4241e-05s (thread); 0s (gc) │ │ │ - -- used 1.609e-05s (cpu); 7.0222e-05s (thread); 0s (gc) │ │ │ - -- used 8.606e-06s (cpu); 5.9061e-05s (thread); 0s (gc) │ │ │ - -- used 5.0224e-05s (cpu); 7.0663e-05s (thread); 0s (gc) │ │ │ + -- used 0.00400225s (cpu); 0.00105592s (thread); 0s (gc) │ │ │ + -- used 2.3515e-05s (cpu); 7.3728e-05s (thread); 0s (gc) │ │ │ + -- used 9.307e-06s (cpu); 6.1696e-05s (thread); 0s (gc) │ │ │ + -- used 8.696e-06s (cpu); 5.8991e-05s (thread); 0s (gc) │ │ │ + -- used 7.955e-06s (cpu); 5.7147e-05s (thread); 0s (gc) │ │ │ + -- used 7.905e-06s (cpu); 5.7618e-05s (thread); 0s (gc) │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ o19 : List │ │ │ │ │ │ i20 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out │ │ │ @@ -2,27 +2,27 @@ │ │ │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ -o3 = {1, 3, 4} │ │ │ +o3 = {3, 5, 6} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i7 : q │ │ │ │ │ │ -o7 = {1, 1, 3, 6, 9} │ │ │ +o7 = {1, 3, 3, 4, 6} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ @@ -6,61 +6,61 @@ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ 0 │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ - -- .0354582s elapsed │ │ │ + -- .0236384s elapsed │ │ │ │ │ │ 5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4 │ │ │ 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 , │ │ │ 2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 2 2 3 4 5 4 2 3 3 2 4 5 │ │ │ x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x } │ │ │ 0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1)) │ │ │ - -- .00151969s elapsed │ │ │ + -- .0010556s elapsed │ │ │ │ │ │ i5 : FF2 = hirzebruchSurface 2; │ │ │ │ │ │ i6 : D2 = 2*FF2_0 + 3 * FF2_1 │ │ │ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ 0 1 │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ i7 : M2 = elapsedTime monomials D2 │ │ │ - -- .0964156s elapsed │ │ │ + -- .110963s elapsed │ │ │ │ │ │ 2 3 2 3 2 3 │ │ │ o7 = {x x , x x , x x x , x x } │ │ │ 1 3 1 2 0 1 2 0 1 │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2)) │ │ │ - -- .00113966s elapsed │ │ │ + -- .000849296s elapsed │ │ │ │ │ │ i9 : X = kleinschmidt (5, {1,2,3}); │ │ │ │ │ │ i10 : D3 = 3*X_0 + 5*X_1 │ │ │ │ │ │ o10 = 3*X + 5*X │ │ │ 0 1 │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ i11 : m3 = elapsedTime # monomials D3 │ │ │ - -- 39.4377s elapsed │ │ │ + -- 43.1615s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .0257955s elapsed │ │ │ + -- .0221523s elapsed │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ @@ -24,19 +24,19 @@ │ │ │ o3 : List │ │ │ │ │ │ i4 : X = normalToricVariety F; │ │ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ - -- used 0.00304489s (cpu); 2.2332e-05s (thread); 0s (gc) │ │ │ + -- used 0.00230841s (cpu); 1.6581e-05s (thread); 0s (gc) │ │ │ │ │ │ o6 = X1 │ │ │ │ │ │ o6 : NormalToricVariety │ │ │ │ │ │ i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}}; │ │ │ - -- used 0.206737s (cpu); 0.0677457s (thread); 0s (gc) │ │ │ + -- used 0.200997s (cpu); 0.0974426s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ @@ -88,15 +88,15 @@ │ │ │ o18 = | 0 1 0 | │ │ │ | 0 0 1 | │ │ │ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ - -- used 0.106449s (cpu); 0.0339456s (thread); 0s (gc) │ │ │ + -- used 0.137947s (cpu); 0.0771225s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 0.0044979s (cpu); 0.00448742s (thread); 0s (gc) │ │ │ + -- used 0.0017407s (cpu); 0.00231854s (thread); 0s (gc) │ │ │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ @@ -181,15 +181,15 @@ │ │ │ │ │ │
      │ │ │

      We end with a slightly larger example.

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -218,20 +218,20 @@ │ │ │ (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 ) │ │ │ 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 │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.298531s (cpu); 0.299707s (thread); 0s (gc)
      │ │ │ + -- used 0.36222s (cpu); 0.363444s (thread); 0s (gc) │ │ │ 
      i15 : A2 = intersectionRing Y;
      │ │ │ 
      i16 : assert (# rays Y === numgens A2)
      │ │ │ 
      i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00144935s (cpu); 0.00144823s (thread); 0s (gc)
│ │ │ - -- used 2.2452e-05s (cpu); 8.0521e-05s (thread); 0s (gc)
│ │ │ - -- used 8.756e-06s (cpu); 6.4241e-05s (thread); 0s (gc)
│ │ │ - -- used 1.609e-05s (cpu); 7.0222e-05s (thread); 0s (gc)
│ │ │ - -- used 8.606e-06s (cpu); 5.9061e-05s (thread); 0s (gc)
│ │ │ - -- used 5.0224e-05s (cpu); 7.0663e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00400225s (cpu); 0.00105592s (thread); 0s (gc)
│ │ │ + -- used 2.3515e-05s (cpu); 7.3728e-05s (thread); 0s (gc)
│ │ │ + -- used 9.307e-06s (cpu); 6.1696e-05s (thread); 0s (gc)
│ │ │ + -- used 8.696e-06s (cpu); 5.8991e-05s (thread); 0s (gc)
│ │ │ + -- used 7.955e-06s (cpu); 5.7147e-05s (thread); 0s (gc)
│ │ │ + -- used 7.905e-06s (cpu); 5.7618e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
      │ │ │
      │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ │ │ o13 : List │ │ │ │ We end with a slightly larger example. │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ │ - -- used 0.298531s (cpu); 0.299707s (thread); 0s (gc) │ │ │ │ + -- used 0.36222s (cpu); 0.363444s (thread); 0s (gc) │ │ │ │ i15 : A2 = intersectionRing Y; │ │ │ │ i16 : assert (# rays Y === numgens A2) │ │ │ │ i17 : ideal A2 │ │ │ │ │ │ │ │ o17 = ideal (t t , t t , t t , t t , t t , t t , t t , t t , t t t , │ │ │ │ 2 3 2 5 4 5 3 6 4 6 1 7 7 9 8 9 0 1 10 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ @@ -130,20 +130,20 @@ │ │ │ │ (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 ) │ │ │ │ 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 │ │ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ i19 : for i to dim Y list time hilbertFunction (i, A2) │ │ │ │ - -- used 0.00144935s (cpu); 0.00144823s (thread); 0s (gc) │ │ │ │ - -- used 2.2452e-05s (cpu); 8.0521e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.756e-06s (cpu); 6.4241e-05s (thread); 0s (gc) │ │ │ │ - -- used 1.609e-05s (cpu); 7.0222e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.606e-06s (cpu); 5.9061e-05s (thread); 0s (gc) │ │ │ │ - -- used 5.0224e-05s (cpu); 7.0663e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00400225s (cpu); 0.00105592s (thread); 0s (gc) │ │ │ │ + -- used 2.3515e-05s (cpu); 7.3728e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.307e-06s (cpu); 6.1696e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.696e-06s (cpu); 5.8991e-05s (thread); 0s (gc) │ │ │ │ + -- used 7.955e-06s (cpu); 5.7147e-05s (thread); 0s (gc) │ │ │ │ + -- used 7.905e-06s (cpu); 5.7618e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total │ │ │ │ coordinate rings (a.k.a. Cox rings) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : setRandomSeed (currentTime ());
    │ │ │ 
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {1, 3, 4}
│ │ │ +o3 = {3, 5, 6}
│ │ │  
│ │ │  o3 : List
    │ │ │ 
    i4 : assert isWellDefined kleinschmidt (4,a)
    │ │ │
    │ │ │ @@ -118,15 +118,15 @@ │ │ │ │ │ │ │ │ │ 
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    │ │ │ │ │ │ │ │ │ 
    i7 : q
│ │ │  
│ │ │ -o7 = {1, 1, 3, 6, 9}
│ │ │ +o7 = {1, 3, 3, 4, 6}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ 
    i8 : assert isWellDefined weightedProjectiveSpace q
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,24 +31,24 @@ │ │ │ │ The first examples illustrate that small projective spaces are well-defined. │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ The second examples show that a randomly selected Kleinschmidt toric variety │ │ │ │ and a weighted projective space are also well-defined. │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ │ │ -o3 = {1, 3, 4} │ │ │ │ +o3 = {3, 5, 6} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j │ │ │ │ -> random (1,9)); │ │ │ │ i7 : q │ │ │ │ │ │ │ │ -o7 = {1, 1, 3, 6, 9} │ │ │ │ +o7 = {1, 3, 3, 4, 6} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ The next ten examples illustrate various ways that two lists can fail to define │ │ │ │ a normal toric variety. By making the current debugging level greater than one, │ │ │ │ one gets some addition information about the nature of the failure. │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ @@ -94,29 +94,29 @@ │ │ │ o2 = 5*PP2 │ │ │ 0 │ │ │ │ │ │ o2 : ToricDivisor on PP2     │ │ │ │ │ │ │ │ │ 
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0354582s elapsed
│ │ │ + -- .0236384s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  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 ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │        0 1 2   0 1 2   0 1 2   0 2   1   0 1   0 1   0 1   0 1   0
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .00151969s elapsed
    │ │ │ + -- .0010556s elapsed     │ │ │ │ │ │ │ │ │
    │ │ │

    Toric varieties of Picard-rank 2 are slightly more interesting.

    │ │ │
    │ │ │ │ │ │ │ │ │ @@ -128,46 +128,46 @@ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ 0 1 │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0964156s elapsed
│ │ │ + -- .110963s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
    │ │ │ 
    i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .00113966s elapsed
    │ │ │ + -- .000849296s elapsed │ │ │ 
    i9 : X = kleinschmidt (5, {1,2,3});
    │ │ │ 
    i10 : D3 = 3*X_0 + 5*X_1
│ │ │  
│ │ │  o10 = 3*X  + 5*X
│ │ │           0      1
│ │ │  
│ │ │  o10 : ToricDivisor on X
    │ │ │ 
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 39.4377s elapsed
│ │ │ + -- 43.1615s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0257955s elapsed
    │ │ │ + -- .0221523s elapsed │ │ │
    │ │ │
    │ │ │

    By exploiting latticePoints, this method function avoids using the basis function.

    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,61 +28,61 @@ │ │ │ │ i2 : D1 = 5*PP2_0 │ │ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ │ 0 │ │ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ │ - -- .0354582s elapsed │ │ │ │ + -- .0236384s elapsed │ │ │ │ │ │ │ │ 5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4 │ │ │ │ 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 , │ │ │ │ 2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 2 2 3 4 5 4 2 3 3 2 4 5 │ │ │ │ x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x } │ │ │ │ 0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0 │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring │ │ │ │ variety D1)) │ │ │ │ - -- .00151969s elapsed │ │ │ │ + -- .0010556s elapsed │ │ │ │ Toric varieties of Picard-rank 2 are slightly more interesting. │ │ │ │ i5 : FF2 = hirzebruchSurface 2; │ │ │ │ i6 : D2 = 2*FF2_0 + 3 * FF2_1 │ │ │ │ │ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ │ 0 1 │ │ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ i7 : M2 = elapsedTime monomials D2 │ │ │ │ - -- .0964156s elapsed │ │ │ │ + -- .110963s elapsed │ │ │ │ │ │ │ │ 2 3 2 3 2 3 │ │ │ │ o7 = {x x , x x , x x x , x x } │ │ │ │ 1 3 1 2 0 1 2 0 1 │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring │ │ │ │ variety D2)) │ │ │ │ - -- .00113966s elapsed │ │ │ │ + -- .000849296s elapsed │ │ │ │ i9 : X = kleinschmidt (5, {1,2,3}); │ │ │ │ i10 : D3 = 3*X_0 + 5*X_1 │ │ │ │ │ │ │ │ o10 = 3*X + 5*X │ │ │ │ 0 1 │ │ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ i11 : m3 = elapsedTime # monomials D3 │ │ │ │ - -- 39.4377s elapsed │ │ │ │ + -- 43.1615s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .0257955s elapsed │ │ │ │ + -- .0221523s elapsed │ │ │ │ By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s │ │ │ │ function. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their │ │ │ │ related groups │ │ │ │ * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox │ │ │ │ ring) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ @@ -120,23 +120,23 @@ │ │ │ │ │ │
    │ │ │

    The recommended method for creating a NormalToricVariety from a fan is normalToricVariety(List,List). In fact, this package avoids using objects from the Polyhedra package whenever possible. Here is a trivial example, namely projective 2-space, illustrating the substantial increase in time resulting from the use of a Polyhedra fan.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 0.00304489s (cpu); 2.2332e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00230841s (cpu); 1.6581e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
    │ │ │ 
    i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.206737s (cpu); 0.0677457s (thread); 0s (gc)
    │ │ │ + -- used 0.200997s (cpu); 0.0974426s (thread); 0s (gc) │ │ │ 
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -49,22 +49,22 @@ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is │ │ │ │ _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from │ │ │ │ the _P_o_l_y_h_e_d_r_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 _P_o_l_y_h_e_d_r_a fan. │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ │ - -- used 0.00304489s (cpu); 2.2332e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00230841s (cpu); 1.6581e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = X1 │ │ │ │ │ │ │ │ o6 : NormalToricVariety │ │ │ │ i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull │ │ │ │ matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}}; │ │ │ │ - -- used 0.206737s (cpu); 0.0677457s (thread); 0s (gc) │ │ │ │ + -- used 0.200997s (cpu); 0.0974426s (thread); 0s (gc) │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra' │ │ │ │ fan │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ @@ -202,19 +202,19 @@ │ │ │ | 0 0 1 | │ │ │ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ 
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.106449s (cpu); 0.0339456s (thread); 0s (gc)
    │ │ │ + -- used 0.137947s (cpu); 0.0771225s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.0044979s (cpu); 0.00448742s (thread); 0s (gc)
    │ │ │ + -- used 0.0017407s (cpu); 0.00231854s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -103,17 +103,17 @@ │ │ │ │ │ │ │ │ o18 = | 0 1 0 | │ │ │ │ | 0 0 1 | │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ │ - -- used 0.106449s (cpu); 0.0339456s (thread); 0s (gc) │ │ │ │ + -- used 0.137947s (cpu); 0.0771225s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 0.0044979s (cpu); 0.00448742s (thread); 0s (gc) │ │ │ │ + -- used 0.0017407s (cpu); 0.00231854s (thread); 0s (gc) │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_M_a_t_r_i_x_) -- make a normal toric variety from a polytope │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_P_o_l_y_h_e_d_r_o_n_) -- make a normal toric variety from a │ │ │ │ 'Polyhedra' polyhedron │ │ ├── ./usr/share/doc/Macaulay2/Normaliz/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ aW50Y2xNb25JZGVhbChJZGVhbCxSaW5nRWxlbWVudCk= │ │ │ #:len=2753 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXphdGlvbiBvZiBSZWVzIGFs │ │ │ Z2VicmEiLCAibGluZW51bSIgPT4gMTg2OSwgSW5wdXRzID0+IHtTUEFOe1NQQU57ImFuICIsVE8y │ │ ├── ./usr/share/doc/Macaulay2/NumericSolutions/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ am9yZGFuRm9ybShNYXRyaXgp │ │ │ #:len=263 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODE0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhqb3JkYW5Gb3JtLE1hdHJpeCksImpvcmRhbkZvcm0o │ │ ├── ./usr/share/doc/Macaulay2/NumericalAlgebraicGeometry/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ aXNTdWJzZXQoTnVtZXJpY2FsVmFyaWV0eSxOdW1lcmljYWxWYXJpZXR5KQ== │ │ │ #:len=1173 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgY29udGFpbm1lbnQiLCAibGlu │ │ │ ZW51bSIgPT4gNzg5LCBJbnB1dHMgPT4ge1NQQU57VFR7IlYifSwiLCAiLCIgb3IgIixTUEFOeyJh │ │ ├── ./usr/share/doc/Macaulay2/NumericalCertification/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=23 │ │ │ aW50ZXJ2YWxDQ2koUlJpLE51bWJlcik= │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDQsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGludGVydmFs │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:38 2025 │ │ │ +# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=42 │ │ │ bnVtZXJpY2FsSGlsYmVydEZ1bmN0aW9uKFJpbmdNYXAsSWRlYWwsWlop │ │ │ #:len=367 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDUwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhudW1lcmljYWxIaWxiZXJ0RnVuY3Rpb24sUmluZ01h │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ @@ -23,19 +23,19 @@ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ Sampling image points ... │ │ │ - -- used .00400102 seconds │ │ │ + -- used .00509194 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00399574 seconds │ │ │ + -- used .00399935 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used 0 seconds │ │ │ + -- used .00399958 seconds │ │ │ Computing numerical kernel ... │ │ │ -- used 0 seconds │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ @@ -13,19 +13,19 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ Sampling image points ... │ │ │ - -- used 0 seconds │ │ │ + -- used .00400068 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0022852 seconds │ │ │ + -- used 0 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00168203 seconds │ │ │ + -- used .00400112 seconds │ │ │ Computing numerical kernel ... │ │ │ -- used 0 seconds │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ 1 3 │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ @@ -13,40 +13,40 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ Sampling image points ... │ │ │ - -- used .011988 seconds │ │ │ + -- used .0114397 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0102348 seconds │ │ │ + -- used .0079962 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00876388 seconds │ │ │ + -- used .00800126 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .00303253 seconds │ │ │ + -- used 0 seconds │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ │ │ i4 : R = CC[x_(1,1)..x_(2,4)]; │ │ │ │ │ │ i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*; │ │ │ │ │ │ i6 : S = numericalImageSample(F, ideal 0_R, 60); │ │ │ │ │ │ i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true) │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00400256 seconds │ │ │ + -- used .00399902 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00697058 seconds │ │ │ + -- used .00799975 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .00394506 seconds │ │ │ + -- used .000016821 seconds │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ @@ -22,12 +22,12 @@ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ - -- used 0.059049s (cpu); 0.0599583s (thread); 0s (gc) │ │ │ + -- used 0.0919986s (cpu); 0.0925198s (thread); 0s (gc) │ │ │ │ │ │ o9 = 69 │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out │ │ │ @@ -31,15 +31,15 @@ │ │ │ o5 : Ideal of R │ │ │ │ │ │ i6 : I = I1 + I2; │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ i7 : elapsedTime p = realPoint(I, Iterations => 100) │ │ │ - -- .72672s elapsed │ │ │ + -- .702197s elapsed │ │ │ │ │ │ o7 = p │ │ │ │ │ │ o7 : Point │ │ │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ @@ -89,19 +89,19 @@ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ │ │ │ 
    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00400102 seconds
│ │ │ +     -- used .00509194 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00399574 seconds
│ │ │ +     -- used .00399935 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used 0 seconds
│ │ │ +     -- used .00399958 seconds
│ │ │  Computing numerical kernel ...
│ │ │       -- used 0 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    │ │ │ ├── html2text {} │ │ │ │ @@ -36,19 +36,19 @@ │ │ │ │ == 0 │ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00400102 seconds │ │ │ │ + -- used .00509194 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00399574 seconds │ │ │ │ + -- used .00399935 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used 0 seconds │ │ │ │ + -- used .00399958 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ -- used 0 seconds │ │ │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ @@ -105,19 +105,19 @@ │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ 
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used 0 seconds
│ │ │ +     -- used .00400068 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0022852 seconds
│ │ │ +     -- used 0 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00168203 seconds
│ │ │ +     -- used .00400112 seconds
│ │ │  Computing numerical kernel ...
│ │ │       -- used 0 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,19 +41,19 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used 0 seconds
│ │ │ │ +     -- used .00400068 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0022852 seconds
│ │ │ │ +     -- used 0 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00168203 seconds
│ │ │ │ +     -- used .00400112 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │       -- used 0 seconds
│ │ │ │  
│ │ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │ │  
│ │ │ │                            1                   3
│ │ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html
│ │ │ @@ -114,21 +114,21 @@
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
    │ │ │ │ │ │ │ │ │ 
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .011988 seconds
│ │ │ +     -- used .0114397 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0102348 seconds
│ │ │ +     -- used .0079962 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00876388 seconds
│ │ │ +     -- used .00800126 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .00303253 seconds
│ │ │ +     -- used 0 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    │ │ │ │ │ │ │ │ │ @@ -146,19 +146,19 @@ │ │ │ │ │ │ │ │ │ 
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    │ │ │ │ │ │ │ │ │ 
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00400256 seconds
│ │ │ +     -- used .00399902 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00697058 seconds
│ │ │ +     -- used .00799975 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .00394506 seconds
│ │ │ +     -- used .000016821 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -60,39 +60,39 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .011988 seconds │ │ │ │ + -- used .0114397 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0102348 seconds │ │ │ │ + -- used .0079962 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00876388 seconds │ │ │ │ + -- used .00800126 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .00303253 seconds │ │ │ │ + -- used 0 seconds │ │ │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ The following example computes the dimension of Plücker quadrics in the │ │ │ │ defining ideal of the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, in the │ │ │ │ ambient space $P^5$. │ │ │ │ i4 : R = CC[x_(1,1)..x_(2,4)]; │ │ │ │ i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*; │ │ │ │ i6 : S = numericalImageSample(F, ideal 0_R, 60); │ │ │ │ i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true) │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00400256 seconds │ │ │ │ + -- used .00399902 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00697058 seconds │ │ │ │ + -- used .00799975 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .00394506 seconds │ │ │ │ + -- used .000016821 seconds │ │ │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ @@ -129,15 +129,15 @@ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ │ │ │ 
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.059049s (cpu); 0.0599583s (thread); 0s (gc)
│ │ │ + -- used 0.0919986s (cpu); 0.0925198s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use numericalImageDim:

    │ │ │ ├── html2text {} │ │ │ │ @@ -46,15 +46,15 @@ │ │ │ │ i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5)))); │ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ │ │ 1 70 │ │ │ │ o8 : Matrix R <-- R │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ │ - -- used 0.059049s (cpu); 0.0599583s (thread); 0s (gc) │ │ │ │ + -- used 0.0919986s (cpu); 0.0925198s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 = 69 │ │ │ │ ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: ********** │ │ │ │ * numericalImageDim(List,Ideal) │ │ │ │ * numericalImageDim(List,Ideal,Point) │ │ │ │ * numericalImageDim(Matrix,Ideal) │ │ │ │ * numericalImageDim(Matrix,Ideal,Point) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ @@ -123,15 +123,15 @@ │ │ │ │ │ │ 
    i6 : I = I1 + I2;
│ │ │  
│ │ │  o6 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .72672s elapsed
│ │ │ + -- .702197s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
    │ │ │ │ │ │ │ │ │ 
    i8 : matrix pack(5, p#Coordinates)
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  i5 : I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1);
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : I = I1 + I2;
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ │ - -- .72672s elapsed
│ │ │ │ + -- .702197s elapsed
│ │ │ │  
│ │ │ │  o7 = p
│ │ │ │  
│ │ │ │  o7 : Point
│ │ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ │ │  
│ │ │ │  o8 = | .722359  .289465  -.295808  .591752  -.454678 |
│ │ ├── ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  bnVtZXJpY2FsS2VybmVsKC4uLixUb2xlcmFuY2U9Pi4uLik=
│ │ │  #:len=352
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjE1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tudW1lcmljYWxLZXJuZWwsVG9sZXJhbmNlXSwibnVt
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  UGllcmlSb290Q291bnQoLi4uLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmVxdWVzdCB2ZXJib3NlIGZlZWRiYWNr
│ │ │  IiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAzMzMsIHN5bWJvbCBEb2N1bWVudFRh
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=37
│ │ │  aGV1cmlzdGljU21vb3RobmVzcyguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=320
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc0NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaGV1cmlzdGljU21vb3RobmVzcyxWZXJib3NlXSwi
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -14,35 +14,35 @@
│ │ │  
│ │ │  i4 : LL7a=select(LL7,L->not knownExample L);#LL7a
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .151522s elapsed
│ │ │ + -- .0706523s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .140684s elapsed
│ │ │ + -- .0605123s elapsed
│ │ │  next gb
│ │ │ - -- .00124739s elapsed
│ │ │ + -- .000553214s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .131949s elapsed
│ │ │ + -- .0633293s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .108241s elapsed
│ │ │ + -- .0608818s elapsed
│ │ │  next gb
│ │ │ - -- .000704851s elapsed
│ │ │ + -- .000361444s elapsed
│ │ │  true
│ │ │ - -- 1.55972s elapsed
│ │ │ + -- .736451s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.14586s elapsed
│ │ │ + -- .730826s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : LL7b=={}
│ │ │  
│ │ │ @@ -75,22 +75,22 @@
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .31452s elapsed
│ │ │ + -- .134398s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .222111s elapsed
│ │ │ + -- .0921477s elapsed
│ │ │  next gb
│ │ │ - -- .00146195s elapsed
│ │ │ + -- .000657969s elapsed
│ │ │  true
│ │ │ - -- .851703s elapsed
│ │ │ -(5, 8,  all semigroups are smoothable) -- .915236s elapsed
│ │ │ + -- .366155s elapsed
│ │ │ +(5, 8,  all semigroups are smoothable) -- .399945s elapsed
│ │ │  
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : L={6,8,9,11}
│ │ │ @@ -100,22 +100,22 @@
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : genus L
│ │ │  
│ │ │  o13 = 8
│ │ │  
│ │ │  i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0789936s elapsed
│ │ │ + -- .0418583s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .446926s elapsed
│ │ │ + -- .197709s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_heuristic__Smoothness.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i3 : setRandomSeed "some singular and some smooth curves";
│ │ │  
│ │ │  i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.07914s elapsed
│ │ │ + -- 1.90367s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Smoothable__Semigroup.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : genus L
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- .771857s elapsed
│ │ │ + -- .546023s elapsed
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.57876s elapsed
│ │ │ + -- 2.06113s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Weierstrass__Semigroup.out
│ │ │ @@ -7,12 +7,12 @@
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : genus L
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 3.84044s elapsed
│ │ │ + -- 2.12333s elapsed
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 6860996532851631556
│ │ │  
│ │ │  i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │ -(6, 7,  all semigroups are smoothable) -- 1.56638s elapsed
│ │ │ +(6, 7,  all semigroups are smoothable) -- .7741s elapsed
│ │ │  
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : LLdifficult={{6, 8, 9, 11}}
│ │ │ @@ -14,61 +14,61 @@
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .380066s elapsed
│ │ │ + -- .24143s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .178994s elapsed
│ │ │ + -- .0875304s elapsed
│ │ │  next gb
│ │ │ - -- .00179905s elapsed
│ │ │ + -- .00130079s elapsed
│ │ │  true
│ │ │ - -- .959568s elapsed
│ │ │ + -- .553599s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .402129s elapsed
│ │ │ + -- .236677s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .190718s elapsed
│ │ │ + -- .100247s elapsed
│ │ │  next gb
│ │ │ - -- .00255459s elapsed
│ │ │ + -- .00185122s elapsed
│ │ │  decompose
│ │ │ - -- .183642s elapsed
│ │ │ + -- .112114s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.51269s elapsed
│ │ │ + -- 1.81031s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .0737587s elapsed
│ │ │ + -- .0659571s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .0919318s elapsed
│ │ │ + -- .0866994s elapsed
│ │ │  next gb
│ │ │ - -- .000445154s elapsed
│ │ │ + -- .000368268s elapsed
│ │ │  true
│ │ │ - -- .609331s elapsed
│ │ │ + -- .40854s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .475404s elapsed
│ │ │ + -- .314834s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .292145s elapsed
│ │ │ + -- .126464s elapsed
│ │ │  next gb
│ │ │ - -- .00644164s elapsed
│ │ │ + -- .002645s elapsed
│ │ │  decompose
│ │ │ - -- 1.13216s elapsed
│ │ │ + -- .514707s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.15341s elapsed
│ │ │ - -- 7.23516s elapsed
│ │ │ + -- 1.54433s elapsed
│ │ │ + -- 4.3169s elapsed
│ │ │  0
│ │ │  
│ │ │ - -- .0000054s elapsed
│ │ │ - -- 7.26994s elapsed
│ │ │ + -- .000003096s elapsed
│ │ │ + -- 4.34576s elapsed
│ │ │  {}
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/___Lab__Book__Protocol.html
│ │ │ @@ -89,36 +89,36 @@
│ │ │                
    i4 : LL7a=select(LL7,L->not knownExample L);#LL7a
│ │ │  
│ │ │  o5 = 2
    
│ │ │            
│ │ │            
│ │ │                
    i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .151522s elapsed
│ │ │ + -- .0706523s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .140684s elapsed
│ │ │ + -- .0605123s elapsed
│ │ │  next gb
│ │ │ - -- .00124739s elapsed
│ │ │ + -- .000553214s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .131949s elapsed
│ │ │ + -- .0633293s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .108241s elapsed
│ │ │ + -- .0608818s elapsed
│ │ │  next gb
│ │ │ - -- .000704851s elapsed
│ │ │ + -- .000361444s elapsed
│ │ │  true
│ │ │ - -- 1.55972s elapsed
│ │ │ + -- .736451s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    
│ │ │            
│ │ │            
│ │ │                
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.14586s elapsed
│ │ │ + -- .730826s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    
│ │ │            
│ │ │            
│ │ │                
    i8 : LL7b=={}
│ │ │ @@ -167,22 +167,22 @@
│ │ │  o10 : Sequence
    
│ │ │            
│ │ │            
│ │ │                
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .31452s elapsed
│ │ │ + -- .134398s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .222111s elapsed
│ │ │ + -- .0921477s elapsed
│ │ │  next gb
│ │ │ - -- .00146195s elapsed
│ │ │ + -- .000657969s elapsed
│ │ │  true
│ │ │ - -- .851703s elapsed
│ │ │ -(5, 8,  all semigroups are smoothable) -- .915236s elapsed
│ │ │ + -- .366155s elapsed
│ │ │ +(5, 8,  all semigroups are smoothable) -- .399945s elapsed
│ │ │  
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    
│ │ │            
│ │ │          
│ │ │ @@ -200,22 +200,22 @@
│ │ │            
│ │ │                
    i13 : genus L
│ │ │  
│ │ │  o13 = 8
    
│ │ │            
│ │ │            
│ │ │                
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0789936s elapsed
│ │ │ + -- .0418583s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .446926s elapsed
│ │ │ + -- .197709s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
    
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,34 +27,34 @@
│ │ │ │  o3 = 39
│ │ │ │  i4 : LL7a=select(LL7,L->not knownExample L);#LL7a
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup
│ │ │ │  (L,0.25,0,Verbose=>true))
│ │ │ │  unfolding
│ │ │ │ - -- .151522s elapsed
│ │ │ │ + -- .0706523s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .140684s elapsed
│ │ │ │ + -- .0605123s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00124739s elapsed
│ │ │ │ + -- .000553214s elapsed
│ │ │ │  true
│ │ │ │  unfolding
│ │ │ │ - -- .131949s elapsed
│ │ │ │ + -- .0633293s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .108241s elapsed
│ │ │ │ + -- .0608818s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000704851s elapsed
│ │ │ │ + -- .000361444s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.55972s elapsed
│ │ │ │ + -- .736451s elapsed
│ │ │ │  
│ │ │ │  o6 = {}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ │ - -- 1.14586s elapsed
│ │ │ │ + -- .730826s elapsed
│ │ │ │  
│ │ │ │  o7 = {}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : LL7b=={}
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │ @@ -93,22 +93,22 @@
│ │ │ │  o10 = (5, 8)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │ │  (13, 1)
│ │ │ │  {5, 8, 11, 12}
│ │ │ │  unfolding
│ │ │ │ - -- .31452s elapsed
│ │ │ │ + -- .134398s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .222111s elapsed
│ │ │ │ + -- .0921477s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00146195s elapsed
│ │ │ │ + -- .000657969s elapsed
│ │ │ │  true
│ │ │ │ - -- .851703s elapsed
│ │ │ │ -(5, 8,  all semigroups are smoothable) -- .915236s elapsed
│ │ │ │ + -- .366155s elapsed
│ │ │ │ +(5, 8,  all semigroups are smoothable) -- .399945s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = {}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  In the verbose mode we get timings of various computation steps and further
│ │ │ │  information. The first line, (13,1), indicates that there 13 semigroups of
│ │ │ │ @@ -121,22 +121,22 @@
│ │ │ │  o12 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : genus L
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ │ - -- .0789936s elapsed
│ │ │ │ + -- .0418583s elapsed
│ │ │ │  6
│ │ │ │  false
│ │ │ │  5
│ │ │ │  false
│ │ │ │  4
│ │ │ │  decompose
│ │ │ │ - -- .446926s elapsed
│ │ │ │ + -- .197709s elapsed
│ │ │ │  number of components: 2
│ │ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │ │  {0, -1}
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  The first integer, 6, tells that in this attempt deformation parameters of
│ │ │ │  degree >= 6 were used and no smooth fiber was found. Finally with all
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_heuristic__Smoothness.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │            
│ │ │            
│ │ │                
    i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.07914s elapsed
│ │ │ + -- 1.90367s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
    
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : setRandomSeed "some singular and some smooth curves";
│ │ │ │  i4 : elapsedTime tally apply(10,i-> (
│ │ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │ │               c=sub(c,x_0=>1);
│ │ │ │               R=kk[support c];c=sub(c,R);
│ │ │ │               heuristicSmoothness c))
│ │ │ │ - -- 3.07914s elapsed
│ │ │ │ + -- 1.90367s elapsed
│ │ │ │  
│ │ │ │  o4 = Tally{false => 6}
│ │ │ │             true => 4
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  ********** WWaayyss ttoo uussee hheeuurriissttiiccSSmmooootthhnneessss:: **********
│ │ │ │      * heuristicSmoothness(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Smoothable__Semigroup.html
│ │ │ @@ -95,21 +95,21 @@
│ │ │            
│ │ │                
    i2 : genus L
│ │ │  
│ │ │  o2 = 8
    
│ │ │            
│ │ │            
│ │ │                
    i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- .771857s elapsed
│ │ │ + -- .546023s elapsed
│ │ │  
│ │ │  o3 = false
    
│ │ │            
│ │ │            
│ │ │                
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.57876s elapsed
│ │ │ + -- 2.06113s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -30,19 +30,19 @@ │ │ │ │ o1 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : genus L │ │ │ │ │ │ │ │ o2 = 8 │ │ │ │ i3 : elapsedTime isSmoothableSemigroup(L,0.30,0) │ │ │ │ - -- .771857s elapsed │ │ │ │ + -- .546023s elapsed │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ │ - -- 4.57876s elapsed │ │ │ │ + -- 2.06113s elapsed │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal │ │ │ │ with positive degree parameters │ │ │ │ * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding │ │ │ │ * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Weierstrass__Semigroup.html │ │ │ @@ -93,15 +93,15 @@ │ │ │ │ │ │ 
    i2 : genus L
│ │ │  
│ │ │  o2 = 8
    │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 3.84044s elapsed
│ │ │ + -- 2.12333s elapsed
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ o1 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : genus L │ │ │ │ │ │ │ │ o2 = 8 │ │ │ │ i3 : elapsedTime isWeierstrassSemigroup(L,0.15) │ │ │ │ - -- 3.84044s elapsed │ │ │ │ + -- 2.12333s elapsed │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal │ │ │ │ with positive degree parameters │ │ │ │ * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding │ │ │ │ * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_non__Weierstrass__Semigroups.html │ │ │ @@ -82,15 +82,15 @@ │ │ │

    Description

    │ │ │
    │ │ │

    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.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -101,61 +101,61 @@ │ │ │ o2 : List │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │ -(6, 7,  all semigroups are smoothable) -- 1.56638s elapsed
│ │ │ +(6, 7,  all semigroups are smoothable) -- .7741s elapsed
│ │ │  
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    │ │ │ 
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .380066s elapsed
│ │ │ + -- .24143s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .178994s elapsed
│ │ │ + -- .0875304s elapsed
│ │ │  next gb
│ │ │ - -- .00179905s elapsed
│ │ │ + -- .00130079s elapsed
│ │ │  true
│ │ │ - -- .959568s elapsed
│ │ │ + -- .553599s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .402129s elapsed
│ │ │ + -- .236677s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .190718s elapsed
│ │ │ + -- .100247s elapsed
│ │ │  next gb
│ │ │ - -- .00255459s elapsed
│ │ │ + -- .00185122s elapsed
│ │ │  decompose
│ │ │ - -- .183642s elapsed
│ │ │ + -- .112114s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.51269s elapsed
│ │ │ + -- 1.81031s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .0737587s elapsed
│ │ │ + -- .0659571s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .0919318s elapsed
│ │ │ + -- .0866994s elapsed
│ │ │  next gb
│ │ │ - -- .000445154s elapsed
│ │ │ + -- .000368268s elapsed
│ │ │  true
│ │ │ - -- .609331s elapsed
│ │ │ + -- .40854s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .475404s elapsed
│ │ │ + -- .314834s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .292145s elapsed
│ │ │ + -- .126464s elapsed
│ │ │  next gb
│ │ │ - -- .00644164s elapsed
│ │ │ + -- .002645s elapsed
│ │ │  decompose
│ │ │ - -- 1.13216s elapsed
│ │ │ + -- .514707s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.15341s elapsed
│ │ │ - -- 7.23516s elapsed
│ │ │ + -- 1.54433s elapsed
│ │ │ + -- 4.3169s elapsed
│ │ │  0
│ │ │  
│ │ │ - -- .0000054s elapsed
│ │ │ - -- 7.26994s elapsed
│ │ │ + -- .000003096s elapsed
│ │ │ + -- 4.34576s elapsed
│ │ │  {}
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,76 +22,76 @@ │ │ │ │ LLdifficult │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ 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. │ │ │ │ i1 : elapsedTime nonWeierstrassSemigroups(6,7) │ │ │ │ -(6, 7, all semigroups are smoothable) -- 1.56638s elapsed │ │ │ │ +(6, 7, all semigroups are smoothable) -- .7741s elapsed │ │ │ │ │ │ │ │ │ │ │ │ o1 = {} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : LLdifficult={{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o2 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true) │ │ │ │ (17, 5) │ │ │ │ {6, 7, 8, 17} │ │ │ │ unfolding │ │ │ │ - -- .380066s elapsed │ │ │ │ + -- .24143s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .178994s elapsed │ │ │ │ + -- .0875304s elapsed │ │ │ │ next gb │ │ │ │ - -- .00179905s elapsed │ │ │ │ + -- .00130079s elapsed │ │ │ │ true │ │ │ │ - -- .959568s elapsed │ │ │ │ + -- .553599s elapsed │ │ │ │ {6, 7, 9, 17} │ │ │ │ unfolding │ │ │ │ - -- .402129s elapsed │ │ │ │ + -- .236677s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .190718s elapsed │ │ │ │ + -- .100247s elapsed │ │ │ │ next gb │ │ │ │ - -- .00255459s elapsed │ │ │ │ + -- .00185122s elapsed │ │ │ │ decompose │ │ │ │ - -- .183642s elapsed │ │ │ │ + -- .112114s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ │ {0, -1} │ │ │ │ - -- 2.51269s elapsed │ │ │ │ + -- 1.81031s elapsed │ │ │ │ {6, 8, 9, 10} │ │ │ │ unfolding │ │ │ │ - -- .0737587s elapsed │ │ │ │ + -- .0659571s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .0919318s elapsed │ │ │ │ + -- .0866994s elapsed │ │ │ │ next gb │ │ │ │ - -- .000445154s elapsed │ │ │ │ + -- .000368268s elapsed │ │ │ │ true │ │ │ │ - -- .609331s elapsed │ │ │ │ + -- .40854s elapsed │ │ │ │ {6, 8, 10, 11, 13} │ │ │ │ unfolding │ │ │ │ - -- .475404s elapsed │ │ │ │ + -- .314834s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .292145s elapsed │ │ │ │ + -- .126464s elapsed │ │ │ │ next gb │ │ │ │ - -- .00644164s elapsed │ │ │ │ + -- .002645s elapsed │ │ │ │ decompose │ │ │ │ - -- 1.13216s elapsed │ │ │ │ + -- .514707s elapsed │ │ │ │ number of components: 1 │ │ │ │ support c, codim c: {(5, 1)} │ │ │ │ {-1} │ │ │ │ - -- 3.15341s elapsed │ │ │ │ - -- 7.23516s elapsed │ │ │ │ + -- 1.54433s elapsed │ │ │ │ + -- 4.3169s elapsed │ │ │ │ 0 │ │ │ │ │ │ │ │ - -- .0000054s elapsed │ │ │ │ - -- 7.26994s elapsed │ │ │ │ + -- .000003096s elapsed │ │ │ │ + -- 4.34576s elapsed │ │ │ │ {} │ │ │ │ │ │ │ │ o3 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ In the verbose mode we get timings of various computation steps and further │ │ │ │ information. The first line, (17,5), indicates that there 17 semigroups of │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=29 │ │ │ dG9TdHJpbmcoUG9seW5vbWlhbE9JQWxnZWJyYSk= │ │ │ #:len=1044 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheSBhIHBvbHlub21pYWwgT0kt │ │ │ YWxnZWJyYSBpbiBjb25kZW5zZWQgZm9ybSIsICJsaW5lbnVtIiA9PiAxNTE2LCBJbnB1dHMgPT4g │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1) │ │ │ - -- used 0.151035s (cpu); 0.0852379s (thread); 0s (gc) │ │ │ + -- used 0.128907s (cpu); 0.0746386s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ o5 : OIResolution │ │ │ │ │ │ i6 : C.dd_0 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.149075s (cpu); 0.0863072s (thread); 0s (gc) │ │ │ + -- used 0.155667s (cpu); 0.102495s (thread); 0s (gc) │ │ │ │ │ │ i6 : C_0 │ │ │ │ │ │ o6 = Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.341566s (cpu); 0.24142s (thread); 0s (gc) │ │ │ + -- used 0.278713s (cpu); 0.182372s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.121853s (cpu); 0.084206s (thread); 0s (gc) │ │ │ + -- used 0.136724s (cpu); 0.08879s (thread); 0s (gc) │ │ │ │ │ │ i6 : describeFull C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ │ @@ -21,11 +21,11 @@ │ │ │ Differential: Source: (e0, {2}, {-2}) Target: (e, {1, 1}, {0, 0}) │ │ │ 1: Module: Basis symbol: e1 │ │ │ Basis element widths: {4, 4} │ │ │ Degree shifts: {-4, -4} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ │ Monomial order: Lex │ │ │ Differential: Source: (e1, {4, 4}, {-4, -4}) Target: (e0, {2}, {-2}) │ │ │ - Basis element images: {-x x e0 + x x e0 + x x e0 - x x e0 , 0} │ │ │ - 2,3 1,1 4,{2, 4},1 2,4 1,1 4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ + Basis element images: {0, -x x e0 + x x e0 + x x e0 - x x e0 } │ │ │ + 2,3 1,1 4,{2, 4},1 2,4 1,1 4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__Free__O__I__Module__Map_rp.out │ │ │ @@ -10,21 +10,18 @@ │ │ │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ │ │ i7 : describe phi │ │ │ │ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ │ - Basis element images: {-x e0 + x e0 + │ │ │ - 2,2 5,{1, 3, 5},1 2,2 5,{1, 3, 4},1 │ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ │ + Basis element images: {x x e0 - x x e0 │ │ │ + 2,3 1,1 5,{2, 4, 5},1 2,4 1,1 5,{2, 3, 5},1 │ │ │ ------------------------------------------------------------------------ │ │ │ - x e0 - x e0 , x x e0 - │ │ │ - 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 2,3 1,1 5,{2, 4, 5},1 │ │ │ + - x x e0 + x x e0 , -x e0 │ │ │ + 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, 5},1 2,2 5,{1, 3, │ │ │ ------------------------------------------------------------------------ │ │ │ - x x e0 - x x e0 + x x e0 │ │ │ - 2,4 1,1 5,{2, 3, 5},1 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, │ │ │ - ------------------------------------------------------------------------ │ │ │ - } │ │ │ - 5},1 │ │ │ + + x e0 + x e0 - x e0 } │ │ │ + 5},1 2,2 5,{1, 3, 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.150135s (cpu); 0.100171s (thread); 0s (gc) │ │ │ + -- used 0.126077s (cpu); 0.0749056s (thread); 0s (gc) │ │ │ │ │ │ i6 : describe C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.349406s (cpu); 0.23824s (thread); 0s (gc) │ │ │ + -- used 0.301099s (cpu); 0.192379s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ 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); │ │ │ │ │ │ i10 : isOIGB {b1, b2} │ │ │ │ │ │ o10 = false │ │ │ │ │ │ i11 : time B = oiGB {b1, b2} │ │ │ - -- used 0.0196182s (cpu); 0.0219962s (thread); 0s (gc) │ │ │ + -- used 0.0159993s (cpu); 0.0177285s (thread); 0s (gc) │ │ │ │ │ │ o11 = {x e + x e , x x e + x x e , │ │ │ 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 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ i5 : installGeneratorsInWidth(F, 3); │ │ │ │ │ │ i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ │ │ 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); │ │ │ │ │ │ i10 : time B = oiGB {b1, b2} │ │ │ - -- used 0.0239924s (cpu); 0.0236563s (thread); 0s (gc) │ │ │ + -- used 0.0199998s (cpu); 0.0202587s (thread); 0s (gc) │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ 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 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ @@ -41,18 +41,18 @@ │ │ │ - x x e } │ │ │ 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : minimizeOIGB C -- an element gets removed │ │ │ │ │ │ - │ │ │ -o14 = {x x e + x x e , x x x e - │ │ │ - 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 │ │ │ + │ │ │ +o14 = {x e + x e , x x e + x x e , │ │ │ + 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 │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 │ │ │ - x x e , x e + x e } │ │ │ - 2,1 1,2 3,{1, 3},3 1,1 1,{1},1 2,1 1,{1},2 │ │ │ + 2 │ │ │ + x x x e - x x e } │ │ │ + 2,3 2,2 1,1 3,{2, 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ o14 : List │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.152193s (cpu); 0.0872943s (thread); 0s (gc) │ │ │ + -- used 0.12806s (cpu); 0.0712107s (thread); 0s (gc) │ │ │ │ │ │ i6 : net C │ │ │ │ │ │ o6 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ │ │ 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); │ │ │ │ │ │ i9 : time oiGB {b1, b2} │ │ │ - -- used 0.0240585s (cpu); 0.0242842s (thread); 0s (gc) │ │ │ + -- used 0.0199672s (cpu); 0.0212784s (thread); 0s (gc) │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.446407s (cpu); 0.288554s (thread); 0s (gc) │ │ │ + -- used 0.301528s (cpu); 0.200939s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2); │ │ │ │ │ │ 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); │ │ │ │ │ │ i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal) │ │ │ - -- used 0.249592s (cpu); 0.189708s (thread); 0s (gc) │ │ │ + -- used 0.148745s (cpu); 0.0934098s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 2,1 1,{1},2 1,1 1,{1},2 1,2 1,1 2,{2},1 2,2 1,2 2,{2},2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ x x e - x x e } │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html │ │ │ @@ -58,15 +58,15 @@ │ │ │ 
    i3 : installGeneratorsInWidth(F, 2);
    │ │ │ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.151035s (cpu); 0.0852379s (thread); 0s (gc)
│ │ │ + -- used 0.128907s (cpu); 0.0746386s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -11,15 +11,15 @@ │ │ │ │ complex, use _i_s_C_o_m_p_l_e_x. To get the $n$th differential in an OI-resolution C, │ │ │ │ use C.dd_n. │ │ │ │ i1 : P = makePolynomialOIAlgebra(2, x, QQ); │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ i5 : time C = oiRes({b}, 1) │ │ │ │ - -- used 0.151035s (cpu); 0.0852379s (thread); 0s (gc) │ │ │ │ + -- used 0.128907s (cpu); 0.0746386s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ │ │ o5 : OIResolution │ │ │ │ i6 : C.dd_0 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ 
    i3 : installGeneratorsInWidth(F, 2);
    │ │ │ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.149075s (cpu); 0.0863072s (thread); 0s (gc)
    │ │ │ + -- used 0.155667s (cpu); 0.102495s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ 
    i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │       Basis element widths: {2}
│ │ │       Degree shifts: {-2}
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Returns the free OI-module of $C$ in homological degree $n$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.149075s (cpu); 0.0863072s (thread); 0s (gc)
│ │ │ │ + -- used 0.155667s (cpu); 0.102495s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -58,15 +58,15 @@
│ │ │                
    i3 : installGeneratorsInWidth(F, 2);
    
│ │ │            
│ │ │            
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.341566s (cpu); 0.24142s (thread); 0s (gc)
│ │ │ + -- used 0.278713s (cpu); 0.182372s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  homological degree $n-1$ to be minimized. Therefore, use TopNonminimal => true
│ │ │ │  for no minimization of the basis in degree $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.341566s (cpu); 0.24142s (thread); 0s (gc)
│ │ │ │ + -- used 0.278713s (cpu); 0.182372s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │                
    i3 : installGeneratorsInWidth(F, 2);
    
│ │ │            
│ │ │            
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.121853s (cpu); 0.084206s (thread); 0s (gc)
    
│ │ │ + -- used 0.136724s (cpu); 0.08879s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ 
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │ @@ -98,16 +98,16 @@
│ │ │       Differential: Source: (e0, {2}, {-2}) Target: (e, {1, 1}, {0, 0})
│ │ │       1: Module: Basis symbol: e1
│ │ │                  Basis element widths: {4, 4}
│ │ │                  Degree shifts: {-4, -4}
│ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │                  Monomial order: Lex
│ │ │       Differential: Source: (e1, {4, 4}, {-4, -4}) Target: (e0, {2}, {-2})
│ │ │ -                   Basis element images: {-x   x   e0           + x   x   e0           + x   x   e0           - x   x   e0          , 0}
│ │ │ -                                            2,3 1,1  4,{2, 4},1    2,4 1,1  4,{2, 3},1    2,3 1,2  4,{1, 4},1    2,4 1,2  4,{1, 3},1
    │ │ │ + Basis element images: {0, -x x e0 + x x e0 + x x e0 - x x e0 } │ │ │ + 2,3 1,1 4,{2, 4},1 2,4 1,1 4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use describeFull:

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -15,30 +15,30 @@ │ │ │ │ Displays the free OI-modules and describes the differentials of an OI- │ │ │ │ resolution. │ │ │ │ i1 : P = makePolynomialOIAlgebra(2, x, QQ); │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ │ - -- used 0.121853s (cpu); 0.084206s (thread); 0s (gc) │ │ │ │ + -- used 0.136724s (cpu); 0.08879s (thread); 0s (gc) │ │ │ │ i6 : describeFull C │ │ │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ │ Basis element widths: {2} │ │ │ │ Degree shifts: {-2} │ │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ │ │ Monomial order: Lex │ │ │ │ Differential: Source: (e0, {2}, {-2}) Target: (e, {1, 1}, {0, 0}) │ │ │ │ 1: Module: Basis symbol: e1 │ │ │ │ Basis element widths: {4, 4} │ │ │ │ Degree shifts: {-4, -4} │ │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ │ │ Monomial order: Lex │ │ │ │ Differential: Source: (e1, {4, 4}, {-4, -4}) Target: (e0, {2}, {-2}) │ │ │ │ - Basis element images: {-x x e0 + x x e0 │ │ │ │ -+ x x e0 - x x e0 , 0} │ │ │ │ - 2,3 1,1 4,{2, 4},1 2,4 1,1 4, │ │ │ │ -{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ + Basis element images: {0, -x x e0 + x x e0 │ │ │ │ ++ x x e0 - x x e0 } │ │ │ │ + 2,3 1,1 4,{2, 4},1 2,4 1,1 │ │ │ │ +4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ ********** WWaayyss ttoo uussee ddeessccrriibbeeFFuullll:: ********** │ │ │ │ * describeFull(OIResolution) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _d_e_s_c_r_i_b_e_F_u_l_l is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__Free__O__I__Module__Map_rp.html │ │ │ @@ -90,26 +90,23 @@ │ │ │ │ │ │ │ │ │ 
      i6 : phi = C.dd_1;
      │ │ │ │ │ │ │ │ │ 
      i7 : describe phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1  
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ +     Basis element images: {x   x   e0              - x   x   e0             
│ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1  
│ │ │ +     - x   x   e0              + x   x   e0             , -x   e0        
│ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   x   e0              - x   x   e0              + x   x   e0        
│ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -         }
│ │ │ -     5},1
      │ │ │ + + x e0 + x e0 - x e0 } │ │ │ + 5},1 2,2 5,{1, 3, 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -19,21 +19,18 @@ │ │ │ │ i2 : F = makeFreeOIModule(e, {1,2}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 3); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2); │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ i7 : describe phi │ │ │ │ │ │ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ │ │ - Basis element images: {-x e0 + x e0 + │ │ │ │ - 2,2 5,{1, 3, 5},1 2,2 5,{1, 3, 4},1 │ │ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ │ │ + Basis element images: {x x e0 - x x e0 │ │ │ │ + 2,3 1,1 5,{2, 4, 5},1 2,4 1,1 5,{2, 3, 5},1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x e0 - x e0 , x x e0 - │ │ │ │ - 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 2,3 1,1 5,{2, 4, 5},1 │ │ │ │ + - x x e0 + x x e0 , -x e0 │ │ │ │ + 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, 5},1 2,2 5,{1, 3, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x e0 - x x e0 + x x e0 │ │ │ │ - 2,4 1,1 5,{2, 3, 5},1 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - } │ │ │ │ - 5},1 │ │ │ │ + + x e0 + x e0 - x e0 } │ │ │ │ + 5},1 2,2 5,{1, 3, 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _d_e_s_c_r_i_b_e_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- display a free OI-module map │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ 
      i3 : installGeneratorsInWidth(F, 2);
      │ │ │ │ │ │ │ │ │ 
      i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
      │ │ │ │ │ │ │ │ │ 
      i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.150135s (cpu); 0.100171s (thread); 0s (gc)
      │ │ │ + -- used 0.126077s (cpu); 0.0749056s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
      i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Displays the free OI-modules and differentials of an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.150135s (cpu); 0.100171s (thread); 0s (gc)
│ │ │ │ + -- used 0.126077s (cpu); 0.0749056s (thread); 0s (gc)
│ │ │ │  i6 : describe C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │                
      i3 : installGeneratorsInWidth(F, 2);
      
│ │ │            
│ │ │            
│ │ │                
      i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
      
│ │ │            
│ │ │            
│ │ │                
      i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.349406s (cpu); 0.23824s (thread); 0s (gc)
│ │ │ + -- used 0.301099s (cpu); 0.192379s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
      
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  option must be either true or false, depending on whether one wants debug
│ │ │ │  information printed.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.349406s (cpu); 0.23824s (thread); 0s (gc)
│ │ │ │ + -- used 0.301099s (cpu); 0.192379s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : isComplex C
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │            
│ │ │                
      i10 : isOIGB {b1, b2}
│ │ │  
│ │ │  o10 = false
      
│ │ │            
│ │ │            
│ │ │                
      i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0196182s (cpu); 0.0219962s (thread); 0s (gc)
│ │ │ + -- used 0.0159993s (cpu); 0.0177285s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          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 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  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);
│ │ │ │  i10 : isOIGB {b1, b2}
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0196182s (cpu); 0.0219962s (thread); 0s (gc)
│ │ │ │ + -- used 0.0159993s (cpu); 0.0177285s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          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
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │                
      i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
      
│ │ │            
│ │ │            
│ │ │                
      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);
      
│ │ │            
│ │ │            
│ │ │                
      i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0239924s (cpu); 0.0236563s (thread); 0s (gc)
│ │ │ + -- used 0.0199998s (cpu); 0.0202587s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          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 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │ @@ -129,21 +129,21 @@
│ │ │        3},3    2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │  o13 : List
      
│ │ │            
│ │ │            
│ │ │                
      i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │ -                                                                        
│ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ -        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  
│ │ │ +                                                                           
│ │ │ +o14 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ +        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 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2
│ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ +                                     2
│ │ │ +      x   x   x   e           - x   x   e          }
│ │ │ +       2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │  o14 : List
      
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    Ways to use minimizeOIGB:

    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1,2}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 1); │ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ i5 : installGeneratorsInWidth(F, 3); │ │ │ │ i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ 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); │ │ │ │ i10 : time B = oiGB {b1, b2} │ │ │ │ - -- used 0.0239924s (cpu); 0.0236563s (thread); 0s (gc) │ │ │ │ + -- used 0.0199998s (cpu); 0.0202587s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ │ 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 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ x x x e - x x x e } │ │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ │ │ @@ -51,19 +51,19 @@ │ │ │ │ - x x e } │ │ │ │ 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ │ │ o13 : List │ │ │ │ i14 : minimizeOIGB C -- an element gets removed │ │ │ │ │ │ │ │ │ │ │ │ -o14 = {x x e + x x e , x x x e - │ │ │ │ - 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 │ │ │ │ +o14 = {x e + x e , x x e + x x e , │ │ │ │ + 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 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 │ │ │ │ - x x e , x e + x e } │ │ │ │ - 2,1 1,2 3,{1, 3},3 1,1 1,{1},1 2,1 1,{1},2 │ │ │ │ + 2 │ │ │ │ + x x x e - x x e } │ │ │ │ + 2,3 2,2 1,1 3,{2, 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ │ │ o14 : List │ │ │ │ ********** WWaayyss ttoo uussee mmiinniimmiizzeeOOIIGGBB:: ********** │ │ │ │ * minimizeOIGB(List) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _m_i_n_i_m_i_z_e_O_I_G_B is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ 
    i3 : installGeneratorsInWidth(F, 2);
    │ │ │ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.152193s (cpu); 0.0872943s (thread); 0s (gc)
    │ │ │ + -- used 0.12806s (cpu); 0.0712107s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ 
    i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -16,14 +16,14 @@ │ │ │ │ Displays the basis element widths and degree shifts of the free OI-modules in │ │ │ │ an OI-resolution. │ │ │ │ i1 : P = makePolynomialOIAlgebra(2, x, QQ); │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ │ - -- used 0.152193s (cpu); 0.0872943s (thread); 0s (gc) │ │ │ │ + -- used 0.12806s (cpu); 0.0712107s (thread); 0s (gc) │ │ │ │ i6 : net C │ │ │ │ │ │ │ │ o6 = 0: (e0, {2}, {-2}) │ │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html │ │ │ @@ -104,15 +104,15 @@ │ │ │ 
    i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
    │ │ │ │ │ │ │ │ │ 
    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);
    │ │ │ │ │ │ │ │ │ 
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0240585s (cpu); 0.0242842s (thread); 0s (gc)
│ │ │ + -- used 0.0199672s (cpu); 0.0212784s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         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 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  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);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.0240585s (cpu); 0.0242842s (thread); 0s (gc)
│ │ │ │ + -- used 0.0199672s (cpu); 0.0212784s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │                
    i3 : installGeneratorsInWidth(F, 2);
    
│ │ │            
│ │ │            
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.446407s (cpu); 0.288554s (thread); 0s (gc)
│ │ │ + -- used 0.301528s (cpu); 0.200939s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  Therefore, use TopNonminimal => true for no minimization of the basis in degree
│ │ │ │  $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.446407s (cpu); 0.288554s (thread); 0s (gc)
│ │ │ │ + -- used 0.301528s (cpu); 0.200939s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │                
    i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
    
│ │ │            
│ │ │            
│ │ │                
    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);
    
│ │ │            
│ │ │            
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.249592s (cpu); 0.189708s (thread); 0s (gc)
│ │ │ + -- used 0.148745s (cpu); 0.0934098s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │ │  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);
│ │ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ │ - -- used 0.249592s (cpu); 0.189708s (thread); 0s (gc)
│ │ │ │ + -- used 0.148745s (cpu); 0.0934098s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2                  2
│ │ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  bWF4Q29uZXM=
│ │ │  #:len=1120
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg
│ │ │  Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gNTIzMiwgSW5wdXRzID0+IHtTUEFOe1RUeyJG
│ │ ├── ./usr/share/doc/Macaulay2/OldToricVectorBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  ZHVhbChUb3JpY1ZlY3RvckJ1bmRsZSk=
│ │ │  #:len=1522
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiIHRoZSBkdWFsIGJ1bmRsZSBvZiBhIHRv
│ │ │  cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjc2MiwgSW5wdXRzID0+IHtTUEFOe1RU
│ │ ├── ./usr/share/doc/Macaulay2/OnlineLookup/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  b2Vpcw==
│ │ │  #:len=753
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT0VJUyBsb29rdXAiLCBEZXNjcmlwdGlv
│ │ │  biA9PiAoRElWe1BBUkF7VEVYeyJUaGlzIGZ1bmN0aW9uIGxvb2tzIHVwIHRoZSBhcmd1bWVudCAo
│ │ ├── ./usr/share/doc/Macaulay2/OpenMath/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  T3Blbk1hdGg=
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3Blbk1hdGggc3VwcG9ydCIsICJsaW5l
│ │ │  bnVtIiA9PiA4NSwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsiU2VlIGFsc28ifSxVTHtMSXtUT0h7
│ │ ├── ./usr/share/doc/Macaulay2/PHCpack/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  ZmFjdG9yV2l0bmVzc1NldCguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=595
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uIHRvIHNwZWNpZnkgd2hldGhl
│ │ │  ciBhZGRpdGlvbmFsIG91dHB1dCBpcyB3YW50ZWQiLCAibGluZW51bSIgPT4gNDEyLCAiZmlsZW5h
│ │ ├── ./usr/share/doc/Macaulay2/PackageCitations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  UGFja2FnZUNpdGF0aW9ucw==
│ │ │  #:len=1928
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZhY2lsaXRhdGluZyBj
│ │ │  aXRhdGlvbiBvZiBNYWNhdWxheTIgcGFja2FnZXMiLCAibGluZW51bSIgPT4gMjY4LCBTZWVBbHNv
│ │ ├── ./usr/share/doc/Macaulay2/PackageTemplate/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  c2Vjb25kRnVuY3Rpb24=
│ │ │  #:len=379
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBzaWxseSBzZWNvbmQgZnVuY3Rpb24i
│ │ │  LCBEZXNjcmlwdGlvbiA9PiAoIlRoaXMgZnVuY3Rpb24gaXMgcHJvdmlkZWQgYnkgdGhlIHBhY2th
│ │ ├── ./usr/share/doc/Macaulay2/Parametrization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  cGFyYW1ldHJpemVDb25pYw==
│ │ │  #:len=1904
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHdoZXRoZXIgdG8gcmF0aW9u
│ │ │  YWxseSBwYXJhbWV0cml6ZSBjb25pY3MuIiwgImxpbmVudW0iID0+IDE2ODYsICJmaWxlbmFtZSIg
│ │ ├── ./usr/share/doc/Macaulay2/Parsing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Y2hhckFuYWx5emVy
│ │ │  #:len=752
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBsZXhpY2FsIGFuYWx5emVyIHRoYXQg
│ │ │  cHJvdmlkZXMgY2hhcmFjdGVycyBmcm9tIGEgc3RyaW5nIG9uZSBhdCBhIHRpbWUiLCAibGluZW51
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  cmFuZG9tRXh0ZW5zaW9uKE1hdHJpeCxNYXRyaXgp
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzE5Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmFuZG9tRXh0ZW5zaW9uLE1hdHJpeCxNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │  i3 : g=3
│ │ │  
│ │ │  o3 = 3
│ │ │  
│ │ │  i4 : kk= ZZ/101;
│ │ │  
│ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.36437s elapsed
│ │ │ + -- 2.32739s elapsed
│ │ │  
│ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │  
│ │ │  o6 : CliffordModule
│ │ │  
│ │ │ @@ -67,30 +67,30 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .537002s elapsed
│ │ │ + -- .72192s elapsed
│ │ │  
│ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │            -4: 16  .     -2: 32  .
│ │ │            -3: 16  .     -1:  .  .
│ │ │            -2:  .  .      0:  .  .
│ │ │            -1:  . 16      1:  . 32
│ │ │             0:  . 16
│ │ │  
│ │ │  o12 : Sequence
│ │ │  
│ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 21.965s elapsed
│ │ │ + -- 28.9739s elapsed
│ │ │  
│ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │  o14 : Matrix S   <-- S
│ │ │  
│ │ │  i15 : r=2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out
│ │ │ @@ -45,30 +45,30 @@
│ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o11 = CliffordModule{...6...}
│ │ │  
│ │ │  o11 : CliffordModule
│ │ │  
│ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.04827s elapsed
│ │ │ + -- 1.32685s elapsed
│ │ │  
│ │ │  i13 : betti res Ulr
│ │ │  
│ │ │               0  1 2
│ │ │  o13 = total: 8 16 8
│ │ │            0: 8 16 8
│ │ │  
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : ann Ulr == ideal qs
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.68875s elapsed
│ │ │ + -- 5.0853s elapsed
│ │ │  
│ │ │  i16 : betti res Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  o3 = 3
    
│ │ │            
│ │ │            
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │            
│ │ │            
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.36437s elapsed
    
│ │ │ + -- 2.32739s elapsed
    │ │ │ │ │ │ │ │ │ 
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │  
│ │ │  o6 : CliffordModule
    │ │ │ @@ -152,15 +152,15 @@ │ │ │ m12=randomExtension(m1.yAction,m2.yAction); │ │ │ V = vectorBundleOnE m12; │ │ │ Ul=tensorProduct(Mor,V); │ │ │ Ul1=tensorProduct(Mor1,V); │ │ │ d0=unique degrees target Ul.yAction; │ │ │ d1=unique degrees target Ul1.yAction; │ │ │ #d1 >=3 or #d0 >=3) do (); │ │ │ - -- .537002s elapsed     │ │ │ + -- .72192s elapsed │ │ │ │ │ │ │ │ │ 
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │            -4: 16  .     -2: 32  .
│ │ │ @@ -169,15 +169,15 @@
│ │ │            -1:  . 16      1:  . 32
│ │ │             0:  . 16
│ │ │  
│ │ │  o12 : Sequence
    │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 21.965s elapsed
    │ │ │ + -- 28.9739s elapsed │ │ │ │ │ │ │ │ │ 
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │  o14 : Matrix S   <-- S
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -56,15 +56,15 @@ │ │ │ │ -- will give an Ulrich bundle, with betti table │ │ │ │ -- 16 32 16 │ │ │ │ i3 : g=3 │ │ │ │ │ │ │ │ o3 = 3 │ │ │ │ i4 : kk= ZZ/101; │ │ │ │ i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g); │ │ │ │ - -- 1.36437s elapsed │ │ │ │ + -- 2.32739s elapsed │ │ │ │ i6 : M=cliffordModule(Mu1,Mu2,R) │ │ │ │ │ │ │ │ o6 = CliffordModule{...6...} │ │ │ │ │ │ │ │ o6 : CliffordModule │ │ │ │ i7 : Mor = vectorBundleOnE M.evenCenter; │ │ │ │ i8 : Mor1= vectorBundleOnE M.oddCenter; │ │ │ │ @@ -76,29 +76,29 @@ │ │ │ │ m12=randomExtension(m1.yAction,m2.yAction); │ │ │ │ V = vectorBundleOnE m12; │ │ │ │ Ul=tensorProduct(Mor,V); │ │ │ │ Ul1=tensorProduct(Mor1,V); │ │ │ │ d0=unique degrees target Ul.yAction; │ │ │ │ d1=unique degrees target Ul1.yAction; │ │ │ │ #d1 >=3 or #d0 >=3) do (); │ │ │ │ - -- .537002s elapsed │ │ │ │ + -- .72192s elapsed │ │ │ │ i12 : betti Ul.yAction, betti Ul1.yAction │ │ │ │ │ │ │ │ 0 1 0 1 │ │ │ │ o12 = (total: 32 32, total: 32 32) │ │ │ │ -4: 16 . -2: 32 . │ │ │ │ -3: 16 . -1: . . │ │ │ │ -2: . . 0: . . │ │ │ │ -1: . 16 1: . 32 │ │ │ │ 0: . 16 │ │ │ │ │ │ │ │ o12 : Sequence │ │ │ │ i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the │ │ │ │ actions of generators │ │ │ │ - -- 21.965s elapsed │ │ │ │ + -- 28.9739s elapsed │ │ │ │ i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S)); │ │ │ │ │ │ │ │ 32 32 │ │ │ │ o14 : Matrix S <-- S │ │ │ │ i15 : r=2 │ │ │ │ │ │ │ │ o15 = 2 │ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html │ │ │ @@ -139,15 +139,15 @@ │ │ │ │ │ │ o11 = CliffordModule{...6...} │ │ │ │ │ │ o11 : CliffordModule │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.04827s elapsed
    │ │ │ + -- 1.32685s elapsed │ │ │ │ │ │ │ │ │ 
    i13 : betti res Ulr
│ │ │  
│ │ │               0  1 2
│ │ │  o13 = total: 8 16 8
│ │ │            0: 8 16 8
│ │ │ @@ -157,15 +157,15 @@
│ │ │            
│ │ │                
    i14 : ann Ulr == ideal qs
│ │ │  
│ │ │  o14 = true
    
│ │ │            
│ │ │            
│ │ │                
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.68875s elapsed
    
│ │ │ + -- 5.0853s elapsed
    │ │ │ │ │ │ │ │ │ 
    i16 : betti res Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,27 +64,27 @@
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o11 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o11 : CliffordModule
│ │ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ │ - -- 1.04827s elapsed
│ │ │ │ + -- 1.32685s elapsed
│ │ │ │  i13 : betti res Ulr
│ │ │ │  
│ │ │ │               0  1 2
│ │ │ │  o13 = total: 8 16 8
│ │ │ │            0: 8 16 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : ann Ulr == ideal qs
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ │ - -- 2.68875s elapsed
│ │ │ │ + -- 5.0853s elapsed
│ │ │ │  i16 : betti res Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Permanents/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  cnlzZXI=
│ │ │  #:len=2107
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBwZXJtYW5lbnQgdXNpbmcg
│ │ │  UnlzZXIncyBmb3JtdWxhIiwgImxpbmVudW0iID0+IDQ0NCwgSW5wdXRzID0+IHtTUEFOe1RUeyJN
│ │ ├── ./usr/share/doc/Macaulay2/Permutations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY2VudHMoUGVybXV0YXRpb24p
│ │ │  #:len=258
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTMxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZXNjZW50cyxQZXJtdXRhdGlvbiksImRlc2NlbnRz
│ │ ├── ./usr/share/doc/Macaulay2/PhylogeneticTrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  cGh5bG9Ub3JpY1F1YWRz
│ │ │  #:len=2848
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgcXVhZHJhdGljIGlu
│ │ │  dmFyaWFudHMgb2YgYSBncm91cC1iYXNlZCBwaHlsb2dlbmV0aWMgbW9kZWwiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/PieriMaps/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c3RhbmRhcmRUYWJsZWF1eChaWixMaXN0KQ==
│ │ │  #:len=268
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTA1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzdGFuZGFyZFRhYmxlYXV4LFpaLExpc3QpLCJzdGFu
│ │ ├── ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  YWRkaXRpb24oSWRlYWwsSWRlYWwsSWRlYWwp
│ │ │  #:len=295
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRpdGlvbixJZGVhbCxJZGVhbCxJZGVhbCksImFk
│ │ ├── ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  cmFuZG9tUG9pbnRzTWF0KFJpbmcsWlop
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODc0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21Qb2ludHNNYXQsUmluZyxaWiksInJhbmRv
│ │ ├── ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
│ │ │ @@ -66,17 +66,17 @@
│ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.39928s elapsed
│ │ │ + -- 2.02362s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.0879s elapsed
│ │ │ + -- 3.72416s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -162,19 +162,19 @@
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.39928s elapsed
    │ │ │ + -- 2.02362s elapsed │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.0879s elapsed
    │ │ │ + -- 3.72416s elapsed │ │ │ │ │ │ │ │ │ 
    i12 : G==H
│ │ │  
│ │ │  o12 = true
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -82,17 +82,17 @@ │ │ │ │ o8 : Matrix K <-- K │ │ │ │ i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3} │ │ │ │ │ │ │ │ o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R); │ │ │ │ - -- 2.39928s elapsed │ │ │ │ + -- 2.02362s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 4.0879s elapsed │ │ │ │ + -- 3.72416s elapsed │ │ │ │ i12 : G==H │ │ │ │ │ │ │ │ o12 = true │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For reduced points, this function may be a bit slower than _a_f_f_i_n_e_P_o_i_n_t_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _a_f_f_i_n_e_F_a_t_P_o_i_n_t_s_B_y_I_n_t_e_r_s_e_c_t_i_o_n_(_M_a_t_r_i_x_,_L_i_s_t_,_R_i_n_g_) -- computes ideal of fat │ │ ├── ./usr/share/doc/Macaulay2/Polyhedra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ bWF4Q29uZXM= │ │ │ #:len=1185 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg │ │ │ Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gODQzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkYi │ │ ├── ./usr/share/doc/Macaulay2/Polymake/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ UG9seW1ha2U= │ │ │ #:len=610 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciBpbnRlcmZhY2lu │ │ │ ZyB3aXRoIHBvbHltYWtlIiwgRGVzY3JpcHRpb24gPT4gKEVNeyJQb2x5bWFrZSJ9LCIgaXMgYSBw │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ cG9seW9JZGVhbChMaXN0KQ== │ │ │ #:len=256 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwb2x5b0lkZWFsLExpc3QpLCJwb2x5b0lkZWFsKExp │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/___Field.out │ │ │ @@ -7,16 +7,16 @@ │ │ │ o1 : GaloisField │ │ │ │ │ │ i2 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}}; │ │ │ │ │ │ i3 : I = polyoIdeal(Q,Field=> F,RingChoice=>1,TermOrder=> GRevLex) │ │ │ │ │ │ o3 = ideal (- x x + x x , - x x + x x , - x x + │ │ │ - 3,2 2,3 3,3 2,2 3,1 2,2 3,2 2,1 3,1 2,3 │ │ │ + 3,1 2,3 3,3 2,1 2,1 1,2 2,2 1,1 3,1 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ x x , - x x + x x , - x x + x x ) │ │ │ - 3,3 2,1 2,1 1,2 2,2 1,1 3,1 1,2 3,2 1,1 │ │ │ + 3,2 1,1 3,2 2,3 3,3 2,2 3,1 2,2 3,2 2,1 │ │ │ │ │ │ o3 : Ideal of F[x , x , x , x , x , x , x , x ] │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/_polyo__Ideal.out │ │ │ @@ -5,44 +5,44 @@ │ │ │ o1 = {{{1, 1}, {2, 2}}, {{2, 1}, {3, 2}}, {{2, 2}, {3, 3}}} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 2,2 1,1 2,1 1,2 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ + 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 3,3 2,1 3,1 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x ) │ │ │ - 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 │ │ │ + 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ 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}}}; │ │ │ │ │ │ i4 : I = polyoIdeal Q │ │ │ │ │ │ o4 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 2,4 1,3 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 3,1 1,2 │ │ │ + 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 4,1 3,2 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 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 1,1 │ │ │ + 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 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 │ │ │ ------------------------------------------------------------------------ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ - 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 │ │ │ + 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 3,3 1,4 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 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 2,1 │ │ │ + 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 4,4 3,2 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 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 │ │ │ ------------------------------------------------------------------------ │ │ │ x x ) │ │ │ - 4,2 3,4 │ │ │ + 3,1 1,2 │ │ │ │ │ │ o4 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ] │ │ │ 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 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/html/___Field.html │ │ │ @@ -55,18 +55,18 @@ │ │ │ │ │ │ 
    i2 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};
    │ │ │ │ │ │ │ │ │ 
    i3 : I = polyoIdeal(Q,Field=> F,RingChoice=>1,TermOrder=> GRevLex)
│ │ │  
│ │ │  o3 = ideal (- x   x    + x   x   , - x   x    + x   x   , - x   x    +
│ │ │ -               3,2 2,3    3,3 2,2     3,1 2,2    3,2 2,1     3,1 2,3  
│ │ │ +               3,1 2,3    3,3 2,1     2,1 1,2    2,2 1,1     3,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   , - x   x    + x   x   , - x   x    + x   x   )
│ │ │ -      3,3 2,1     2,1 1,2    2,2 1,1     3,1 1,2    3,2 1,1
│ │ │ +      3,2 1,1     3,2 2,3    3,3 2,2     3,1 2,2    3,2 2,1
│ │ │  
│ │ │  o3 : Ideal of F[x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                   3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -12,18 +12,18 @@ │ │ │ │ o1 = F │ │ │ │ │ │ │ │ o1 : GaloisField │ │ │ │ i2 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}}; │ │ │ │ i3 : I = polyoIdeal(Q,Field=> F,RingChoice=>1,TermOrder=> GRevLex) │ │ │ │ │ │ │ │ o3 = ideal (- x x + x x , - x x + x x , - x x + │ │ │ │ - 3,2 2,3 3,3 2,2 3,1 2,2 3,2 2,1 3,1 2,3 │ │ │ │ + 3,1 2,3 3,3 2,1 2,1 1,2 2,2 1,1 3,1 1,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x , - x x + x x , - x x + x x ) │ │ │ │ - 3,3 2,1 2,1 1,2 2,2 1,1 3,1 1,2 3,2 1,1 │ │ │ │ + 3,2 1,1 3,2 2,3 3,3 2,2 3,1 2,2 3,2 2,1 │ │ │ │ │ │ │ │ o3 : Ideal of F[x , x , x , x , x , x , x , x ] │ │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _p_o_l_y_o_I_d_e_a_l -- Ideal of inner 2-minors of a collection of cells │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd FFiieelldd:: ********** │ │ │ │ * polyoIdeal(...,Field=>...) │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/html/_polyo__Ideal.html │ │ │ @@ -85,50 +85,50 @@ │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ 
    i2 : I = polyoIdeal Q
│ │ │  
│ │ │  o2 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             2,2 1,1    2,1 1,2   3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2 
│ │ │ +             3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2   3,3 2,1    3,1 2,3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   )
│ │ │ -      3,3 2,1    3,1 2,3   3,2 2,1    3,1 2,2
│ │ │ +      3,2 2,1    3,1 2,2   2,2 1,1    2,1 1,2
│ │ │  
│ │ │  o2 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
    │ │ │ │ │ │ │ │ │

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    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}}};
    │ │ │ 
    i4 : I = polyoIdeal Q
│ │ │  
│ │ │  o4 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             2,4 1,3    2,3 1,4   4,4 3,3    4,3 3,4   3,2 1,1    3,1 1,2 
│ │ │ +             4,4 2,3    4,3 2,4   2,2 1,1    2,1 1,2   4,2 3,1    4,1 3,2 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      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 1,1
│ │ │ +      2,3 1,1    2,1 1,3   4,3 3,1    4,1 3,3   4,4 1,3    4,3 1,4   3,4 2,3
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        2,1 1,3   4,3 3,1    4,1 3,3   4,4 1,3    4,3 1,4   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  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -      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 
│ │ │ +      2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2    4,2 3,3   3,4 1,3    3,3 1,4 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      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 2,1
│ │ │ +      3,2 2,1    3,1 2,2   4,2 1,1    4,1 1,2   2,4 1,2    2,2 1,4   4,4 3,2
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        3,1 2,2   4,2 1,1    4,1 1,2   2,4 1,2    2,2 1,4   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  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   )
│ │ │ -      4,2 3,4
│ │ │ +      3,1 1,2
│ │ │  
│ │ │  o4 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    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
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -32,47 +32,47 @@ │ │ │ │ │ │ │ │ o1 = {{{1, 1}, {2, 2}}, {{2, 1}, {3, 2}}, {{2, 2}, {3, 3}}} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 2,2 1,1 2,1 1,2 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ │ + 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 3,3 2,1 3,1 2,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x ) │ │ │ │ - 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 │ │ │ │ + 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ │ │ │ │ │ │ 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}}}; │ │ │ │ i4 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o4 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 2,4 1,3 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 3,1 1,2 │ │ │ │ + 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 4,1 3,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 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 1,1 │ │ │ │ + 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ │ - 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 │ │ │ │ + 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 3,3 1,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 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 2,1 │ │ │ │ + 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 4,4 3,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ - 4,2 3,4 │ │ │ │ + 3,1 1,2 │ │ │ │ │ │ │ │ o4 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x │ │ │ │ , x , x , x , x , x ] │ │ │ │ 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 │ │ │ │ ********** WWaayyss ttoo uussee ppoollyyooIIddeeaall:: ********** │ │ │ │ * polyoIdeal(List) │ │ ├── ./usr/share/doc/Macaulay2/Posets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=17 │ │ │ bWF4aW1hbEFudGljaGFpbnM= │ │ │ #:len=1127 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYWxsIG1heGltYWwgYW50 │ │ │ aWNoYWlucyBvZiBhIHBvc2V0IiwgImxpbmVudW0iID0+IDQ4NzgsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ @@ -31,27 +31,27 @@ │ │ │ o5 = CacheTable{name => P} │ │ │ │ │ │ i6 : C == P │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : time isDistributive C │ │ │ - -- used 0.000627246s (cpu); 7.434e-06s (thread); 0s (gc) │ │ │ + -- used 0.00378664s (cpu); 7.724e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 6.1046s (cpu); 4.14657s (thread); 0s (gc) │ │ │ + -- used 8.55574s (cpu); 5.58938s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 0.000291406s (cpu); 6.492e-06s (thread); 0s (gc) │ │ │ + -- used 0.000194174s (cpu); 3.597e-06s (thread); 0s (gc) │ │ │ │ │ │ o10 = true │ │ │ │ │ │ i11 : peek C'.cache │ │ │ │ │ │ o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}} } │ │ │ coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}} │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ @@ -7,22 +7,22 @@ │ │ │ o2 = Partition{4, 2} │ │ │ │ │ │ o2 : Partition │ │ │ │ │ │ i3 : D = dominanceLattice 6; │ │ │ │ │ │ i4 : time greeneKleitmanPartition(D, Strategy => "antichains") │ │ │ - -- used 0.454112s (cpu); 0.262257s (thread); 0s (gc) │ │ │ + -- used 0.505912s (cpu); 0.30656s (thread); 0s (gc) │ │ │ │ │ │ o4 = Partition{9, 2} │ │ │ │ │ │ o4 : Partition │ │ │ │ │ │ i5 : time greeneKleitmanPartition(D, Strategy => "chains") │ │ │ - -- used 0.000245891s (cpu); 1.076e-05s (thread); 0s (gc) │ │ │ + -- used 0.000155412s (cpu); 8.536e-06s (thread); 0s (gc) │ │ │ │ │ │ o5 = Partition{9, 2} │ │ │ │ │ │ o5 : Partition │ │ │ │ │ │ i6 : greeneKleitmanPartition chain 10 │ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ @@ -87,35 +87,35 @@ │ │ │ │ │ │ 
    i6 : C == P
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ 
    i7 : time isDistributive C
│ │ │ - -- used 0.000627246s (cpu); 7.434e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00378664s (cpu); 7.724e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 6.1046s (cpu); 4.14657s (thread); 0s (gc)
│ │ │ + -- used 8.55574s (cpu); 5.58938s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │
    │ │ │

    We also know that the dual of a distributive lattice is again a distributive lattice. Other information is copied when possible.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : C' = dual C;
    │ │ │ 
    i10 : time isDistributive C'
│ │ │ - -- used 0.000291406s (cpu); 6.492e-06s (thread); 0s (gc)
│ │ │ + -- used 0.000194174s (cpu); 3.597e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    │ │ │ 
    i11 : peek C'.cache
│ │ │  
│ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}                                      }
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 0.000627246s (cpu); 7.434e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.00378664s (cpu); 7.724e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.1046s (cpu); 4.14657s (thread); 0s (gc)
│ │ │ │ + -- used 8.55574s (cpu); 5.58938s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 0.000291406s (cpu); 6.492e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.000194174s (cpu); 3.597e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : peek C'.cache
│ │ │ │  
│ │ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}
│ │ │ │  }
│ │ │ │                   coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4},
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html
│ │ │ @@ -95,23 +95,23 @@
│ │ │          
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i3 : D = dominanceLattice 6;
    
│ │ │      		          
                  
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.454112s (cpu); 0.262257s (thread); 0s (gc)
│ │ │ + -- used 0.505912s (cpu); 0.30656s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │      		          
                  
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 0.000245891s (cpu); 1.076e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000155412s (cpu); 8.536e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │      		          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,21 +29,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.454112s (cpu); 0.262257s (thread); 0s (gc)
│ │ │ │ + -- used 0.505912s (cpu); 0.30656s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 0.000245891s (cpu); 1.076e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000155412s (cpu); 8.536e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o5 : Partition
│ │ │ │  The Greene-Kleitman partition of the $n$ _c_h_a_i_n is the partition of $n$ with $1$
│ │ │ │  part.
│ │ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/PositivityToricBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZHJhd1BhcmxpYW1lbnQyRHRpa3o=
│ │ │  #:len=2626
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmlzdWFsaXNlcyB0aGUgcGFybGlhbWVu
│ │ │  dCBvZiBwb2x5dG9wZXMgZm9yIGEgdmVjdG9yIGJ1bmRsZSBvbiBhIHRvcmljIHN1cmZhY2UgdXNp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  cHJpbWFyeUNvbXBvbmVudCguLi4sU3RyYXRlZ3k9Pi4uLik=
│ │ │  #:len=316
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzQ5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcmltYXJ5Q29tcG9uZW50LFN0cmF0ZWd5XSwicHJp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out
│ │ │ @@ -24,35 +24,35 @@
│ │ │                | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │  
│ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .225134s elapsed
│ │ │ + -- .186633s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
│ │ │  
│ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0849113s elapsed
│ │ │ + -- .0712993s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
│ │ │  
│ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0294912s elapsed
│ │ │ + -- .0155793s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       x x )
│ │ │        0 4
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0877917s elapsed
│ │ │ + -- .0808574s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : R = QQ[h,l,s,x,y,z]
│ │ │ @@ -41,15 +41,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .0078238s elapsed
│ │ │ + -- .00797509s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -101,37 +101,37 @@
│ │ │                | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
    │ │ │ 
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .225134s elapsed
│ │ │ + -- .186633s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
    │ │ │ 
    i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0849113s elapsed
│ │ │ + -- .0712993s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
    │ │ │ 
    i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0294912s elapsed
│ │ │ + -- .0155793s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
    │ │ │ ├── html2text {} │ │ │ │ @@ -42,39 +42,39 @@ │ │ │ │ | │ │ │ │ | 0 0 0 0 x_1^5- │ │ │ │ x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o3 : R-module, quotient of R │ │ │ │ i4 : elapsedTime kernelOfLocalization(M, I1) │ │ │ │ - -- .225134s elapsed │ │ │ │ + -- .186633s elapsed │ │ │ │ │ │ │ │ o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 1 0 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 1 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o4 : R-module, subquotient of R │ │ │ │ i5 : elapsedTime kernelOfLocalization(M, I2) │ │ │ │ - -- .0849113s elapsed │ │ │ │ + -- .0712993s elapsed │ │ │ │ │ │ │ │ o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 0 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 1 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o5 : R-module, subquotient of R │ │ │ │ i6 : elapsedTime kernelOfLocalization(M, I3) │ │ │ │ - -- .0294912s elapsed │ │ │ │ + -- .0155793s elapsed │ │ │ │ │ │ │ │ o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 1 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 0 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ x x ) │ │ │ 0 4 │ │ │ │ │ │ o2 : Ideal of R │ │ │ 
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0877917s elapsed
│ │ │ + -- .0808574s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │
    │ │ │ @@ -140,15 +140,15 @@ │ │ │ │ │ │ 
    i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .0078238s elapsed
│ │ │ + -- .00797509s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ 2 7 0 7 3 6 2 6 1 6 0 6 2 5 0 5 3 4 2 4 1 4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 4 │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime regSeqInIdeal I │ │ │ │ - -- .0877917s elapsed │ │ │ │ + -- .0808574s elapsed │ │ │ │ │ │ │ │ o3 = ideal (x x , x x + x x , x x + x x , x x + x x ) │ │ │ │ 2 7 3 6 0 7 2 5 0 7 1 4 0 7 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ If I is the unit ideal, then an ideal of variables of the ring is returned. │ │ │ │ If the codimension of I is already known, then one can specify this, along with │ │ │ │ @@ -71,15 +71,15 @@ │ │ │ │ l , s ) │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3 │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1) │ │ │ │ - -- .0078238s elapsed │ │ │ │ + -- .00797509s elapsed │ │ │ │ │ │ │ │ 2 3 2 2 8 3 2 2 │ │ │ │ o7 = ideal (h*l - l - 4l*s + h*y, h + l s - h x, s + h + l s - h x) │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_d_i_c_a_l -- the radical of an ideal │ │ ├── ./usr/share/doc/Macaulay2/Probability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ dERpc3RyaWJ1dGlvbihOdW1iZXIp │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIyMiwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodERpc3RyaWJ1dGlvbixOdW1iZXIpLCJ0RGlzdHJp │ │ ├── ./usr/share/doc/Macaulay2/PruneComplex/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ cHJ1bmVDb21wbGV4KC4uLixQcnVuaW5nTWFwPT4uLi4p │ │ │ #:len=276 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQ3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcnVuZUNvbXBsZXgsUHJ1bmluZ01hcF0sInBydW5l │ │ ├── ./usr/share/doc/Macaulay2/PseudomonomialPrimaryDecomposition/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ UHNldWRvbW9ub21pYWxQcmltYXJ5RGVjb21wb3NpdGlvbg== │ │ │ #:len=1413 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbWFyeSBkZWNvbXBvc2l0aW9uIG9m │ │ │ IGEgc3F1YXJlIGZyZWUgcHNldWRvbW9ub21pYWwgaWRlYWwiLCBEZXNjcmlwdGlvbiA9PiAoRElW │ │ ├── ./usr/share/doc/Macaulay2/Pullback/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ aW50ZXJuYWxVc2VEaXJlY3RTdW0oUmluZyxSaW5nKQ== │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpbnRlcm5hbFVzZURpcmVjdFN1bSxSaW5nLFJpbmcp │ │ ├── ./usr/share/doc/Macaulay2/PushForward/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ cHVzaEZ3ZChNb2R1bGUp │ │ │ #:len=267 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwdXNoRndkLE1vZHVsZSksInB1c2hGd2QoTW9kdWxl │ │ ├── ./usr/share/doc/Macaulay2/Python/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ UHl0aG9uT2JqZWN0IC8gUHl0aG9uT2JqZWN0 │ │ │ #:len=267 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgLyxQeXRob25PYmplY3QsUHl0aG9uT2Jq │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_iterator_lp__Python__Object_rp.out │ │ │ @@ -4,12 +4,12 @@ │ │ │ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ i2 : i = iterator x │ │ │ │ │ │ -o2 = │ │ │ +o2 = │ │ │ │ │ │ o2 : PythonObject of class range_iterator │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ i2 : i = iterator x │ │ │ │ │ │ -o2 = │ │ │ +o2 = │ │ │ │ │ │ o2 : PythonObject of class range_iterator │ │ │ │ │ │ i3 : next i │ │ │ │ │ │ o3 = 0 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_to__Python.out │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ o12 = m2sqrt │ │ │ │ │ │ o12 : FunctionClosure │ │ │ │ │ │ i13 : pysqrt = toPython m2sqrt │ │ │ │ │ │ -o13 = │ │ │ +o13 = │ │ │ │ │ │ o13 : PythonObject of class builtin_function_or_method │ │ │ │ │ │ i14 : pysqrt 2 │ │ │ calling Macaulay2 code from Python! │ │ │ │ │ │ o14 = 1.4142135623730951 │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ │ │ │ 
    i2 : i = iterator x
│ │ │  
│ │ │ -o2 = <range_iterator object at 0x7fc71203b450>
│ │ │ +o2 = <range_iterator object at 0x7f09c3c17390>
│ │ │  
│ │ │  o2 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -18,14 +18,14 @@ │ │ │ │ i1 : x = pythonValue "range(3)" │ │ │ │ │ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ i2 : i = iterator x │ │ │ │ │ │ │ │ -o2 = │ │ │ │ +o2 = │ │ │ │ │ │ │ │ o2 : PythonObject of class range_iterator │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html │ │ │ @@ -73,15 +73,15 @@ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ │ │ │ 
    i2 : i = iterator x
│ │ │  
│ │ │ -o2 = <range_iterator object at 0x7fc4db76f420>
│ │ │ +o2 = <range_iterator object at 0x7ff7810bf390>
│ │ │  
│ │ │  o2 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ 
    i3 : next i
│ │ │  
│ │ │  o3 = 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i1 : x = pythonValue "range(3)"
│ │ │ │  
│ │ │ │  o1 = range(0, 3)
│ │ │ │  
│ │ │ │  o1 : PythonObject of class range
│ │ │ │  i2 : i = iterator x
│ │ │ │  
│ │ │ │ -o2 = 
│ │ │ │ +o2 = 
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range_iterator
│ │ │ │  i3 : next i
│ │ │ │  
│ │ │ │  o3 = 0
│ │ │ │  
│ │ │ │  o3 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -156,15 +156,15 @@
│ │ │  o12 = m2sqrt
│ │ │  
│ │ │  o12 : FunctionClosure
    │ │ │ │ │ │ │ │ │ 
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f76c3e22890>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7feb9f892890>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    │ │ │ │ │ │ │ │ │ 
    i14 : pysqrt 2
│ │ │  calling Macaulay2 code from Python!
│ │ │ ├── html2text {}
│ │ │ │ @@ -73,15 +73,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/QthPower/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  bWluaW1pemF0aW9u
│ │ │  #:len=2755
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hhbmdlIHRvIGEgYmV0dGVyIE5vZXRo
│ │ │  ZXIgbm9ybWFsaXphdGlvbiBzdWdnZXN0ZWQgYnkgdGhlIGluZHVjZWQgd2VpZ2h0cyIsICJsaW5l
│ │ ├── ./usr/share/doc/Macaulay2/QuadraticIdealExamplesByRoos/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aGlnaGVyRGVwdGhUYWJsZQ==
│ │ │  #:len=793
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlcyBoYXNodGFibGUgb2YgSmFu
│ │ │  LUVyaWsgUm9vcycgZXhhbXBsZXMgb2YgcXVhZHJhdGljIGlkZWFscyB3aXRoIHBvc2l0aXZlIGRl
│ │ ├── ./usr/share/doc/Macaulay2/Quasidegrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  cXVhc2lkZWdyZWVzTG9jYWxDb2hvbW9sb2d5
│ │ │  #:len=3881
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgcXVhc2lkZWdyZWUg
│ │ │  c2V0cyBvZiBsb2NhbCBjb2hvbW9sb2d5IG1vZHVsZXMiLCAibGluZW51bSIgPT4gNzczLCBJbnB1
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  W1FRXQ==
│ │ │  #:len=544
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUXVhdGVybmFyeSBRdWFydGljIEZvcm1z
│ │ │  IGFuZCBHb3JlbnN0ZWluIHJpbmdzIChLYXB1c3RrYSwgS2FwdXN0a2EsIFJhbmVzdGFkLCBTY2hl
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -179,15 +179,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 3.03312s elapsed
│ │ │ + -- 4.69806s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : I = pointsIdeal randomPoints(S, 6)
│ │ │  
│ │ │                               2                              2   2          
│ │ │  o24 = ideal (a*c - 7b*c - 49c  + 40a*d - 42b*d + 12c*d + 28d , b  - 36b*c -
│ │ │ @@ -301,15 +301,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.63738s elapsed
│ │ │ + -- 3.59341s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │  
│ │ │ @@ -319,37 +319,37 @@
│ │ │  
│ │ │  i43 : compsL441/dim == {16, 14}
│ │ │  
│ │ │  o43 = true
│ │ │  
│ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │ +o44 = | -49 11 -25 17 44 -16 -27 40 -34 20 -19 29 41 19 -40 42 -13 5 17 39 31
│ │ │        -----------------------------------------------------------------------
│ │ │ -      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │ +      10 45 26 43 49 -32 -29 19 -50 15 18 37 -10 34 -28 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
│ │ │  
│ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │                2              2                              2               
│ │ │ -o45 = ideal (a  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │ +o45 = ideal (a  - 40b*c + 20c  + 29a*d - 16b*d - 25c*d - 49d , a*b + 43b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                          2   2              2                  
│ │ │ -      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*d +
│ │ │ +         2                              2   2              2                
│ │ │ +      17c  + 31a*d + 41b*d - 34c*d + 11d , b  + 34b*c - 29c  + 37a*d + 10b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                             2     2  
│ │ │ -      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*d + 19c*d - 22d , b*c  -
│ │ │ +                   2                   2                              2     2
│ │ │ +      + 42c*d - 27d , a*c + 18b*c + 49c  + 19a*d + 39b*d + 19c*d + 44d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │ +                     2         2        2        2      3   3            
│ │ │ +      - 50b*c*d + 45c d - 32a*d  - 13b*d  - 19c*d  + 17d , c  - 28b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │ +         2         2        2       2      3
│ │ │ +      15c d - 10a*d  + 26b*d  + 5c*d  + 40d )
│ │ │  
│ │ │  o45 : Ideal of S
│ │ │  
│ │ │  i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │  o46 = total: 1 6 8 3
│ │ │ @@ -357,84 +357,81 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o46 : BettiTally
│ │ │  
│ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 5d, b - 33d, a - 21d)                                                                            |
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      2              2                                                      |
│ │ │ -      |ideal (b + 45c + 49d, a - 22c - 26d, c  + 49c*d + 42d )                                                     |
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2                      2                           2   2                     2 |
│ │ │ -      |ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d + 27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 19d, b - 10d, a + 6d)                                                                                                                                   |
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2              2                      2   3                2        2        2      3     2                2         2        2      3 |
│ │ │ +      |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 )|
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                             
│ │ │ -o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │ +                                                                     2  
│ │ │ +o48 = {ideal (c + 19d, b - 10d, a + 6d), ideal (a + 18b + 49c - 3d, b  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -       2              2                                2                  
│ │ │ -      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │ +                 2                      2   3                2        2  
│ │ │ +      34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                           2   2                     2
│ │ │ -      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │ +           2      3     2                2         2        2      3
│ │ │ +      10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = b + 45c + 49d
│ │ │ +o49 = a + 18b + 49c - 3d
│ │ │  
│ │ │  o49 : S
│ │ │  
│ │ │  i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 2, 3}
│ │ │ +o50 = {1, 5}
│ │ │  
│ │ │  o50 : List
│ │ │  
│ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true, false}
│ │ │ +o51 = {false, true}
│ │ │  
│ │ │  o51 : List
│ │ │  
│ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 2
│ │ │ +o52 = 5
│ │ │  
│ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ +o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ +      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
│ │ │  
│ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │ +              2              2                              2               
│ │ │ +o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                 
│ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ +        2                              2   2              2                  
│ │ │ +      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                 2                             2     2            
│ │ │ -      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │ +         2                   2                              2     2          
│ │ │ +      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │ +          2         2       2        2     3   3               2        2  
│ │ │ +      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ +           2       2      3
│ │ │ +      47b*d  - 4c*d  + 27d )
│ │ │  
│ │ │  o54 : Ideal of S
│ │ │  
│ │ │  i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │  o55 = total: 1 6 8 3
│ │ │ @@ -442,114 +439,80 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o55 : BettiTally
│ │ │  
│ │ │  i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                                                                                                                               |
│ │ │ +o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )                                                                                                                                                              |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                             2                                 2                                   2   2                            2                                   2   2                             2 |
│ │ │ +      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d , a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b + 16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                         2   2                      2                                                                                                             |
│ │ │ -o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b  + 28b*d - 32c*d + 16d )                                                                                                            |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                      2                                                                                                           |
│ │ │ -      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b  - 20b*d + 29c*d + 38d )                                                                                                          |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                     2                                                                                                            |
│ │ │ -      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b  - 8b*d - 12c*d - 46d )                                                                                                           |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     3      2         2      3                                                                                                                                                      |
│ │ │ -      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )                                                                                                                                                     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                      2   2      2                      2   3      2         2        2      3                                                                                |
│ │ │ -      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c  + 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )                                                                               |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                   2                      2   2      2                      2   3      2         2        2     3                                                                                   |
│ │ │ -      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  + 34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )                                                                                  |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                            2                                 2                                 2   2                              2                                  2   2                              2 |
│ │ │ -      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                     2   2      2                      2   3      2         2        2     3                                                                                  |
│ │ │ -      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c  + 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )                                                                                 |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                   2   2                              2                                 2   2                           2  |
│ │ │ -      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                  2   2                      2                                   2   2                              2      |
│ │ │ -      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o57 = ++
│ │ │ +      ++
│ │ │  
│ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │ +o58 = | 44 15 26 -28 -4 33 49 -19 -6 -18 -37 -34 18 -42 -23 23 38 -40 27 26
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │ +      -39 -24 -21 -13 38 39 -33 15 8 -18 18 -22 -2 -29 -23 46 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
│ │ │  
│ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ +o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ +      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │  
│ │ │                 1       36
│ │ │  o59 : Matrix kk  <-- kk
│ │ │  
│ │ │  i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │ +              2              2                              2               
│ │ │ +o60 = ideal (a  - 23b*c - 18c  - 34a*d + 33b*d + 26c*d + 44d , a*b + 38b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │ +         2                             2   2              2                 
│ │ │ +      27c  - 39a*d + 18b*d - 6c*d + 15d , b  - 23b*c + 15c  - 2a*d - 24b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                  2                             2     2  
│ │ │ -      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │ +                 2                   2                            2     2  
│ │ │ +      23c*d + 49d , a*c - 22b*c + 39c  + 8a*d + 26b*d - 42c*d - 4d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      18b*c*d - 21c d - 33a*d  + 38b*d  - 37c*d  - 28d , c  + 46b*c*d + 18c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2        2        2      3
│ │ │ -      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │ +             2        2        2      3
│ │ │ +      - 29a*d  - 13b*d  - 40c*d  - 19d )
│ │ │  
│ │ │  o60 : Ideal of S
│ │ │  
│ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ +              2            2                             2               
│ │ │ +o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ +         2                              2   2              2                
│ │ │ +      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2         
│ │ │ -      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │ +           2                  2                             2     2         
│ │ │ +      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2        2        2      3   3                2        2
│ │ │ -      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │ +           2         2       2       2      3   3                2         2
│ │ │ +      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ +             2        2      3
│ │ │ +      + 33b*d  - 14c*d  + 33d )
│ │ │  
│ │ │  o61 : Ideal of S
│ │ │  
│ │ │  i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │  o62 = total: 1 6 8 3
│ │ │ @@ -567,38 +530,34 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o63 : BettiTally
│ │ │  
│ │ │  i64 : netList decompose I0
│ │ │  
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 40d, b - 10d, a + 32d)                                                                      |
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      2              2                                                 |
│ │ │ -      |ideal (b + 10c + 25d, a + 27c - 50d, c  - 34c*d - 17d )                                                |
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ -      |                            2                    2                          2   2                    2 |
│ │ │ -      |ideal (a - 13b - 2c + 29d, c  - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d + 8d , b  - b*d + 15c*d + 40d )|
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ +      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o64 = |ideal (c + 8d, b + 5d, a - 25d)                                                                                                                             |
│ │ │ +      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                             2              2              2   3                2         2        2      3     2                2         2        2     3 |
│ │ │ +      |ideal (a - 22b + 39c + 50d, b  - 23b*c + 15c  + 33b*d + 48d , c  + 46b*c*d + 18c d - 45b*d  - 20c*d  + 17d , b*c  - 18b*c*d - 21c d + 19b*d  + 38c*d  + 6d )|
│ │ │ +      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i65 : netList decompose I1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o65 = |ideal (c + 32d, b + 18d, a - 33d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ -      +------------------------------------------------------+
│ │ │ +      +---------------------------------+
│ │ │ +o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ +      +---------------------------------+
│ │ │  
│ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │  
│ │ │  o66 : Ideal of T
│ │ │  
│ │ │  i67 : C = res(I, FastNonminimal => true)
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -291,15 +291,15 @@
│ │ │  o21 : Ideal of T
    │ │ │ │ │ │ │ │ │ 
    i22 : assert(dim L == 18)
    │ │ │ │ │ │ │ │ │ 
    i23 : elapsedTime isPrime L
│ │ │ - -- 3.03312s elapsed
│ │ │ + -- 4.69806s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │
    │ │ │

    The Schreyer resolution and minimal Betti numbers

    │ │ │
    │ │ │ @@ -469,15 +469,15 @@ │ │ │ │ │ │ 
    i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
    │ │ │ │ │ │ │ │ │ 
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.63738s elapsed
    │ │ │ + -- 3.59341s elapsed │ │ │ │ │ │ │ │ │ 
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    │ │ │ │ │ │ │ │ │ @@ -496,38 +496,38 @@ │ │ │
    │ │ │

    Both components are rational, and here are random points, one on each component:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │ +o44 = | -49 11 -25 17 44 -16 -27 40 -34 20 -19 29 41 19 -40 42 -13 5 17 39 31
│ │ │        -----------------------------------------------------------------------
│ │ │ -      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │ +      10 45 26 43 49 -32 -29 19 -50 15 18 37 -10 34 -28 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
    │ │ │ 
    i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │                2              2                              2               
│ │ │ -o45 = ideal (a  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │ +o45 = ideal (a  - 40b*c + 20c  + 29a*d - 16b*d - 25c*d - 49d , a*b + 43b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                          2   2              2                  
│ │ │ -      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*d +
│ │ │ +         2                              2   2              2                
│ │ │ +      17c  + 31a*d + 41b*d - 34c*d + 11d , b  + 34b*c - 29c  + 37a*d + 10b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                             2     2  
│ │ │ -      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*d + 19c*d - 22d , b*c  -
│ │ │ +                   2                   2                              2     2
│ │ │ +      + 42c*d - 27d , a*c + 18b*c + 49c  + 19a*d + 39b*d + 19c*d + 44d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │ +                     2         2        2        2      3   3            
│ │ │ +      - 50b*c*d + 45c d - 32a*d  - 13b*d  - 19c*d  + 17d , c  - 28b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │ +         2         2        2       2      3
│ │ │ +      15c d - 10a*d  + 26b*d  + 5c*d  + 40d )
│ │ │  
│ │ │  o45 : Ideal of S
    │ │ │ 
    i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │ @@ -537,93 +537,90 @@
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o46 : BettiTally
    │ │ │ 
    i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 5d, b - 33d, a - 21d)                                                                            |
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      2              2                                                      |
│ │ │ -      |ideal (b + 45c + 49d, a - 22c - 26d, c  + 49c*d + 42d )                                                     |
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2                      2                           2   2                     2 |
│ │ │ -      |ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d + 27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------+
    │ │ │ + +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ +o47 = |ideal (c + 19d, b - 10d, a + 6d) | │ │ │ + +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 3 2 2 2 3 2 2 2 2 3 | │ │ │ + |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 )| │ │ │ + +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ 
    i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                             
│ │ │ -o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │ +                                                                     2  
│ │ │ +o48 = {ideal (c + 19d, b - 10d, a + 6d), ideal (a + 18b + 49c - 3d, b  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -       2              2                                2                  
│ │ │ -      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │ +                 2                      2   3                2        2  
│ │ │ +      34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                           2   2                     2
│ │ │ -      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │ +           2      3     2                2         2        2      3
│ │ │ +      10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )}
│ │ │  
│ │ │  o48 : List
    │ │ │ 
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = b + 45c + 49d
│ │ │ +o49 = a + 18b + 49c - 3d
│ │ │  
│ │ │  o49 : S
    │ │ │ 
    i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 2, 3}
│ │ │ +o50 = {1, 5}
│ │ │  
│ │ │  o50 : List
    │ │ │ 
    i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true, false}
│ │ │ +o51 = {false, true}
│ │ │  
│ │ │  o51 : List
    │ │ │ 
    i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 2
    │ │ │ +o52 = 5 │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ +o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ +      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
    │ │ │ 
    i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │ +              2              2                              2               
│ │ │ +o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                 
│ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ +        2                              2   2              2                  
│ │ │ +      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                 2                             2     2            
│ │ │ -      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │ +         2                   2                              2     2          
│ │ │ +      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │ +          2         2       2        2     3   3               2        2  
│ │ │ +      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ +           2       2      3
│ │ │ +      47b*d  - 4c*d  + 27d )
│ │ │  
│ │ │  o54 : Ideal of S
    │ │ │ 
    i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │ @@ -633,126 +630,92 @@
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o55 : BettiTally
    │ │ │ 
    i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ -      +-------------------------------------------------------+
    │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ +o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c + 12c*d - 37d ) | │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d , a*c + 43a*d + 44b*d - 48c*d + 24d , b - 4a*d - 40b*d - 9c*d - 39d , a*b + 16a*d + 26b*d + 37c*d - 24d , a - 25a*d + 47b*d + 34c*d + 8d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ 
    i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                         2   2                      2                                                                                                             |
│ │ │ -o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b  + 28b*d - 32c*d + 16d )                                                                                                            |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                      2                                                                                                           |
│ │ │ -      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b  - 20b*d + 29c*d + 38d )                                                                                                          |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                     2                                                                                                            |
│ │ │ -      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b  - 8b*d - 12c*d - 46d )                                                                                                           |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     3      2         2      3                                                                                                                                                      |
│ │ │ -      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )                                                                                                                                                     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                      2   2      2                      2   3      2         2        2      3                                                                                |
│ │ │ -      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c  + 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )                                                                               |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                   2                      2   2      2                      2   3      2         2        2     3                                                                                   |
│ │ │ -      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  + 34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )                                                                                  |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                            2                                 2                                 2   2                              2                                  2   2                              2 |
│ │ │ -      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                     2   2      2                      2   3      2         2        2     3                                                                                  |
│ │ │ -      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c  + 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )                                                                                 |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                   2   2                              2                                 2   2                           2  |
│ │ │ -      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                  2   2                      2                                   2   2                              2      |
│ │ │ -      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ +o57 = ++ │ │ │ + ++ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │ +o58 = | 44 15 26 -28 -4 33 49 -19 -6 -18 -37 -34 18 -42 -23 23 38 -40 27 26
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │ +      -39 -24 -21 -13 38 39 -33 15 8 -18 18 -22 -2 -29 -23 46 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
    │ │ │ 
    i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ +o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ +      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │  
│ │ │                 1       36
│ │ │  o59 : Matrix kk  <-- kk
    │ │ │
    │ │ │
    │ │ │

    We compute the ideal of the corresponding zero dimensional scheme with length 6, corresponding to the points pt0, pt1 in Hilb.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │ +              2              2                              2               
│ │ │ +o60 = ideal (a  - 23b*c - 18c  - 34a*d + 33b*d + 26c*d + 44d , a*b + 38b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │ +         2                             2   2              2                 
│ │ │ +      27c  - 39a*d + 18b*d - 6c*d + 15d , b  - 23b*c + 15c  - 2a*d - 24b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                  2                             2     2  
│ │ │ -      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │ +                 2                   2                            2     2  
│ │ │ +      23c*d + 49d , a*c - 22b*c + 39c  + 8a*d + 26b*d - 42c*d - 4d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      18b*c*d - 21c d - 33a*d  + 38b*d  - 37c*d  - 28d , c  + 46b*c*d + 18c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2        2        2      3
│ │ │ -      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │ +             2        2        2      3
│ │ │ +      - 29a*d  - 13b*d  - 40c*d  - 19d )
│ │ │  
│ │ │  o60 : Ideal of S
    │ │ │ 
    i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ +              2            2                             2               
│ │ │ +o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ +         2                              2   2              2                
│ │ │ +      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2         
│ │ │ -      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │ +           2                  2                             2     2         
│ │ │ +      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2        2        2      3   3                2        2
│ │ │ -      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │ +           2         2       2       2      3   3                2         2
│ │ │ +      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ +             2        2      3
│ │ │ +      + 33b*d  - 14c*d  + 33d )
│ │ │  
│ │ │  o61 : Ideal of S
    │ │ │ 
    i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │ @@ -775,39 +738,35 @@
│ │ │  o63 : BettiTally
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i64 : netList decompose I0
│ │ │  
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 40d, b - 10d, a + 32d)                                                                      |
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      2              2                                                 |
│ │ │ -      |ideal (b + 10c + 25d, a + 27c - 50d, c  - 34c*d - 17d )                                                |
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
│ │ │ -      |                            2                    2                          2   2                    2 |
│ │ │ -      |ideal (a - 13b - 2c + 29d, c  - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d + 8d , b  - b*d + 15c*d + 40d )|
│ │ │ -      +-------------------------------------------------------------------------------------------------------+
    │ │ │ + +------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ +o64 = |ideal (c + 8d, b + 5d, a - 25d) | │ │ │ + +------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 3 2 2 2 3 2 2 2 2 3 | │ │ │ + |ideal (a - 22b + 39c + 50d, b - 23b*c + 15c + 33b*d + 48d , c + 46b*c*d + 18c d - 45b*d - 20c*d + 17d , b*c - 18b*c*d - 21c d + 19b*d + 38c*d + 6d )| │ │ │ + +------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ 
    i65 : netList decompose I1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o65 = |ideal (c + 32d, b + 18d, a - 33d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ -      +------------------------------------------------------+
    │ │ │ + +---------------------------------+ │ │ │ +o65 = |ideal (c - 9d, b + 15d, a + 27d) | │ │ │ + +---------------------------------+ │ │ │ + |ideal (c + 48d, b + 11d, a - 37d)| │ │ │ + +---------------------------------+ │ │ │ + |ideal (c + 29d, b + 46d, a + 18d)| │ │ │ + +---------------------------------+ │ │ │ + |ideal (c + 24d, b + 46d, a + 33d)| │ │ │ + +---------------------------------+ │ │ │ + |ideal (c + 22d, b + 38d, a - 50d)| │ │ │ + +---------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │  
│ │ │  o66 : Ideal of T
    │ │ │ ├── html2text {} │ │ │ │ @@ -250,15 +250,15 @@ │ │ │ │ | 31 33 32 34 35 36 | │ │ │ │ +--------------------------------------------------------------+ │ │ │ │ i21 : L = trim groebnerStratum F; │ │ │ │ │ │ │ │ o21 : Ideal of T │ │ │ │ i22 : assert(dim L == 18) │ │ │ │ i23 : elapsedTime isPrime L │ │ │ │ - -- 3.03312s elapsed │ │ │ │ + -- 4.69806s elapsed │ │ │ │ │ │ │ │ o23 = true │ │ │ │ ********** TThhee SScchhrreeyyeerr rreessoolluuttiioonn aanndd mmiinniimmaall BBeettttii nnuummbbeerrss ********** │ │ │ │ Schreyer's construction of a nonminimal free resolution starts with a Groebner │ │ │ │ basis. First, one constructs the SScchhrreeyyeerr ffrraammee (see La Scala, Stillman). This │ │ │ │ is determined solely from the initial ideal $J$ and its minimal generators (but │ │ │ │ depends on some choices of ordering, but otherwise is combinatorial). This │ │ │ │ @@ -414,15 +414,15 @@ │ │ │ │ We now compute the locus in $V(L)$ where the Betti diagram has no cancellation. │ │ │ │ This is a closed subscheme of $V(L)$, which is a closed subscheme of the │ │ │ │ Hilbert scheme. Notice that there are two components. │ │ │ │ i39 : L441 = trim(L + ideal M1); │ │ │ │ │ │ │ │ o39 : Ideal of T │ │ │ │ i40 : elapsedTime compsL441 = decompose L441; │ │ │ │ - -- 2.63738s elapsed │ │ │ │ + -- 3.59341s elapsed │ │ │ │ i41 : #compsL441 │ │ │ │ │ │ │ │ o41 = 2 │ │ │ │ i42 : compsL441/dim -- two components, of dimensions 14 and 16. │ │ │ │ │ │ │ │ o42 = {16, 14} │ │ │ │ │ │ │ │ @@ -430,36 +430,36 @@ │ │ │ │ i43 : compsL441/dim == {16, 14} │ │ │ │ │ │ │ │ o43 = true │ │ │ │ Both components are rational, and here are random points, one on each │ │ │ │ component: │ │ │ │ i44 : pta = randomPointOnRationalVariety compsL441_0 │ │ │ │ │ │ │ │ -o44 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6 │ │ │ │ +o44 = | -49 11 -25 17 44 -16 -27 40 -34 20 -19 29 41 19 -40 42 -13 5 17 39 31 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 | │ │ │ │ + 10 45 26 43 49 -32 -29 19 -50 15 18 37 -10 34 -28 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o44 : Matrix kk <-- kk │ │ │ │ i45 : Fa = sub(F, (vars S) | pta) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o45 = ideal (a + 14b*c - 16c - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c + │ │ │ │ +o45 = ideal (a - 40b*c + 20c + 29a*d - 16b*d - 25c*d - 49d , a*b + 43b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 14c + 6a*d + 29b*d - c*d - 4d , b + 45b*c + 15c + 49a*d + 10b*d + │ │ │ │ + 2 2 2 2 │ │ │ │ + 17c + 31a*d + 41b*d - 34c*d + 11d , b + 34b*c - 29c + 37a*d + 10b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 23c*d - 30d , a*c + 26b*c + 37c + 5a*d + 29b*d + 19c*d - 22d , b*c - │ │ │ │ + 2 2 2 2 │ │ │ │ + + 42c*d - 27d , a*c + 18b*c + 49c + 19a*d + 39b*d + 19c*d + 44d , b*c │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 10b*c*d - 32c d + 34a*d - 21b*d + 45c*d + 44d , c - 28b*c*d - 29c d │ │ │ │ + 2 2 2 2 3 3 │ │ │ │ + - 50b*c*d + 45c d - 32a*d - 13b*d - 19c*d + 17d , c - 28b*c*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - - 50a*d + 18b*d + 19c*d + 13d ) │ │ │ │ + 2 2 2 2 3 │ │ │ │ + 15c d - 10a*d + 26b*d + 5c*d + 40d ) │ │ │ │ │ │ │ │ o45 : Ideal of S │ │ │ │ i46 : betti res Fa │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o46 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ @@ -467,256 +467,172 @@ │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o46 : BettiTally │ │ │ │ i47 : netList decompose Fa -- this one is 5 points on a plane, and another │ │ │ │ point │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------------------------------+ │ │ │ │ -o47 = |ideal (c + 5d, b - 33d, a - 21d) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -| │ │ │ │ - |ideal (b + 45c + 49d, a - 22c - 26d, c + 49c*d + 42d ) │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------+ │ │ │ │ +o47 = |ideal (c + 19d, b - 10d, a + 6d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 | │ │ │ │ - |ideal (a + 26b + 37c + 36d, c - 21b*d + 43c*d + 27d , b*c - 30b*d + │ │ │ │ -16c*d + 26d , b - 3b*d - 24c*d - 36d )| │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------+ │ │ │ │ + | 2 2 2 3 │ │ │ │ +2 2 2 3 2 2 2 2 3 | │ │ │ │ + |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 )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------------------------------+ │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------+ │ │ │ │ i48 : CFa = minimalPrimes Fa │ │ │ │ │ │ │ │ - │ │ │ │ -o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d, │ │ │ │ + 2 │ │ │ │ +o48 = {ideal (c + 19d, b - 10d, a + 6d), ideal (a + 18b + 49c - 3d, b + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - c + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c - 21b*d + 43c*d + │ │ │ │ + 2 2 3 2 2 │ │ │ │ + 34b*c - 29c - 50b*d + 47c*d - 17d , c - 28b*c*d + 15c d + 4b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 27d , b*c - 30b*d + 16c*d + 26d , b - 3b*d - 24c*d - 36d )} │ │ │ │ + 2 3 2 2 2 2 3 │ │ │ │ + 10c*d + 10d , b*c - 50b*c*d + 45c d - 43b*d + 34c*d + 22d )} │ │ │ │ │ │ │ │ o48 : List │ │ │ │ i49 : lin = CFa_1_0 -- a linear form, defining a plane. │ │ │ │ │ │ │ │ -o49 = b + 45c + 49d │ │ │ │ +o49 = a + 18b + 49c - 3d │ │ │ │ │ │ │ │ o49 : S │ │ │ │ i50 : CFa/degree │ │ │ │ │ │ │ │ -o50 = {1, 2, 3} │ │ │ │ +o50 = {1, 5} │ │ │ │ │ │ │ │ o50 : List │ │ │ │ i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane. │ │ │ │ │ │ │ │ -o51 = {false, true, false} │ │ │ │ +o51 = {false, true} │ │ │ │ │ │ │ │ o51 : List │ │ │ │ i52 : degree(Fa : (Fa : lin)) -- somewhat simpler(?) way to see the ideal of │ │ │ │ the 5 points │ │ │ │ │ │ │ │ -o52 = 2 │ │ │ │ +o52 = 5 │ │ │ │ i53 : ptb = randomPointOnRationalVariety compsL441_1 │ │ │ │ │ │ │ │ -o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31 │ │ │ │ +o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 | │ │ │ │ + 18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o53 : Matrix kk <-- kk │ │ │ │ i54 : Fb = sub(F, (vars S) | ptb) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o54 = ideal (a + 50b*c - 42c - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o54 = ideal (a + 22b*c + 45c + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 42c + 13a*d + 14b*d - 3c*d + 42d , b - 9b*c - 29c + 31b*d + 5c*d + │ │ │ │ + 2 2 2 2 │ │ │ │ + 9c + 43a*d - 41b*d + 40c*d + 12d , b - 37b*c + 17c + 18b*d - 49c*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 43d , a*c + 9b*c - 4c - 23a*d + 47b*d + 2c*d + 19d , b*c + 21b*c*d - │ │ │ │ + 2 2 2 2 │ │ │ │ + 32d , a*c + 47b*c + 21c - 20a*d - 23b*d - 13c*d - 19d , b*c + 21b*c*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ │ - 37c d + 38a*d + 36b*d - 29c*d + 25d , c - 47b*c*d + 17c d + 21a*d │ │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ │ + - 9c d + 38a*d - 4b*d - 13c*d + 9d , c + 9b*c*d - 29c d + 2a*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 3 │ │ │ │ - - 23b*d - 13c*d - 7d ) │ │ │ │ + 2 2 3 │ │ │ │ + 47b*d - 4c*d + 27d ) │ │ │ │ │ │ │ │ o54 : Ideal of S │ │ │ │ i55 : betti res Fb │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o55 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o55 : BettiTally │ │ │ │ i56 : netList decompose Fb -- │ │ │ │ │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ -o56 = |ideal (c - 45d, b + 16d, a + 38d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 43d, b + 10d, a + 8d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 34d, b + 15d, a + 28d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 11d, b + 39d, a + 23d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b - 32c + 42d, a - 19c - 16d, c - 28c*d - 40d )| │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ -i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2) │ │ │ │ - │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 │ │ │ │ -| │ │ │ │ -o57 = |ideal (a - 7b + 32c + d, c + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + │ │ │ │ -18d , b + 28b*d - 32c*d + 16d ) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 │ │ │ │ -| │ │ │ │ - |ideal (a - 7b + 32c + d, c + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d │ │ │ │ -+ 39d , b - 20b*d + 29c*d + 38d ) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 │ │ │ │ -| │ │ │ │ - |ideal (a - 7b + 32c + d, c - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d │ │ │ │ -- 32d , b - 8b*d - 12c*d - 46d ) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 3 2 2 3 │ │ │ │ -| │ │ │ │ - |ideal (b + 23c - 11d, a - 9c + 25d, c - 13c d - 14c*d + 23d ) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 2 2 │ │ │ │ -2 3 2 2 2 3 │ │ │ │ -| │ │ │ │ - |ideal (a + 48b - 40c - 20d, b*c - 32c + 43b*d - 21c*d - 12d , b - 14c │ │ │ │ -+ 14b*d + 18c*d + 36d , c + 28c d - 20b*d + 42c*d - 50d ) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 2 2 │ │ │ │ -2 3 2 2 2 3 │ │ │ │ -| │ │ │ │ - |ideal (a + b + 50c + 26d, b*c - 32c + 34b*d - 36c*d + 14d , b - 14c + │ │ │ │ -34b*d - 16c*d - 33d , c + 28c d + 39b*d - 28c*d + 4d ) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 2 │ │ │ │ -2 2 2 2 2 │ │ │ │ -2 | │ │ │ │ - |ideal (c - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d │ │ │ │ -, a*c - 6a*d + 35b*d - 39c*d - 2d , b - 46a*d + 22b*d + 42c*d + 43d , a*b + │ │ │ │ -3a*d - 12b*d - 49c*d + 40d , a + 28a*d - 13b*d - 25c*d - 35d )| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 2 2 │ │ │ │ -2 3 2 2 2 3 │ │ │ │ +--------------------------------------------------------------+ │ │ │ │ + | 2 2 │ │ │ │ | │ │ │ │ - |ideal (a - 46b + 39c - 29d, b*c - 32c + 11b*d - 7c*d - 43d , b - 14c │ │ │ │ -+ 29b*d + 43c*d - 41d , c + 28c d + 46b*d - 50c*d - 5d ) │ │ │ │ +o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c + 12c*d - 37d ) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 2 │ │ │ │ -2 2 2 | │ │ │ │ - |ideal (c + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d │ │ │ │ -, a*c - 10a*d + 45b*d + 20c*d - 23d , b - 23a*d + 15b*d + 31c*d - 13d , a*b - │ │ │ │ -6a*d - 40b*d + 8c*d + 18d , a - 8a*d - 24b*d + c*d - 22d ) | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 2 │ │ │ │ -2 2 2 | │ │ │ │ - |ideal (c + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d │ │ │ │ -, a*c - 14a*d + 27b*d - 35c*d - 8d , b - 33b*d + 19c*d + 27d , a*b - 15a*d - │ │ │ │ -30b*d - 40c*d - 24d , a - 44a*d + 16b*d + 11c*d + 12d ) | │ │ │ │ +--------------------------------------------------------------+ │ │ │ │ + | 2 2 │ │ │ │ +2 2 2 2 │ │ │ │ +2 2 2 | │ │ │ │ + |ideal (c + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d │ │ │ │ +, a*c + 43a*d + 44b*d - 48c*d + 24d , b - 4a*d - 40b*d - 9c*d - 39d , a*b + │ │ │ │ +16a*d + 26b*d + 37c*d - 24d , a - 25a*d + 47b*d + 34c*d + 8d )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ --------------------------------------------------------------+ │ │ │ │ +--------------------------------------------------------------+ │ │ │ │ +i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2) │ │ │ │ + │ │ │ │ +o57 = ++ │ │ │ │ + ++ │ │ │ │ i58 : pt0 = randomPointOnRationalVariety(compsL441_0) │ │ │ │ │ │ │ │ -o58 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24 │ │ │ │ +o58 = | 44 15 26 -28 -4 33 49 -19 -6 -18 -37 -34 18 -42 -23 23 38 -40 27 26 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 | │ │ │ │ + -39 -24 -21 -13 38 39 -33 15 8 -18 18 -22 -2 -29 -23 46 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o58 : Matrix kk <-- kk │ │ │ │ i59 : pt1 = randomPointOnRationalVariety(compsL441_1) │ │ │ │ │ │ │ │ -o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49 │ │ │ │ +o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 | │ │ │ │ + -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o59 : Matrix kk <-- kk │ │ │ │ We compute the ideal of the corresponding zero dimensional scheme with length │ │ │ │ 6, corresponding to the points pt0, pt1 in Hilb. │ │ │ │ i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o60 = ideal (a + 35b*c + 8c - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o60 = ideal (a - 23b*c - 18c - 34a*d + 33b*d + 26c*d + 44d , a*b + 38b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 2 │ │ │ │ - 32c - 8a*d + 49b*d - 50c*d - 24d , b - 21b*c + 18c + 39a*d - 24b*d + │ │ │ │ + 27c - 39a*d + 18b*d - 6c*d + 15d , b - 23b*c + 15c - 2a*d - 24b*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 49c*d + 36d , a*c - 13b*c - 2c - 40a*d + 15b*d + 8c*d - 49d , b*c - │ │ │ │ + 2 2 2 2 │ │ │ │ + 23c*d + 49d , a*c - 22b*c + 39c + 8a*d + 26b*d - 42c*d - 4d , b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 29b*c*d - 33c d - 23a*d + 6b*d + 31c*d + 5d , c + 46b*c*d + 15c d - │ │ │ │ + 2 2 2 2 3 3 2 │ │ │ │ + 18b*c*d - 21c d - 33a*d + 38b*d - 37c*d - 28d , c + 46b*c*d + 18c d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - 18a*d - 22b*d - 42c*d - 36d ) │ │ │ │ + 2 2 2 3 │ │ │ │ + - 29a*d - 13b*d - 40c*d - 19d ) │ │ │ │ │ │ │ │ o60 : Ideal of S │ │ │ │ i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o61 = ideal (a - 21b*c + 45c - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o61 = ideal (a + 30b*c + c - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 17c - 23a*d + 37b*d - 5c*d + 18d , b - 15b*c + 10c + 15b*d + 28c*d + │ │ │ │ + 2 2 2 2 │ │ │ │ + 27c + 15a*d + 32b*d - 23c*d + 41d , b - 50b*c - 18c - 25b*d - 22c*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 50d , a*c + 32b*c - 14c - 31a*d - 49b*d - 35c*d + 21d , b*c + 3b*c*d │ │ │ │ + 2 2 2 2 │ │ │ │ + - 38d , a*c - 49b*c + 3c - 9a*d + 37b*d + 46c*d + 50d , b*c + 3b*c*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ │ - - 50c d + 21a*d + 27b*d - 47c*d + 38d , c + 33b*c*d - 18c d + 3a*d │ │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ │ + - 15c d + 21a*d - 2b*d - 2c*d - 44d , c + 32b*c*d + 10c d - 35a*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 3 │ │ │ │ - + 37b*d + 46c*d - 8d ) │ │ │ │ + 2 2 3 │ │ │ │ + + 33b*d - 14c*d + 33d ) │ │ │ │ │ │ │ │ o61 : Ideal of S │ │ │ │ i62 : betti res I0 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o62 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ @@ -732,45 +648,42 @@ │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o63 : BettiTally │ │ │ │ i64 : netList decompose I0 │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ --------------------------------+ │ │ │ │ -o64 = |ideal (c - 40d, b - 10d, a + 32d) │ │ │ │ -| │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -| │ │ │ │ - |ideal (b + 10c + 25d, a + 27c - 50d, c - 34c*d - 17d ) │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +-----+ │ │ │ │ +o64 = |ideal (c + 8d, b + 5d, a - 25d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ --------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 | │ │ │ │ - |ideal (a - 13b - 2c + 29d, c - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d │ │ │ │ -+ 8d , b - b*d + 15c*d + 40d )| │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +-----+ │ │ │ │ + | 2 2 2 3 │ │ │ │ +2 2 2 3 2 2 2 2 3 | │ │ │ │ + |ideal (a - 22b + 39c + 50d, b - 23b*c + 15c + 33b*d + 48d , c + │ │ │ │ +46b*c*d + 18c d - 45b*d - 20c*d + 17d , b*c - 18b*c*d - 21c d + 19b*d + │ │ │ │ +38c*d + 6d )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ --------------------------------+ │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +-----+ │ │ │ │ i65 : netList decompose I1 │ │ │ │ │ │ │ │ - +------------------------------------------------------+ │ │ │ │ -o65 = |ideal (c + 32d, b + 18d, a - 33d) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - |ideal (c + 29d, b - 8d, a + 50d) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - |ideal (c + 16d, b + 39d, a - 32d) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - |ideal (c + 5d, b - 14d, a + 7d) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b - 40c + 5d, a - 47c + 24d, c - 27c*d + 15d )| │ │ │ │ - +------------------------------------------------------+ │ │ │ │ + +---------------------------------+ │ │ │ │ +o65 = |ideal (c - 9d, b + 15d, a + 27d) | │ │ │ │ + +---------------------------------+ │ │ │ │ + |ideal (c + 48d, b + 11d, a - 37d)| │ │ │ │ + +---------------------------------+ │ │ │ │ + |ideal (c + 29d, b + 46d, a + 18d)| │ │ │ │ + +---------------------------------+ │ │ │ │ + |ideal (c + 24d, b + 46d, a + 33d)| │ │ │ │ + +---------------------------------+ │ │ │ │ + |ideal (c + 22d, b + 38d, a - 50d)| │ │ │ │ + +---------------------------------+ │ │ │ │ i66 : L430 = (trim minors(2, M1)) + groebnerStratum F; │ │ │ │ │ │ │ │ o66 : Ideal of T │ │ │ │ i67 : C = res(I, FastNonminimal => true) │ │ │ │ │ │ │ │ 1 4 5 2 │ │ │ │ o67 = S <-- S <-- S <-- S <-- 0 │ │ ├── ./usr/share/doc/Macaulay2/QuillenSuslin/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ aG9ycm9ja3M= │ │ │ #:len=4664 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBsb2NhbCBzb2x1dGlv │ │ │ biB0byB0aGUgdW5pbW9kdWxhciByb3cgcHJvYmxlbSBvdmVyIGEgbG9jYWxpemF0aW9uIGF0IGEg │ │ ├── ./usr/share/doc/Macaulay2/RInterface/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ bmV3IFJPYmplY3QgZnJvbSBDQw== │ │ │ #:len=251 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFJPYmplY3QsQ0MpLCJuZXcg │ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ cmFuZG9tQ2Fub25pY2FsQ3VydmU= │ │ │ #:len=218 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzAsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InJhbmRvbUNh │ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : g=14; │ │ │ │ │ │ i3 : FF=ZZ/10007; │ │ │ │ │ │ i4 : R=FF[x_0..x_(g-1)]; │ │ │ │ │ │ i5 : time betti(I=(random canonicalCurve)(g,R)) │ │ │ - -- used 7.86379s (cpu); 5.87077s (thread); 0s (gc) │ │ │ + -- used 10.9851s (cpu); 8.58118s (thread); 0s (gc) │ │ │ │ │ │ 0 1 │ │ │ o5 = total: 1 66 │ │ │ 0: 1 . │ │ │ 1: . 66 │ │ │ │ │ │ o5 : BettiTally │ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ 
    i3 : FF=ZZ/10007;
    │ │ │ 
    i4 : R=FF[x_0..x_(g-1)];
    │ │ │ 
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 7.86379s (cpu); 5.87077s (thread); 0s (gc)
│ │ │ + -- used 10.9851s (cpu); 8.58118s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,15 +17,15 @@ │ │ │ │ Compute a random canonical curve of genus $g \le{} 14$, based on the proofs of │ │ │ │ unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra. │ │ │ │ i1 : setRandomSeed "alpha"; │ │ │ │ i2 : g=14; │ │ │ │ i3 : FF=ZZ/10007; │ │ │ │ i4 : R=FF[x_0..x_(g-1)]; │ │ │ │ i5 : time betti(I=(random canonicalCurve)(g,R)) │ │ │ │ - -- used 7.86379s (cpu); 5.87077s (thread); 0s (gc) │ │ │ │ + -- used 10.9851s (cpu); 8.58118s (thread); 0s (gc) │ │ │ │ │ │ │ │ 0 1 │ │ │ │ o5 = total: 1 66 │ │ │ │ 0: 1 . │ │ │ │ 1: . 66 │ │ │ │ │ │ │ │ o5 : BettiTally │ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bWF4aW1hbEVudHJ5KENoYWluQ29tcGxleCk= │ │ │ #:len=280 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYXhpbWFsRW50cnksQ2hhaW5Db21wbGV4KSwibWF4 │ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ @@ -6,42 +6,42 @@ │ │ │ │ │ │ o2 = (10, 20) │ │ │ │ │ │ o2 : Sequence │ │ │ │ │ │ i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11) │ │ │ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00138046, 0) │ │ │ +o3 = ({5, 2.91596e52, 9}, 0, 0) │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100) │ │ │ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00618369, .0388145) │ │ │ +o4 = ({50, 2.30853e454, 98}, .0071902, .0294253) │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ -o5 = {{.00415423, .0129743}, {.00772311, .0045526}, {.00327175, .00800128}, │ │ │ +o5 = {{.00325437, .0119979}, {.00400018, .00400053}, {.00400019, .00400104}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00449345, .012301}, {.00573879, .0131784}, {.00740736, .0112207}, │ │ │ + {.00399913, .00799843}, {.00400038, .0119972}, {.00400124, .0119974}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00400166, .00799792}, {.00798013, .00399967}, {.00800154, .00399867}, │ │ │ + {0, .00399857}, {.0040008, .0367455}, {.00399943, .00400034}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00800417, .00799329}} │ │ │ + {.00400125, .00799759}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ -o6 = .006077619700000004 │ │ │ +o6 = .003525697699999997 │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ -o7 = .008621774699999963 │ │ │ +o7 = .0104734578 │ │ │ │ │ │ o7 : RR (of precision 53) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ @@ -92,49 +92,49 @@ │ │ │ o2 = (10, 20) │ │ │ │ │ │ o2 : Sequence │ │ │ 
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00138046, 0)
│ │ │ +o3 = ({5, 2.91596e52, 9}, 0, 0)
│ │ │  
│ │ │  o3 : Sequence
    │ │ │ 
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00618369, .0388145)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .0071902, .0294253)
│ │ │  
│ │ │  o4 : Sequence
    │ │ │ 
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00415423, .0129743}, {.00772311, .0045526}, {.00327175, .00800128},
│ │ │ +o5 = {{.00325437, .0119979}, {.00400018, .00400053}, {.00400019, .00400104},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00449345, .012301}, {.00573879, .0131784}, {.00740736, .0112207},
│ │ │ +     {.00399913, .00799843}, {.00400038, .0119972}, {.00400124, .0119974},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00400166, .00799792}, {.00798013, .00399967}, {.00800154, .00399867},
│ │ │ +     {0, .00399857}, {.0040008, .0367455}, {.00399943, .00400034},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00800417, .00799329}}
│ │ │ +     {.00400125, .00799759}}
│ │ │  
│ │ │  o5 : List
    │ │ │ 
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .006077619700000004
│ │ │ +o6 = .003525697699999997
│ │ │  
│ │ │  o6 : RR (of precision 53)
    │ │ │ 
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008621774699999963
│ │ │ +o7 = .0104734578
│ │ │  
│ │ │  o7 : RR (of precision 53)
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use testTimeForLLLonSyzygies:

    │ │ │ ├── html2text {} │ │ │ │ @@ -25,40 +25,40 @@ │ │ │ │ i2 : r=10,n=20 │ │ │ │ │ │ │ │ o2 = (10, 20) │ │ │ │ │ │ │ │ o2 : Sequence │ │ │ │ i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11) │ │ │ │ │ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00138046, 0) │ │ │ │ +o3 = ({5, 2.91596e52, 9}, 0, 0) │ │ │ │ │ │ │ │ o3 : Sequence │ │ │ │ i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100) │ │ │ │ │ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00618369, .0388145) │ │ │ │ +o4 = ({50, 2.30853e454, 98}, .0071902, .0294253) │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ │ │ -o5 = {{.00415423, .0129743}, {.00772311, .0045526}, {.00327175, .00800128}, │ │ │ │ +o5 = {{.00325437, .0119979}, {.00400018, .00400053}, {.00400019, .00400104}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00449345, .012301}, {.00573879, .0131784}, {.00740736, .0112207}, │ │ │ │ + {.00399913, .00799843}, {.00400038, .0119972}, {.00400124, .0119974}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00400166, .00799792}, {.00798013, .00399967}, {.00800154, .00399867}, │ │ │ │ + {0, .00399857}, {.0040008, .0367455}, {.00399943, .00400034}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00800417, .00799329}} │ │ │ │ + {.00400125, .00799759}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ │ │ -o6 = .006077619700000004 │ │ │ │ +o6 = .003525697699999997 │ │ │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ │ │ -o7 = .008621774699999963 │ │ │ │ +o7 = .0104734578 │ │ │ │ │ │ │ │ o7 : RR (of precision 53) │ │ │ │ ********** WWaayyss ttoo uussee tteessttTTiimmeeFFoorrLLLLLLoonnSSyyzzyyggiieess:: ********** │ │ │ │ * testTimeForLLLonSyzygies(ZZ,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _t_e_s_t_T_i_m_e_F_o_r_L_L_L_o_n_S_y_z_y_g_i_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ UmFuZG9tQ3VydmVz │ │ │ #:len=775 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmFuZG9tIGN1cnZlcyIsIERlc2NyaXB0 │ │ │ aW9uID0+IDE6KERJVntQQVJBe1RFWHsiVGhpcyBwYWNrYWdlIGxvYWRzIHRoZSAifSxUT3tuZXcg │ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ c21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1KC4uLixQcmludGluZz0+Li4uKQ== │ │ │ #:len=370 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTI4Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1LFBy │ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 11549527689790345152 │ │ │ │ │ │ i1 : time ICan = smoothCanonicalCurve(11,5); │ │ │ - -- used 1.26634s (cpu); 1.02757s (thread); 0s (gc) │ │ │ + -- used 1.58877s (cpu); 1.38252s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o1 : Ideal of --[t ..t ] │ │ │ 5 0 10 │ │ │ │ │ │ i2 : (dim ICan, genus ICan, degree ICan) │ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html │ │ │ @@ -85,15 +85,15 @@ │ │ │

    For g=15 the curves are constructed via matrix factorizations.

    │ │ │

    If the option Printing is set to true then printings about the current step in the construction are displayed.

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -37,30 +37,28 @@ │ │ │ │ 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 < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1]. │ │ │ │ i1 : setRandomSeed(currentTime()) │ │ │ │ │ │ │ │ -o1 = 1739147131 │ │ │ │ +o1 = 1752898290 │ │ │ │ i2 : S=ZZ/2[vars(0..3)] │ │ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ i3 : J = monomialIdeal"ab,ad, bcd" │ │ │ │ │ │ │ │ o3 = monomialIdeal (a*b, a*d, b*c*d) │ │ │ │ │ │ │ │ o3 : MonomialIdeal of S │ │ │ │ i4 : randomSquareFreeStep J │ │ │ │ │ │ │ │ -o4 = {monomialIdeal (a*b, a*c, a*d, b*c*d), {a*b, a*c, a*d, b*c*d}, {c*d, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - b*d, b*c, a}} │ │ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ 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. │ │ │ │ i5 : setRandomSeed(1) │ │ │ │ │ │ │ │ @@ -73,15 +71,15 @@ │ │ │ │ i7 : J = monomialIdeal 0_S │ │ │ │ │ │ │ │ o7 = monomialIdeal () │ │ │ │ │ │ │ │ o7 : MonomialIdeal of S │ │ │ │ i8 : time T=tally for t from 1 to 5000 list first (J=rsfs │ │ │ │ (J,AlexanderProbability => .01)); │ │ │ │ - -- used 4.308s (cpu); 2.68151s (thread); 0s (gc) │ │ │ │ + -- used 7.38417s (cpu); 5.02413s (thread); 0s (gc) │ │ │ │ i9 : #T │ │ │ │ │ │ │ │ o9 = 168 │ │ │ │ i10 : T │ │ │ │ │ │ │ │ o10 = Tally{monomialIdeal () => 45 } │ │ │ │ monomialIdeal (a*b*c, a*b*d) => 33 │ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ @@ -47,31 +47,31 @@ │ │ │
    │ │ │

    This package can be used to make experiments, trying many ideals, perhaps over small fields. For example...what would you expect the regularities of "typical" 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.

    │ │ │
    │ │ │ 
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.26634s (cpu); 1.02757s (thread); 0s (gc)
│ │ │ + -- used 1.58877s (cpu); 1.38252s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    │ │ │ 
    i2 : (dim ICan, genus ICan, degree ICan)
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  For g<=10 the curves are constructed via plane models.
│ │ │ │  For g<=13 the curves are constructed via space models.
│ │ │ │  For g=14 the curves are constructed by Verra's method.
│ │ │ │  For g=15 the curves are constructed via matrix factorizations.
│ │ │ │  If the option Printing is set to true then printings about the current step in
│ │ │ │  the construction are displayed.
│ │ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ │ - -- used 1.26634s (cpu); 1.02757s (thread); 0s (gc)
│ │ │ │ + -- used 1.58877s (cpu); 1.38252s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o1 : Ideal of --[t ..t  ]
│ │ │ │                 5  0   10
│ │ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ │ │  
│ │ │ │  o2 = (2, 11, 20)
│ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  Y2Fub25pY2FsQ3VydmVHZW51czE0
│ │ │  #:len=1011
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBhIHJhbmRvbSBjdXJ2ZSBv
│ │ │  ZiBnZW51cyAxNCBpbiBpdHMgY2Fub25pY2FsIGVtYmVkZGluZyIsICJsaW5lbnVtIiA9PiAyMDEs
│ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : setRandomSeed("alpha");
│ │ │  
│ │ │  i2 : FF=ZZ/10007;
│ │ │  
│ │ │  i3 : S=FF[x_0..x_6];
│ │ │  
│ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.77032s (cpu); 1.47337s (thread); 0s (gc)
│ │ │ + -- used 2.00727s (cpu); 1.87101s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : betti res I
│ │ │  
│ │ │              0  1  2  3  4 5
│ │ │  o5 = total: 1 13 45 56 25 2
│ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
    i2 : FF=ZZ/10007;
    │ │ │ 
    i3 : S=FF[x_0..x_6];
    │ │ │ 
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.77032s (cpu); 1.47337s (thread); 0s (gc)
│ │ │ + -- used 2.00727s (cpu); 1.87101s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
    │ │ │ 
    i5 : betti res I
│ │ │  
│ │ │              0  1  2  3  4 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  the unirationality of $M_{14}$. It proves the unirationality of $M_{14}$ for
│ │ │ │  fields of the chosen finite characteristic 10007, for fields of characteristic
│ │ │ │  0 by semi-continuity, and, hence, for all but finitely many primes $p$.
│ │ │ │  i1 : setRandomSeed("alpha");
│ │ │ │  i2 : FF=ZZ/10007;
│ │ │ │  i3 : S=FF[x_0..x_6];
│ │ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ │ - -- used 1.77032s (cpu); 1.47337s (thread); 0s (gc)
│ │ │ │ + -- used 2.00727s (cpu); 1.87101s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : betti res I
│ │ │ │  
│ │ │ │              0  1  2  3  4 5
│ │ │ │  o5 = total: 1 13 45 56 25 2
│ │ │ │           0: 1  .  .  .  . .
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  cmFuZG9tU2hlbGxhYmxlSWRlYWxDaGFpbg==
│ │ │  #:len=1807
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUHJvZHVjZXMgYSBjaGFpbiBvZiBpZGVh
│ │ │  bHMgZnJvbSBhIHJhbmRvbSBzaGVsbGluZyIsICJsaW5lbnVtIiA9PiA3NjksIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147038
│ │ │ +o1 = 1752898181
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 1.72944s (cpu); 1.22524s (thread); 0s (gc)
│ │ │ + -- used 2.55741s (cpu); 2.03658s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 32 }
│ │ │ -           5 => 219
│ │ │ -           6 => 166
│ │ │ -           7 => 68
│ │ │ -           8 => 14
│ │ │ +o4 = Tally{4 => 37 }
│ │ │ +           5 => 203
│ │ │ +           6 => 183
│ │ │ +           7 => 60
│ │ │ +           8 => 16
│ │ │             9 => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147084
│ │ │ +o1 = 1752898235
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -14,13 +14,13 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      3
│ │ │ -o4 = a
│ │ │ +      2
│ │ │ +o4 = a b
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147148
│ │ │ +o1 = 1752898316
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -21,18 +21,18 @@
│ │ │  o4 = {3, 5, 7}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*d*e)
│ │ │ +o5 = ideal(a*b*c)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*c, d*e, b*d, b*c, b*e)
│ │ │ +o6 = ideal (a*c, c*e, b*c, a*b, d*e)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147131
│ │ │ +o1 = 1752898290
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -14,17 +14,15 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c, a*d, b*c*d), {a*b, a*c, a*d, b*c*d}, {c*d,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     b*d, b*c, a}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │ @@ -37,15 +35,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.308s (cpu); 2.68151s (thread); 0s (gc)
│ │ │ + -- used 7.38417s (cpu); 5.02413s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147084
    
│ │ │ +o1 = 1752898235
│ │ │      		          
                  
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
    
│ │ │ @@ -91,16 +91,16 @@
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │      		          
                  
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      3
│ │ │ -o4 = a
│ │ │ +      2
│ │ │ +o4 = a b
│ │ │  
│ │ │  o4 : S
    
│ │ │      		          
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,29 +13,29 @@
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │  
│ │ │ │ -o1 = 1739147084
│ │ │ │ +o1 = 1752898235
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -      3
│ │ │ │ -o4 = a
│ │ │ │ +      2
│ │ │ │ +o4 = a b
│ │ │ │  
│ │ │ │  o4 : S
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_M_o_n_o_m_i_a_l_I_d_e_a_l -- random monomial ideal with given degree generators
│ │ │ │      * _r_a_n_d_o_m_S_q_u_a_r_e_F_r_e_e_M_o_n_o_m_i_a_l_I_d_e_a_l -- random square-free monomial ideal with
│ │ │ │        given degree generators
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommMMoonnoommiiaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147148
    
│ │ │ +o1 = 1752898316
│ │ │      		          
                  
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
    
│ │ │ @@ -99,22 +99,22 @@
│ │ │  
│ │ │  o4 : List
│ │ │      		          
                  
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*d*e)
│ │ │ +o5 = ideal(a*b*c)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │      		          
                  
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*c, d*e, b*d, b*c, b*e)
│ │ │ +o6 = ideal (a*c, c*e, b*c, a*b, d*e)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o I, an _i_d_e_a_l, square-free monomial ideal with generators of
│ │ │ │              specified degrees
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Choose a random square-free monomial ideal whose generators have the degrees
│ │ │ │  specified by the list or sequence L.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │  
│ │ │ │ -o1 = 1739147148
│ │ │ │ +o1 = 1752898316
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = {3, 5, 7}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │ -o5 = ideal(a*d*e)
│ │ │ │ +o5 = ideal(a*b*c)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*c, d*e, b*d, b*c, b*e)
│ │ │ │ +o6 = ideal (a*c, c*e, b*c, a*b, d*e)
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The ideal is constructed degree by degree, starting from the lowest degree
│ │ │ │  specified. If there are not enough monomials of the next specified degree that
│ │ │ │  are not already in the ideal, the function prints a warning and returns an
│ │ │ │  ideal containing all the generators of that degree.
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    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.
    
│ │ │            
    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 "next" 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 < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147131
    
│ │ │ +o1 = 1752898290
│ │ │      		          
                  
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │ @@ -101,17 +101,15 @@
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │      		          
                  
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c, a*d, b*c*d), {a*b, a*c, a*d, b*c*d}, {c*d,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     b*d, b*c, a}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │  
│ │ │  o4 : List
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │ @@ -133,15 +131,15 @@
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    │ │ │ 
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.308s (cpu); 2.68151s (thread); 0s (gc)
    │ │ │ + -- used 7.38417s (cpu); 5.02413s (thread); 0s (gc) │ │ │ 
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147038
    │ │ │ +o1 = 1752898181 │ │ │ 
    i2 : kk=ZZ/101;
    │ │ │ 
    i3 : S=kk[vars(0..5)];
    │ │ │ 
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 1.72944s (cpu); 1.22524s (thread); 0s (gc)
│ │ │ + -- used 2.55741s (cpu); 2.03658s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 32 }
│ │ │ -           5 => 219
│ │ │ -           6 => 166
│ │ │ -           7 => 68
│ │ │ -           8 => 14
│ │ │ +o4 = Tally{4 => 37 }
│ │ │ +           5 => 203
│ │ │ +           6 => 183
│ │ │ +           7 => 60
│ │ │ +           8 => 16
│ │ │             9 => 1
│ │ │  
│ │ │  o4 : Tally
    │ │ │
    │ │ │
    │ │ │

    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.

    │ │ │ ├── html2text {} │ │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ small fields. For example...what would you expect the regularities of "typical" │ │ │ │ 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. │ │ │ │ i1 : setRandomSeed(currentTime()) │ │ │ │ │ │ │ │ -o1 = 1739147038 │ │ │ │ +o1 = 1752898181 │ │ │ │ i2 : kk=ZZ/101; │ │ │ │ i3 : S=kk[vars(0..5)]; │ │ │ │ i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S) │ │ │ │ - -- used 1.72944s (cpu); 1.22524s (thread); 0s (gc) │ │ │ │ + -- used 2.55741s (cpu); 2.03658s (thread); 0s (gc) │ │ │ │ │ │ │ │ -o4 = Tally{4 => 32 } │ │ │ │ - 5 => 219 │ │ │ │ - 6 => 166 │ │ │ │ - 7 => 68 │ │ │ │ - 8 => 14 │ │ │ │ +o4 = Tally{4 => 37 } │ │ │ │ + 5 => 203 │ │ │ │ + 6 => 183 │ │ │ │ + 7 => 60 │ │ │ │ + 8 => 16 │ │ │ │ 9 => 1 │ │ │ │ │ │ │ │ o4 : Tally │ │ │ │ How does this compare with the case of binomial ideals? or pure binomial │ │ │ │ ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" │ │ │ │ above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger │ │ │ │ numbers of examples. Click the link "Finding Extreme Examples" below to see │ │ ├── ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ VmFyaWFibGVOYW1l │ │ │ #:len=1241 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl │ │ │ IHRoZSBpbmRleGVkIHZhcmlhYmxlIG5hbWUgZm9yIHRoZSBwb2x5bm9taWFsIHJpbmciLCAibGlu │ │ ├── ./usr/share/doc/Macaulay2/RandomObjects/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ QXR0ZW1wdHM= │ │ │ #:len=502 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibnVtYmVyIG9mIGF0dGVtcHRzIGluIHRo │ │ │ ZSBjb25zdHJ1Y3Rpb24gb2YgYSByYW5kb20gb2JqZWN0IiwgImxpbmVudW0iID0+IDI2MiwgImZp │ │ ├── ./usr/share/doc/Macaulay2/RandomPlaneCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=49 │ │ │ Y29tcGxldGVMaW5lYXJTeXN0ZW1Pbk5vZGFsUGxhbmVDdXJ2ZShJZGVhbCxMaXN0KQ== │ │ │ #:len=382 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc2LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV0ZUxpbmVhclN5c3RlbU9uTm9kYWxQbGFu │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ ZmluZEFOb25aZXJvTWlub3IoLi4uLFBvaW50Q2hlY2tBdHRlbXB0cz0+Li4uKQ== │ │ │ #:len=315 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg5MSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEFOb25aZXJvTWlub3IsUG9pbnRDaGVja0F0 │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out │ │ │ @@ -5,17 +5,17 @@ │ │ │ i2 : S=kk[y_0..y_8]; │ │ │ │ │ │ i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2; │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : elapsedTime dimViaBezout(I) │ │ │ - -- 1.72465s elapsed │ │ │ + -- 1.89486s elapsed │ │ │ │ │ │ o4 = 4 │ │ │ │ │ │ i5 : elapsedTime dim I │ │ │ - -- 2.94822s elapsed │ │ │ + -- 4.02503s elapsed │ │ │ │ │ │ o5 = 4 │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out │ │ │ @@ -35,15 +35,15 @@ │ │ │ i8 : i = 0; │ │ │ │ │ │ i9 : J = I; │ │ │ │ │ │ o9 : Ideal of T │ │ │ │ │ │ i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) ); │ │ │ - -- 2.03564s elapsed │ │ │ + -- 2.16388s elapsed │ │ │ │ │ │ i11 : dim J │ │ │ │ │ │ o11 = 1 │ │ │ │ │ │ i12 : i │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ @@ -27,24 +27,24 @@ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable) │ │ │ - -- 3.17305s elapsed │ │ │ + -- 3.60362s elapsed │ │ │ │ │ │ o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0, │ │ │ ------------------------------------------------------------------------ │ │ │ -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}} │ │ │ │ │ │ o8 : List │ │ │ │ │ │ i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose) │ │ │ - -- 2.24874s elapsed │ │ │ + -- 2.81786s elapsed │ │ │ │ │ │ o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16, │ │ │ ------------------------------------------------------------------------ │ │ │ 11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}} │ │ │ │ │ │ o9 : List │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ @@ -94,21 +94,21 @@ │ │ │ │ │ │ 
    i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │  
│ │ │  o3 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.72465s elapsed
│ │ │ + -- 1.89486s elapsed
│ │ │  
│ │ │  o4 = 4
    │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime dim I
│ │ │ - -- 2.94822s elapsed
│ │ │ + -- 4.02503s elapsed
│ │ │  
│ │ │  o5 = 4
    │ │ │ │ │ │ │ │ │
    │ │ │

    The user may set the MinimumFieldSize 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.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -33,19 +33,19 @@ │ │ │ │ examples, the built in dim function is much faster. │ │ │ │ i1 : kk=ZZ/101; │ │ │ │ i2 : S=kk[y_0..y_8]; │ │ │ │ i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2; │ │ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ i4 : elapsedTime dimViaBezout(I) │ │ │ │ - -- 1.72465s elapsed │ │ │ │ + -- 1.89486s elapsed │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : elapsedTime dim I │ │ │ │ - -- 2.94822s elapsed │ │ │ │ + -- 4.02503s elapsed │ │ │ │ │ │ │ │ o5 = 4 │ │ │ │ The user may set the MinimumFieldSize 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. │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ │ │ 
    i9 : J = I;
│ │ │  
│ │ │  o9 : Ideal of T
    │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.03564s elapsed
    │ │ │ + -- 2.16388s elapsed │ │ │ │ │ │ │ │ │ 
    i11 : dim J
│ │ │  
│ │ │  o11 = 1
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ o7 : Matrix T <-- T │ │ │ │ i8 : i = 0; │ │ │ │ i9 : J = I; │ │ │ │ │ │ │ │ o9 : Ideal of T │ │ │ │ i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = │ │ │ │ extendIdealByNonZeroMinor(4, M, J)) ); │ │ │ │ - -- 2.03564s elapsed │ │ │ │ + -- 2.16388s elapsed │ │ │ │ i11 : dim J │ │ │ │ │ │ │ │ o11 = 1 │ │ │ │ i12 : i │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ In this particular example, there tend to be about 5 associated primes when │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ @@ -141,25 +141,25 @@ │ │ │ │ │ │ 
    i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.17305s elapsed
│ │ │ + -- 3.60362s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │ 
    i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.24874s elapsed
│ │ │ + -- 2.81786s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -67,24 +67,24 @@ │ │ │ │ first in rings with more variables. │ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>MultiplicationTable) │ │ │ │ - -- 3.17305s elapsed │ │ │ │ + -- 3.60362s elapsed │ │ │ │ │ │ │ │ o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>Decompose) │ │ │ │ - -- 2.24874s elapsed │ │ │ │ + -- 2.81786s elapsed │ │ │ │ │ │ │ │ o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ ********** WWaayyss ttoo uussee rraannddoommPPooiinnttss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/RandomSpaceCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=38 │ │ │ a25vd25VbmlyYXRpb25hbENvbXBvbmVudE9mU3BhY2VDdXJ2ZXM= │ │ │ #:len=2298 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgd2hldGhlciB0aGVyZSBpcyBh │ │ │ IHVuaXJhdGlvbmFsIGNvbnN0cnVjdGlvbiBmb3IgYSBjb21wb25lbnQgb2YgdGhlIEhpbGJlcnQg │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ aXNCaXJhdGlvbmFsT250b0ltYWdlKC4uLixBc3N1bWVEb21pbmFudD0+Li4uKQ== │ │ │ #:len=321 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgwNywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaXNCaXJhdGlvbmFsT250b0ltYWdlLEFzc3VtZURv │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ i12 : Q=QQ[x,y,z,t,u]; │ │ │ │ │ │ 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}}); │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ │ │ i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0) │ │ │ - -- used 0.571395s (cpu); 0.3667s (thread); 0s (gc) │ │ │ + -- used 1.55369s (cpu); 0.6396s (thread); 0s (gc) │ │ │ │ │ │ 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 │ │ │ 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} │ │ │ │ │ │ o14 : RationalMapping │ │ │ │ │ │ i15 : R=QQ[x,y,z,t]/(z-2*t); │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ @@ -171,15 +171,15 @@ │ │ │ │ │ │ 
    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}});
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
    │ │ │ │ │ │ │ │ │ 
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.571395s (cpu); 0.3667s (thread); 0s (gc)
│ │ │ + -- used 1.55369s (cpu); 0.6396s (thread); 0s (gc)
│ │ │  
│ │ │                                  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
│ │ │  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}
│ │ │  
│ │ │  o14 : RationalMapping
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ o11 : Ideal of blowUpSubvar │ │ │ │ The next example is a birational map on $\mathbb{P}^4$. │ │ │ │ i12 : Q=QQ[x,y,z,t,u]; │ │ │ │ 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}}); │ │ │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0) │ │ │ │ - -- used 0.571395s (cpu); 0.3667s (thread); 0s (gc) │ │ │ │ + -- used 1.55369s (cpu); 0.6396s (thread); 0s (gc) │ │ │ │ │ │ │ │ 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 │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ U29ydEdlbnM= │ │ │ #:len=246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTb3J0R2VucyIsIlNvcnRHZW5zIiwiUmF0aW9uYWxQ │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ bmV0KFByb2plY3RpdmVQb2ludCk= │ │ │ #:len=205 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTcsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KG5ldCxQcm9q │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ @@ -48,15 +48,15 @@ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ ZZ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ i14 : time rationalPoints(I, Amount => true) │ │ │ - -- used 0.0024039s (cpu); 0.0049125s (thread); 0s (gc) │ │ │ + -- used 0.00407818s (cpu); 0.00267448s (thread); 0s (gc) │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ │ │ o16 : Ideal of QQ[x..z] │ │ │ │ │ │ @@ -141,24 +141,24 @@ │ │ │ o30 : Ideal of R │ │ │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ - -- used 0.963533s (cpu); 0.744977s (thread); 0s (gc) │ │ │ + -- used 1.55189s (cpu); 1.32883s (thread); 0s (gc) │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ │ │ i33 : #oo │ │ │ │ │ │ o33 = 31 │ │ │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ - -- used 0.286815s (cpu); 0.206311s (thread); 0s (gc) │ │ │ + -- used 0.377933s (cpu); 0.32083s (thread); 0s (gc) │ │ │ │ │ │ o35 = 31 │ │ │ │ │ │ i36 : │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ @@ -167,15 +167,15 @@ │ │ │ │ │ │ ZZ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ │ │ │ 
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.0024039s (cpu); 0.0049125s (thread); 0s (gc)
│ │ │ + -- used 0.00407818s (cpu); 0.00267448s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    │ │ │ │ │ │ │ │ │
    │ │ │

    Over number fields

    │ │ │
    │ │ │ @@ -308,15 +308,15 @@ │ │ │ │ │ │ 
    i31 : nodes = I + ideal jacobian I;
│ │ │  
│ │ │  o31 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │ - -- used 0.963533s (cpu); 0.744977s (thread); 0s (gc)
│ │ │ + -- used 1.55189s (cpu); 1.32883s (thread); 0s (gc)
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
    │ │ │ │ │ │ │ │ │ 
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    │ │ │ │ │ │ @@ -328,15 +328,15 @@ │ │ │ │ │ │ 
    i34 : nodes' = baseChange_32003 nodes;
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
    │ │ │ │ │ │ │ │ │ 
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.286815s (cpu); 0.206311s (thread); 0s (gc)
│ │ │ + -- used 0.377933s (cpu); 0.32083s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Caveat

    │ │ │ ├── html2text {} │ │ │ │ @@ -90,15 +90,15 @@ │ │ │ │ o13 = ideal(u + u + u + u + u + u + u + u + u + u + u ) │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ │ 101 0 10 │ │ │ │ i14 : time rationalPoints(I, Amount => true) │ │ │ │ - -- used 0.0024039s (cpu); 0.0049125s (thread); 0s (gc) │ │ │ │ + -- used 0.00407818s (cpu); 0.00267448s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ ******** OOvveerr nnuummbbeerr ffiieellddss ******** │ │ │ │ Over a number field one can use the option Bound to specify a maximal │ │ │ │ multiplicative height given by $(x_0:\dots:x_n)\mapsto \prod_{v}\max_i|x_i|_v ^ │ │ │ │ {d_v/d}$ (this is also available as a method _g_l_o_b_a_l_H_e_i_g_h_t). │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ @@ -197,25 +197,25 @@ │ │ │ │ (2z-qw)(4(x2+y2-z2)+(1+3(5-q2))w2)2"; │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ │ - -- used 0.963533s (cpu); 0.744977s (thread); 0s (gc) │ │ │ │ + -- used 1.55189s (cpu); 1.32883s (thread); 0s (gc) │ │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ i33 : #oo │ │ │ │ │ │ │ │ o33 = 31 │ │ │ │ Still it runs a lot faster when reduced to a positive characteristic. │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ │ - -- used 0.286815s (cpu); 0.206311s (thread); 0s (gc) │ │ │ │ + -- used 0.377933s (cpu); 0.32083s (thread); 0s (gc) │ │ │ │ │ │ │ │ o35 = 31 │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For a number field other than QQ, the enumeration of elements with bounded │ │ │ │ height depends on an algorithm by Doyle–Krumm, which is currently only │ │ │ │ implemented in Sage. │ │ │ │ ********** WWaayyss ttoo uussee rraattiioonnaallPPooiinnttss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/ReactionNetworks/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ bW9kaWZpY2F0aW9uT2ZUd29TdWJzdHJhdGVzSCgp │ │ │ #:len=344 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20gezE6KG1vZGlmaWNhdGlvbk9mVHdvU3Vic3RyYXRlc0gp │ │ ├── ./usr/share/doc/Macaulay2/RealRoots/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ U3lsdmVzdGVyQ291bnQ= │ │ │ #:len=2145 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGRpZmZlcmVuY2UgaW4gdmFyaWF0 │ │ │ aW9ucyBvZiB0aGUgU3lsdmVzdGVyIHNlcXVlbmNlIG9mIHR3byByYXRpb25hbCB1bml2YXJpYXRl │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ aXNMaW5lYXJUeXBlKE1vZHVsZSk= │ │ │ #:len=257 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIxMCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNMaW5lYXJUeXBlLE1vZHVsZSksImlzTGluZWFy │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out │ │ │ @@ -331,15 +331,15 @@ │ │ │ 2 2 2 2 2 2 2 │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ - -- used 0.858404s (cpu); 0.724585s (thread); 0s (gc) │ │ │ + -- used 1.11597s (cpu); 0.920221s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ 32003 0 2 0 1 │ │ │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_expected__Rees__Ideal.out │ │ │ @@ -57,15 +57,15 @@ │ │ │ o5 : Matrix S <-- S │ │ │ │ │ │ i6 : I = minors(n-1, M); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec. │ │ │ - -- used 1.09519s (cpu); 0.772014s (thread); 0s (gc) │ │ │ + -- used 1.2093s (cpu); 0.984319s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S[w ..w ] │ │ │ 0 4 │ │ │ │ │ │ i8 : kk = ZZ/101; │ │ │ │ │ │ i9 : S = kk[x,y,z]; │ │ │ @@ -76,19 +76,19 @@ │ │ │ o10 : Matrix S <-- S │ │ │ │ │ │ i11 : I = minors(3,m); │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time reesIdeal (I, I_0); │ │ │ - -- used 1.47858s (cpu); 1.13496s (thread); 0s (gc) │ │ │ + -- used 1.90712s (cpu); 1.50629s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ - -- used 1.59206s (cpu); 1.20982s (thread); 0s (gc) │ │ │ + -- used 1.68094s (cpu); 1.42792s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i14 : │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ @@ -13,21 +13,21 @@ │ │ │ 3 2 │ │ │ - x x x , x - x x ) │ │ │ 0 2 4 1 0 4 │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : time V1 = reesIdeal i; │ │ │ - -- used 0.0296671s (cpu); 0.026976s (thread); 0s (gc) │ │ │ + -- used 0.146548s (cpu); 0.0237621s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ - -- used 0.120321s (cpu); 0.122344s (thread); 0s (gc) │ │ │ + -- used 0.188336s (cpu); 0.135312s (thread); 0s (gc) │ │ │ │ │ │ o5 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i6 : S=kk[a,b,c] │ │ │ │ │ │ o6 = S │ │ │ @@ -47,21 +47,21 @@ │ │ │ │ │ │ 2 2 │ │ │ o8 = ideal (a , a*b, b ) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : time I1 = reesIdeal i; │ │ │ - -- used 0.0194455s (cpu); 0.0184988s (thread); 0s (gc) │ │ │ + -- used 0.363757s (cpu); 0.0473346s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ - -- used 0.00778222s (cpu); 0.0084035s (thread); 0s (gc) │ │ │ + -- used 0.0528772s (cpu); 0.0100092s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html │ │ │ @@ -485,15 +485,15 @@ │ │ │ │ │ │
    │ │ │

    We compute the singular locus once again:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -162,22 +162,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.858404s (cpu); 0.724585s (thread); 0s (gc)
│ │ │ + -- used 1.11597s (cpu); 0.920221s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
    │ │ │ 
    i49 : saturate(sing2, sub(irrelTot, ring sing2))
│ │ │ ├── html2text {}
│ │ │ │ @@ -325,15 +325,15 @@
│ │ │ │                 2 2    2   2             2 2    2
│ │ │ │        - p w , p y  - p , p w y - p p , p w  - p )
│ │ │ │           2 1   0      1   0 0     1 2   0 0    2
│ │ │ │  
│ │ │ │  o47 : Ideal of B2
│ │ │ │  We compute the singular locus once again:
│ │ │ │  i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ │ - -- used 0.858404s (cpu); 0.724585s (thread); 0s (gc)
│ │ │ │ + -- used 1.11597s (cpu); 0.920221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │ │                 32003  0   2   0   1
│ │ │ │  i49 : saturate(sing2, sub(irrelTot, ring sing2))
│ │ │ │  
│ │ │ │  o49 = ideal 1
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html
│ │ │ @@ -138,15 +138,15 @@
│ │ │            
    i6 : I = minors(n-1, M);
│ │ │  
│ │ │  o6 : Ideal of S
    │ │ │ 
    i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.09519s (cpu); 0.772014s (thread); 0s (gc)
│ │ │ + -- used 1.2093s (cpu); 0.984319s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    │ │ │ 
    i8 : kk = ZZ/101;
    │ │ │ 
    i11 : I = minors(3,m);
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ 
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.47858s (cpu); 1.13496s (thread); 0s (gc)
│ │ │ + -- used 1.90712s (cpu); 1.50629s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │ 
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.59206s (cpu); 1.20982s (thread); 0s (gc)
│ │ │ + -- used 1.68094s (cpu); 1.42792s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -86,34 +86,34 @@ │ │ │ │ │ │ │ │ 5 4 │ │ │ │ o5 : Matrix S <-- S │ │ │ │ i6 : I = minors(n-1, M); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec. │ │ │ │ - -- used 1.09519s (cpu); 0.772014s (thread); 0s (gc) │ │ │ │ + -- used 1.2093s (cpu); 0.984319s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S[w ..w ] │ │ │ │ 0 4 │ │ │ │ i8 : kk = ZZ/101; │ │ │ │ i9 : S = kk[x,y,z]; │ │ │ │ i10 : m = random(S^3, S^{4:-2}); │ │ │ │ │ │ │ │ 3 4 │ │ │ │ o10 : Matrix S <-- S │ │ │ │ i11 : I = minors(3,m); │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time reesIdeal (I, I_0); │ │ │ │ - -- used 1.47858s (cpu); 1.13496s (thread); 0s (gc) │ │ │ │ + -- used 1.90712s (cpu); 1.50629s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ │ - -- used 1.59206s (cpu); 1.20982s (thread); 0s (gc) │ │ │ │ + -- used 1.68094s (cpu); 1.42792s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or │ │ │ │ module │ │ │ │ * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ @@ -113,22 +113,22 @@ │ │ │ - x x x , x - x x ) │ │ │ 0 2 4 1 0 4 │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ 
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0296671s (cpu); 0.026976s (thread); 0s (gc)
│ │ │ + -- used 0.146548s (cpu); 0.0237621s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ 
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.120321s (cpu); 0.122344s (thread); 0s (gc)
│ │ │ + -- used 0.188336s (cpu); 0.135312s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │
    │ │ │

    The following example shows how we handle degrees

    │ │ │ @@ -157,22 +157,22 @@ │ │ │ 2 2 │ │ │ o8 = ideal (a , a*b, b ) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ │ │ │ 
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0194455s (cpu); 0.0184988s (thread); 0s (gc)
│ │ │ + -- used 0.363757s (cpu); 0.0473346s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    │ │ │ │ │ │ │ │ │ 
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00778222s (cpu); 0.0084035s (thread); 0s (gc)
│ │ │ + -- used 0.0528772s (cpu); 0.0100092s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
    │ │ │ │ │ │ │ │ │ 
    i11 : transpose gens I1
│ │ │ ├── html2text {}
│ │ │ │ @@ -52,20 +52,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                  3    2
│ │ │ │       - x x x , x  - x x )
│ │ │ │          0 2 4   1    0 4
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time V1 = reesIdeal i;
│ │ │ │ - -- used 0.0296671s (cpu); 0.026976s (thread); 0s (gc)
│ │ │ │ + -- used 0.146548s (cpu); 0.0237621s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.120321s (cpu); 0.122344s (thread); 0s (gc)
│ │ │ │ + -- used 0.188336s (cpu); 0.135312s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  The following example shows how we handle degrees
│ │ │ │  i6 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o6 = S
│ │ │ │ @@ -82,20 +82,20 @@
│ │ │ │  i8 : i=minors(2,m)
│ │ │ │  
│ │ │ │               2        2
│ │ │ │  o8 = ideal (a , a*b, b )
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : time I1 = reesIdeal i;
│ │ │ │ - -- used 0.0194455s (cpu); 0.0184988s (thread); 0s (gc)
│ │ │ │ + -- used 0.363757s (cpu); 0.0473346s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of S[w ..w ]
│ │ │ │                   0   2
│ │ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.00778222s (cpu); 0.0084035s (thread); 0s (gc)
│ │ │ │ + -- used 0.0528772s (cpu); 0.0100092s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of S[w ..w ]
│ │ │ │                    0   2
│ │ │ │  i11 : transpose gens I1
│ │ │ │  
│ │ │ │  o11 = {-1, -3} | aw_1-bw_2    |
│ │ │ │        {-1, -3} | aw_0-bw_1    |
│ │ ├── ./usr/share/doc/Macaulay2/ReflexivePolytopesDB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=7
│ │ │  S1NFbnRyeQ==
│ │ │  #:len=2433
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZW50cnkgZnJvbSB0aGUgS3JldXpl
│ │ │  ci1Ta2Fya2UgZGF0YWJhc2Ugb2YgZGltZW5zaW9uIDMgYW5kIDQgcmVmbGV4aXZlIHBvbHl0b3Bl
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  UmVndWxhcml0eQ==
│ │ │  #:len=797
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBDYXN0ZWxudW92by1NdW1m
│ │ │  b3JkIHJlZ3VsYXJpdHkgb2YgYSBob21vZ2VuZW91cyBpZGVhbCIsIERlc2NyaXB0aW9uID0+IChQ
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out
│ │ │ @@ -71,15 +71,15 @@
│ │ │       x x x , x  + x x  - x x  - x x x , x  + x  - x x )
│ │ │        0 1 3   0    0 1    1 2    0 2 5   0    2    0 5
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2602446350000001
│ │ │ +o8 = .2536823330000001
│ │ │  
│ │ │  o8 : RR (of precision 53)
│ │ │  
│ │ │  i9 : R = QQ[x_0..x_5]
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │ @@ -87,17 +87,17 @@
│ │ │  
│ │ │  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);
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .0691025631714286
│ │ │ +o11 = .0762167599518072
│ │ │  
│ │ │  o11 : RR (of precision 53)
│ │ │  
│ │ │  i12 : time regularity I2  
│ │ │ - -- used 0.00416868s (cpu); 0.00256224s (thread); 0s (gc)
│ │ │ + -- used 0.00400005s (cpu); 0.00193564s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html
│ │ │ @@ -165,15 +165,15 @@
│ │ │        0 1 3   0    0 1    1 2    0 2 5   0    2    0 5
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2602446350000001
│ │ │ +o8 = .2536823330000001
│ │ │  
│ │ │  o8 : RR (of precision 53)
    │ │ │ │ │ │ │ │ │

    This is an example where regularity is faster than mRegularity.

    │ │ │ │ │ │ │ │ │ @@ -187,21 +187,21 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    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);
│ │ │  
│ │ │  o10 : Ideal of R
    │ │ │ 
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .0691025631714286
│ │ │ +o11 = .0762167599518072
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.00416868s (cpu); 0.00256224s (thread); 0s (gc)
│ │ │ + -- used 0.00400005s (cpu); 0.00193564s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
    │ │ │
    │ │ │

    This symbol is provided by the package Regularity.

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -95,34 +95,34 @@ │ │ │ │ 3 2 2 3 3 2 │ │ │ │ x x x , x + x x - x x - x x x , x + x - x x ) │ │ │ │ 0 1 3 0 0 1 1 2 0 2 5 0 2 0 5 │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ i8 : benchmark "mRegularity I1" │ │ │ │ │ │ │ │ -o8 = .2602446350000001 │ │ │ │ +o8 = .2536823330000001 │ │ │ │ │ │ │ │ o8 : RR (of precision 53) │ │ │ │ This is an example where regularity is faster than mRegularity. │ │ │ │ i9 : R = QQ[x_0..x_5] │ │ │ │ │ │ │ │ o9 = R │ │ │ │ │ │ │ │ o9 : PolynomialRing │ │ │ │ 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); │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : benchmark " mRegularity I2" │ │ │ │ │ │ │ │ -o11 = .0691025631714286 │ │ │ │ +o11 = .0762167599518072 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.00416868s (cpu); 0.00256224s (thread); 0s (gc) │ │ │ │ + -- used 0.00400005s (cpu); 0.00193564s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ This symbol is provided by the package Regularity. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_u_l_a_r_i_t_y -- compute the Castelnuovo-Mumford regularity │ │ │ │ ********** WWaayyss ttoo uussee mmRReegguullaarriittyy:: ********** │ │ │ │ * mRegularity(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ Y2Fub25pY2FsTXVsdGlwbGllcnM= │ │ │ #:len=1705 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNhbm9uaWNhbCBt │ │ │ dWx0aXBsaWVycyBvZiBhIHJhdGlvbmFsIGN1cnZlcyB3aXRoIG5vZGVzIiwgImxpbmVudW0iID0+ │ │ ├── ./usr/share/doc/Macaulay2/ResLengthThree/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ bXVsdFRhYmxlT25lVHdvKFJpbmcp │ │ │ #:len=273 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjMwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0VGFibGVPbmVUd28sUmluZyksIm11bHRUYWJs │ │ ├── ./usr/share/doc/Macaulay2/ResidualIntersections/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ Z2VuZXJpY1Jlc2lkdWFs │ │ │ #:len=1739 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgZ2VuZXJpYyByZXNpZHVh │ │ │ bCBpbnRlcnNlY3Rpb25zIG9mIGFuIGlkZWFsIiwgImxpbmVudW0iID0+IDg4NiwgSW5wdXRzID0+ │ │ ├── ./usr/share/doc/Macaulay2/ResolutionsOfStanleyReisnerRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=41 │ │ │ YmFyeWNlbnRyaWNTdWJkaXZpc2lvbihTaW1wbGljaWFsQ29tcGxleCk= │ │ │ #:len=381 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiYXJ5Y2VudHJpY1N1YmRpdmlzaW9uLFNpbXBsaWNp │ │ ├── ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aHVyd2l0ekZvcm0oLi4uLFNpbmd1bGFyTG9jdXM9Pi4uLik= │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMzMCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaHVyd2l0ekZvcm0sU2luZ3VsYXJMb2N1c10sImh1 │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ @@ -5,18 +5,18 @@ │ │ │ o2 = ideal(x x - x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ - -- used 0.142755s (cpu); 0.0805606s (thread); 0s (gc) │ │ │ + -- used 0.0902452s (cpu); 0.0472935s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.05218s (cpu); 0.0522233s (thread); 0s (gc) │ │ │ + -- used 0.0737947s (cpu); 0.0747475s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o4 = (ideal (x x - x x , x x - x x , x x - x x , │ │ │ 0,3 1,2 0,2 1,3 1,0 1,1 1,2 1,3 0,3 1,1 0,1 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ @@ -37,17 +37,17 @@ │ │ │ 2 2 │ │ │ 4x x x x - 2x x x x + x x )) │ │ │ 0,0 0,1 1,2 1,3 0,2 0,3 1,2 1,3 0,2 1,3 │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : time cayleyTrick(P1xP1,1,Duality=>true); │ │ │ - -- used 0.149721s (cpu); 0.103387s (thread); 0s (gc) │ │ │ + -- used 0.152743s (cpu); 0.0736586s (thread); 0s (gc) │ │ │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ - -- used 0.146483s (cpu); 0.0877464s (thread); 0s (gc) │ │ │ + -- used 0.182141s (cpu); 0.136407s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ o2 = ideal (x + x + x + x , x x + x x + x x ) │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ i3 : -- Chow equations of C │ │ │ time eqsC = chowEquations chowForm C │ │ │ - -- used 0.106786s (cpu); 0.0541456s (thread); 0s (gc) │ │ │ + -- used 0.141058s (cpu); 0.0946273s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 4 2 2 2 2 │ │ │ o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + │ │ │ 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 2 3 3 2 2 2 2 │ │ │ x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x - x x x + x x │ │ │ @@ -72,15 +72,15 @@ │ │ │ o5 = ideal (x - x x , x - x x x , x x - x x ) │ │ │ 1 0 2 2 0 1 3 1 2 0 3 │ │ │ │ │ │ o5 : Ideal of P3 │ │ │ │ │ │ i6 : -- Chow equations of D │ │ │ time eqsD = chowEquations chowForm D │ │ │ - -- used 0.0298427s (cpu); 0.0334105s (thread); 0s (gc) │ │ │ + -- used 0.0719996s (cpu); 0.0740332s (thread); 0s (gc) │ │ │ │ │ │ 4 3 2 3 2 2 3 2 2 2 2 2 2 │ │ │ o6 = ideal (x x - x x , x x x - x x x , x x x - x x x , x x x - x x x , │ │ │ 2 3 1 3 1 2 3 0 1 3 0 2 3 0 1 3 1 2 3 0 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 3 3 2 4 2 2 2 3 │ │ │ x x x x - x x , x x x - x x , x x - 4x x x x + 3x x x , x x x - │ │ │ @@ -117,24 +117,24 @@ │ │ │ o9 = ideal(x x + x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o9 : Ideal of P3 │ │ │ │ │ │ i10 : -- tangential Chow forms of Q │ │ │ time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2)) │ │ │ - -- used 0.144745s (cpu); 0.0765926s (thread); 0s (gc) │ │ │ + -- used 0.183428s (cpu); 0.131307s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o10 = (x x + x x , x - 4x x + 2x x + x , x x + │ │ │ 0 1 2 3 0,1 0,2 1,3 0,1 2,3 2,3 0,1,2 0,1,3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x ) │ │ │ 0,2,3 1,2,3 │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2)) │ │ │ - -- used 0.126093s (cpu); 0.0638828s (thread); 0s (gc) │ │ │ + -- used 0.148884s (cpu); 0.0994172s (thread); 0s (gc) │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ @@ -16,15 +16,15 @@ │ │ │ │ │ │ ZZ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ 3331 0 5 │ │ │ │ │ │ i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian) │ │ │ time ChowV = chowForm(V,AffineChartGrass=>{1,2,3}) │ │ │ - -- used 4.91471s (cpu); 4.47543s (thread); 0s (gc) │ │ │ + -- used 5.4004s (cpu); 5.08707s (thread); 0s (gc) │ │ │ │ │ │ 4 2 2 2 2 │ │ │ o3 = x + 2x x x + x x - 2x x x + │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 │ │ │ x x - x x x + x x x x + │ │ │ @@ -143,19 +143,19 @@ │ │ │ 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 │ │ │ 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 , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x - x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x + x x - x x + x x , x x - x x + x x , x x - x x + x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x - x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x ) │ │ │ 2,3,5 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 │ │ │ │ │ │ i4 : -- equivalently (but faster)... │ │ │ time assert(ChowV === chowForm f) │ │ │ - -- used 1.1541s (cpu); 1.04468s (thread); 0s (gc) │ │ │ + -- used 1.218s (cpu); 1.16436s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.218349s (cpu); 0.168032s (thread); 0s (gc) │ │ │ + -- used 0.312455s (cpu); 0.189813s (thread); 0s (gc) │ │ │ │ │ │ i6 : -- three generic ternary quadrics │ │ │ F = genericPolynomials({2,2,2},ZZ/3331) │ │ │ │ │ │ 2 2 2 2 2 │ │ │ o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + │ │ │ 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 │ │ │ @@ -164,12 +164,12 @@ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : -- resultant of the three forms │ │ │ time resF = resultant F; │ │ │ - -- used 0.309447s (cpu); 0.186941s (thread); 0s (gc) │ │ │ + -- used 0.390298s (cpu); 0.156038s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres)) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ @@ -4,30 +4,30 @@ │ │ │ │ │ │ 2 2 │ │ │ o2 = a*x + b*x*y + c*y │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ i3 : time discriminant F │ │ │ - -- used 0.0078513s (cpu); 0.00792964s (thread); 0s (gc) │ │ │ + -- used 0.011999s (cpu); 0.0130189s (thread); 0s (gc) │ │ │ │ │ │ 2 │ │ │ o3 = - b + 4a*c │ │ │ │ │ │ o3 : ZZ[a..c] │ │ │ │ │ │ i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3 │ │ │ │ │ │ 3 2 2 3 │ │ │ o5 = a*x + b*x y + c*x*y + d*y │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ i6 : time discriminant F │ │ │ - -- used 0.0840155s (cpu); 0.0312505s (thread); 0s (gc) │ │ │ + -- used 0.0737907s (cpu); 0.0236709s (thread); 0s (gc) │ │ │ │ │ │ 2 2 3 3 2 2 │ │ │ o6 = - b c + 4a*c + 4b d - 18a*b*c*d + 27a d │ │ │ │ │ │ o6 : ZZ[a..d] │ │ │ │ │ │ i7 : x=symbol x; R=ZZ/331[x_0..x_3] │ │ │ @@ -59,15 +59,15 @@ │ │ │ 4 3 4 4 3 4 │ │ │ o12 = (t + t )x - t x x + t x + (t - t )x + t x x + t x │ │ │ 0 1 0 1 0 1 0 1 0 1 2 1 2 3 0 3 │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ i13 : time D=discriminant pencil │ │ │ - -- used 0.422412s (cpu); 0.42392s (thread); 0s (gc) │ │ │ + -- used 0.439326s (cpu); 0.437955s (thread); 0s (gc) │ │ │ │ │ │ 108 106 2 102 6 100 8 98 10 96 12 │ │ │ o13 = - 62t + 19t t + 160t t + 91t t + 129t t + 117t t + │ │ │ 0 0 1 0 1 0 1 0 1 0 1 │ │ │ ----------------------------------------------------------------------- │ │ │ 94 14 92 16 90 18 88 20 86 22 84 24 │ │ │ 161t t + 124t t - 82t t - 21t t - 49t t - 123t t + │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ @@ -9,25 +9,25 @@ │ │ │ x x ) │ │ │ 0 3 │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 5 │ │ │ │ │ │ i2 : time V' = dualVariety V │ │ │ - -- used 0.180744s (cpu); 0.130805s (thread); 0s (gc) │ │ │ + -- used 0.294961s (cpu); 0.127308s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 │ │ │ o2 = ideal(x x - x x x + x x + x x - 4x x x ) │ │ │ 2 3 1 2 4 0 4 1 5 0 3 5 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 5 │ │ │ │ │ │ i3 : time V == dualVariety V' │ │ │ - -- used 0.199797s (cpu); 0.140703s (thread); 0s (gc) │ │ │ + -- used 0.249635s (cpu); 0.144745s (thread); 0s (gc) │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331) │ │ │ │ │ │ 3 2 2 3 2 2 2 │ │ │ o4 = a x + a x x + a x x + a x + a x x + a x x x + a x x + a x x + │ │ │ @@ -38,22 +38,22 @@ │ │ │ 8 1 2 9 2 │ │ │ │ │ │ ZZ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ i5 : time discF = ideal discriminant F; │ │ │ - -- used 0.0624866s (cpu); 0.0609208s (thread); 0s (gc) │ │ │ + -- used 0.070908s (cpu); 0.0702981s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ 3331 0 9 │ │ │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ - -- used 0.621853s (cpu); 0.55823s (thread); 0s (gc) │ │ │ + -- used 0.66942s (cpu); 0.626341s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ 3331 0 9 │ │ │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ @@ -6,15 +6,15 @@ │ │ │ o1 = ideal (x - x x , x x - x x , x - x x ) │ │ │ 2 1 3 1 2 0 3 1 0 2 │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i2 : time fromPluckerToStiefel dualize chowForm C │ │ │ - -- used 0.114714s (cpu); 0.0536053s (thread); 0s (gc) │ │ │ + -- used 0.0875735s (cpu); 0.0457231s (thread); 0s (gc) │ │ │ │ │ │ 3 3 2 2 2 2 2 3 │ │ │ o2 = - x x + x x x x - x x x x + x x x - │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 2 │ │ │ x x x x + 2x x x x + x x x x x x - │ │ │ @@ -56,15 +56,15 @@ │ │ │ x x x x - 2x x x x - x x x x + x x │ │ │ 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 │ │ │ │ │ │ o2 : QQ[x ..x ] │ │ │ 0,0 1,3 │ │ │ │ │ │ i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1}) │ │ │ - -- used 0.03594s (cpu); 0.0388574s (thread); 0s (gc) │ │ │ + -- used 0.0356639s (cpu); 0.035488s (thread); 0s (gc) │ │ │ │ │ │ 3 2 2 │ │ │ o3 = - x x + x x x - x x x + x x + 3x x x - │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 │ │ │ 2x x + x + x │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o4 : QQ[a ..a ] │ │ │ 0,0 1,1 │ │ │ │ │ │ i5 : w = chowForm C; │ │ │ │ │ │ i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s)) │ │ │ - -- used 0.0871858s (cpu); 0.0336167s (thread); 0s (gc) │ │ │ + -- used 0.0679509s (cpu); 0.0242628s (thread); 0s (gc) │ │ │ │ │ │ 3 2 3 2 │ │ │ o6 = {ideal(- x x + x x x - x - 3x x x + 2x x + │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,2 0,3 1,2 0,2 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 3 2 │ │ │ x x x - x x + x ), ideal(x x - 2x x x x + │ │ │ @@ -130,14 +130,14 @@ │ │ │ 2 3 2 │ │ │ 2x x - x + x )} │ │ │ 0,0 1,1 1,1 1,0 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : time apply(U,u->dim singularLocus u) │ │ │ - -- used 0.0154955s (cpu); 0.0148463s (thread); 0s (gc) │ │ │ + -- used 0.014661s (cpu); 0.013321s (thread); 0s (gc) │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ --p + p p + -p p + -p p + -p p + 7p ) │ │ │ 10 3 0 4 4 1 4 2 2 4 3 3 4 4 │ │ │ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i2 : time hurwitzForm Q │ │ │ - -- used 0.110524s (cpu); 0.0675797s (thread); 0s (gc) │ │ │ + -- used 0.092074s (cpu); 0.0461657s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o2 = 143100p + 267300p p + 96525p - 56700p p - 56100p p │ │ │ 0,1 0,1 0,2 0,2 0,1 1,2 0,2 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ + 900p + 140400p p + 111780p p + 133380p - 8100p p │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ @@ -22,15 +22,15 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o1 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i2 : time isCoisotropic w │ │ │ - -- used 0.00799966s (cpu); 0.00727879s (thread); 0s (gc) │ │ │ + -- used 0.0406609s (cpu); 0.0405998s (thread); 0s (gc) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ 7 2 2 3 2 5 │ │ │ @@ -52,12 +52,12 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o3 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : time isCoisotropic w' │ │ │ - -- used 0.00400191s (cpu); 0.00650056s (thread); 0s (gc) │ │ │ + -- used 0.00798371s (cpu); 0.00829385s (thread); 0s (gc) │ │ │ │ │ │ o4 = false │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ @@ -31,12 +31,12 @@ │ │ │ 4 5 │ │ │ │ │ │ ZZ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I)) │ │ │ - -- used 0.0228258s (cpu); 0.0201847s (thread); 0s (gc) │ │ │ + -- used 0.0479977s (cpu); 0.0478608s (thread); 0s (gc) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ @@ -13,15 +13,15 @@ │ │ │ 2 2 2 3 │ │ │ c x x x + c x x + c x x + c x x + c x } │ │ │ 4 0 1 2 7 1 2 5 0 2 8 1 2 9 2 │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time (D,D') = macaulayFormula F │ │ │ - -- used 0.00215734s (cpu); 0.00350587s (thread); 0s (gc) │ │ │ + -- used 0.00400078s (cpu); 0.00300471s (thread); 0s (gc) │ │ │ │ │ │ o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0 0 0 0 0 0 0 0 0 0 0 │ │ │ | 0 a_0 0 a_1 a_2 0 a_3 a_4 a_5 0 0 0 0 0 0 0 0 │ │ │ | 0 0 a_0 0 a_1 a_2 0 a_3 a_4 a_5 0 0 0 0 0 0 0 │ │ │ | 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 0 0 │ │ │ | 0 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 0 │ │ │ | 0 0 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 │ │ │ @@ -78,15 +78,15 @@ │ │ │ 6 2 2 3 │ │ │ -p p + 2p p + 6p } │ │ │ 7 0 2 1 2 2 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : time (D,D') = macaulayFormula F │ │ │ - -- used 0.00399943s (cpu); 0.00217211s (thread); 0s (gc) │ │ │ + -- used 0.00123595s (cpu); 0.00199651s (thread); 0s (gc) │ │ │ │ │ │ o4 = (| 9/2 1/2 9/4 1/2 1 3/4 0 0 0 0 0 0 0 0 0 │ │ │ | 0 9/2 0 1/2 9/4 0 1/2 1 3/4 0 0 0 0 0 0 │ │ │ | 0 0 9/2 0 1/2 9/4 0 1/2 1 3/4 0 0 0 0 0 │ │ │ | 0 0 0 9/2 0 0 1/2 9/4 0 0 1/2 1 3/4 0 0 │ │ │ | 0 0 0 0 9/2 0 0 1/2 9/4 0 0 1/2 1 3/4 0 │ │ │ | 0 0 0 0 0 9/2 0 0 1/2 9/4 0 0 1/2 1 3/4 │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ @@ -9,29 +9,29 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ 664x ) │ │ │ 4 │ │ │ │ │ │ o3 : Ideal of P4 │ │ │ │ │ │ i4 : time p = plucker L │ │ │ - -- used 0.00396538s (cpu); 0.00437586s (thread); 0s (gc) │ │ │ + -- used 0.00400241s (cpu); 0.00584151s (thread); 0s (gc) │ │ │ │ │ │ o4 = ideal (x + 8480x , x - 6727x , x + 15777x , x + │ │ │ 2,4 3,4 1,4 3,4 0,4 3,4 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 11656x , x - 14853x , x + 664x , x + 13522x , x + │ │ │ 3,4 1,3 3,4 0,3 3,4 1,2 3,4 0,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 11804x , x + 14854x ) │ │ │ 3,4 0,1 3,4 │ │ │ │ │ │ o4 : Ideal of G'1'4 │ │ │ │ │ │ i5 : time L' = plucker p │ │ │ - -- used 0.0973721s (cpu); 0.039945s (thread); 0s (gc) │ │ │ + -- used 0.117661s (cpu); 0.0734802s (thread); 0s (gc) │ │ │ │ │ │ o5 = ideal (x + 8480x - 11656x , x - 6727x + 14853x , x + 15777x - │ │ │ 2 3 4 1 3 4 0 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 664x ) │ │ │ 4 │ │ │ │ │ │ @@ -40,25 +40,25 @@ │ │ │ i6 : assert(L' == L) │ │ │ │ │ │ i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve │ │ │ │ │ │ o7 : Ideal of G'1'4 │ │ │ │ │ │ i8 : time W = plucker Y; -- surface swept out by the lines of Y │ │ │ - -- used 0.0280079s (cpu); 0.0301314s (thread); 0s (gc) │ │ │ + -- used 0.0839972s (cpu); 0.0834231s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of P4 │ │ │ │ │ │ i9 : (codim W,degree W) │ │ │ │ │ │ o9 = (2, 5) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : time Y' = plucker(W,1); -- variety of lines contained in W │ │ │ - -- used 0.147833s (cpu); 0.149666s (thread); 0s (gc) │ │ │ + -- used 0.3189s (cpu); 0.318989s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of G'1'4 │ │ │ │ │ │ i11 : assert(Y' == Y) │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ @@ -35,15 +35,15 @@ │ │ │ 3 2 9 7 2 9 3 1 8 4 │ │ │ -b)y*w + (-a + -b)z*w + (-a + 2b)w , 2x + -y + -z + -w} │ │ │ 4 8 8 7 4 3 5 │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time resultant(F,Algorithm=>"Poisson2") │ │ │ - -- used 0.239497s (cpu); 0.149163s (thread); 0s (gc) │ │ │ + -- used 0.352888s (cpu); 0.249339s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o3 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -56,15 +56,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o3 : QQ[a..b] │ │ │ │ │ │ i4 : time resultant(F,Algorithm=>"Macaulay2") │ │ │ - -- used 0.128223s (cpu); 0.0770729s (thread); 0s (gc) │ │ │ + -- used 0.110133s (cpu); 0.0644408s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o4 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -77,15 +77,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o4 : QQ[a..b] │ │ │ │ │ │ i5 : time resultant(F,Algorithm=>"Poisson") │ │ │ - -- used 0.287256s (cpu); 0.289426s (thread); 0s (gc) │ │ │ + -- used 0.307491s (cpu); 0.309999s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o5 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -98,15 +98,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o5 : QQ[a..b] │ │ │ │ │ │ i6 : time resultant(F,Algorithm=>"Macaulay") │ │ │ - -- used 0.608867s (cpu); 0.559075s (thread); 0s (gc) │ │ │ + -- used 0.776348s (cpu); 0.725524s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o6 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ 2 2 3 2 4 │ │ │ o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time resultant F │ │ │ - -- used 0.0344967s (cpu); 0.0339164s (thread); 0s (gc) │ │ │ + -- used 0.0479909s (cpu); 0.0495041s (thread); 0s (gc) │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ ------------------------------------------------------------------------ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -64,15 +64,15 @@ │ │ │ 3 │ │ │ + c x } │ │ │ 9 2 │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : time resultant F │ │ │ - -- used 2.8419s (cpu); 2.11491s (thread); 0s (gc) │ │ │ + -- used 2.37705s (cpu); 1.88069s (thread); 0s (gc) │ │ │ │ │ │ 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 │ │ │ o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - │ │ │ 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 │ │ │ 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c - │ │ │ @@ -1690,12 +1690,12 @@ │ │ │ 2 2 2 2 │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : time # terms resultant F │ │ │ - -- used 0.503578s (cpu); 0.380091s (thread); 0s (gc) │ │ │ + -- used 0.439201s (cpu); 0.388865s (thread); 0s (gc) │ │ │ │ │ │ o7 = 21894 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ @@ -8,15 +8,15 @@ │ │ │ 1 2 0 3 1 3 0 4 3 2 4 │ │ │ │ │ │ o2 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form) │ │ │ time tangentialChowForm(S,0) │ │ │ - -- used 0.0984014s (cpu); 0.0465418s (thread); 0s (gc) │ │ │ + -- used 0.0994219s (cpu); 0.0524418s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o3 = p p - p p p - p p p + p p p + p p + │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 │ │ │ p p - 2p p p - p p p │ │ │ @@ -26,15 +26,15 @@ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 │ │ │ o3 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (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 ) │ │ │ 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 │ │ │ │ │ │ i4 : -- 1-th associated hypersurface of S in G(2,4) │ │ │ time tangentialChowForm(S,1) │ │ │ - -- used 0.0522902s (cpu); 0.0514451s (thread); 0s (gc) │ │ │ + -- used 0.0705868s (cpu); 0.0724036s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 3 2 2 │ │ │ o4 = p p + p p - 2p p + p p - │ │ │ 1,2,3 1,2,4 0,2,4 1,2,4 0,2,3 1,2,4 0,2,4 0,3,4 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 3 3 │ │ │ 4p p - 4p p - 2p p + │ │ │ @@ -68,32 +68,32 @@ │ │ │ 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 │ │ │ o4 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (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 ) │ │ │ 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 │ │ │ │ │ │ i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S) │ │ │ time tangentialChowForm(S,2) │ │ │ - -- used 0.0332139s (cpu); 0.033128s (thread); 0s (gc) │ │ │ + -- used 0.0240011s (cpu); 0.0267962s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o5 = p p - p p p + p p │ │ │ 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 │ │ │ │ │ │ o5 : QQ[p ..p , p , p , p ] │ │ │ 0,1,2,3 0,1,2,4 0,1,3,4 0,2,3,4 1,2,3,4 │ │ │ │ │ │ i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing │ │ │ time S' = ideal dualize tangentialChowForm(S,2) │ │ │ - -- used 0.111927s (cpu); 0.0524302s (thread); 0s (gc) │ │ │ + -- used 0.103348s (cpu); 0.0599139s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = ideal(p p - p p p + p p ) │ │ │ 1 2 0 1 3 0 4 │ │ │ │ │ │ o6 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i7 : -- we then can recover S │ │ │ time assert(dualize tangentialChowForm(S',3) == S) │ │ │ - -- used 0.128042s (cpu); 0.0814872s (thread); 0s (gc) │ │ │ + -- used 0.279752s (cpu); 0.0764804s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ @@ -87,22 +87,22 @@ │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ 
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.142755s (cpu); 0.0805606s (thread); 0s (gc)
    │ │ │ + -- used 0.0902452s (cpu); 0.0472935s (thread); 0s (gc) │ │ │ │ │ │ │ │ │

    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.

    │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.05218s (cpu); 0.0522233s (thread); 0s (gc)
│ │ │ + -- used 0.0737947s (cpu); 0.0747475s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -127,22 +127,22 @@
│ │ │  o4 : Sequence
    │ │ │
    │ │ │

    If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.149721s (cpu); 0.103387s (thread); 0s (gc)
    │ │ │ + -- used 0.152743s (cpu); 0.0736586s (thread); 0s (gc) │ │ │ 
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    │ │ │ 
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.146483s (cpu); 0.0877464s (thread); 0s (gc)
    │ │ │ + -- used 0.182141s (cpu); 0.136407s (thread); 0s (gc) │ │ │ 
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -39,20 +39,20 @@ │ │ │ │ │ │ │ │ o2 = ideal(x x - x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ │ - -- used 0.142755s (cpu); 0.0805606s (thread); 0s (gc) │ │ │ │ + -- used 0.0902452s (cpu); 0.0472935s (thread); 0s (gc) │ │ │ │ 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. │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ │ - -- used 0.05218s (cpu); 0.0522233s (thread); 0s (gc) │ │ │ │ + -- used 0.0737947s (cpu); 0.0747475s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ o4 = (ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ 0,3 1,2 0,2 1,3 1,0 1,1 1,2 1,3 0,3 1,1 0,1 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ @@ -74,18 +74,18 @@ │ │ │ │ 4x x x x - 2x x x x + x x )) │ │ │ │ 0,0 0,1 1,2 1,3 0,2 0,3 1,2 1,3 0,2 1,3 │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ If the option Duality is set to true, then the method applies the so-called │ │ │ │ "dual Cayley trick". │ │ │ │ i5 : time cayleyTrick(P1xP1,1,Duality=>true); │ │ │ │ - -- used 0.149721s (cpu); 0.103387s (thread); 0s (gc) │ │ │ │ + -- used 0.152743s (cpu); 0.0736586s (thread); 0s (gc) │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ │ - -- used 0.146483s (cpu); 0.0877464s (thread); 0s (gc) │ │ │ │ + -- used 0.182141s (cpu); 0.136407s (thread); 0s (gc) │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety │ │ │ │ ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: ********** │ │ │ │ * cayleyTrick(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ │ │ │ 
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.106786s (cpu); 0.0541456s (thread); 0s (gc)
│ │ │ + -- used 0.141058s (cpu); 0.0946273s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -153,15 +153,15 @@
│ │ │               1    0 2   2    0 1 3   1 2    0 3
│ │ │  
│ │ │  o5 : Ideal of P3
    │ │ │ │ │ │ │ │ │ 
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.0298427s (cpu); 0.0334105s (thread); 0s (gc)
│ │ │ + -- used 0.0719996s (cpu); 0.0740332s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -205,28 +205,28 @@
│ │ │              0 1    2 3
│ │ │  
│ │ │  o9 : Ideal of P3
    │ │ │ │ │ │ │ │ │ 
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.144745s (cpu); 0.0765926s (thread); 0s (gc)
│ │ │ + -- used 0.183428s (cpu); 0.131307s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
    │ │ │ │ │ │ │ │ │ 
    i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.126093s (cpu); 0.0638828s (thread); 0s (gc)
│ │ │ + -- used 0.148884s (cpu); 0.0994172s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    │ │ │ │ │ │ │ │ │

    Note that chowEquations(W,0) is not the same as chowEquations W.

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ 2 2 2 2 │ │ │ │ o2 = ideal (x + x + x + x , x x + x x + x x ) │ │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ i3 : -- Chow equations of C │ │ │ │ time eqsC = chowEquations chowForm C │ │ │ │ - -- used 0.106786s (cpu); 0.0541456s (thread); 0s (gc) │ │ │ │ + -- used 0.141058s (cpu); 0.0946273s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 4 2 2 2 2 │ │ │ │ o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + │ │ │ │ 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 2 3 3 2 2 2 2 │ │ │ │ x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x - x x x + x x │ │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ 2 3 2 2 │ │ │ │ o5 = ideal (x - x x , x - x x x , x x - x x ) │ │ │ │ 1 0 2 2 0 1 3 1 2 0 3 │ │ │ │ │ │ │ │ o5 : Ideal of P3 │ │ │ │ i6 : -- Chow equations of D │ │ │ │ time eqsD = chowEquations chowForm D │ │ │ │ - -- used 0.0298427s (cpu); 0.0334105s (thread); 0s (gc) │ │ │ │ + -- used 0.0719996s (cpu); 0.0740332s (thread); 0s (gc) │ │ │ │ │ │ │ │ 4 3 2 3 2 2 3 2 2 2 2 2 2 │ │ │ │ o6 = ideal (x x - x x , x x x - x x x , x x x - x x x , x x x - x x x , │ │ │ │ 2 3 1 3 1 2 3 0 1 3 0 2 3 0 1 3 1 2 3 0 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 3 3 2 4 2 2 2 3 │ │ │ │ x x x x - x x , x x x - x x , x x - 4x x x x + 3x x x , x x x - │ │ │ │ @@ -136,27 +136,27 @@ │ │ │ │ o9 = ideal(x x + x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o9 : Ideal of P3 │ │ │ │ i10 : -- tangential Chow forms of Q │ │ │ │ time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm │ │ │ │ (Q,1),tangentialChowForm(Q,2)) │ │ │ │ - -- used 0.144745s (cpu); 0.0765926s (thread); 0s (gc) │ │ │ │ + -- used 0.183428s (cpu); 0.131307s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o10 = (x x + x x , x - 4x x + 2x x + x , x x + │ │ │ │ 0 1 2 3 0,1 0,2 1,3 0,1 2,3 2,3 0,1,2 0,1,3 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ x x ) │ │ │ │ 0,2,3 1,2,3 │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations │ │ │ │ (W2,2)) │ │ │ │ - -- used 0.126093s (cpu); 0.0638828s (thread); 0s (gc) │ │ │ │ + -- used 0.148884s (cpu); 0.0994172s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ Note that chowEquations(W,0) is not the same as chowEquations W. │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: ********** │ │ │ │ * chowEquations(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ ZZ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ 3331 0 5 │ │ │ │ │ │ │ │ │ 
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 4.91471s (cpu); 4.47543s (thread); 0s (gc)
│ │ │ + -- used 5.4004s (cpu); 5.08707s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        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  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -227,20 +227,20 @@
│ │ │  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     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      - x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      - x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     )
│ │ │         2,3,5 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
    │ │ │ │ │ │ │ │ │ 
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.1541s (cpu); 1.04468s (thread); 0s (gc)
    │ │ │ + -- used 1.218s (cpu); 1.16436s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.218349s (cpu); 0.168032s (thread); 0s (gc)
    │ │ │ + -- used 0.312455s (cpu); 0.189813s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │           2               2                        2     2               2  
│ │ │  o6 = {a x  + a x x  + a x  + a x x  + a x x  + a x , b x  + b x x  + b x  +
│ │ │ @@ -251,15 +251,15 @@
│ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ 
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.309447s (cpu); 0.186941s (thread); 0s (gc)
    │ │ │ + -- used 0.390298s (cpu); 0.156038s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -42,15 +42,15 @@ │ │ │ │ │ │ │ │ ZZ │ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 5 │ │ │ │ i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an │ │ │ │ affine chart of the Grassmannian) │ │ │ │ time ChowV = chowForm(V,AffineChartGrass=>{1,2,3}) │ │ │ │ - -- used 4.91471s (cpu); 4.47543s (thread); 0s (gc) │ │ │ │ + -- used 5.4004s (cpu); 5.08707s (thread); 0s (gc) │ │ │ │ │ │ │ │ 4 2 2 2 2 │ │ │ │ o3 = x + 2x x x + x x - 2x x x + │ │ │ │ 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 │ │ │ │ x x - x x x + x x x x + │ │ │ │ @@ -235,33 +235,33 @@ │ │ │ │ 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 │ │ │ │ i4 : -- equivalently (but faster)... │ │ │ │ time assert(ChowV === chowForm f) │ │ │ │ - -- used 1.1541s (cpu); 1.04468s (thread); 0s (gc) │ │ │ │ + -- used 1.218s (cpu); 1.16436s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.218349s (cpu); 0.168032s (thread); 0s (gc) │ │ │ │ + -- used 0.312455s (cpu); 0.189813s (thread); 0s (gc) │ │ │ │ i6 : -- three generic ternary quadrics │ │ │ │ F = genericPolynomials({2,2,2},ZZ/3331) │ │ │ │ │ │ │ │ 2 2 2 2 2 │ │ │ │ o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + │ │ │ │ 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : -- resultant of the three forms │ │ │ │ time resF = resultant F; │ │ │ │ - -- used 0.309447s (cpu); 0.186941s (thread); 0s (gc) │ │ │ │ + -- used 0.390298s (cpu); 0.156038s (thread); 0s (gc) │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring │ │ │ │ Xres)) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: ********** │ │ │ │ * chowForm(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ 2 2 │ │ │ o2 = a*x + b*x*y + c*y │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ │ │ │ 
    i3 : time discriminant F
│ │ │ - -- used 0.0078513s (cpu); 0.00792964s (thread); 0s (gc)
│ │ │ + -- used 0.011999s (cpu); 0.0130189s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    │ │ │ │ │ │ │ │ │ @@ -100,15 +100,15 @@ │ │ │ 3 2 2 3 │ │ │ o5 = a*x + b*x y + c*x*y + d*y │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ │ │ │ 
    i6 : time discriminant F
│ │ │ - -- used 0.0840155s (cpu); 0.0312505s (thread); 0s (gc)
│ │ │ + -- used 0.0737907s (cpu); 0.0236709s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    │ │ │ │ │ │ │ │ │ @@ -149,15 +149,15 @@ │ │ │ o12 = (t + t )x - t x x + t x + (t - t )x + t x x + t x │ │ │ 0 1 0 1 0 1 0 1 0 1 2 1 2 3 0 3 │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ │ │ │ 
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.422412s (cpu); 0.42392s (thread); 0s (gc)
│ │ │ + -- used 0.439326s (cpu); 0.437955s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,28 +24,28 @@
│ │ │ │  i1 : ZZ[a,b,c][x,y]; F = a*x^2+b*x*y+c*y^2
│ │ │ │  
│ │ │ │          2              2
│ │ │ │  o2 = a*x  + b*x*y + c*y
│ │ │ │  
│ │ │ │  o2 : ZZ[a..c][x..y]
│ │ │ │  i3 : time discriminant F
│ │ │ │ - -- used 0.0078513s (cpu); 0.00792964s (thread); 0s (gc)
│ │ │ │ + -- used 0.011999s (cpu); 0.0130189s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2
│ │ │ │  o3 = - b  + 4a*c
│ │ │ │  
│ │ │ │  o3 : ZZ[a..c]
│ │ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │ │  
│ │ │ │          3      2         2      3
│ │ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │ │  
│ │ │ │  o5 : ZZ[a..d][x..y]
│ │ │ │  i6 : time discriminant F
│ │ │ │ - -- used 0.0840155s (cpu); 0.0312505s (thread); 0s (gc)
│ │ │ │ + -- used 0.0737907s (cpu); 0.0236709s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2       3     3                   2 2
│ │ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │ │  
│ │ │ │  o6 : ZZ[a..d]
│ │ │ │  The next example illustrates how computing the intersection of a pencil
│ │ │ │  generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with
│ │ │ │ @@ -75,15 +75,15 @@
│ │ │ │  
│ │ │ │                  4        3      4             4        3      4
│ │ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │ │  
│ │ │ │  o12 : R'
│ │ │ │  i13 : time D=discriminant pencil
│ │ │ │ - -- used 0.422412s (cpu); 0.42392s (thread); 0s (gc)
│ │ │ │ + -- used 0.439326s (cpu); 0.437955s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -91,26 +91,26 @@
│ │ │        0 3
│ │ │  
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ 
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.180744s (cpu); 0.130805s (thread); 0s (gc)
│ │ │ + -- used 0.294961s (cpu); 0.127308s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ 
    i3 : time V == dualVariety V'
│ │ │ - -- used 0.199797s (cpu); 0.140703s (thread); 0s (gc)
│ │ │ + -- used 0.249635s (cpu); 0.144745s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │

    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 $\mathbb{P}^9$

    │ │ │ │ │ │ │ │ │ @@ -126,23 +126,23 @@ │ │ │ │ │ │ ZZ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.0624866s (cpu); 0.0609208s (thread); 0s (gc)
│ │ │ + -- used 0.070908s (cpu); 0.0702981s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    │ │ │ 
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.621853s (cpu); 0.55823s (thread); 0s (gc)
│ │ │ + -- used 0.66942s (cpu); 0.626341s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    │ │ │ 
    i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,24 +32,24 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x )
│ │ │ │        0 3
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i2 : time V' = dualVariety V
│ │ │ │ - -- used 0.180744s (cpu); 0.130805s (thread); 0s (gc)
│ │ │ │ + -- used 0.294961s (cpu); 0.127308s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              2                 2    2
│ │ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i3 : time V == dualVariety V'
│ │ │ │ - -- used 0.199797s (cpu); 0.140703s (thread); 0s (gc)
│ │ │ │ + -- used 0.249635s (cpu); 0.144745s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  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 $\mathbb{P}^9$
│ │ │ │  i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331)
│ │ │ │  
│ │ │ │ @@ -61,21 +61,21 @@
│ │ │ │       a x x  + a x
│ │ │ │        8 1 2    9 2
│ │ │ │  
│ │ │ │        ZZ
│ │ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │ │       3331  0   9   0   2
│ │ │ │  i5 : time discF = ideal discriminant F;
│ │ │ │ - -- used 0.0624866s (cpu); 0.0609208s (thread); 0s (gc)
│ │ │ │ + -- used 0.070908s (cpu); 0.0702981s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o5 : Ideal of ----[a ..a ]
│ │ │ │                3331  0   9
│ │ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ │ - -- used 0.621853s (cpu); 0.55823s (thread); 0s (gc)
│ │ │ │ + -- used 0.66942s (cpu); 0.626341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o6 : Ideal of ----[x ..x ]
│ │ │ │                3331  0   9
│ │ │ │  i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
│ │ │ │  
│ │ │ │  o7 = true
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │               2    1 3   1 2    0 3   1    0 2
│ │ │  
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
    │ │ │ 
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.114714s (cpu); 0.0536053s (thread); 0s (gc)
│ │ │ + -- used 0.0875735s (cpu); 0.0457231s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          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  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -136,15 +136,15 @@
│ │ │        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
│ │ │  
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    │ │ │ 
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.03594s (cpu); 0.0388574s (thread); 0s (gc)
│ │ │ + -- used 0.0356639s (cpu); 0.035488s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          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  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -171,15 +171,15 @@
│ │ │          
    As another application, we check that the singular locus of the Chow form of the twisted cubic has dimension 2 (on each standard chart).
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i5 : w = chowForm C;
    
│ │ │      		          
                  
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0871858s (cpu); 0.0336167s (thread); 0s (gc)
│ │ │ + -- used 0.0679509s (cpu); 0.0242628s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -217,15 +217,15 @@
│ │ │       2x   x    - x    + x   )}
│ │ │         0,0 1,1    1,1    1,0
│ │ │  
│ │ │  o6 : List
    
│ │ │      		          
                  
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.0154955s (cpu); 0.0148463s (thread); 0s (gc)
│ │ │ + -- used 0.014661s (cpu); 0.013321s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
    
│ │ │      		          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │               2                       2
│ │ │ │  o1 = ideal (x  - x x , x x  - x x , x  - x x )
│ │ │ │               2    1 3   1 2    0 3   1    0 2
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   3
│ │ │ │  i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ │ - -- used 0.114714s (cpu); 0.0536053s (thread); 0s (gc)
│ │ │ │ + -- used 0.0875735s (cpu); 0.0457231s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3   3          2   2              2       2          2   3
│ │ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │ │          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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2       2               2   2
│ │ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │            2       2       2           2      2           2      3   3
│ │ │ │       x   x   x   x    - 2x   x   x   x    - x   x   x   x    + x   x
│ │ │ │        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
│ │ │ │  
│ │ │ │  o2 : QQ[x   ..x   ]
│ │ │ │           0,0   1,3
│ │ │ │  i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ │ - -- used 0.03594s (cpu); 0.0388574s (thread); 0s (gc)
│ │ │ │ + -- used 0.0356639s (cpu); 0.035488s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3          2                         2
│ │ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │ │          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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2      3      2
│ │ │ │       2x   x    + x    + x
│ │ │ │ @@ -106,15 +106,15 @@
│ │ │ │  o4 : QQ[a   ..a   ]
│ │ │ │           0,0   1,1
│ │ │ │  As another application, we check that the singular locus of the Chow form of
│ │ │ │  the twisted cubic has dimension 2 (on each standard chart).
│ │ │ │  i5 : w = chowForm C;
│ │ │ │  i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel
│ │ │ │  (w,AffineChartGrass=>s))
│ │ │ │ - -- used 0.0871858s (cpu); 0.0336167s (thread); 0s (gc)
│ │ │ │ + -- used 0.0679509s (cpu); 0.0242628s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     3          2          3                       2
│ │ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2      2            2   3               2
│ │ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ │ @@ -150,15 +150,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2      3      2
│ │ │ │       2x   x    - x    + x   )}
│ │ │ │         0,0 1,1    1,1    1,0
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time apply(U,u->dim singularLocus u)
│ │ │ │ - -- used 0.0154955s (cpu); 0.0148463s (thread); 0s (gc)
│ │ │ │ + -- used 0.014661s (cpu); 0.013321s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: **********
│ │ │ │      * fromPluckerToStiefel(Ideal)
│ │ │ │      * fromPluckerToStiefel(Matrix)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │       10 3    0 4   4 1 4   2 2 4   3 3 4     4
│ │ │  
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
    │ │ │ 
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.110524s (cpu); 0.0675797s (thread); 0s (gc)
│ │ │ + -- used 0.092074s (cpu); 0.0461657s (thread); 0s (gc)
│ │ │  
│ │ │              2                            2                                  
│ │ │  o2 = 143100p    + 267300p   p    + 96525p    - 56700p   p    - 56100p   p   
│ │ │              0,1          0,1 0,2         0,2         0,1 1,2         0,2 1,2
│ │ │       ------------------------------------------------------------------------
│ │ │             2                                              2                 
│ │ │       + 900p    + 140400p   p    + 111780p   p    + 133380p    - 8100p   p
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │        7 2          7       1       7         2
│ │ │ │       --p  + p p  + -p p  + -p p  + -p p  + 7p )
│ │ │ │       10 3    0 4   4 1 4   2 2 4   3 3 4     4
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i2 : time hurwitzForm Q
│ │ │ │ - -- used 0.110524s (cpu); 0.0675797s (thread); 0s (gc)
│ │ │ │ + -- used 0.092074s (cpu); 0.0461657s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              2                            2
│ │ │ │  o2 = 143100p    + 267300p   p    + 96525p    - 56700p   p    - 56100p   p
│ │ │ │              0,1          0,1 0,2         0,2         0,1 1,2         0,2 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2                                              2
│ │ │ │       + 900p    + 140400p   p    + 111780p   p    + 133380p    - 8100p   p
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │  o1 : --------------------------------------
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ 
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00799966s (cpu); 0.00727879s (thread); 0s (gc)
│ │ │ + -- used 0.0406609s (cpu); 0.0405998s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    │ │ │ 
    i3 : -- random quadric in G(1,3)
│ │ │       w' = random(2,Grass(1,3))
│ │ │  
│ │ │ @@ -131,15 +131,15 @@
│ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │  o3 : --------------------------------------
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ 
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00400191s (cpu); 0.00650056s (thread); 0s (gc)
│ │ │ + -- used 0.00798371s (cpu); 0.00829385s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use isCoisotropic:

    │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o1 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i2 : time isCoisotropic w │ │ │ │ - -- used 0.00799966s (cpu); 0.00727879s (thread); 0s (gc) │ │ │ │ + -- used 0.0406609s (cpu); 0.0405998s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ │ │ 7 2 2 3 2 5 │ │ │ │ o3 = -p + 7p p + p + -p p + 2p p + 5p + -p p + │ │ │ │ @@ -69,14 +69,14 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o3 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i4 : time isCoisotropic w' │ │ │ │ - -- used 0.00400191s (cpu); 0.00650056s (thread); 0s (gc) │ │ │ │ + -- used 0.00798371s (cpu); 0.00829385s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = false │ │ │ │ ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: ********** │ │ │ │ * isCoisotropic(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ @@ -114,15 +114,15 @@ │ │ │ │ │ │ ZZ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ │ │ │ 
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0228258s (cpu); 0.0201847s (thread); 0s (gc)
│ │ │ + -- used 0.0479977s (cpu); 0.0478608s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -55,15 +55,15 @@ │ │ │ │ 2380x + 9482x ) │ │ │ │ 4 5 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ │ 33331 0 5 │ │ │ │ i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I)) │ │ │ │ - -- used 0.0228258s (cpu); 0.0201847s (thread); 0s (gc) │ │ │ │ + -- used 0.0479977s (cpu); 0.0478608s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace │ │ │ │ ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: ********** │ │ │ │ * isInCoisotropic(Ideal,Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ @@ -84,15 +84,15 @@ │ │ │ c x x x + c x x + c x x + c x x + c x } │ │ │ 4 0 1 2 7 1 2 5 0 2 8 1 2 9 2 │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ 
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00215734s (cpu); 0.00350587s (thread); 0s (gc)
│ │ │ + -- used 0.00400078s (cpu); 0.00300471s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -151,15 +151,15 @@
│ │ │       -p p  + 2p p  + 6p }
│ │ │       7 0 2     1 2     2
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ 
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00399943s (cpu); 0.00217211s (thread); 0s (gc)
│ │ │ + -- used 0.00123595s (cpu); 0.00199651s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 1/2  9/4  1/2 1    3/4  0   0   0   0   0   0    0    0    0  
│ │ │        | 0   9/2  0    1/2 9/4  0    1/2 1   3/4 0   0   0    0    0    0  
│ │ │        | 0   0    9/2  0   1/2  9/4  0   1/2 1   3/4 0   0    0    0    0  
│ │ │        | 0   0    0    9/2 0    0    1/2 9/4 0   0   1/2 1    3/4  0    0  
│ │ │        | 0   0    0    0   9/2  0    0   1/2 9/4 0   0   1/2  1    3/4  0  
│ │ │        | 0   0    0    0   0    9/2  0   0   1/2 9/4 0   0    1/2  1    3/4
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     2          2        2      3
│ │ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00215734s (cpu); 0.00350587s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400078s (cpu); 0.00300471s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0
│ │ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0
│ │ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0
│ │ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0
│ │ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0
│ │ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0
│ │ │ │ @@ -92,15 +92,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       6   2       2     3
│ │ │ │       -p p  + 2p p  + 6p }
│ │ │ │       7 0 2     1 2     2
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00399943s (cpu); 0.00217211s (thread); 0s (gc)
│ │ │ │ + -- used 0.00123595s (cpu); 0.00199651s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 1/2  9/4  1/2 1    3/4  0   0   0   0   0   0    0    0    0
│ │ │ │        | 0   9/2  0    1/2 9/4  0    1/2 1   3/4 0   0   0    0    0    0
│ │ │ │        | 0   0    9/2  0   1/2  9/4  0   1/2 1   3/4 0   0    0    0    0
│ │ │ │        | 0   0    0    9/2 0    0    1/2 9/4 0   0   1/2 1    3/4  0    0
│ │ │ │        | 0   0    0    0   9/2  0    0   1/2 9/4 0   0   1/2  1    3/4  0
│ │ │ │        | 0   0    0    0   0    9/2  0   0   1/2 9/4 0   0    1/2  1    3/4
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -91,30 +91,30 @@
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │  o3 : Ideal of P4
    │ │ │ │ │ │ │ │ │ 
    i4 : time p = plucker L
│ │ │ - -- used 0.00396538s (cpu); 0.00437586s (thread); 0s (gc)
│ │ │ + -- used 0.00400241s (cpu); 0.00584151s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │       11804x   , x    + 14854x   )
│ │ │             3,4   0,1         3,4
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ 
    i5 : time L' = plucker p
│ │ │ - -- used 0.0973721s (cpu); 0.039945s (thread); 0s (gc)
│ │ │ + -- used 0.117661s (cpu); 0.0734802s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -129,15 +129,15 @@
│ │ │            
│ │ │                
    i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    
│ │ │            
│ │ │            
│ │ │                
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0280079s (cpu); 0.0301314s (thread); 0s (gc)
│ │ │ + -- used 0.0839972s (cpu); 0.0834231s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    
│ │ │            
│ │ │            
│ │ │                
    i9 : (codim W,degree W)
│ │ │  
│ │ │  o9 = (2, 5)
│ │ │ @@ -145,15 +145,15 @@
│ │ │  o9 : Sequence
    
│ │ │            
│ │ │          
│ │ │          
    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 $W$.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.147833s (cpu); 0.149666s (thread); 0s (gc)
│ │ │ + -- used 0.3189s (cpu); 0.318989s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
    
│ │ │      		          
                  
    i11 : assert(Y' == Y)
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │  o3 : Ideal of P4
│ │ │ │  i4 : time p = plucker L
│ │ │ │ - -- used 0.00396538s (cpu); 0.00437586s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400241s (cpu); 0.00584151s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11804x   , x    + 14854x   )
│ │ │ │             3,4   0,1         3,4
│ │ │ │  
│ │ │ │  o4 : Ideal of G'1'4
│ │ │ │  i5 : time L' = plucker p
│ │ │ │ - -- used 0.0973721s (cpu); 0.039945s (thread); 0s (gc)
│ │ │ │ + -- used 0.117661s (cpu); 0.0734802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │ @@ -61,26 +61,26 @@
│ │ │ │  $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points
│ │ │ │  of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$
│ │ │ │  over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.
│ │ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │ │  
│ │ │ │  o7 : Ideal of G'1'4
│ │ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ │ - -- used 0.0280079s (cpu); 0.0301314s (thread); 0s (gc)
│ │ │ │ + -- used 0.0839972s (cpu); 0.0834231s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of P4
│ │ │ │  i9 : (codim W,degree W)
│ │ │ │  
│ │ │ │  o9 = (2, 5)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  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 $W$.
│ │ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ │ - -- used 0.147833s (cpu); 0.149666s (thread); 0s (gc)
│ │ │ │ + -- used 0.3189s (cpu); 0.318989s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │       4          8    8          7                4    3    5
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ 
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.239497s (cpu); 0.149163s (thread); 0s (gc)
│ │ │ + -- used 0.352888s (cpu); 0.249339s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -121,15 +121,15 @@
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o3 : QQ[a..b]
    │ │ │ │ │ │ │ │ │ 
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.128223s (cpu); 0.0770729s (thread); 0s (gc)
│ │ │ + -- used 0.110133s (cpu); 0.0644408s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -143,15 +143,15 @@
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o4 : QQ[a..b]
    │ │ │ │ │ │ │ │ │ 
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.287256s (cpu); 0.289426s (thread); 0s (gc)
│ │ │ + -- used 0.307491s (cpu); 0.309999s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -165,15 +165,15 @@
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o5 : QQ[a..b]
    │ │ │ │ │ │ │ │ │ 
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.608867s (cpu); 0.559075s (thread); 0s (gc)
│ │ │ + -- used 0.776348s (cpu); 0.725524s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,15 +59,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │ │       4          8    8          7                4    3    5
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ │ - -- used 0.239497s (cpu); 0.149163s (thread); 0s (gc)
│ │ │ │ + -- used 0.352888s (cpu); 0.249339s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o3 : QQ[a..b]
│ │ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ │ - -- used 0.128223s (cpu); 0.0770729s (thread); 0s (gc)
│ │ │ │ + -- used 0.110133s (cpu); 0.0644408s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -99,15 +99,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o4 : QQ[a..b]
│ │ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ │ - -- used 0.287256s (cpu); 0.289426s (thread); 0s (gc)
│ │ │ │ + -- used 0.307491s (cpu); 0.309999s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -119,15 +119,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o5 : QQ[a..b]
│ │ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ │ - -- used 0.608867s (cpu); 0.559075s (thread); 0s (gc)
│ │ │ │ + -- used 0.776348s (cpu); 0.725524s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │         2               2                            3      2         4
│ │ │  o2 = {x  + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y  + t*x y*z + u*z }
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ 
    i3 : time resultant F
│ │ │ - -- used 0.0344967s (cpu); 0.0339164s (thread); 0s (gc)
│ │ │ + -- used 0.0479909s (cpu); 0.0495041s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -149,15 +149,15 @@
│ │ │       + c x }
│ │ │          9 2
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ 
    i5 : time resultant F
│ │ │ - -- used 2.8419s (cpu); 2.11491s (thread); 0s (gc)
│ │ │ + -- used 2.37705s (cpu); 1.88069s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1780,15 +1780,15 @@
│ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ 
    i7 : time # terms resultant F
│ │ │ - -- used 0.503578s (cpu); 0.380091s (thread); 0s (gc)
│ │ │ + -- used 0.439201s (cpu); 0.388865s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -33,15 +33,15 @@ │ │ │ │ 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} │ │ │ │ │ │ │ │ 2 2 3 2 4 │ │ │ │ o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time resultant F │ │ │ │ - -- used 0.0344967s (cpu); 0.0339164s (thread); 0s (gc) │ │ │ │ + -- used 0.0479909s (cpu); 0.0495041s (thread); 0s (gc) │ │ │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -87,15 +87,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 │ │ │ │ + c x } │ │ │ │ 9 2 │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : time resultant F │ │ │ │ - -- used 2.8419s (cpu); 2.11491s (thread); 0s (gc) │ │ │ │ + -- used 2.37705s (cpu); 1.88069s (thread); 0s (gc) │ │ │ │ │ │ │ │ 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 │ │ │ │ o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - │ │ │ │ 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 │ │ │ │ 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c - │ │ │ │ @@ -1713,15 +1713,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : time # terms resultant F │ │ │ │ - -- used 0.503578s (cpu); 0.380091s (thread); 0s (gc) │ │ │ │ + -- used 0.439201s (cpu); 0.388865s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 21894 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * resultant(List) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o2 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ │ │ │ 
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0984014s (cpu); 0.0465418s (thread); 0s (gc)
│ │ │ + -- used 0.0994219s (cpu); 0.0524418s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        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  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -121,15 +121,15 @@
│ │ │  o3 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (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   )
│ │ │         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
    │ │ │ │ │ │ │ │ │ 
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.0522902s (cpu); 0.0514451s (thread); 0s (gc)
│ │ │ + -- used 0.0705868s (cpu); 0.0724036s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -164,39 +164,39 @@
│ │ │  o4 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (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     )
│ │ │         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
    │ │ │ │ │ │ │ │ │ 
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.0332139s (cpu); 0.033128s (thread); 0s (gc)
│ │ │ + -- used 0.0240011s (cpu); 0.0267962s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        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
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
    │ │ │ │ │ │ │ │ │ 
    i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.111927s (cpu); 0.0524302s (thread); 0s (gc)
│ │ │ + -- used 0.103348s (cpu); 0.0599139s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
    │ │ │ │ │ │ │ │ │ 
    i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.128042s (cpu); 0.0814872s (thread); 0s (gc)
    │ │ │ + -- used 0.279752s (cpu); 0.0764804s (thread); 0s (gc) │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -64,15 +64,15 @@ │ │ │ │ o2 = ideal (- p p + p p , - p p + p p , - p + p p ) │ │ │ │ 1 2 0 3 1 3 0 4 3 2 4 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[p ..p ] │ │ │ │ 0 4 │ │ │ │ i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form) │ │ │ │ time tangentialChowForm(S,0) │ │ │ │ - -- used 0.0984014s (cpu); 0.0465418s (thread); 0s (gc) │ │ │ │ + -- used 0.0994219s (cpu); 0.0524418s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o3 = p p - p p p - p p p + p p p + p p + │ │ │ │ 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ p p - 2p p p - p p p │ │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ - p p + p p , p p - p p + p p , p p - p p + p │ │ │ │ p ) │ │ │ │ 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 │ │ │ │ i4 : -- 1-th associated hypersurface of S in G(2,4) │ │ │ │ time tangentialChowForm(S,1) │ │ │ │ - -- used 0.0522902s (cpu); 0.0514451s (thread); 0s (gc) │ │ │ │ + -- used 0.0705868s (cpu); 0.0724036s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 3 2 2 │ │ │ │ o4 = p p + p p - 2p p + p p - │ │ │ │ 1,2,3 1,2,4 0,2,4 1,2,4 0,2,3 1,2,4 0,2,4 0,3,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 3 │ │ │ │ 4p p - 4p p - 2p p + │ │ │ │ @@ -139,35 +139,35 @@ │ │ │ │ p + p p , p p - p p + p p ) │ │ │ │ 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 │ │ │ │ i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent │ │ │ │ hyperplanes to S) │ │ │ │ time tangentialChowForm(S,2) │ │ │ │ - -- used 0.0332139s (cpu); 0.033128s (thread); 0s (gc) │ │ │ │ + -- used 0.0240011s (cpu); 0.0267962s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o5 = p p - p p p + p p │ │ │ │ 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 │ │ │ │ │ │ │ │ o5 : QQ[p ..p , p , p , p ] │ │ │ │ 0,1,2,3 0,1,2,4 0,1,3,4 0,2,3,4 1,2,3,4 │ │ │ │ i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing │ │ │ │ time S' = ideal dualize tangentialChowForm(S,2) │ │ │ │ - -- used 0.111927s (cpu); 0.0524302s (thread); 0s (gc) │ │ │ │ + -- used 0.103348s (cpu); 0.0599139s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = ideal(p p - p p p + p p ) │ │ │ │ 1 2 0 1 3 0 4 │ │ │ │ │ │ │ │ o6 : Ideal of QQ[p ..p ] │ │ │ │ 0 4 │ │ │ │ i7 : -- we then can recover S │ │ │ │ time assert(dualize tangentialChowForm(S',3) == S) │ │ │ │ - -- used 0.128042s (cpu); 0.0814872s (thread); 0s (gc) │ │ │ │ + -- used 0.279752s (cpu); 0.0764804s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_C_o_i_s_o_t_r_o_p_i_c -- whether a hypersurface of a Grassmannian is a tangential │ │ │ │ Chow form │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee ttaannggeennttiiaallCChhoowwFFoorrmm:: ********** │ │ │ │ * tangentialChowForm(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ aXNFeHRlcm5hbE0yUGFyZW50 │ │ │ #:len=1043 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW5kaWNhdGUgaWYgdGhpcyBwcm9jZXNz │ │ │ IGlzIGEgcGFyZW50IHByb2Nlc3Mgb3Igbm90IiwgImxpbmVudW0iID0+IDg5NCwgImZpbGVuYW1l │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ @@ -3,16 +3,16 @@ │ │ │ i1 : run("ulimit -a") │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ -locked memory(kbytes) 2047000 │ │ │ -process 63802 │ │ │ +locked memory(kbytes) 8192 │ │ │ +process 256974 │ │ │ nofiles 512 │ │ │ vmemory(kbytes) unlimited │ │ │ locks unlimited │ │ │ rtprio 0 │ │ │ │ │ │ o1 = 0 │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ @@ -1,23 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 2927978066455787395 │ │ │ │ │ │ i1 : fn=temporaryFileName()|".m2" │ │ │ │ │ │ -o1 = /tmp/M2-76770-0/0.m2 │ │ │ +o1 = /tmp/M2-134173-0/0.m2 │ │ │ │ │ │ i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-76770-0/1.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/1.m2" >"/tmp/M2-134173-0/1.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i7 : h │ │ │ │ │ │ o7 = HashTable{"answer file" => null} │ │ │ "exit code" => 0 │ │ │ "output file" => null │ │ │ @@ -33,105 +33,105 @@ │ │ │ o8 = true │ │ │ │ │ │ i9 : h#"exit code"===0 │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : h=runExternalM2(fn,"justexit",()); │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/2.m2" >"/tmp/M2-76770-0/2.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/2.m2" >"/tmp/M2-134173-0/2.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i11 : h │ │ │ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-76770-0/2.ans} │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-134173-0/2.ans} │ │ │ "exit code" => 27 │ │ │ - "output file" => /tmp/M2-76770-0/2.out │ │ │ + "output file" => /tmp/M2-134173-0/2.out │ │ │ "return code" => 6912 │ │ │ "statistics" => null │ │ │ - "time used" => 1 │ │ │ + "time used" => 2 │ │ │ value => null │ │ │ │ │ │ o11 : HashTable │ │ │ │ │ │ i12 : fileExists(h#"output file") │ │ │ │ │ │ o12 = true │ │ │ │ │ │ i13 : fileExists(h#"answer file") │ │ │ │ │ │ o13 = false │ │ │ │ │ │ i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2"); │ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/3.m2" >"/tmp/M2-76770-0/3.out" 2>&1 )) │ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/3.m2" >"/tmp/M2-134173-0/3.out" 2>&1 )) │ │ │ Killed │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i15 : h │ │ │ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-76770-0/3.ans} │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-134173-0/3.ans} │ │ │ "exit code" => 0 │ │ │ - "output file" => /tmp/M2-76770-0/3.out │ │ │ + "output file" => /tmp/M2-134173-0/3.out │ │ │ "return code" => 9 │ │ │ "statistics" => null │ │ │ - "time used" => 3 │ │ │ + "time used" => 2 │ │ │ value => null │ │ │ │ │ │ o15 : HashTable │ │ │ │ │ │ i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file") │ │ │ │ │ │ o16 = │ │ │ - i1 : -- Script /tmp/M2-76770-0/3.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-134173-0/3.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-76770-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-134173-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/3.ans",spin (10)); │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-134173-0/3.ans",spin (10)); │ │ │ Spinning!! │ │ │ │ │ │ │ │ │ i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file") │ │ │ │ │ │ i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true); │ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/4.m2" >"/tmp/M2-76770-0/4.out" 2>&1') >"/tmp/M2-76770-0/4.stat" 2>&1 )) │ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/4.m2" >"/tmp/M2-134173-0/4.out" 2>&1') >"/tmp/M2-134173-0/4.stat" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i19 : h#"statistics" │ │ │ │ │ │ -o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/4.m2" >"/tmp/M2-76770-0/4.out" 2>&1" │ │ │ - User time (seconds): 4.41 │ │ │ - System time (seconds): 0.09 │ │ │ - Percent of CPU this job got: 89% │ │ │ - Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.03 │ │ │ +o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/4.m2" >"/tmp/M2-134173-0/4.out" 2>&1" │ │ │ + User time (seconds): 4.49 │ │ │ + System time (seconds): 0.26 │ │ │ + Percent of CPU this job got: 112% │ │ │ + Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.24 │ │ │ Average shared text size (kbytes): 0 │ │ │ Average unshared data size (kbytes): 0 │ │ │ Average stack size (kbytes): 0 │ │ │ Average total size (kbytes): 0 │ │ │ - Maximum resident set size (kbytes): 260880 │ │ │ + Maximum resident set size (kbytes): 336780 │ │ │ Average resident set size (kbytes): 0 │ │ │ Major (requiring I/O) page faults: 0 │ │ │ - Minor (reclaiming a frame) page faults: 9418 │ │ │ - Voluntary context switches: 2569 │ │ │ - Involuntary context switches: 2025 │ │ │ + Minor (reclaiming a frame) page faults: 11173 │ │ │ + Voluntary context switches: 8564 │ │ │ + Involuntary context switches: 773 │ │ │ Swaps: 0 │ │ │ File system inputs: 0 │ │ │ - File system outputs: 0 │ │ │ + File system outputs: 16 │ │ │ Socket messages sent: 0 │ │ │ Socket messages received: 0 │ │ │ Signals delivered: 0 │ │ │ Page size (bytes): 4096 │ │ │ Exit status: 0 │ │ │ │ │ │ │ │ │ i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///; │ │ │ │ │ │ i21 : (runExternalM2(fn,identity,v))#value===v │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/6.m2" >"/tmp/M2-76770-0/6.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/6.m2" >"/tmp/M2-134173-0/6.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ o21 = true │ │ │ │ │ │ i22 : R=QQ[x,y]; │ │ │ │ │ │ i23 : v=coker random(R^2,R^{3:-1}) │ │ │ @@ -139,54 +139,54 @@ │ │ │ o23 = cokernel | 9/2x+1/2y x+3/4y 7/4x+7/9y | │ │ │ | 9/4x+1/2y 3/2x+3/4y 7/10x+1/2y | │ │ │ │ │ │ 2 │ │ │ o23 : R-module, quotient of R │ │ │ │ │ │ i24 : h=runExternalM2(fn,identity,v) │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/7.m2" >"/tmp/M2-76770-0/7.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/7.m2" >"/tmp/M2-134173-0/7.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-76770-0/7.ans} │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-134173-0/7.ans} │ │ │ "exit code" => 1 │ │ │ - "output file" => /tmp/M2-76770-0/7.out │ │ │ + "output file" => /tmp/M2-134173-0/7.out │ │ │ "return code" => 256 │ │ │ "statistics" => null │ │ │ "time used" => 1 │ │ │ value => null │ │ │ │ │ │ o24 : HashTable │ │ │ │ │ │ i25 : get(h#"output file") │ │ │ │ │ │ o25 = │ │ │ - i1 : -- Script /tmp/M2-76770-0/7.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-134173-0/7.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-76770-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-134173-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(1/2)*y, x+(3/4)*y, (7/4)*x+(7/9)*y}, {(9/4)*x+(1/2)*y, (3/2)*x+(3/4)*y, (7/10)*x+(1/2)*y}})))); │ │ │ - stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects: │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-134173-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(1/2)*y, x+(3/4)*y, (7/4)*x+(7/9)*y}, {(9/4)*x+(1/2)*y, (3/2)*x+(3/4)*y, (7/10)*x+(1/2)*y}})))); │ │ │ + stdio:4:75:(3):[1]: error: no method for binary operator ^ applied to objects: │ │ │ R (of class Symbol) │ │ │ ^ 2 (of class ZZ) │ │ │ │ │ │ │ │ │ i26 : fn<"/tmp/M2-76770-0/8.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/8.m2" >"/tmp/M2-134173-0/8.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ o27 = true │ │ │ │ │ │ i28 : v=R; │ │ │ │ │ │ i29 : h=runExternalM2(fn,identity,v); │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-76770-0/9.m2" >"/tmp/M2-76770-0/9.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-134173-0/9.m2" >"/tmp/M2-134173-0/9.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i30 : h#value │ │ │ │ │ │ o30 = QQ[x..y] │ │ │ │ │ │ o30 : PolynomialRing │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ @@ -66,16 +66,16 @@ │ │ │ 
      i1 : run("ulimit -a")
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │ -locked memory(kbytes) 2047000
│ │ │ -process              63802
│ │ │ +locked memory(kbytes) 8192
│ │ │ +process              256974
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
      │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -34,16 +34,16 @@ │ │ │ │ i1 : run("ulimit -a") │ │ │ │ time(seconds) 700 │ │ │ │ file(blocks) unlimited │ │ │ │ data(kbytes) unlimited │ │ │ │ stack(kbytes) 8192 │ │ │ │ coredump(blocks) unlimited │ │ │ │ memory(kbytes) 850000 │ │ │ │ -locked memory(kbytes) 2047000 │ │ │ │ -process 63802 │ │ │ │ +locked memory(kbytes) 8192 │ │ │ │ +process 256974 │ │ │ │ nofiles 512 │ │ │ │ vmemory(kbytes) unlimited │ │ │ │ locks unlimited │ │ │ │ rtprio 0 │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ This starts a new shell and executes the command given, which in this case │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ @@ -91,15 +91,15 @@ │ │ │

      The hash table h stores the exit code of the created Macaulay2 process, the return code of the created Macaulay2 process (see run for details; this is usually 256 times the exit code, plus information about any signals received by the child), the wall-clock time used (as opposed to the CPU time), the name of the output file (unless it was deleted), the name of the answer file (unless it was deleted), any statistics recorded about the resource usage, and the value returned by the function func. If the child process terminates abnormally, then usually the exit code is nonzero and the value returned is null.

      │ │ │

      For example, we can write a few functions to a temporary file:

      │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -112,15 +112,15 @@ │ │ │ 
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-76770-0/0.m2
    │ │ │ +o1 = /tmp/M2-134173-0/0.m2 │ │ │ 
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    │ │ │ 
    i3 : fn<</// justexit = () -> ( exit(27); ); ///<<endl;
    │ │ │
    │ │ │
    │ │ │

    and then call them:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -309,35 +309,35 @@ │ │ │

    To view the error message:

    │ │ │ │ │ │ 
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/1.m2" >"/tmp/M2-76770-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/1.m2" >"/tmp/M2-134173-0/1.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │ 
    i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │ @@ -146,27 +146,27 @@
│ │ │          
    
│ │ │            
    
│ │ │            
    An abnormal program exit will have a nonzero exit code; also, the value will be null, the output file should exist, but the answer file may not exist unless the routine finished successfully.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
                  
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/2.m2" >"/tmp/M2-76770-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/2.m2" >"/tmp/M2-134173-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │      		          
                  
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-76770-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-134173-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-76770-0/2.out
│ │ │ +                "output file" => /tmp/M2-134173-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 1
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    
│ │ │      		          
                  
    i12 : fileExists(h#"output file")
│ │ │  
│ │ │ @@ -180,79 +180,79 @@
│ │ │          
    
│ │ │          
    
│ │ │            
    Here, we use resource limits to limit the routine to 2 seconds of computational time, while the system is asked to use 10 seconds of computational time:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/3.m2" >"/tmp/M2-76770-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/3.m2" >"/tmp/M2-134173-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │      		          
                  
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-76770-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-134173-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-76770-0/3.out
│ │ │ +                "output file" => /tmp/M2-134173-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 3
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    
│ │ │      		          
                  
    i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │ -      i1 : -- Script /tmp/M2-76770-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-134173-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-134173-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-134173-0/3.ans",spin (10));
│ │ │        Spinning!!
    
│ │ │      		          
                  
    i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    
│ │ │            
    We can get quite a lot of detail on the resources used with the KeepStatistics command:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/4.m2" >"/tmp/M2-76770-0/4.out" 2>&1') >"/tmp/M2-76770-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/4.m2" >"/tmp/M2-134173-0/4.out" 2>&1') >"/tmp/M2-134173-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │      		          
                  
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/4.m2" >"/tmp/M2-76770-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 4.41
│ │ │ -              System time (seconds): 0.09
│ │ │ -              Percent of CPU this job got: 89%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.03
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/4.m2" >"/tmp/M2-134173-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.49
│ │ │ +              System time (seconds): 0.26
│ │ │ +              Percent of CPU this job got: 112%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.24
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 260880
│ │ │ +              Maximum resident set size (kbytes): 336780
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 9418
│ │ │ -              Voluntary context switches: 2569
│ │ │ -              Involuntary context switches: 2025
│ │ │ +              Minor (reclaiming a frame) page faults: 11173
│ │ │ +              Voluntary context switches: 8564
│ │ │ +              Involuntary context switches: 773
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    
│ │ │      		          
    
│ │ │ @@ -262,15 +262,15 @@
│ │ │          
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    
│ │ │      		          
                  
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/6.m2" >"/tmp/M2-76770-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/6.m2" >"/tmp/M2-134173-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    Some care is required, however:
    
│ │ │ @@ -286,21 +286,21 @@
│ │ │                 | 9/4x+1/2y 3/2x+3/4y 7/10x+1/2y |
│ │ │  
│ │ │                               2
│ │ │  o23 : R-module, quotient of R
    │ │ │ 
    i24 : h=runExternalM2(fn,identity,v)
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/7.m2" >"/tmp/M2-76770-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/7.m2" >"/tmp/M2-134173-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-76770-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-134173-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-76770-0/7.out
│ │ │ +                "output file" => /tmp/M2-134173-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    │ │ │
    │ │ │ │ │ │ │ │ │ 
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-76770-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-134173-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-134173-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(1/2)*y, x+(3/4)*y, (7/4)*x+(7/9)*y}, {(9/4)*x+(1/2)*y, (3/2)*x+(3/4)*y, (7/10)*x+(1/2)*y}}))));
│ │ │ -      stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-134173-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(1/2)*y, x+(3/4)*y, (7/4)*x+(7/9)*y}, {(9/4)*x+(1/2)*y, (3/2)*x+(3/4)*y, (7/10)*x+(1/2)*y}}))));
│ │ │ +      stdio:4:75:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    │ │ │
    │ │ │
    │ │ │

    Keep in mind that the object you are passing must make sense in the context of the file containing your function! For instance, here we need to define the ring:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    │ │ │ 
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/8.m2" >"/tmp/M2-76770-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/8.m2" >"/tmp/M2-134173-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    │ │ │
    │ │ │
    │ │ │

    This problem can be avoided by following some suggestions for using RunExternalM2.

    │ │ │ @@ -345,15 +345,15 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : v=R;
    │ │ │ 
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-76770-0/9.m2" >"/tmp/M2-76770-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-134173-0/9.m2" >"/tmp/M2-134173-0/9.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │ 
    i30 : h#value
│ │ │  
│ │ │  o30 = QQ[x..y]
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  the output file (unless it was deleted), the name of the answer file (unless it
│ │ │ │  was deleted), any statistics recorded about the resource usage, and the value
│ │ │ │  returned by the function func. If the child process terminates abnormally, then
│ │ │ │  usually the exit code is nonzero and the value returned is _n_u_l_l.
│ │ │ │  For example, we can write a few functions to a temporary file:
│ │ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-76770-0/0.m2
│ │ │ │ +o1 = /tmp/M2-134173-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-76770-0/1.out" 2>&1 ))
│ │ │ │ +M2-134173-0/1.m2" >"/tmp/M2-134173-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │ @@ -80,167 +80,167 @@
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  An abnormal program exit will have a nonzero exit code; also, the value will be
│ │ │ │  null, the output file should exist, but the answer file may not exist unless
│ │ │ │  the routine finished successfully.
│ │ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-76770-0/2.m2" >"/tmp/M2-76770-0/2.out" 2>&1 ))
│ │ │ │ +M2-134173-0/2.m2" >"/tmp/M2-134173-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-76770-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-134173-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-76770-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-134173-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 1
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-76770-0/3.m2" >"/tmp/M2-76770-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-134173-0/3.m2" >"/tmp/M2-134173-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-76770-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-134173-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-76770-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-134173-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 3
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │  (h#"output file")
│ │ │ │  
│ │ │ │  o16 =
│ │ │ │ -      i1 : -- Script /tmp/M2-76770-0/3.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-134173-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-134173-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-134173-0/3.ans",spin (10));
│ │ │ │        Spinning!!
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-76770-0/4.m2" >"/tmp/M2-76770-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-76770-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-134173-0/4.m2" >"/tmp/M2-134173-0/4.out"
│ │ │ │ +2>&1') >"/tmp/M2-134173-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-76770-0/4.m2" >"/tmp/M2-76770-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 4.41
│ │ │ │ -              System time (seconds): 0.09
│ │ │ │ -              Percent of CPU this job got: 89%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.03
│ │ │ │ +--silent  -q  <"/tmp/M2-134173-0/4.m2" >"/tmp/M2-134173-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 4.49
│ │ │ │ +              System time (seconds): 0.26
│ │ │ │ +              Percent of CPU this job got: 112%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.24
│ │ │ │                Average shared text size (kbytes): 0
│ │ │ │                Average unshared data size (kbytes): 0
│ │ │ │                Average stack size (kbytes): 0
│ │ │ │                Average total size (kbytes): 0
│ │ │ │ -              Maximum resident set size (kbytes): 260880
│ │ │ │ +              Maximum resident set size (kbytes): 336780
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 9418
│ │ │ │ -              Voluntary context switches: 2569
│ │ │ │ -              Involuntary context switches: 2025
│ │ │ │ +              Minor (reclaiming a frame) page faults: 11173
│ │ │ │ +              Voluntary context switches: 8564
│ │ │ │ +              Involuntary context switches: 773
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 16
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-76770-0/6.m2" >"/tmp/M2-76770-0/6.out" 2>&1 ))
│ │ │ │ +M2-134173-0/6.m2" >"/tmp/M2-134173-0/6.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  Some care is required, however:
│ │ │ │  i22 : R=QQ[x,y];
│ │ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ │  
│ │ │ │  o23 = cokernel | 9/2x+1/2y x+3/4y    7/4x+7/9y  |
│ │ │ │                 | 9/4x+1/2y 3/2x+3/4y 7/10x+1/2y |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o23 : R-module, quotient of R
│ │ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-76770-0/7.m2" >"/tmp/M2-76770-0/7.out" 2>&1 ))
│ │ │ │ +M2-134173-0/7.m2" >"/tmp/M2-134173-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-76770-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-134173-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-76770-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-134173-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-76770-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-134173-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-134173-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/7.ans",identity (cokernel
│ │ │ │ -(map(R^2,R^{3:{-1}},{{(9/2)*x+(1/2)*y, x+(3/4)*y, (7/4)*x+(7/9)*y}, {(9/4)*x+
│ │ │ │ -(1/2)*y, (3/2)*x+(3/4)*y, (7/10)*x+(1/2)*y}}))));
│ │ │ │ -      stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-134173-0/7.ans",identity
│ │ │ │ +(cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(1/2)*y, x+(3/4)*y, (7/4)*x+(7/9)*y}, {
│ │ │ │ +(9/4)*x+(1/2)*y, (3/2)*x+(3/4)*y, (7/10)*x+(1/2)*y}}))));
│ │ │ │ +      stdio:4:75:(3):[1]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-76770-0/8.out" 2>&1 ))
│ │ │ │ +M2-134173-0/8.m2" >"/tmp/M2-134173-0/8.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o27 = true
│ │ │ │  This problem can be avoided by following some _s_u_g_g_e_s_t_i_o_n_s_ _f_o_r_ _u_s_i_n_g
│ │ │ │  _R_u_n_E_x_t_e_r_n_a_l_M_2.
│ │ │ │  The objects may unavoidably lose some internal references, though:
│ │ │ │  i28 : v=R;
│ │ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-76770-0/9.m2" >"/tmp/M2-76770-0/9.out" 2>&1 ))
│ │ │ │ +M2-134173-0/9.m2" >"/tmp/M2-134173-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SCMAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsTW9kdWxlKElkZWFsKQ==
│ │ │  #:len=266
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzE1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjYW5vbmljYWxNb2R1bGUsSWRlYWwpLCJjYW5vbmlj
│ │ ├── ./usr/share/doc/Macaulay2/SCSCP/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  UmVtb3RlT2JqZWN0IGFuZCBSZW1vdGVPYmplY3Q=
│ │ │  #:len=217
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wg
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  dHJhbnNwb3NlKEdhdGVNYXRyaXgp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTQ5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cmFuc3Bvc2UsR2F0ZU1hdHJpeCksInRyYW5zcG9z
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
│ │ │ @@ -30,23 +30,23 @@
│ │ │                                              )
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
│ │ │  
│ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.00272579s (cpu); 0.000187722s (thread); 0s (gc)
│ │ │ + -- used 0.00178726s (cpu); 0.000166362s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 4.85636s (cpu); 3.34076s (thread); 0s (gc)
│ │ │ + -- used 4.02758s (cpu); 2.91964s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -86,25 +86,25 @@
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    │ │ │ 
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.00272579s (cpu); 0.000187722s (thread); 0s (gc)
│ │ │ + -- used 0.00178726s (cpu); 0.000166362s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    │ │ │ 
    i7 : ZZ[y];
    │ │ │ 
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 4.85636s (cpu); 3.34076s (thread); 0s (gc)
│ │ │ + -- used 4.02758s (cpu); 2.91964s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │                                              output nodes: 1
│ │ │ │                                              )
│ │ │ │  
│ │ │ │                            "variable positions" => {-1}
│ │ │ │  
│ │ │ │  o5 : InterpretedSLProgram
│ │ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ │ - -- used 0.00272579s (cpu); 0.000187722s (thread); 0s (gc)
│ │ │ │ + -- used 0.00178726s (cpu); 0.000166362s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 4.85636s (cpu); 3.34076s (thread); 0s (gc)
│ │ │ │ + -- used 4.02758s (cpu); 2.91964s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2xFcXVpdmFyaWFudFZlY3RvckJ1bmRsZQ==
│ │ │  #:len=4962
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBTTC1lcXVpdmFyaWFu
│ │ │  dCB2ZWN0b3IgYnVuZGxlIG92ZXIgc29tZSBwcm9qZWN0aXZlIHNwYWNlIiwgImxpbmVudW0iID0+
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RXhhbXBsZSBmaXJzdCBvcmRlciBkZWZvcm1hdGlvbg==
│ │ │  #:len=3174
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXhhbXBsZSBhY2Nlc3NpbmcgdGhlIGRh
│ │ │  dGEgc3RvcmVkIGluIGEgZmlyc3Qgb3JkZXIgZGVmb3JtYXRpb24uIiwgImxpbmVudW0iID0+IDQy
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/example-output/___Co__Complex.out
│ │ │ @@ -56,15 +56,15 @@
│ │ │  
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i8 : cC=complement C
│ │ │  
│ │ │  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  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i9 : dualize cC
│ │ │  
│ │ │  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
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/example-output/___Complex.out
│ │ │ @@ -56,15 +56,15 @@
│ │ │  
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i8 : complement C
│ │ │  
│ │ │  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  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i9 : R=QQ[x_0..x_5]
│ │ │  
│ │ │  o9 = R
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/html/___Co__Complex.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    │ │ │ 
    i8 : cC=complement C
│ │ │  
│ │ │  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  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    │ │ │ 
    i9 : dualize cC
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │           0 1 2  0 1 4  0 3 4  1 2 3  1 2 5  1 4 5  2 3 4  3 4 5
│ │ │ │  
│ │ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │ │  i8 : cC=complement C
│ │ │ │  
│ │ │ │  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
│ │ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │ │  
│ │ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │ │  i9 : dualize cC
│ │ │ │  
│ │ │ │  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
│ │ │ │           3 4 5  2 3 5  1 2 5  0 4 5  0 3 4  0 2 3  0 1 5  0 1 2
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/html/___Complex.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    │ │ │ 
    i8 : complement C
│ │ │  
│ │ │  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  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    │ │ │
    │ │ │
    │ │ │

    │ │ │ ├── html2text {} │ │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 │ │ │ │ │ │ │ │ o7 : co-complex of dim 2 embedded in dim 5 (printing facets) │ │ │ │ equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 │ │ │ │ i8 : complement C │ │ │ │ │ │ │ │ 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 │ │ │ │ - 3 4 5 1 4 5 2 1 5 2 3 4 3 0 4 1 0 4 2 3 1 2 1 0 │ │ │ │ + 4 5 3 1 4 5 1 5 2 4 2 3 0 4 3 1 0 4 1 2 3 1 0 2 │ │ │ │ │ │ │ │ o8 : co-complex of dim 2 embedded in dim 5 (printing facets) │ │ │ │ equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 │ │ │ │ i9 : R=QQ[x_0..x_5] │ │ │ │ │ │ │ │ o9 = R │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ U1ZEQ29tcGxleA== │ │ │ #:len=3135 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZSB0aGUgU1ZEIGRlY29tcG9z │ │ │ aXRpb24gb2YgYSBjaGFpbkNvbXBsZXggb3ZlciBSUiIsICJsaW5lbnVtIiA9PiAxMTAyLCBJbnB1 │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ i3 : r={5,11,3,2} │ │ │ │ │ │ o3 = {5, 11, 3, 2} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>4) │ │ │ - -- .00698521s elapsed │ │ │ + -- .00794349s elapsed │ │ │ │ │ │ 6 19 19 7 3 │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o4 : ChainComplex │ │ │ @@ -51,15 +51,15 @@ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ - -- .00230115s elapsed │ │ │ + -- .00191398s elapsed │ │ │ │ │ │ i8 : h │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ 0 => 3 │ │ │ 1 => 5 │ │ │ 2 => 2 │ │ │ @@ -95,15 +95,15 @@ │ │ │ i12 : maximalEntry chainComplex errors │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ - -- .00467146s elapsed │ │ │ + -- .00458827s elapsed │ │ │ │ │ │ i14 : hL === h │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : SigmaL =source U; │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ i3 : r={4,3,3} │ │ │ │ │ │ o3 = {4, 3, 3} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .00438766s elapsed │ │ │ + -- .00238951s elapsed │ │ │ │ │ │ 5 10 11 5 │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o4 : ChainComplex │ │ │ @@ -47,25 +47,25 @@ │ │ │ 53 53 53 53 │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ - -- .000909523s elapsed │ │ │ + -- .000527335s elapsed │ │ │ │ │ │ o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) }) │ │ │ 1 => 3 2 => (37.9214, 30.3707, 1.61954e-14) │ │ │ 2 => 5 3 => (14.972, 8.57847, 3.90646e-15) │ │ │ 3 => 2 │ │ │ │ │ │ o7 : Sequence │ │ │ │ │ │ i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian) │ │ │ - -- .0020757s elapsed │ │ │ + -- .00113243s elapsed │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ @@ -17,15 +17,15 @@ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ - -- .00344524s elapsed │ │ │ + -- .00417378s elapsed │ │ │ │ │ │ 6 10 13 8 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ @@ -17,15 +17,15 @@ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .00282751s elapsed │ │ │ + -- .00230848s elapsed │ │ │ │ │ │ 6 10 11 5 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ @@ -17,15 +17,15 @@ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .00288191s elapsed │ │ │ + -- .00242931s elapsed │ │ │ │ │ │ 6 10 11 5 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ @@ -103,15 +103,15 @@ │ │ │ │ │ │ o3 = {5, 11, 3, 2} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00698521s elapsed
│ │ │ + -- .00794349s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
    │ │ │ @@ -142,15 +142,15 @@ │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00230115s elapsed
    │ │ │ + -- .00191398s elapsed │ │ │ │ │ │ │ │ │ 
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │                 0 => 3
│ │ │                 1 => 5
│ │ │ @@ -192,15 +192,15 @@
│ │ │  
│ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00467146s elapsed
    │ │ │ + -- .00458827s elapsed │ │ │ │ │ │ │ │ │ 
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,15 +38,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : r={5,11,3,2} │ │ │ │ │ │ │ │ o3 = {5, 11, 3, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>4) │ │ │ │ - -- .00698521s elapsed │ │ │ │ + -- .00794349s elapsed │ │ │ │ │ │ │ │ 6 19 19 7 3 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o4 : ChainComplex │ │ │ │ @@ -71,15 +71,15 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ │ - -- .00230115s elapsed │ │ │ │ + -- .00191398s elapsed │ │ │ │ i8 : h │ │ │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ │ 0 => 3 │ │ │ │ 1 => 5 │ │ │ │ 2 => 2 │ │ │ │ 3 => 1 │ │ │ │ @@ -110,15 +110,15 @@ │ │ │ │ 1)*Sigma.dd_ell*transpose U_ell); │ │ │ │ i12 : maximalEntry chainComplex errors │ │ │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ │ - -- .00467146s elapsed │ │ │ │ + -- .00458827s elapsed │ │ │ │ i14 : hL === h │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : SigmaL =source U; │ │ │ │ i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i) │ │ │ │ │ │ │ │ o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15} │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ @@ -105,15 +105,15 @@ │ │ │ │ │ │ o3 = {4, 3, 3} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00438766s elapsed
│ │ │ + -- .00238951s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
    │ │ │ @@ -140,26 +140,26 @@ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000909523s elapsed
│ │ │ + -- .000527335s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
    │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .0020757s elapsed
│ │ │ + -- .00113243s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
    │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : r={4,3,3} │ │ │ │ │ │ │ │ o3 = {4, 3, 3} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00438766s elapsed │ │ │ │ + -- .00238951s elapsed │ │ │ │ │ │ │ │ 5 10 11 5 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o4 : ChainComplex │ │ │ │ @@ -70,24 +70,24 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ │ - -- .000909523s elapsed │ │ │ │ + -- .000527335s elapsed │ │ │ │ │ │ │ │ o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) }) │ │ │ │ 1 => 3 2 => (37.9214, 30.3707, 1.61954e-14) │ │ │ │ 2 => 5 3 => (14.972, 8.57847, 3.90646e-15) │ │ │ │ 3 => 2 │ │ │ │ │ │ │ │ o7 : Sequence │ │ │ │ i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian) │ │ │ │ - -- .0020757s elapsed │ │ │ │ + -- .00113243s elapsed │ │ │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ @@ -104,15 +104,15 @@ │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00344524s elapsed
│ │ │ + -- .00417378s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ │ - -- .00344524s elapsed │ │ │ │ + -- .00417378s elapsed │ │ │ │ │ │ │ │ 6 10 13 8 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00282751s elapsed
│ │ │ + -- .00230848s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00282751s elapsed │ │ │ │ + -- .00230848s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00288191s elapsed
│ │ │ + -- .00242931s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00288191s elapsed │ │ │ │ + -- .00242931s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SagbiGbDetection/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ d2VpZ2h0VmVjdG9yc1JlYWxpemluZ1NBR0JJ │ │ │ #:len=1819 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIG1haW4gZnVuY3Rpb24gZm9yIGRl │ │ │ dGVjdGluZyBTQUdCSSBiYXNlcyIsICJsaW5lbnVtIiA9PiAyMTUsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ c2F0dXJhdGUoSWRlYWwsTGlzdCk= │ │ │ #:len=207 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOSwgInVuZG9jdW1lbnRlZCIgPT4gdHJ1 │ │ │ ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2F0dXJhdGUs │ │ ├── ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ @@ -38,33 +38,33 @@ │ │ │ o5 : Ideal of S │ │ │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ - -- used 0.289625s (cpu); 0.290581s (thread); 0s (gc) │ │ │ + -- used 0.368251s (cpu); 0.365743s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ - -- used 0.82801s (cpu); 0.755559s (thread); 0s (gc) │ │ │ + -- used 0.616781s (cpu); 0.557287s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : S = ZZ/101[vars(0..4)]; │ │ │ │ │ │ i10 : I = ideal vars S; │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ i11 : time quotient(I^5, I^3, Strategy => Iterate); │ │ │ - -- used 0.0265088s (cpu); 0.024945s (thread); 0s (gc) │ │ │ + -- used 0.016s (cpu); 0.0181278s (thread); 0s (gc) │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ - -- used 0.00796558s (cpu); 0.00809413s (thread); 0s (gc) │ │ │ + -- used 0.00520218s (cpu); 0.00662177s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ @@ -106,21 +106,21 @@ │ │ │ │ │ │ 
    i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5));
│ │ │  
│ │ │  o6 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.289625s (cpu); 0.290581s (thread); 0s (gc)
│ │ │ + -- used 0.368251s (cpu); 0.365743s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.82801s (cpu); 0.755559s (thread); 0s (gc)
│ │ │ + -- used 0.616781s (cpu); 0.557287s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    │ │ │ │ │ │ │ │ │
    │ │ │

    Strategy => Quotient is faster in other cases:

    │ │ │
    │ │ │ @@ -131,21 +131,21 @@ │ │ │ │ │ │ 
    i10 : I = ideal vars S;
│ │ │  
│ │ │  o10 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0265088s (cpu); 0.024945s (thread); 0s (gc)
│ │ │ + -- used 0.016s (cpu); 0.0181278s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ │ │ │ │ │ │ 
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00796558s (cpu); 0.00809413s (thread); 0s (gc)
│ │ │ + -- used 0.00520218s (cpu); 0.00662177s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Further information

    │ │ │ ├── html2text {} │ │ │ │ @@ -57,32 +57,32 @@ │ │ │ │ i5 : I = monomialCurveIdeal(S, 1..n-1); │ │ │ │ │ │ │ │ o5 : Ideal of S │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ │ - -- used 0.289625s (cpu); 0.290581s (thread); 0s (gc) │ │ │ │ + -- used 0.368251s (cpu); 0.365743s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ │ - -- used 0.82801s (cpu); 0.755559s (thread); 0s (gc) │ │ │ │ + -- used 0.616781s (cpu); 0.557287s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ Strategy => Quotient is faster in other cases: │ │ │ │ i9 : S = ZZ/101[vars(0..4)]; │ │ │ │ i10 : I = ideal vars S; │ │ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ i11 : time quotient(I^5, I^3, Strategy => Iterate); │ │ │ │ - -- used 0.0265088s (cpu); 0.024945s (thread); 0s (gc) │ │ │ │ + -- used 0.016s (cpu); 0.0181278s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ │ - -- used 0.00796558s (cpu); 0.00809413s (thread); 0s (gc) │ │ │ │ + -- used 0.00520218s (cpu); 0.00662177s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: null │ │ │ │ * Function: _q_u_o_t_i_e_n_t -- quotient or division │ │ │ │ * Option key: _S_t_r_a_t_e_g_y -- an optional argument │ │ │ │ ********** RReeffeerreenncceess ********** │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ aW5jbHVzaW9uKC4uLixTdWJEaW1lbnNpb249Pi4uLik= │ │ │ #:len=260 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg1NSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW5jbHVzaW9uLFN1YkRpbWVuc2lvbl0sImluY2x1 │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ @@ -40,23 +40,23 @@ │ │ │ ) │ │ │ │ │ │ o6 = f │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ - -- used 0.00610602s (cpu); 0.0042014s (thread); 0s (gc) │ │ │ - -- used 0.00729864s (cpu); 0.00737338s (thread); 0s (gc) │ │ │ - -- used 0.00603282s (cpu); 0.00957053s (thread); 0s (gc) │ │ │ - -- used 0.0161226s (cpu); 0.017607s (thread); 0s (gc) │ │ │ - -- used 0.0304987s (cpu); 0.030588s (thread); 0s (gc) │ │ │ - -- used 0.0511511s (cpu); 0.0533558s (thread); 0s (gc) │ │ │ - -- used 0.0897187s (cpu); 0.0906512s (thread); 0s (gc) │ │ │ - -- used 0.145204s (cpu); 0.146137s (thread); 0s (gc) │ │ │ - -- used 0.316997s (cpu); 0.245415s (thread); 0s (gc) │ │ │ + -- used 0.00400207s (cpu); 0.00405974s (thread); 0s (gc) │ │ │ + -- used 0.00305747s (cpu); 0.00503857s (thread); 0s (gc) │ │ │ + -- used 0.00599153s (cpu); 0.00815647s (thread); 0s (gc) │ │ │ + -- used 0.013819s (cpu); 0.0145021s (thread); 0s (gc) │ │ │ + -- used 0.0752968s (cpu); 0.0758669s (thread); 0s (gc) │ │ │ + -- used 0.0953928s (cpu); 0.0980919s (thread); 0s (gc) │ │ │ + -- used 0.117275s (cpu); 0.121098s (thread); 0s (gc) │ │ │ + -- used 0.276155s (cpu); 0.277463s (thread); 0s (gc) │ │ │ + -- used 0.538407s (cpu); 0.423264s (thread); 0s (gc) │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ ------------------------------------------------------------------------ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ o7 : List │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ @@ -106,23 +106,23 @@ │ │ │ │ │ │ o6 = f │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ │ │ │ 
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00610602s (cpu); 0.0042014s (thread); 0s (gc)
│ │ │ - -- used 0.00729864s (cpu); 0.00737338s (thread); 0s (gc)
│ │ │ - -- used 0.00603282s (cpu); 0.00957053s (thread); 0s (gc)
│ │ │ - -- used 0.0161226s (cpu); 0.017607s (thread); 0s (gc)
│ │ │ - -- used 0.0304987s (cpu); 0.030588s (thread); 0s (gc)
│ │ │ - -- used 0.0511511s (cpu); 0.0533558s (thread); 0s (gc)
│ │ │ - -- used 0.0897187s (cpu); 0.0906512s (thread); 0s (gc)
│ │ │ - -- used 0.145204s (cpu); 0.146137s (thread); 0s (gc)
│ │ │ - -- used 0.316997s (cpu); 0.245415s (thread); 0s (gc)
│ │ │ + -- used 0.00400207s (cpu); 0.00405974s (thread); 0s (gc)
│ │ │ + -- used 0.00305747s (cpu); 0.00503857s (thread); 0s (gc)
│ │ │ + -- used 0.00599153s (cpu); 0.00815647s (thread); 0s (gc)
│ │ │ + -- used 0.013819s (cpu); 0.0145021s (thread); 0s (gc)
│ │ │ + -- used 0.0752968s (cpu); 0.0758669s (thread); 0s (gc)
│ │ │ + -- used 0.0953928s (cpu); 0.0980919s (thread); 0s (gc)
│ │ │ + -- used 0.117275s (cpu); 0.121098s (thread); 0s (gc)
│ │ │ + -- used 0.276155s (cpu); 0.277463s (thread); 0s (gc)
│ │ │ + -- used 0.538407s (cpu); 0.423264s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -56,23 +56,23 @@ │ │ │ │ integral chern symmetricPower_(2*n-3) last bundles G │ │ │ │ ) │ │ │ │ │ │ │ │ o6 = f │ │ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ │ - -- used 0.00610602s (cpu); 0.0042014s (thread); 0s (gc) │ │ │ │ - -- used 0.00729864s (cpu); 0.00737338s (thread); 0s (gc) │ │ │ │ - -- used 0.00603282s (cpu); 0.00957053s (thread); 0s (gc) │ │ │ │ - -- used 0.0161226s (cpu); 0.017607s (thread); 0s (gc) │ │ │ │ - -- used 0.0304987s (cpu); 0.030588s (thread); 0s (gc) │ │ │ │ - -- used 0.0511511s (cpu); 0.0533558s (thread); 0s (gc) │ │ │ │ - -- used 0.0897187s (cpu); 0.0906512s (thread); 0s (gc) │ │ │ │ - -- used 0.145204s (cpu); 0.146137s (thread); 0s (gc) │ │ │ │ - -- used 0.316997s (cpu); 0.245415s (thread); 0s (gc) │ │ │ │ + -- used 0.00400207s (cpu); 0.00405974s (thread); 0s (gc) │ │ │ │ + -- used 0.00305747s (cpu); 0.00503857s (thread); 0s (gc) │ │ │ │ + -- used 0.00599153s (cpu); 0.00815647s (thread); 0s (gc) │ │ │ │ + -- used 0.013819s (cpu); 0.0145021s (thread); 0s (gc) │ │ │ │ + -- used 0.0752968s (cpu); 0.0758669s (thread); 0s (gc) │ │ │ │ + -- used 0.0953928s (cpu); 0.0980919s (thread); 0s (gc) │ │ │ │ + -- used 0.117275s (cpu); 0.121098s (thread); 0s (gc) │ │ │ │ + -- used 0.276155s (cpu); 0.277463s (thread); 0s (gc) │ │ │ │ + -- used 0.538407s (cpu); 0.423264s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ Note: in characteristic zero, using Bertini's theorem, the numbers computed can │ │ ├── ./usr/share/doc/Macaulay2/SchurComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ c2NodXJDb21wbGV4 │ │ │ #:len=3101 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NodXIgZnVuY3RvcnMgb2YgY2hhaW4g │ │ │ Y29tcGxleGVzIiwgImxpbmVudW0iID0+IDU1MiwgSW5wdXRzID0+IHtTUEFOe1RUeyJsYW1iZGEi │ │ ├── ./usr/share/doc/Macaulay2/SchurFunctors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ c3BsaXRDaGFyYWN0ZXI= │ │ │ #:len=913 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRGVjb21wb3NlcyBhIHN5bW1ldHJpYyBw │ │ │ b2x5bm9taWFsIGFzIGEgc3VtIG9mIFNjaHVyIGZ1bmN0aW9ucyIsICJsaW5lbnVtIiA9PiA0MDEs │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ c2NodXJSZXNvbHV0aW9uKFJpbmdFbGVtZW50LExpc3Qp │ │ │ #:len=286 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjQ0OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2NodXJSZXNvbHV0aW9uLFJpbmdFbGVtZW50LExp │ │ ├── ./usr/share/doc/Macaulay2/SchurVeronese/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ bWFrZUJldHRpVGFsbHk= │ │ │ #:len=1334 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydHMgYSBoYXNoIHRhYmxlIHJl │ │ │ cHJlc2VudGluZyBhIEJldHRpIHRhYmxlIHRvIGEgQmV0dGkgdGFsbHkiLCAibGluZW51bSIgPT4g │ │ ├── ./usr/share/doc/Macaulay2/SectionRing/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ bVJlZ3VsYXI= │ │ │ #:len=1136 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibVJlZ3VsYXIoRixHKSBjb21wdXRlcyB0 │ │ │ aGUgcmVndWxhcml0eSBvZiBGIHdpdGggcmVzcGVjdCB0byBHIChnbG9iYWxseSBnZW5lcmF0ZWQp │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk= │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUHJvZHVjdFJpbmcsUmluZyxMaXN0KSwibWFr │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ @@ -53,15 +53,15 @@ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : X=((W)*ideal(y)+ideal(f)); │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : time isComponentContained(X,Y) │ │ │ - -- used 4.6241s (cpu); 3.15815s (thread); 0s (gc) │ │ │ + -- used 13.2922s (cpu); 4.478s (thread); 0s (gc) │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : print "we could confirm this with the computation:" │ │ │ we could confirm this with the computation: │ │ │ │ │ │ i13 : B=ideal(x)*ideal(y)*ideal(z) │ │ │ @@ -71,12 +71,12 @@ │ │ │ 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, │ │ │ ----------------------------------------------------------------------- │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time isSubset(saturate(Y,B),saturate(X,B)) │ │ │ - -- used 53.5862s (cpu); 49.5289s (thread); 0s (gc) │ │ │ + -- used 68.7848s (cpu); 61.9266s (thread); 0s (gc) │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ @@ -23,24 +23,24 @@ │ │ │ i5 : A = makeChowRing(R) │ │ │ │ │ │ o5 = A │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ i6 : time s = segreDimX(X,Y,A) │ │ │ - -- used 0.233953s (cpu); 0.139307s (thread); 0s (gc) │ │ │ + -- used 0.953026s (cpu); 0.157082s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = 2H + 4H H + 2H │ │ │ 1 1 2 2 │ │ │ │ │ │ o6 : A │ │ │ │ │ │ i7 : time segre(X,Y,A) │ │ │ - -- used 0.340288s (cpu); 0.0954096s (thread); 0s (gc) │ │ │ + -- used 0.591932s (cpu); 0.187423s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ o7 : A │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ @@ -144,15 +144,15 @@ │ │ │ │ │ │ 
    i10 : X=((W)*ideal(y)+ideal(f));
│ │ │  
│ │ │  o10 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.6241s (cpu); 3.15815s (thread); 0s (gc)
│ │ │ + -- used 13.2922s (cpu); 4.478s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    │ │ │ │ │ │ │ │ │ 
    i12 : print "we could confirm this with the computation:"
│ │ │  we could confirm this with the computation:
    │ │ │ │ │ │ @@ -165,15 +165,15 @@ │ │ │ ----------------------------------------------------------------------- │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ │ │ │ 
    i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 53.5862s (cpu); 49.5289s (thread); 0s (gc)
│ │ │ + -- used 68.7848s (cpu); 61.9266s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use isComponentContained:

    │ │ │ ├── html2text {} │ │ │ │ @@ -69,29 +69,29 @@ │ │ │ │ i9 : Y=ideal (z_0*W_0-z_1*W_1)+ideal(f); │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : X=((W)*ideal(y)+ideal(f)); │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : time isComponentContained(X,Y) │ │ │ │ - -- used 4.6241s (cpu); 3.15815s (thread); 0s (gc) │ │ │ │ + -- used 13.2922s (cpu); 4.478s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ i12 : print "we could confirm this with the computation:" │ │ │ │ we could confirm this with the computation: │ │ │ │ i13 : B=ideal(x)*ideal(y)*ideal(z) │ │ │ │ │ │ │ │ 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, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 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, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time isSubset(saturate(Y,B),saturate(X,B)) │ │ │ │ - -- used 53.5862s (cpu); 49.5289s (thread); 0s (gc) │ │ │ │ + -- used 68.7848s (cpu); 61.9266s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: ********** │ │ │ │ * isComponentContained(Ideal,Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ @@ -111,25 +111,25 @@ │ │ │ │ │ │ o5 = A │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ │ │ │ 
    i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.233953s (cpu); 0.139307s (thread); 0s (gc)
│ │ │ + -- used 0.953026s (cpu); 0.157082s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
    │ │ │ │ │ │ │ │ │ 
    i7 : time segre(X,Y,A)
│ │ │ - -- used 0.340288s (cpu); 0.0954096s (thread); 0s (gc)
│ │ │ + -- used 0.591932s (cpu); 0.187423s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -49,23 +49,23 @@ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : A = makeChowRing(R) │ │ │ │ │ │ │ │ o5 = A │ │ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ i6 : time s = segreDimX(X,Y,A) │ │ │ │ - -- used 0.233953s (cpu); 0.139307s (thread); 0s (gc) │ │ │ │ + -- used 0.953026s (cpu); 0.157082s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = 2H + 4H H + 2H │ │ │ │ 1 1 2 2 │ │ │ │ │ │ │ │ o6 : A │ │ │ │ i7 : time segre(X,Y,A) │ │ │ │ - -- used 0.340288s (cpu); 0.0954096s (thread); 0s (gc) │ │ │ │ + -- used 0.591932s (cpu); 0.187423s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o7 : A │ │ │ │ ********** WWaayyss ttoo uussee sseeggrreeDDiimmXX:: ********** │ │ ├── ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ b3B0aW1pemUoLi4uLFZlcmJvc2l0eT0+Li4uKQ== │ │ │ #:len=300 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvcHRpbWl6ZSxWZXJib3NpdHldLCJvcHRpbWl6ZSgu │ │ ├── ./usr/share/doc/Macaulay2/Seminormalization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ YmV0dGVyTm9ybWFsaXphdGlvbk1hcA== │ │ │ #:len=1762 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXplcyBub24gZG9tYWlucyIs │ │ │ ICJsaW5lbnVtIiA9PiA4ODIsIElucHV0cyA9PiB7U1BBTntUVHsiUyJ9LCIsICIsU1BBTnsiYSAi │ │ ├── ./usr/share/doc/Macaulay2/Serialization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ U2VyaWFsaXphdGlvbg== │ │ │ #:len=554 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV2ZXJzaWJsZSBjb252ZXJzaW9uIG9m │ │ │ IGFsbCBNYWNhdWxheTIgb2JqZWN0cyB0byBzdHJpbmdzIiwgRGVzY3JpcHRpb24gPT4gMTooRElW │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ YXJYaXYoU3RyaW5nLFN0cmluZyk= │ │ │ #:len=236 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhclhpdixTdHJpbmcsU3RyaW5nKSwiYXJYaXYoU3Ry │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ @@ -1,6 +1,6 @@ │ │ │ -- -*- M2-comint -*- hash: 1331702921222 │ │ │ │ │ │ i1 : check SimpleDoc │ │ │ - -- capturing check(0, "SimpleDoc") -- .14669s elapsed │ │ │ + -- capturing check(0, "SimpleDoc") -- .100602s elapsed │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ @@ -69,15 +69,15 @@ │ │ │ │ │ │
    │ │ │

    The check method executes all package tests defined this way.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .14669s elapsed
    │ │ │ + -- capturing check(0, "SimpleDoc") -- .100602s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ (String) (missing documentation) to write a test case. │ │ │ │ │ │ │ │ TEST /// -* test for simpleDocFrob *- │ │ │ │ assert(simpleDocFrob(2,matrix{{1,2}}) == matrix{{1,2,0,0},{0,0,1,2}}) │ │ │ │ /// │ │ │ │ The _c_h_e_c_k method executes all package tests defined this way. │ │ │ │ i1 : check SimpleDoc │ │ │ │ - -- capturing check(0, "SimpleDoc") -- .14669s elapsed │ │ │ │ + -- capturing check(0, "SimpleDoc") -- .100602s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _T_E_S_T -- add a test for a package │ │ │ │ * _c_h_e_c_k -- perform tests of a package │ │ │ │ * _p_a_c_k_a_g_e_T_e_m_p_l_a_t_e -- a template for a package │ │ │ │ * _d_o_c_E_x_a_m_p_l_e -- an example of a documentation string │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _t_e_s_t_E_x_a_m_p_l_e is a _s_t_r_i_n_g. │ │ ├── ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ ZWxlbWVudGFyeUNvbGxhcHNl │ │ │ #:len=365 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU1Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiZWxlbWVudGFyeUNvbGxhcHNlIiwiZWxlbWVudGFy │ │ ├── ./usr/share/doc/Macaulay2/SimplicialDecomposability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ c2hlbGxpbmdPcmRlciguLi4sUmFuZG9tPT4uLi4p │ │ │ #:len=311 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODk0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzaGVsbGluZ09yZGVyLFJhbmRvbV0sInNoZWxsaW5n │ │ ├── ./usr/share/doc/Macaulay2/SimplicialPosets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ aXNCb29sZWFu │ │ │ #:len=1246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRGV0ZXJtaW5lIGlmIGEgcG9zZXQgaXMg │ │ │ YSBib29sZWFuIGFsZ2VicmEuIiwgImxpbmVudW0iID0+IDMyNywgSW5wdXRzID0+IHtTUEFOe1RU │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ dW5pdmVyc2FsSWRlYWw= │ │ │ #:len=2182 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIHVuaXZlcnNhbCBy │ │ │ ZWFsaXphdGlvbiBpZGVhbCBvZiBhIG1hdHJvaWQiLCAibGluZW51bSIgPT4gMjE0MiwgSW5wdXRz │ │ ├── ./usr/share/doc/Macaulay2/SpaceCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ aXNTbW9vdGhBQ01CZXR0aQ== │ │ │ #:len=902 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSBCZXR0aSB0 │ │ │ YWJsZSBpcyB0aGF0IG9mIGEgc21vb3RoIEFDTSBjdXJ2ZSIsICJsaW5lbnVtIiA9PiAxMjY5LCBJ │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=47 │ │ │ TXVsdGlkaW1lbnNpb25hbE1hdHJpeCAqIE11bHRpZGltZW5zaW9uYWxNYXRyaXg= │ │ │ #:len=1779 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvZHVjdCBvZiBtdWx0aWRpbWVuc2lv │ │ │ bmFsIG1hdHJpY2VzIiwgImxpbmVudW0iID0+IDE0MjYsIElucHV0cyA9PiB7U1BBTntUVHsiTSJ9 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ @@ -3,24 +3,24 @@ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time degreeDeterminant n │ │ │ - -- used 0.00162956s (cpu); 6.6033e-05s (thread); 0s (gc) │ │ │ + -- used 0.00278936s (cpu); 4.6277e-05s (thread); 0s (gc) │ │ │ │ │ │ o2 = 6 │ │ │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ 0,0,0 1,2,1 │ │ │ │ │ │ i4 : time degree determinant M │ │ │ - -- used 0.145443s (cpu); 0.0875454s (thread); 0s (gc) │ │ │ + -- used 0.506025s (cpu); 0.210314s (thread); 0s (gc) │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ @@ -1,13 +1,13 @@ │ │ │ -- -*- M2-comint -*- hash: 17130321902108223178 │ │ │ │ │ │ i1 : (d,n) := (2,3); │ │ │ │ │ │ i2 : time Disc = denseDiscriminant(d,n) │ │ │ - -- used 0.293083s (cpu); 0.188819s (thread); 0s (gc) │ │ │ + -- used 0.593241s (cpu); 0.346695s (thread); 0s (gc) │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |) │ │ │ | 0 0 0 1 1 2 0 0 1 0 | │ │ │ | 0 1 2 0 1 0 0 1 0 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ @@ -9,18 +9,18 @@ │ │ │ 2 │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ i2 : time denseResultant(f0,f1,f2); -- using Poisson formula │ │ │ - -- used 0.0838277s (cpu); 0.0863562s (thread); 0s (gc) │ │ │ + -- used 0.179961s (cpu); 0.178106s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.289099s (cpu); 0.231488s (thread); 0s (gc) │ │ │ + -- used 0.758079s (cpu); 0.412225s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.366071s (cpu); 0.319698s (thread); 0s (gc) │ │ │ + -- used 0.569088s (cpu); 0.510561s (thread); 0s (gc) │ │ │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2, │ │ │ ------------------------------------------------------------------------ │ │ │ 3}}}} │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ i2 : time det M │ │ │ - -- used 0.156073s (cpu); 0.0871313s (thread); 0s (gc) │ │ │ + -- used 1.44933s (cpu); 0.264196s (thread); 0s (gc) │ │ │ │ │ │ o2 = 9698337990421512192 │ │ │ │ │ │ i3 : M = randomMultidimensionalMatrix(2,2,2,2,5) │ │ │ │ │ │ o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -24,13 +24,13 @@ │ │ │ 7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6, │ │ │ ------------------------------------------------------------------------ │ │ │ 4, 8, 4, 2}}}}} │ │ │ │ │ │ o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ │ │ │ │ │ │ i4 : time det M │ │ │ - -- used 0.520079s (cpu); 0.398527s (thread); 0s (gc) │ │ │ + -- used 0.487605s (cpu); 0.438353s (thread); 0s (gc) │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ 9257139493926586400187927813888 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ a x y z + a x y z + a x y z │ │ │ 1,1,1 1 1 1 1,2,0 1 2 0 1,2,1 1 2 1 │ │ │ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ i2 : time sparseDiscriminant f │ │ │ - -- used 2.32213s (cpu); 1.95828s (thread); 0s (gc) │ │ │ + -- used 4.30239s (cpu); 3.93566s (thread); 0s (gc) │ │ │ │ │ │ 2 │ │ │ o2 = a a a a a a - a a a a a - │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 │ │ │ a a a a + a a a a a - │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 16228363821945730064 │ │ │ │ │ │ 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}}) │ │ │ - -- used 0.486545s (cpu); 0.422968s (thread); 0s (gc) │ │ │ + -- used 0.812636s (cpu); 0.775587s (thread); 0s (gc) │ │ │ │ │ │ o1 = Res │ │ │ │ │ │ o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |}) │ │ │ | 0 0 1 1 | | 1 0 1 2 | | 0 1 0 1 | │ │ │ │ │ │ i2 : QQ[c_(1,1)..c_(3,4)][x,y]; │ │ │ @@ -18,15 +18,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ c x*y + c x + c y + c ) │ │ │ 3,3 3,4 3,2 3,1 │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : time Res(f,g,h) │ │ │ - -- used 0.0085602s (cpu); 0.0088889s (thread); 0s (gc) │ │ │ + -- used 0.126846s (cpu); 0.0831587s (thread); 0s (gc) │ │ │ │ │ │ 2 4 2 2 4 │ │ │ o4 = - c c c c c c c + c c c c c c + │ │ │ 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 2 3 2 3 │ │ │ c c c c c c - 2c c c c c c c c + │ │ │ @@ -730,30 +730,30 @@ │ │ │ │ │ │ o4 : QQ[c ..c ] │ │ │ 1,1 3,4 │ │ │ │ │ │ i5 : assert(Res(f,g,h) == sparseResultant(f,g,h)) │ │ │ │ │ │ i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331); │ │ │ - -- used 0.0283122s (cpu); 0.0283181s (thread); 0s (gc) │ │ │ + -- used 1.77925s (cpu); 0.279211s (thread); 0s (gc) │ │ │ │ │ │ o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331) │ │ │ | 0 1 0 1 | │ │ │ │ │ │ i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y]; │ │ │ │ │ │ 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) │ │ │ │ │ │ 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 ) │ │ │ 3 1 2 0 3 1 2 0 3 1 2 0 │ │ │ │ │ │ o8 : Sequence │ │ │ │ │ │ i9 : time Res(f,g,h) │ │ │ - -- used 0.000612348s (cpu); 0.00298562s (thread); 0s (gc) │ │ │ + -- used 0.00400075s (cpu); 0.0029303s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ o9 = a b b c - a a b b c - a a b b c + a a b c - a b b c c - │ │ │ 3 1 2 0 2 3 1 3 0 1 3 2 3 0 1 2 3 0 3 0 2 0 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ 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 + │ │ │ @@ -822,15 +822,15 @@ │ │ │ 2 │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o11 : Sequence │ │ │ │ │ │ i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true)); │ │ │ - -- used 0.522148s (cpu); 0.449632s (thread); 0s (gc) │ │ │ + -- used 2.43223s (cpu); 1.08352s (thread); 0s (gc) │ │ │ │ │ │ i13 : quotientRemainder(UnmixedRes,MixedRes) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o13 = (b c - b b c c + b b c + b c c - 2b b c c - b b c c + b c , 0) │ │ │ 5 2 4 5 2 4 2 5 4 4 2 5 2 5 2 5 2 4 4 5 2 5 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ @@ -73,27 +73,27 @@ │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ 
      i2 : time degreeDeterminant n
│ │ │ - -- used 0.00162956s (cpu); 6.6033e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00278936s (cpu); 4.6277e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
      │ │ │ │ │ │ │ │ │ 
      i3 : M = genericMultidimensionalMatrix n;
│ │ │  
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
      │ │ │ │ │ │ │ │ │ 
      i4 : time degree determinant M
│ │ │ - -- used 0.145443s (cpu); 0.0875454s (thread); 0s (gc)
│ │ │ + -- used 0.506025s (cpu); 0.210314s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
      │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -16,23 +16,23 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : time degreeDeterminant n │ │ │ │ - -- used 0.00162956s (cpu); 6.6033e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00278936s (cpu); 4.6277e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 6 │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ │ 0,0,0 1,2,1 │ │ │ │ i4 : time degree determinant M │ │ │ │ - -- used 0.145443s (cpu); 0.0875454s (thread); 0s (gc) │ │ │ │ + -- used 0.506025s (cpu); 0.210314s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ @@ -82,15 +82,15 @@ │ │ │

    Description

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ * Outputs: │ │ │ │ o for (d,n), this is the same as _s_p_a_r_s_e_D_i_s_c_r_i_m_i_n_a_n_t _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x │ │ │ │ ""ggeenneerriicc ppoollyynnoommiiaall ooff ddeeggrreeee dd iinn nn vvaarriiaabblleess"";; │ │ │ │ o for f, this is the same as _a_f_f_i_n_e_D_i_s_c_r_i_m_i_n_a_n_t(f). │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : (d,n) := (2,3); │ │ │ │ i2 : time Disc = denseDiscriminant(d,n) │ │ │ │ - -- used 0.293083s (cpu); 0.188819s (thread); 0s (gc) │ │ │ │ + -- used 0.593241s (cpu); 0.346695s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 │ │ │ │ 2 |) │ │ │ │ | 0 0 0 1 1 2 0 0 1 │ │ │ │ 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ @@ -92,23 +92,23 @@ │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : (d,n) := (2,3);
    │ │ │ 
    i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.293083s (cpu); 0.188819s (thread); 0s (gc)
│ │ │ + -- used 0.593241s (cpu); 0.346695s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
    │ │ │ 
    i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.0838277s (cpu); 0.0863562s (thread); 0s (gc)
    │ │ │ + -- used 0.179961s (cpu); 0.178106s (thread); 0s (gc) │ │ │ 
    i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.289099s (cpu); 0.231488s (thread); 0s (gc)
    │ │ │ + -- used 0.758079s (cpu); 0.412225s (thread); 0s (gc) │ │ │ 
    i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.366071s (cpu); 0.319698s (thread); 0s (gc)
    │ │ │ + -- used 0.569088s (cpu); 0.510561s (thread); 0s (gc) │ │ │ 
    i5 : assert(o2 == o3 and o3 == o4)
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,20 +29,20 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ c x x + c x + c x + c x + c ) │ │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ │ │ o1 : Sequence │ │ │ │ i2 : time denseResultant(f0,f1,f2); -- using Poisson formula │ │ │ │ - -- used 0.0838277s (cpu); 0.0863562s (thread); 0s (gc) │ │ │ │ + -- used 0.179961s (cpu); 0.178106s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.289099s (cpu); 0.231488s (thread); 0s (gc) │ │ │ │ + -- used 0.758079s (cpu); 0.412225s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.366071s (cpu); 0.319698s (thread); 0s (gc) │ │ │ │ + -- used 0.569088s (cpu); 0.510561s (thread); 0s (gc) │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant) │ │ │ │ * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant │ │ │ │ * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant) │ │ │ │ * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials │ │ │ │ * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ @@ -84,15 +84,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ 3}}}} │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ │ │ │ 
    i2 : time det M
│ │ │ - -- used 0.156073s (cpu); 0.0871313s (thread); 0s (gc)
│ │ │ + -- used 1.44933s (cpu); 0.264196s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
    │ │ │ │ │ │ │ │ │ 
    i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │  
│ │ │  o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5,
│ │ │ @@ -105,15 +105,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       4, 8, 4, 2}}}}}
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
    │ │ │ │ │ │ │ │ │ 
    i4 : time det M
│ │ │ - -- used 0.520079s (cpu); 0.398527s (thread); 0s (gc)
│ │ │ + -- used 0.487605s (cpu); 0.438353s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ │ │ │ │ o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3}}}} │ │ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ i2 : time det M │ │ │ │ - -- used 0.156073s (cpu); 0.0871313s (thread); 0s (gc) │ │ │ │ + -- used 1.44933s (cpu); 0.264196s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 9698337990421512192 │ │ │ │ i3 : M = randomMultidimensionalMatrix(2,2,2,2,5) │ │ │ │ │ │ │ │ o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6}}}, {{{3, 5, 7, 7, 9}, {4, 5, 0, 4, 3}}, {{1, 8, 9, 1, 2}, {9, 6, 6, │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 4, 8, 4, 2}}}}} │ │ │ │ │ │ │ │ o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ │ │ │ │ i4 : time det M │ │ │ │ - -- used 0.520079s (cpu); 0.398527s (thread); 0s (gc) │ │ │ │ + -- used 0.487605s (cpu); 0.438353s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ │ 9257139493926586400187927813888 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x -- the class of all multidimensional matrices │ │ │ │ * _d_e_g_r_e_e_D_e_t_e_r_m_i_n_a_n_t -- degree of the hyperdeterminant of a generic │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ 1,1,1 1 1 1 1,2,0 1 2 0 1,2,1 1 2 1 │ │ │ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ │ │ │ 
    i2 : time sparseDiscriminant f
│ │ │ - -- used 2.32213s (cpu); 1.95828s (thread); 0s (gc)
│ │ │ + -- used 4.30239s (cpu); 3.93566s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        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  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,15 +38,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │ │  
│ │ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │ │  i2 : time sparseDiscriminant f
│ │ │ │ - -- used 2.32213s (cpu); 1.95828s (thread); 0s (gc)
│ │ │ │ + -- used 4.30239s (cpu); 3.93566s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    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 exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
    
│ │ │          
    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 $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │ @@ -100,15 +100,15 @@
│ │ │       c   x*y + c   x + c   y + c   )
│ │ │        3,3       3,4     3,2     3,1
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    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}})
│ │ │ - -- used 0.486545s (cpu); 0.422968s (thread); 0s (gc)
│ │ │ + -- used 0.812636s (cpu); 0.775587s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
    
│ │ │      		          
              
                  
    i4 : time Res(f,g,h)
│ │ │ - -- used 0.0085602s (cpu); 0.0088889s (thread); 0s (gc)
│ │ │ + -- used 0.126846s (cpu); 0.0831587s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          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  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -817,15 +817,15 @@
│ │ │  
    		              
    i5 : assert(Res(f,g,h) == sparseResultant(f,g,h))
    
│ │ │      		          
    
│ │ │          
    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 $\mathbb{Z}/3331$.
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ @@ -835,15 +835,15 @@
│ │ │  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 )
│ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0283122s (cpu); 0.0283181s (thread); 0s (gc)
│ │ │ + -- used 1.77925s (cpu); 0.279211s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
    
│ │ │      		          
                  
    i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
    
│ │ │      		          
              
                  
    i9 : time Res(f,g,h)
│ │ │ - -- used 0.000612348s (cpu); 0.00298562s (thread); 0s (gc)
│ │ │ + -- used 0.00400075s (cpu); 0.0029303s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       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  +
│ │ │ @@ -918,15 +918,15 @@
│ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │  
│ │ │  o11 : Sequence
    
│ │ │      		          
                  
    i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.522148s (cpu); 0.449632s (thread); 0s (gc)
    
│ │ │ + -- used 2.43223s (cpu); 1.08352s (thread); 0s (gc)
│ │ │      		          
                  
    i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  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 $f = c_{(1,1)}+c_{(1,2)} x y+c_{
│ │ │ │  (1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{
│ │ │ │  (2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
│ │ │ │  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}})
│ │ │ │ - -- used 0.486545s (cpu); 0.422968s (thread); 0s (gc)
│ │ │ │ + -- used 0.812636s (cpu); 0.775587s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = Res
│ │ │ │  
│ │ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1
│ │ │ │  2 2 |, | 0 0 1 1 |})
│ │ │ │                                                              | 0 0 1 1 |  | 1 0
│ │ │ │  1 2 |  | 0 1 0 1 |
│ │ │ │ @@ -56,15 +56,15 @@
│ │ │ │         1,3       1,2       1,4     1,1   2,2        2,3       2,4     2,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       c   x*y + c   x + c   y + c   )
│ │ │ │        3,3       3,4     3,2     3,1
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : time Res(f,g,h)
│ │ │ │ - -- used 0.0085602s (cpu); 0.0088889s (thread); 0s (gc)
│ │ │ │ + -- used 0.126846s (cpu); 0.0831587s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2                       4      2   2               4
│ │ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │ │          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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3       2       3               2                   3
│ │ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ │ @@ -772,29 +772,29 @@
│ │ │ │  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
│ │ │ │  $\mathbb{Z}/3331$.
│ │ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},
│ │ │ │  {0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ │ - -- used 0.0283122s (cpu); 0.0283181s (thread); 0s (gc)
│ │ │ │ + -- used 1.77925s (cpu); 0.279211s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over
│ │ │ │  ZZ/3331)
│ │ │ │                                                               | 0 1 0 1 |
│ │ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │ │  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)
│ │ │ │  
│ │ │ │  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 )
│ │ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : time Res(f,g,h)
│ │ │ │ - -- used 0.000612348s (cpu); 0.00298562s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400075s (cpu); 0.0029303s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2            2            2        2 2    2
│ │ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2                       2
│ │ │ │       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  +
│ │ │ │ @@ -864,15 +864,15 @@
│ │ │ │                    2
│ │ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o11 : Sequence
│ │ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant
│ │ │ │  (f,g,h,Unmixed=>true));
│ │ │ │ - -- used 0.522148s (cpu); 0.449632s (thread); 0s (gc)
│ │ │ │ + -- used 2.43223s (cpu); 1.08352s (thread); 0s (gc)
│ │ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │ │  
│ │ │ │          2 2                   2    2                               2 2
│ │ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  c2NodXJQb2x5bm9taWFsKC4uLixBc0V4cHJlc3Npb249Pi4uLik=
│ │ │  #:len=287
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMwNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2NodXJQb2x5bm9taWFsLEFzRXhwcmVzc2lvbl0s
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o4 = | 0 1 |
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
│ │ │  
│ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.000884418s (cpu); 0.00130217s (thread); 0s (gc)
│ │ │ + -- used 0.00128616s (cpu); 0.00104511s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  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  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 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  -
│ │ │ @@ -46,15 +46,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0021735s (cpu); 0.00108661s (thread); 0s (gc)
│ │ │ + -- used 0.00166903s (cpu); 0.000894327s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  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  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 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  -
│ │ │ @@ -67,15 +67,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o6 : R
│ │ │  
│ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00181311s (cpu); 0.00189266s (thread); 0s (gc)
│ │ │ + -- used 0.000860514s (cpu); 0.00173213s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : tal := tally apply (H,h->conjugacyClass h);
│ │ │  
│ │ │  i4 : partis = partitions 6;
│ │ │  
│ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.256962s (cpu); 0.201718s (thread); 0s (gc)
│ │ │ + -- used 0.459066s (cpu); 0.420364s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : genList = {{1,2,3,0,5,4},{0,4,2,5,1,3}}
│ │ │  
│ │ │  o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time seco = secondaryInvariants(genList,R);
│ │ │ - -- used 0.605979s (cpu); 0.435231s (thread); 0s (gc)
│ │ │ + -- used 1.11655s (cpu); 0.992502s (thread); 0s (gc)
│ │ │  (Partition{6}, Ambient_Dimension, 1, Rank, 1)
│ │ │  (Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
    
│ │ │      		          
                  
    i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.000884418s (cpu); 0.00130217s (thread); 0s (gc)
│ │ │ + -- used 0.00128616s (cpu); 0.00104511s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  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  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 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  -
│ │ │ @@ -144,15 +144,15 @@
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o5 : R
    
│ │ │      		          
                  
    i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0021735s (cpu); 0.00108661s (thread); 0s (gc)
│ │ │ + -- used 0.00166903s (cpu); 0.000894327s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  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  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 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  -
│ │ │ @@ -166,15 +166,15 @@
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o6 : R
    
│ │ │      		          
                  
    i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00181311s (cpu); 0.00189266s (thread); 0s (gc)
│ │ │ + -- used 0.000860514s (cpu); 0.00173213s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.000884418s (cpu); 0.00130217s (thread); 0s (gc)
│ │ │ │ + -- used 0.00128616s (cpu); 0.00104511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  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  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 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  -
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ │ - -- used 0.0021735s (cpu); 0.00108661s (thread); 0s (gc)
│ │ │ │ + -- used 0.00166903s (cpu); 0.000894327s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  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  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 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  -
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o6 : R
│ │ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ │ - -- used 0.00181311s (cpu); 0.00189266s (thread); 0s (gc)
│ │ │ │ + -- used 0.000860514s (cpu); 0.00173213s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │  o2 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │ │       (This is a classical example of rational fourfold)
│ │ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ │ - -- used 2.59548s (cpu); 0.993183s (thread); 0s (gc)
│ │ │ │ + -- used 7.19095s (cpu); 2.40393s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^7 cut out by 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 2.32728s (cpu); 1.00597s (thread); 0s (gc)
│ │ │ │ + -- used 9.72896s (cpu); 2.39258s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,15 +44,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 8.32356s (cpu); 4.38639s (thread); 0s (gc)
│ │ │ │ + -- used 17.1913s (cpu); 8.25251s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : partis = partitions 6;
    
│ │ │      		          
                  
    i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.256962s (cpu); 0.201718s (thread); 0s (gc)
│ │ │ + -- used 0.459066s (cpu); 0.420364s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │  representations of $H$ in each irreducible representation of $S_6$. We take
│ │ │ │  into account that there are multiple copies of each representation by
│ │ │ │  multiplying the values with the number of copies which is given by the
│ │ │ │  hookLengthFormula.
│ │ │ │  i4 : partis = partitions 6;
│ │ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity
│ │ │ │  (tal,p))
│ │ │ │ - -- used 0.256962s (cpu); 0.201718s (thread); 0s (gc)
│ │ │ │ + -- used 0.459066s (cpu); 0.420364s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}
│ │ │  
│ │ │  o2 : List
    
│ │ │      		          
                  
    i3 : time seco = secondaryInvariants(genList,R);
│ │ │ - -- used 0.605979s (cpu); 0.435231s (thread); 0s (gc)
│ │ │ + -- used 1.11655s (cpu); 0.992502s (thread); 0s (gc)
│ │ │  (Partition{6}, Ambient_Dimension, 1, Rank, 1)
│ │ │  (Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : genList = {{1,2,3,0,5,4},{0,4,2,5,1,3}}
│ │ │ │  
│ │ │ │  o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time seco = secondaryInvariants(genList,R);
│ │ │ │ - -- used 0.605979s (cpu); 0.435231s (thread); 0s (gc)
│ │ │ │ + -- used 1.11655s (cpu); 0.992502s (thread); 0s (gc)
│ │ │ │  (Partition{6}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │  (Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  SW50ZXJzZWN0aW9uT2ZUaHJlZVF1YWRyaWNzSW5QNw==
│ │ │  #:len=625
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNsYXNzIG9mIGFsbCBzcGVjaWFs
│ │ │  IGludGVyc2VjdGlvbiBvZiB0aHJlZSBxdWFkcmljcyBpbiBQXjciLCAibGluZW51bSIgPT4gMzY1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       of discriminant 31 = det| 8 1 |
│ │ │                               | 1 4 |
│ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
│ │ │  
│ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.59548s (cpu); 0.993183s (thread); 0s (gc)
│ │ │ + -- used 7.19095s (cpu); 2.40393s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 2.32728s (cpu); 1.00597s (thread); 0s (gc)
│ │ │ + -- used 9.72896s (cpu); 2.39258s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 8.32356s (cpu); 4.38639s (thread); 0s (gc)
│ │ │ + -- used 17.1913s (cpu); 8.25251s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 4.72888s (cpu); 2.04389s (thread); 0s (gc)
│ │ │ + -- used 7.15636s (cpu); 3.70662s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │  
│ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │  
│ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │  
│ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.454906s (cpu); 0.271329s (thread); 0s (gc)
│ │ │ + -- used 1.12874s (cpu); 0.605253s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  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))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface in PP^8 of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 16.7755s (cpu); 7.55644s (thread); 0s (gc)
│ │ │ + -- used 24.7063s (cpu); 6.18577s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937, 13402, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.427032s (cpu); 0.240066s (thread); 0s (gc)
│ │ │ + -- used 0.487928s (cpu); 0.207225s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1729890813579561111
│ │ │  
│ │ │  i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 0.331789s (cpu); 0.131229s (thread); 0s (gc)
│ │ │ + -- used 0.568223s (cpu); 0.0817205s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1730220932418738713
│ │ │  
│ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 1.14069s (cpu); 0.479565s (thread); 0s (gc)
│ │ │ + -- used 5.08014s (cpu); 0.980607s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.38407s (cpu); 0.289966s (thread); 0s (gc)
│ │ │ + -- used 0.405372s (cpu); 0.33203s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  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));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.634373s (cpu); 0.352601s (thread); 0s (gc)
│ │ │ + -- used 1.0125s (cpu); 0.633498s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  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));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.30052s (cpu); 2.70046s (thread); 0s (gc)
│ │ │ + -- used 5.48569s (cpu); 3.90588s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
│ │ │  
│ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.72858s (cpu); 0.706037s (thread); 0s (gc)
│ │ │ + -- used 2.74286s (cpu); 1.28565s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  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);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │  
│ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.00797778s (cpu); 0.00864695s (thread); 0s (gc)
│ │ │ + -- used 0.00801837s (cpu); 0.00732559s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 0.664875s (cpu); 0.18344s (thread); 0s (gc)
│ │ │ + -- used 2.53079s (cpu); 0.353743s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i7 : assert(F == X)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^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);
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │  
│ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.5216s (cpu); 1.68427s (thread); 0s (gc)
│ │ │ + -- used 3.19895s (cpu); 2.50875s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.45039s (cpu); 2.75319s (thread); 0s (gc)
│ │ │ + -- used 15.6202s (cpu); 5.86826s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  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));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.09444s (cpu); 2.47453s (thread); 0s (gc)
│ │ │ + -- used 5.89382s (cpu); 2.07587s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  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);
│ │ │  
│ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.99465s (cpu); 2.69046s (thread); 0s (gc)
│ │ │ + -- used 14.4182s (cpu); 4.91307s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.00984s (cpu); 0.44616s (thread); 0s (gc)
│ │ │ + -- used 2.53993s (cpu); 1.07614s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │                               | 1 4 |
│ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
    
│ │ │      		          
                  
    i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.59548s (cpu); 0.993183s (thread); 0s (gc)
│ │ │ + -- used 7.19095s (cpu); 2.40393s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
    
│ │ │      		          
                  
    i4 : (mu,U,C,f) = building U';
    
│ │ │      		          
              
                  
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 2.32728s (cpu); 1.00597s (thread); 0s (gc)
│ │ │ + -- used 9.72896s (cpu); 2.39258s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │      		          
                  
    i4 : (mu,U,C,f) = building U';
    
│ │ │      		          
              
                  
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 8.32356s (cpu); 4.38639s (thread); 0s (gc)
│ │ │ + -- used 17.1913s (cpu); 8.25251s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │      		          
                  
    i4 : (mu,U,C,f) = building U';
    
│ │ │      		          
              
                  
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 4.72888s (cpu); 2.04389s (thread); 0s (gc)
│ │ │ + -- used 7.15636s (cpu); 3.70662s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
    
│ │ │      		          
              
                  
    i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.454906s (cpu); 0.271329s (thread); 0s (gc)
│ │ │ + -- used 1.12874s (cpu); 0.605253s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
    
│ │ │      		          
                  
    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))
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,28 +31,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 4.72888s (cpu); 2.04389s (thread); 0s (gc)
│ │ │ │ + -- used 7.15636s (cpu); 3.70662s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the cubic map and passing through a
│ │ │ │  general point: 8
│ │ │ │  number 2-secant lines = 7
│ │ │ │  number 5-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │ │  
│ │ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ │ - -- used 0.454906s (cpu); 0.271329s (thread); 0s (gc)
│ │ │ │ + -- used 1.12874s (cpu); 0.605253s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  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))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
    │ │ │ │ │ │ │ │ │ 
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 16.7755s (cpu); 7.55644s (thread); 0s (gc)
│ │ │ + -- used 24.7063s (cpu); 6.18577s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
    │ │ │ │ │ │ │ │ │ @@ -114,15 +114,15 @@ │ │ │ │ │ │ o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937, 13402, 1] │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^8 │ │ │ │ │ │ │ │ │ 
    i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.427032s (cpu); 0.240066s (thread); 0s (gc)
│ │ │ + -- used 0.487928s (cpu); 0.207225s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
    │ │ │ │ │ │ │ │ │ 
    i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
    │ │ │ ├── html2text {} │ │ │ │ @@ -37,15 +37,15 @@ │ │ │ │ o2 = Special Gushel-Mukai fourfold of discriminant 20 │ │ │ │ containing a surface in PP^8 of degree 9 and sectional genus 2 │ │ │ │ cut out by 19 hypersurfaces of degree 2 │ │ │ │ and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2) │ │ │ │ Type: ordinary │ │ │ │ (case 17 of Table 1 in arXiv:2002.07026) │ │ │ │ i3 : time f = detectCongruence(X,Verbose=>true); │ │ │ │ - -- used 16.7755s (cpu); 7.55644s (thread); 0s (gc) │ │ │ │ + -- used 24.7063s (cpu); 6.18577s (thread); 0s (gc) │ │ │ │ number lines contained in the image of the quadratic map and passing through a │ │ │ │ general point: 7 │ │ │ │ number 1-secant lines = 6 │ │ │ │ number 3-secant conics = 1 │ │ │ │ │ │ │ │ o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8 │ │ │ │ i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ i5 : p := point Y -- random point on Y │ │ │ │ │ │ │ │ o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937, │ │ │ │ 13402, 1] │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^8 │ │ │ │ i6 : time C = f p; -- 3-secant conic to the surface │ │ │ │ - -- used 0.427032s (cpu); 0.240066s (thread); 0s (gc) │ │ │ │ + -- used 0.487928s (cpu); 0.207225s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y) │ │ │ │ i7 : S = surface X; │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y) │ │ │ │ i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3 │ │ │ │ and isSubset(p,C) and isSubset(C,Y)) │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ │ │ 
    i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
    │ │ │ │ │ │ │ │ │ 
    i2 : time discriminant X
│ │ │ - -- used 0.331789s (cpu); 0.131229s (thread); 0s (gc)
│ │ │ + -- used 0.568223s (cpu); 0.0817205s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -21,15 +21,15 @@ │ │ │ │ thanks to the functions _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm │ │ │ │ allows you to select the method). │ │ │ │ i1 : X = specialCubicFourfold "quintic del Pezzo surface"; │ │ │ │ │ │ │ │ o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and │ │ │ │ sectional genus 1 │ │ │ │ i2 : time discriminant X │ │ │ │ - -- used 0.331789s (cpu); 0.131229s (thread); 0s (gc) │ │ │ │ + -- used 0.568223s (cpu); 0.0817205s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 14 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special │ │ │ │ Gushel-Mukai fourfold │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * discriminant(HodgeSpecialFourfold) │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ │ │ 
    i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
    │ │ │ │ │ │ │ │ │ 
    i2 : time discriminant X
│ │ │ - -- used 1.14069s (cpu); 0.479565s (thread); 0s (gc)
│ │ │ + -- used 5.08014s (cpu); 0.980607s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -21,15 +21,15 @@ │ │ │ │ the functions _c_y_c_l_e_C_l_a_s_s, _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm │ │ │ │ allows you to select the method). │ │ │ │ i1 : X = specialGushelMukaiFourfold "tau-quadric"; │ │ │ │ │ │ │ │ o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and │ │ │ │ sectional genus 0 │ │ │ │ i2 : time discriminant X │ │ │ │ - -- used 1.14069s (cpu); 0.479565s (thread); 0s (gc) │ │ │ │ + -- used 5.08014s (cpu); 0.980607s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 10 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_) -- discriminant of a special cubic │ │ │ │ fourfold │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ │ │ 
    i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
    │ │ │ │ │ │ │ │ │ 
    i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.38407s (cpu); 0.289966s (thread); 0s (gc)
│ │ │ + -- used 0.405372s (cpu); 0.33203s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  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));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  i1 : K = ZZ/33331; S = PP_K^(1,5);
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ │ - -- used 0.38407s (cpu); 0.289966s (thread); 0s (gc)
│ │ │ │ + -- used 0.405372s (cpu); 0.33203s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  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));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │            
│ │ │                
    i3 : X = specialCubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.634373s (cpu); 0.352601s (thread); 0s (gc)
│ │ │ + -- used 1.0125s (cpu); 0.633498s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  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));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = specialCubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.634373s (cpu); 0.352601s (thread); 0s (gc)
│ │ │ │ + -- used 1.0125s (cpu); 0.633498s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  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));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │            
│ │ │                
    i3 : X = specialGushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.30052s (cpu); 2.70046s (thread); 0s (gc)
│ │ │ + -- used 5.48569s (cpu); 3.90588s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 4.30052s (cpu); 2.70046s (thread); 0s (gc)
│ │ │ │ + -- used 5.48569s (cpu); 3.90588s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │                
    i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.72858s (cpu); 0.706037s (thread); 0s (gc)
│ │ │ + -- used 2.74286s (cpu); 1.28565s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9
│ │ │ │  i3 : ? X
│ │ │ │  
│ │ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │ │       1^2 2^5
│ │ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ │ - -- used 1.72858s (cpu); 0.706037s (thread); 0s (gc)
│ │ │ │ + -- used 2.74286s (cpu); 1.28565s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html
│ │ │ @@ -93,23 +93,23 @@
│ │ │            
│ │ │                
    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);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.00797778s (cpu); 0.00864695s (thread); 0s (gc)
│ │ │ + -- used 0.00801837s (cpu); 0.00732559s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
    
│ │ │            
│ │ │            
│ │ │                
    i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 0.664875s (cpu); 0.18344s (thread); 0s (gc)
│ │ │ + -- used 2.53079s (cpu); 0.353743s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
    
│ │ │            
│ │ │            
│ │ │                
    i7 : assert(F == X)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -116,24 +116,24 @@
│ │ │ │  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);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ │ - -- used 0.00797778s (cpu); 0.00864695s (thread); 0s (gc)
│ │ │ │ + -- used 0.00801837s (cpu); 0.00732559s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and
│ │ │ │  sectional genus 0
│ │ │ │  i6 : time describe F
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 0.664875s (cpu); 0.18344s (thread); 0s (gc)
│ │ │ │ + -- used 2.53079s (cpu); 0.353743s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i7 : assert(F == X)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_(_E_m_b_e_d_d_e_d_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- random special cubic
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html
│ │ │ @@ -90,23 +90,23 @@
│ │ │            
│ │ │                
    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);
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
    
│ │ │            
│ │ │            
│ │ │                
    i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.5216s (cpu); 1.68427s (thread); 0s (gc)
│ │ │ + -- used 3.19895s (cpu); 2.50875s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
    
│ │ │            
│ │ │            
│ │ │                
    i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.45039s (cpu); 2.75319s (thread); 0s (gc)
│ │ │ + -- used 15.6202s (cpu); 5.86826s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,24 +34,24 @@
│ │ │ │  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);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ │ - -- used 2.5216s (cpu); 1.68427s (thread); 0s (gc)
│ │ │ │ + -- used 3.19895s (cpu); 2.50875s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i6 : time describe F
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 5.45039s (cpu); 2.75319s (thread); 0s (gc)
│ │ │ │ + -- used 15.6202s (cpu); 5.86826s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.09444s (cpu); 2.47453s (thread); 0s (gc)
│ │ │ + -- used 5.89382s (cpu); 2.07587s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
    
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 5.09444s (cpu); 2.47453s (thread); 0s (gc)
│ │ │ │ + -- used 5.89382s (cpu); 2.07587s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces
│ │ │ │  of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.99465s (cpu); 2.69046s (thread); 0s (gc)
│ │ │ + -- used 14.4182s (cpu); 4.91307s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
    
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.99465s (cpu); 2.69046s (thread); 0s (gc)
│ │ │ │ + -- used 14.4182s (cpu); 4.91307s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │            
│ │ │                
    i3 : X = specialCubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
    
│ │ │            
│ │ │            
│ │ │                
    i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.00984s (cpu); 0.44616s (thread); 0s (gc)
│ │ │ + -- used 2.53993s (cpu); 1.07614s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
    
│ │ │            
│ │ │            
│ │ │                
    i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = specialCubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 1.00984s (cpu); 0.44616s (thread); 0s (gc)
│ │ │ │ + -- used 2.53993s (cpu); 1.07614s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  U3BlY3RyYWxTZXF1ZW5jZSBeIEluZmluaXRlTnVtYmVy
│ │ │  #:len=967
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGluZmluaXR5IHBhZ2Ugb2YgYSBz
│ │ │  cGVjdHJhbCBzZXF1ZW5jZSIsICJsaW5lbnVtIiA9PiAzNDk4LCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNMb29wbGVzcyhEaWdyYXBoKQ==
│ │ │  #:len=247
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0xvb3BsZXNzLERpZ3JhcGgpLCJpc0xvb3BsZXNz
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_graph_lp__Mixed__Graph_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │                                b => {a, c}
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
│ │ │  
│ │ │  i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Bigraph, Graph, Digraph}
│ │ │ +o3 = {Digraph, Bigraph, Graph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_to__String_lp__Mixed__Graph_rp.out
│ │ │ @@ -11,12 +11,12 @@
│ │ │                  Graph => Graph{1 => {3}}
│ │ │                                 3 => {1}
│ │ │  
│ │ │  o1 : MixedGraph
│ │ │  
│ │ │  i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ -     Bigraph => bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Graph => graph ({3,
│ │ │ -     1}, {{1, 3}})}
│ │ │ +o2 = new HashTable from {Bigraph => bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}),
│ │ │ +     Graph => graph ({3, 1}, {{1, 3}}), Digraph => digraph ({1, 2, 3}, {{1,
│ │ │ +     2}, {2, 3}})}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
    
│ │ │            
│ │ │            
│ │ │                
    i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Bigraph, Graph, Digraph}
│ │ │ +o3 = {Digraph, Bigraph, Graph}
│ │ │  
│ │ │  o3 : List
    
│ │ │            
│ │ │            
│ │ │                
    i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │                 Graph => Graph{a => {b}   }
│ │ │ │                                b => {a, c}
│ │ │ │                                c => {b}
│ │ │ │  
│ │ │ │  o2 : HashTable
│ │ │ │  i3 : keys (graph G)
│ │ │ │  
│ │ │ │ -o3 = {Bigraph, Graph, Digraph}
│ │ │ │ +o3 = {Digraph, Bigraph, Graph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_to__String_lp__Mixed__Graph_rp.html
│ │ │ @@ -86,17 +86,17 @@
│ │ │                                 3 => {1}
│ │ │  
│ │ │  o1 : MixedGraph
    
│ │ │            
│ │ │            
│ │ │                
    i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ -     Bigraph => bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Graph => graph ({3,
│ │ │ -     1}, {{1, 3}})}
    
│ │ │ +o2 = new HashTable from {Bigraph => bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}),
│ │ │ +     Graph => graph ({3, 1}, {{1, 3}}), Digraph => digraph ({1, 2, 3}, {{1,
│ │ │ +     2}, {2, 3}})}
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -24,16 +24,16 @@ │ │ │ │ 3 => {} │ │ │ │ Graph => Graph{1 => {3}} │ │ │ │ 3 => {1} │ │ │ │ │ │ │ │ o1 : MixedGraph │ │ │ │ i2 : toString G │ │ │ │ │ │ │ │ -o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}), │ │ │ │ - Bigraph => bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Graph => graph ({3, │ │ │ │ - 1}, {{1, 3}})} │ │ │ │ +o2 = new HashTable from {Bigraph => bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), │ │ │ │ + Graph => graph ({3, 1}, {{1, 3}}), Digraph => digraph ({1, 2, 3}, {{1, │ │ │ │ + 2}, {2, 3}})} │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges │ │ │ │ * _n_e_t_(_M_i_x_e_d_G_r_a_p_h_) -- print a mixed graph as a net │ │ │ │ * _S_t_r_i_n_g -- the class of all strings │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _t_o_S_t_r_i_n_g_(_M_i_x_e_d_G_r_a_p_h_) -- print a mixed graph as a string │ │ ├── ./usr/share/doc/Macaulay2/StatePolytope/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ aW5pdGlhbElkZWFscw== │ │ │ #:len=950 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsbHMgZ2ZhbiBhbmQgcmV0dXJucyB0 │ │ │ aGUgbGlzdCBvZiBpbml0aWFsIGlkZWFscyIsICJsaW5lbnVtIiA9PiAxNDAsIElucHV0cyA9PiB7 │ │ ├── ./usr/share/doc/Macaulay2/StronglyStableIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ bWFjYXVsYXlEZWNvbXBvc2l0aW9uKFJpbmdFbGVtZW50KQ== │ │ │ #:len=329 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWNhdWxheURlY29tcG9zaXRpb24sUmluZ0VsZW1l │ │ ├── ./usr/share/doc/Macaulay2/Style/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=5 │ │ │ U3R5bGU= │ │ │ #:len=344 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3R5bGUgc2hlZXRzIGFuZCBpbWFnZXMg │ │ │ Zm9yIHRoZSBkb2N1bWVudGF0aW9uIiwgRGVzY3JpcHRpb24gPT4gMTooIlRoaXMgcGFja2FnZSBp │ │ ├── ./usr/share/doc/Macaulay2/SubalgebraBases/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=24 │ │ │ c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p │ │ │ #:len=300 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU3OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2FnYmksU3RyYXRlZ3ldLCJzYWdiaSguLi4sU3Ry │ │ ├── ./usr/share/doc/Macaulay2/SumsOfSquares/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ bG93ZXJCb3VuZCguLi4sVmVyYm9zaXR5PT4uLi4p │ │ │ #:len=302 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tsb3dlckJvdW5kLFZlcmJvc2l0eV0sImxvd2VyQm91 │ │ ├── ./usr/share/doc/Macaulay2/SuperLinearAlgebra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=6 │ │ │ cGFyaXR5 │ │ │ #:len=1717 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicGFyaXR5IG9mIGFuIGVsZW1lbnQgb2Yg │ │ │ YSBzdXBlciByaW5nLiIsICJsaW5lbnVtIiA9PiA2MTAsIElucHV0cyA9PiB7U1BBTntUVHsiZiJ9 │ │ ├── ./usr/share/doc/Macaulay2/SwitchingFields/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ ZmllbGRCYXNlQ2hhbmdlKFJpbmcsR2Fsb2lzRmllbGQp │ │ │ #:len=300 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmaWVsZEJhc2VDaGFuZ2UsUmluZyxHYWxvaXNGaWVs │ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ bm9QYWNrZWRBbGxTdWJz │ │ │ #:len=1151 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgYWxsIHN1YnN0aXR1dGlvbnMg │ │ │ b2YgdmFyaWFibGVzIGJ5IDEgYW5kL29yIDAgZm9yIHdoaWNoIGlkZWFsIGlzIG5vdCBLb25pZy4i │ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ @@ -31,15 +31,15 @@ │ │ │ o5 : Ideal of QQ[x..z] │ │ │ │ │ │ i6 : isHomogeneous P │ │ │ │ │ │ o6 = false │ │ │ │ │ │ i7 : time symbolicPower(P,4); │ │ │ - -- used 0.268743s (cpu); 0.210039s (thread); 0s (gc) │ │ │ + -- used 0.264144s (cpu); 0.216232s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of QQ[x..z] │ │ │ │ │ │ i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5}) │ │ │ │ │ │ 2 3 2 2 │ │ │ o8 = ideal (y - x*z, x - y*z, x y - z ) │ │ │ @@ -47,12 +47,12 @@ │ │ │ o8 : Ideal of QQ[x..z] │ │ │ │ │ │ i9 : isHomogeneous Q │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : time symbolicPower(Q,4); │ │ │ - -- used 0.128023s (cpu); 0.0677668s (thread); 0s (gc) │ │ │ + -- used 0.0784835s (cpu); 0.037094s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of QQ[x..z] │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ @@ -132,15 +132,15 @@ │ │ │ │ │ │ 
      i6 : isHomogeneous P
│ │ │  
│ │ │  o6 = false
      │ │ │ │ │ │ │ │ │ 
      i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.268743s (cpu); 0.210039s (thread); 0s (gc)
│ │ │ + -- used 0.264144s (cpu); 0.216232s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
      │ │ │ │ │ │ │ │ │ 
      i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │  
│ │ │               2         3         2     2
│ │ │ @@ -151,15 +151,15 @@
│ │ │            
│ │ │                
      i9 : isHomogeneous Q
│ │ │  
│ │ │  o9 = true
      
│ │ │            
│ │ │            
│ │ │                
      i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.128023s (cpu); 0.0677668s (thread); 0s (gc)
│ │ │ + -- used 0.0784835s (cpu); 0.037094s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
      
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -60,28 +60,28 @@ │ │ │ │ o5 = ideal (y - x*z, x y - z , x - y*z) │ │ │ │ │ │ │ │ o5 : Ideal of QQ[x..z] │ │ │ │ i6 : isHomogeneous P │ │ │ │ │ │ │ │ o6 = false │ │ │ │ i7 : time symbolicPower(P,4); │ │ │ │ - -- used 0.268743s (cpu); 0.210039s (thread); 0s (gc) │ │ │ │ + -- used 0.264144s (cpu); 0.216232s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of QQ[x..z] │ │ │ │ i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5}) │ │ │ │ │ │ │ │ 2 3 2 2 │ │ │ │ o8 = ideal (y - x*z, x - y*z, x y - z ) │ │ │ │ │ │ │ │ o8 : Ideal of QQ[x..z] │ │ │ │ i9 : isHomogeneous Q │ │ │ │ │ │ │ │ o9 = true │ │ │ │ i10 : time symbolicPower(Q,4); │ │ │ │ - -- used 0.128023s (cpu); 0.0677668s (thread); 0s (gc) │ │ │ │ + -- used 0.0784835s (cpu); 0.037094s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of QQ[x..z] │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_b_P_o_w_e_r_P_r_i_m_e_P_o_s_C_h_a_r │ │ │ │ ********** WWaayyss ttoo uussee ssyymmbboolliiccPPoowweerr:: ********** │ │ │ │ * symbolicPower(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/SymmetricPolynomials/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:38 2025 │ │ │ +# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ YnVpbGRTeW1tZXRyaWNHQihQb2x5bm9taWFsUmluZyk= │ │ │ #:len=936 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiR3JvZWJuZXIgYmFzaXMgb2YgZWxlbWVu │ │ │ dGFyeSBzeW1tZXRyaWMgcG9seW5vbWlhbHMgYWxnZWJyYSIsICJsaW5lbnVtIiA9PiAxNzAsIElu │ │ ├── ./usr/share/doc/Macaulay2/TSpreadIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ Y291bnRUTGV4TW9uKC4uLixGaXhlZE1heD0+Li4uKQ== │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM4Miwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY291bnRUTGV4TW9uLEZpeGVkTWF4XSwiY291bnRU │ │ ├── ./usr/share/doc/Macaulay2/TangentCone/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=11 │ │ │ VGFuZ2VudENvbmU= │ │ │ #:len=312 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFuZ2VudCBjb25lcyIsIERlc2NyaXB0 │ │ │ aW9uID0+IDE6KCJUaGlzIHBhY2thZ2UgcHJvdmlkZXMgYSBzaW5nbGUgZnVuY3Rpb24gdGhhdCBj │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=39 │ │ │ YWN0aW9uT25EaXJlY3RJbWFnZShJZGVhbCxDaGFpbkNvbXBsZXgp │ │ │ #:len=318 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA2Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYWN0aW9uT25EaXJlY3RJbWFnZSxJZGVhbCxDaGFp │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ o3 = 0 <-- E <-- 0 │ │ │ │ │ │ -1 0 1 │ │ │ │ │ │ o3 : ChainComplex │ │ │ │ │ │ i4 : time T=tateExtension W; │ │ │ - -- used 0.403151s (cpu); 0.208015s (thread); 0s (gc) │ │ │ + -- used 0.439062s (cpu); 0.153183s (thread); 0s (gc) │ │ │ │ │ │ i5 : cohomologyMatrix(T,-{3,3},{3,3}) │ │ │ │ │ │ o5 = | 8h 4h 0 4 8 12 16 | │ │ │ | 6h 3h 0 3 6 9 12 | │ │ │ | 4h 2h 0 2 4 6 8 | │ │ │ | 2h h 0 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ -1 0 1 │ │ │ │ │ │ o3 : ChainComplex     │ │ │ │ │ │ │ │ │ 
    i4 : time T=tateExtension W;
│ │ │ - -- used 0.403151s (cpu); 0.208015s (thread); 0s (gc)
    │ │ │ + -- used 0.439062s (cpu); 0.153183s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 0.403151s (cpu); 0.208015s (thread); 0s (gc)
│ │ │ │ + -- used 0.439062s (cpu); 0.153183s (thread); 0s (gc)
│ │ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │ │  
│ │ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ │       | 2h  h   0 1  2  3  4  |
│ │ │ │       | 0   0   0 0  0  0  0  |
│ │ ├── ./usr/share/doc/Macaulay2/TensorComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  bWlub3JzTWFwKE1hdHJpeCxMYWJlbGVkTW9kdWxlKQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkzNywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWlub3JzTWFwLE1hdHJpeCxMYWJlbGVkTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/TerraciniLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  dGVycmFjaW5pTG9jdXMoWlosUmluZ01hcCk=
│ │ │  #:len=278
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXJyYWNpbmlMb2N1cyxaWixSaW5nTWFwKSwidGVy
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  ZnJvYmVuaXVzUHJlaW1hZ2U=
│ │ │  #:len=934
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgdGhlIGlkZWFsIG9mIGVsZW1l
│ │ │  bnRzIG1hcHBlZCBpbnRvIGEgZ2l2ZW4gaWRlYWwsIHVuZGVyIGFsbCAkcF57LWV9JC1saW5lYXIg
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.77346s (cpu); 0.598028s (thread); 0s (gc)
│ │ │ + -- used 0.882524s (cpu); 0.735583s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.43564s (cpu); 1.98855s (thread); 0s (gc)
│ │ │ + -- used 2.71356s (cpu); 2.40552s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J1 == J2
│ │ │  
│ │ │  o19 = true
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │  
│ │ │  o3 : RingMap T <-- S
│ │ │  
│ │ │  i4 : R = S/(ker g);
│ │ │  
│ │ │  i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00181995s (cpu); 0.00172336s (thread); 0s (gc)
│ │ │ + -- used 0.00148371s (cpu); 0.00154069s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00365471s (cpu); 0.00378473s (thread); 0s (gc)
│ │ │ + -- used 0.000971581s (cpu); 0.00353922s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │  
│ │ │  i8 : isCohenMacaulay(R)
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out
│ │ │ @@ -60,49 +60,49 @@
│ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : time isFInjective(R)
│ │ │ - -- used 0.0239992s (cpu); 0.0237627s (thread); 0s (gc)
│ │ │ + -- used 0.0838948s (cpu); 0.0814579s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.15989s (cpu); 1.16923s (thread); 0s (gc)
│ │ │ + -- used 2.93272s (cpu); 1.82411s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │  
│ │ │  i23 : time isFInjective(R)
│ │ │ - -- used 0.155409s (cpu); 0.0979975s (thread); 0s (gc)
│ │ │ + -- used 0.161113s (cpu); 0.107699s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.0700561s (cpu); 0.0715188s (thread); 0s (gc)
│ │ │ + -- used 0.0969768s (cpu); 0.100488s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │  
│ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │  
│ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │  
│ │ │  o27 : RingMap EP1 <-- S
│ │ │  
│ │ │  i28 : R = S/(ker f);
│ │ │  
│ │ │  i29 : time isFInjective(R)
│ │ │ - -- used 0.847953s (cpu); 0.66496s (thread); 0s (gc)
│ │ │ + -- used 1.43316s (cpu); 1.00401s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.352773s (cpu); 0.235218s (thread); 0s (gc)
│ │ │ + -- used 0.401099s (cpu); 0.30867s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -79,20 +79,20 @@
│ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ - -- used 0.120933s (cpu); 0.0586845s (thread); 0s (gc)
│ │ │ + -- used 0.0954204s (cpu); 0.0466078s (thread); 0s (gc)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.220328s (cpu); 0.153633s (thread); 0s (gc)
│ │ │ + -- used 0.309374s (cpu); 0.265214s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : debugLevel = 0;
│ │ │  
│ │ │  i30 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out
│ │ │ @@ -81,21 +81,21 @@
│ │ │  i22 : testIdeal({3/4, 2/3, 3/5}, L)
│ │ │  
│ │ │  o22 = ideal (y, x)
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.266662s (cpu); 0.138587s (thread); 0s (gc)
│ │ │ + -- used 0.420843s (cpu); 0.270252s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.39027s (cpu); 0.214669s (thread); 0s (gc)
│ │ │ + -- used 0.538261s (cpu); 0.402858s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -205,21 +205,21 @@
│ │ │            
│ │ │                
    i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │            
│ │ │            
│ │ │                
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.77346s (cpu); 0.598028s (thread); 0s (gc)
│ │ │ + -- used 0.882524s (cpu); 0.735583s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    
│ │ │            
│ │ │            
│ │ │                
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.43564s (cpu); 1.98855s (thread); 0s (gc)
│ │ │ + -- used 2.71356s (cpu); 2.40552s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
    
│ │ │            
│ │ │            
│ │ │                
    i19 : J1 == J2
│ │ │  
│ │ │  o19 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -107,19 +107,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 0.77346s (cpu); 0.598028s (thread); 0s (gc)
│ │ │ │ + -- used 0.882524s (cpu); 0.735583s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.43564s (cpu); 1.98855s (thread); 0s (gc)
│ │ │ │ + -- used 2.71356s (cpu); 2.40552s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -89,21 +89,21 @@
│ │ │  o3 : RingMap T <-- S
    │ │ │ │ │ │ │ │ │ 
    i4 : R = S/(ker g);
    │ │ │ │ │ │ │ │ │ 
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00181995s (cpu); 0.00172336s (thread); 0s (gc)
│ │ │ + -- used 0.00148371s (cpu); 0.00154069s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    │ │ │ │ │ │ │ │ │ 
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00365471s (cpu); 0.00378473s (thread); 0s (gc)
│ │ │ + -- used 0.000971581s (cpu); 0.00353922s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
    │ │ │ ├── html2text {} │ │ │ │ @@ -24,19 +24,19 @@ │ │ │ │ i1 : T = ZZ/5[x,y]; │ │ │ │ i2 : S = ZZ/5[a,b,c,d]; │ │ │ │ i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3}); │ │ │ │ │ │ │ │ o3 : RingMap T <-- S │ │ │ │ i4 : R = S/(ker g); │ │ │ │ i5 : time isCohenMacaulay(R) │ │ │ │ - -- used 0.00181995s (cpu); 0.00172336s (thread); 0s (gc) │ │ │ │ + -- used 0.00148371s (cpu); 0.00154069s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : time isCohenMacaulay(R, AtOrigin => true) │ │ │ │ - -- used 0.00365471s (cpu); 0.00378473s (thread); 0s (gc) │ │ │ │ + -- used 0.000971581s (cpu); 0.00353922s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v); │ │ │ │ i8 : isCohenMacaulay(R) │ │ │ │ │ │ │ │ o8 = false │ │ │ │ The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring, │ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ @@ -182,41 +182,41 @@ │ │ │ │ │ │ o19 = R │ │ │ │ │ │ o19 : QuotientRing │ │ │ 
    i20 : time isFInjective(R)
│ │ │ - -- used 0.0239992s (cpu); 0.0237627s (thread); 0s (gc)
│ │ │ + -- used 0.0838948s (cpu); 0.0814579s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    │ │ │ 
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.15989s (cpu); 1.16923s (thread); 0s (gc)
│ │ │ + -- used 2.93272s (cpu); 1.82411s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    │ │ │
    │ │ │
    │ │ │

    If the option AtOrigin (default value false) is set to true, isFInjective 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.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    │ │ │ 
    i23 : time isFInjective(R)
│ │ │ - -- used 0.155409s (cpu); 0.0979975s (thread); 0s (gc)
│ │ │ + -- used 0.161113s (cpu); 0.107699s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    │ │ │ 
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.0700561s (cpu); 0.0715188s (thread); 0s (gc)
│ │ │ + -- used 0.0969768s (cpu); 0.100488s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    │ │ │
    │ │ │
    │ │ │

    If the option AssumeCM (default value false) is set to true, then isFInjective 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 $\mathbb{P}^1$.

    │ │ │
    │ │ │ @@ -233,21 +233,21 @@ │ │ │ o27 : RingMap EP1 <-- S │ │ │ │ │ │ │ │ │ 
    i28 : R = S/(ker f);
    │ │ │ │ │ │ │ │ │ 
    i29 : time isFInjective(R)
│ │ │ - -- used 0.847953s (cpu); 0.66496s (thread); 0s (gc)
│ │ │ + -- used 1.43316s (cpu); 1.00401s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    │ │ │ │ │ │ │ │ │ 
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.352773s (cpu); 0.235218s (thread); 0s (gc)
│ │ │ + -- used 0.401099s (cpu); 0.30867s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    │ │ │ │ │ │ │ │ │
    │ │ │

    If the option AssumedReduced is set to true (its default behavior), then the bottom local cohomology is avoided (this means the Frobenius action on the top potentially nonzero Ext is not computed).

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -82,52 +82,52 @@ │ │ │ │ much faster. │ │ │ │ i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3) │ │ │ │ │ │ │ │ o19 = R │ │ │ │ │ │ │ │ o19 : QuotientRing │ │ │ │ i20 : time isFInjective(R) │ │ │ │ - -- used 0.0239992s (cpu); 0.0237627s (thread); 0s (gc) │ │ │ │ + -- used 0.0838948s (cpu); 0.0814579s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 = true │ │ │ │ i21 : time isFInjective(R, CanonicalStrategy => null) │ │ │ │ - -- used 2.15989s (cpu); 1.16923s (thread); 0s (gc) │ │ │ │ + -- used 2.93272s (cpu); 1.82411s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = true │ │ │ │ If the option AtOrigin (default value false) is set to true, isFInjective 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. │ │ │ │ i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5); │ │ │ │ i23 : time isFInjective(R) │ │ │ │ - -- used 0.155409s (cpu); 0.0979975s (thread); 0s (gc) │ │ │ │ + -- used 0.161113s (cpu); 0.107699s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = false │ │ │ │ i24 : time isFInjective(R, AtOrigin => true) │ │ │ │ - -- used 0.0700561s (cpu); 0.0715188s (thread); 0s (gc) │ │ │ │ + -- used 0.0969768s (cpu); 0.100488s (thread); 0s (gc) │ │ │ │ │ │ │ │ o24 = true │ │ │ │ If the option AssumeCM (default value false) is set to true, then isFInjective │ │ │ │ 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 $\mathbb{P}^1$. │ │ │ │ i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt]; │ │ │ │ i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2); │ │ │ │ i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t}); │ │ │ │ │ │ │ │ o27 : RingMap EP1 <-- S │ │ │ │ i28 : R = S/(ker f); │ │ │ │ i29 : time isFInjective(R) │ │ │ │ - -- used 0.847953s (cpu); 0.66496s (thread); 0s (gc) │ │ │ │ + -- used 1.43316s (cpu); 1.00401s (thread); 0s (gc) │ │ │ │ │ │ │ │ o29 = false │ │ │ │ i30 : time isFInjective(R, AssumeCM => true) │ │ │ │ - -- used 0.352773s (cpu); 0.235218s (thread); 0s (gc) │ │ │ │ + -- used 0.401099s (cpu); 0.30867s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = true │ │ │ │ If the option AssumedReduced is set to true (its default behavior), then the │ │ │ │ bottom local cohomology is avoided (this means the Frobenius action on the top │ │ │ │ potentially nonzero Ext is not computed). │ │ │ │ If the option AssumeNormal (default value false) is set to true, then the │ │ │ │ bottom two local cohomology modules (or, rather, their duals) need not be │ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ @@ -230,22 +230,22 @@ │ │ │ o25 : Ideal of S │ │ │ │ │ │ │ │ │ 
    i26 : debugLevel = 1;
    │ │ │ │ │ │ │ │ │ 
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ - -- used 0.120933s (cpu); 0.0586845s (thread); 0s (gc)
│ │ │ + -- used 0.0954204s (cpu); 0.0466078s (thread); 0s (gc)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │  
│ │ │  o27 = false
    │ │ │ │ │ │ │ │ │ 
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.220328s (cpu); 0.153633s (thread); 0s (gc)
│ │ │ + -- used 0.309374s (cpu); 0.265214s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    │ │ │ │ │ │ │ │ │ 
    i29 : debugLevel = 0;
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -113,21 +113,21 @@ │ │ │ │ also use the option DepthOfSearch to increase the depth of search. │ │ │ │ i24 : S = ZZ/7[x,y,z,u,v,w]; │ │ │ │ i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}}); │ │ │ │ │ │ │ │ o25 : Ideal of S │ │ │ │ i26 : debugLevel = 1; │ │ │ │ i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1) │ │ │ │ - -- used 0.120933s (cpu); 0.0586845s (thread); 0s (gc) │ │ │ │ + -- used 0.0954204s (cpu); 0.0466078s (thread); 0s (gc) │ │ │ │ isFRegular: This ring does not appear to be F-regular. Increasing │ │ │ │ DepthOfSearch will let the function search more deeply. │ │ │ │ │ │ │ │ o27 = false │ │ │ │ i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2) │ │ │ │ - -- used 0.220328s (cpu); 0.153633s (thread); 0s (gc) │ │ │ │ + -- used 0.309374s (cpu); 0.265214s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = true │ │ │ │ i29 : debugLevel = 0; │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring │ │ │ │ * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational │ │ │ │ ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: ********** │ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ @@ -219,23 +219,23 @@ │ │ │ │ │ │
    │ │ │

    It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.266662s (cpu); 0.138587s (thread); 0s (gc)
│ │ │ + -- used 0.420843s (cpu); 0.270252s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    │ │ │ 
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.39027s (cpu); 0.214669s (thread); 0s (gc)
│ │ │ + -- used 0.538261s (cpu); 0.402858s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -101,21 +101,21 @@ │ │ │ │ o22 = ideal (y, x) │ │ │ │ │ │ │ │ o22 : Ideal of R │ │ │ │ It is often more efficient to pass a list, as opposed to finding a common │ │ │ │ denominator and passing a single element, since testIdeal can do things in a │ │ │ │ more intelligent way for such a list. │ │ │ │ i23 : time testIdeal({3/4, 2/3, 3/5}, L) │ │ │ │ - -- used 0.266662s (cpu); 0.138587s (thread); 0s (gc) │ │ │ │ + -- used 0.420843s (cpu); 0.270252s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = ideal (y, x) │ │ │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36) │ │ │ │ - -- used 0.39027s (cpu); 0.214669s (thread); 0s (gc) │ │ │ │ + -- used 0.538261s (cpu); 0.402858s (thread); 0s (gc) │ │ │ │ │ │ │ │ o24 = ideal (y, x) │ │ │ │ │ │ │ │ o24 : Ideal of R │ │ │ │ The option AssumeDomain (default value false) is used when finding a test │ │ │ │ element. The option FrobeniusRootStrategy (default value Substitution) is │ │ │ │ passed to internal _f_r_o_b_e_n_i_u_s_R_o_o_t calls. │ │ ├── ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ bmV3IFRPSCBmcm9tIFRoaW5n │ │ │ #:len=219 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFRPSCxUaGluZyksIm5ldyBU │ │ ├── ./usr/share/doc/Macaulay2/ThinSincereQuivers/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ aXNUaWdodChUb3JpY1F1aXZlcik= │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA1OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNUaWdodCxUb3JpY1F1aXZlciksImlzVGlnaHQo │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ dGdiKExpc3Qp │ │ │ #:len=213 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0Z2IsTGlzdCksInRnYihMaXN0KSIsIlRocmVhZGVk │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ @@ -3,30 +3,33 @@ │ │ │ i1 : S = ZZ/101[a,b,c]; │ │ │ │ │ │ i2 : allowableThreads= 2; │ │ │ │ │ │ i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true) │ │ │ │ │ │ o3 = LineageTable{((0, 2), 0) => null} │ │ │ + ((0, 2), 1) => null │ │ │ 2 │ │ │ ((1, 2), 0) => c │ │ │ (0, 1) => null │ │ │ (0, 2) => null │ │ │ (1, 2) => a*c │ │ │ 0 => null │ │ │ 1 => null │ │ │ 2 │ │ │ 2 => b │ │ │ │ │ │ o3 : LineageTable │ │ │ │ │ │ i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2") │ │ │ │ │ │ - 3 │ │ │ -o4 = LineageTable{((0, 2), 0) => -c } │ │ │ + 3 │ │ │ +o4 = LineageTable{(((0, 1), 0), 0) => -c } │ │ │ + 2 │ │ │ + ((0, 1), 0) => -a*c │ │ │ 2 │ │ │ ((1, 2), 0) => -c │ │ │ 2 │ │ │ (0, 1) => a c │ │ │ 2 │ │ │ (0, 2) => b*c │ │ │ (1, 2) => -a*c │ │ │ @@ -37,15 +40,16 @@ │ │ │ 2 │ │ │ 2 => b │ │ │ │ │ │ o4 : LineageTable │ │ │ │ │ │ i5 : minimize T │ │ │ │ │ │ -o5 = LineageTable{((0, 2), 0) => null} │ │ │ +o5 = LineageTable{(((0, 1), 0), 0) => null} │ │ │ + ((0, 1), 0) => null │ │ │ 2 │ │ │ ((1, 2), 0) => c │ │ │ (0, 1) => null │ │ │ (0, 2) => null │ │ │ (1, 2) => a*c │ │ │ 0 => null │ │ │ 1 => null │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ @@ -22,46 +22,28 @@ │ │ │ │ │ │ i3 : allowableThreads = 4 │ │ │ │ │ │ o3 = 4 │ │ │ │ │ │ i4 : g = tgb(rnc) │ │ │ │ │ │ - 5 2 2 5 │ │ │ -o4 = LineageTable{(((((1, 2), 1), 1), 6), 6) => x x - x x } │ │ │ - 0 5 0 2 │ │ │ - 4 2 3 │ │ │ - ((((1, 2), 1), 1), 6) => x x x - x x x │ │ │ - 0 5 4 0 3 2 │ │ │ - 3 2 4 │ │ │ - (((1, 2), 1), 1) => x x - x x │ │ │ - 0 4 0 2 │ │ │ - 2 2 │ │ │ - ((1, 2), 1) => x x x - x x x │ │ │ - 0 5 2 0 3 2 │ │ │ - 2 3 │ │ │ - ((1, 2), 8) => x x x - x x │ │ │ - 0 5 2 3 2 │ │ │ + 2 │ │ │ +o4 = LineageTable{((2, 3), 6) => x x - x x x } │ │ │ + 0 5 0 3 2 │ │ │ 3 │ │ │ (1, 2) => - x x x + x │ │ │ 0 4 2 2 │ │ │ 2 │ │ │ (1, 4) => - x x x + x x │ │ │ 0 5 2 3 2 │ │ │ - 2 2 │ │ │ - (1, 7) => - x x + x x │ │ │ - 0 4 4 2 │ │ │ 2 │ │ │ (2, 3) => x x - x │ │ │ 0 4 2 │ │ │ (4, 6) => x x - x x │ │ │ 0 5 3 2 │ │ │ - 2 2 │ │ │ - (6, 7) => - x x + x x │ │ │ - 0 5 4 2 │ │ │ 2 3 │ │ │ (8, 9) => - x x + x │ │ │ 5 2 4 │ │ │ 2 │ │ │ 0 => - x + x x │ │ │ 1 0 2 │ │ │ 1 => - x x + x x │ │ │ @@ -112,28 +94,22 @@ │ │ │ 0 4 2 2 │ │ │ │ │ │ o7 : QQ[x , x , x , x , x , x ] │ │ │ 1 0 3 5 4 2 │ │ │ │ │ │ i8 : minimize g │ │ │ │ │ │ -o8 = LineageTable{(((((1, 2), 1), 1), 6), 6) => null} │ │ │ - ((((1, 2), 1), 1), 6) => null │ │ │ - (((1, 2), 1), 1) => null │ │ │ - ((1, 2), 1) => null │ │ │ - ((1, 2), 8) => null │ │ │ +o8 = LineageTable{((2, 3), 6) => null } │ │ │ (1, 2) => null │ │ │ (1, 4) => null │ │ │ - (1, 7) => null │ │ │ 2 │ │ │ (2, 3) => x x - x │ │ │ 0 4 2 │ │ │ (4, 6) => x x - x x │ │ │ 0 5 3 2 │ │ │ - (6, 7) => null │ │ │ 2 3 │ │ │ (8, 9) => x x - x │ │ │ 5 2 4 │ │ │ 2 │ │ │ 0 => x - x x │ │ │ 1 0 2 │ │ │ 1 => x x - x x │ │ │ @@ -155,28 +131,22 @@ │ │ │ 9 => x x - x │ │ │ 3 5 4 │ │ │ │ │ │ o8 : LineageTable │ │ │ │ │ │ i9 : gRed = reduce g │ │ │ │ │ │ -o9 = LineageTable{(((((1, 2), 1), 1), 6), 6) => null} │ │ │ - ((((1, 2), 1), 1), 6) => null │ │ │ - (((1, 2), 1), 1) => null │ │ │ - ((1, 2), 1) => null │ │ │ - ((1, 2), 8) => null │ │ │ +o9 = LineageTable{((2, 3), 6) => null } │ │ │ (1, 2) => null │ │ │ (1, 4) => null │ │ │ - (1, 7) => null │ │ │ 2 │ │ │ (2, 3) => x x - x │ │ │ 0 4 2 │ │ │ (4, 6) => x x - x x │ │ │ 0 5 3 2 │ │ │ - (6, 7) => null │ │ │ 2 3 │ │ │ (8, 9) => x x - x │ │ │ 5 2 4 │ │ │ 2 │ │ │ 0 => x - x x │ │ │ 1 0 2 │ │ │ 1 => x x - x x │ │ │ @@ -228,25 +198,30 @@ │ │ │ │ │ │ i13 : allowableThreads = 2; │ │ │ │ │ │ i14 : T = tgb(I,Verbose=>true) │ │ │ Scheduling a task for lineage (0,1) │ │ │ Scheduling a task for lineage (0,2) │ │ │ Scheduling a task for lineage (1,2) │ │ │ -Scheduling task for lineage ((0,1),0) │ │ │ -Scheduling task for lineage ((0,1),1) │ │ │ -Scheduling task for lineage ((0,1),2) │ │ │ -Adding the following remainder to GB: Adding the following remainder to GB: -1 from lineage (1,2) │ │ │ --a^3*b-a*b^2*d+c^2 from lineage (0,1) │ │ │ Scheduling task for lineage ((0,2),0) │ │ │ Scheduling task for lineage ((0,2),1) │ │ │ -Scheduling task for lineage ((0,2),(0,1)) │ │ │ Scheduling task for lineage ((0,2),2) │ │ │ -Scheduling task for lineage ((0,2),(1,2)) │ │ │ Adding the following remainder to GB: -a^3*b-a*b^2*d from lineage (0,2) │ │ │ +Scheduling task for lineage ((0,1),0) │ │ │ +Scheduling task for lineage ((0,1),1) │ │ │ +Scheduling task for lineage ((0,1),2) │ │ │ +Scheduling task for lineage ((0,1),(0,2)) │ │ │ +Adding the following remainder to GB: c^2 from lineage (0,1) │ │ │ +Adding the following remainder to GB: -1 from lineage (1,2) │ │ │ +/usr/share/Macaulay2/Core/methods.m2:170:23:(1):[3]: error: no method found for applying leadTerm to: │ │ │ + argument 1 : 1 (of class ZZ) │ │ │ + argument 2 : 2 (of class ZZ) │ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:308:46:(2):[2]: --back trace-- │ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:271:104:(2):[1]: --back trace-- │ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:270:10:(2): --back trace-- │ │ │ Found a unit in the Groebner basis; reducing now. │ │ │ │ │ │ o14 = LineageTable{(0, 1) => null} │ │ │ (0, 2) => null │ │ │ (1, 2) => 1 │ │ │ 0 => null │ │ │ 1 => null │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ @@ -2,17 +2,16 @@ │ │ │ │ │ │ i1 : R = ZZ/101[a,b,c]; │ │ │ │ │ │ i2 : allowableThreads= 2; │ │ │ │ │ │ i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2") │ │ │ │ │ │ -o3 = LineageTable{((0, 2), 0) => null} │ │ │ 2 │ │ │ - ((1, 2), 0) => c │ │ │ +o3 = LineageTable{((1, 2), 0) => c } │ │ │ (0, 1) => null │ │ │ (0, 2) => null │ │ │ (1, 2) => a*c │ │ │ 0 => null │ │ │ 1 => null │ │ │ 2 │ │ │ 2 => b │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize.out │ │ │ @@ -2,18 +2,18 @@ │ │ │ │ │ │ i1 : R = ZZ/101[a,b,c]; │ │ │ │ │ │ i2 : allowableThreads= 2; │ │ │ │ │ │ i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2") │ │ │ │ │ │ + 2 │ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c } │ │ │ 3 │ │ │ -o3 = LineageTable{((0, 2), 0) => -c } │ │ │ - 2 │ │ │ - ((0, 2), 1) => a*c │ │ │ + ((0, 2), 0) => -c │ │ │ 2 │ │ │ ((1, 2), 0) => -c │ │ │ 2 │ │ │ (0, 1) => a c │ │ │ 2 │ │ │ (0, 2) => b*c │ │ │ (1, 2) => -a*c │ │ │ @@ -24,16 +24,16 @@ │ │ │ 2 │ │ │ 2 => b │ │ │ │ │ │ o3 : LineageTable │ │ │ │ │ │ i4 : minimize T │ │ │ │ │ │ -o4 = LineageTable{((0, 2), 0) => null} │ │ │ - ((0, 2), 1) => null │ │ │ +o4 = LineageTable{((0, 1), 0) => null} │ │ │ + ((0, 2), 0) => null │ │ │ 2 │ │ │ ((1, 2), 0) => c │ │ │ (0, 1) => null │ │ │ (0, 2) => null │ │ │ (1, 2) => a*c │ │ │ 0 => null │ │ │ 1 => null │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ @@ -2,16 +2,18 @@ │ │ │ │ │ │ i1 : R = ZZ/101[a,b,c]; │ │ │ │ │ │ i2 : allowableThreads= 2; │ │ │ │ │ │ i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2" │ │ │ │ │ │ - 3 │ │ │ -o3 = LineageTable{((0, 2), 0) => -c } │ │ │ + 3 │ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => -c } │ │ │ + 2 │ │ │ + ((0, 1), 0) => -a*c │ │ │ 2 │ │ │ ((1, 2), 0) => -c │ │ │ 2 │ │ │ (0, 1) => a c │ │ │ 2 │ │ │ (0, 2) => b*c │ │ │ (1, 2) => -a*c │ │ │ @@ -22,15 +24,16 @@ │ │ │ 2 │ │ │ 2 => b │ │ │ │ │ │ o3 : LineageTable │ │ │ │ │ │ i4 : reduce T │ │ │ │ │ │ -o4 = LineageTable{((0, 2), 0) => null} │ │ │ +o4 = LineageTable{(((0, 1), 0), 0) => null} │ │ │ + ((0, 1), 0) => null │ │ │ 2 │ │ │ ((1, 2), 0) => c │ │ │ (0, 1) => null │ │ │ (0, 2) => null │ │ │ (1, 2) => a*c │ │ │ 0 => null │ │ │ 1 => null │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ @@ -6,44 +6,32 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : allowableThreads = 4; │ │ │ │ │ │ i4 : H = tgb I │ │ │ │ │ │ - 2 5 │ │ │ -o4 = LineageTable{(((0, 3), 2), (0, 2)) => -22y z } │ │ │ - 3 5 │ │ │ - (((0, 3), 2), (0, 3)) => 39y z │ │ │ - 3 8 │ │ │ - (((0, 3), 2), 1) => 8y z │ │ │ - 3 6 │ │ │ - (((0, 3), 2), 2) => 9y z │ │ │ + 2 9 │ │ │ +o4 = LineageTable{(((0, 1), 3), 2) => 9y z } │ │ │ 4 4 3 7 │ │ │ ((0, 1), 2) => 9y z - 6y z │ │ │ - 3 9 3 8 │ │ │ - ((0, 1), 3) => - 14y z - 38y z │ │ │ - 3 9 │ │ │ - ((0, 2), 1) => -47y z │ │ │ - 3 9 3 8 │ │ │ - ((0, 2), 3) => - 38y z - 6y z │ │ │ - 3 10 3 9 │ │ │ - ((0, 3), 1) => 43y z - 42y z │ │ │ - 4 4 3 6 │ │ │ - ((0, 3), 2) => 9y z - 27y z │ │ │ - 2 4 │ │ │ - ((2, 3), ((0, 1), 2)) => -47y z │ │ │ - 2 4 │ │ │ - ((2, 3), ((0, 3), 2)) => -47y z │ │ │ + 3 7 3 6 │ │ │ + ((0, 1), 3) => - 6y z + 27y z │ │ │ + 2 5 2 4 │ │ │ + ((0, 3), (0, 1)) => 36y z + 40y z │ │ │ + 2 4 │ │ │ + ((1, 2), (0, 3)) => -10y z │ │ │ + 2 8 │ │ │ + ((1, 2), 2) => 9y z │ │ │ 5 2 3 4 │ │ │ (0, 1) => - 25y z - 19y z │ │ │ - 5 3 2 4 │ │ │ - (0, 2) => 5y z + 9y z │ │ │ - 5 2 5 │ │ │ - (0, 3) => 5y z + 28y z │ │ │ + 5 3 4 │ │ │ + (0, 3) => 28y z - 24y z │ │ │ + 3 6 │ │ │ + (1, 2) => -19y z │ │ │ 3 4 2 4 │ │ │ (2, 3) => 7y z - 9y z │ │ │ 2 │ │ │ 0 => 2x + 10y z │ │ │ 2 3 │ │ │ 1 => 8x y + 10x*y*z │ │ │ 3 2 3 │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ @@ -65,14 +65,15 @@ │ │ │ │ │ │ 
    i2 : allowableThreads= 2;
    │ │ │ │ │ │ │ │ │ 
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │  o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ @@ -85,16 +86,18 @@
│ │ │          
    
│ │ │            
    By default, the option is false. The basis can also be minimized after the distributed computation is finished:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ -          
│ │ │ -
│ │ │ +          
│ │ │ +
│ │ │ -          
│ │ │ -
│ │ │ +          
│ │ │ +
│ │ │ -          
│ │ │ -
│ │ │ +          
│ │ │ +
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                   3
│ │ │ -o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │ +                                        3
│ │ │ +o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ +                                     2
│ │ │ +                  ((0, 1), 0) => -a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -106,15 +109,16 @@
│ │ │                    2 => b
│ │ │  
│ │ │  o4 : LineageTable
    
│ │ │      		          
                  
    i5 : minimize T
│ │ │  
│ │ │ -o5 = LineageTable{((0, 2), 0) => null}
│ │ │ +o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ +                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,14 +14,15 @@
│ │ │ │  that were added to the basis during the distributed computation are saved, with
│ │ │ │  the corresponding entry in the table being null.
│ │ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │ │  
│ │ │ │  o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +                  ((0, 2), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │ @@ -29,16 +30,18 @@
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  By default, the option is false. The basis can also be minimized after the
│ │ │ │  distributed computation is finished:
│ │ │ │  i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -                                   3
│ │ │ │ -o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │ +                                        3
│ │ │ │ +o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ +                                     2
│ │ │ │ +                  ((0, 1), 0) => -a*c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -48,15 +51,16 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o4 : LineageTable
│ │ │ │  i5 : minimize T
│ │ │ │  
│ │ │ │ -o5 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ +                  ((0, 1), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -84,17 +84,16 @@
│ │ │  
    		          
                  
    i2 : allowableThreads= 2;
    
│ │ │      		          
                  
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,16 @@
│ │ │ │  This simple function just returns the Gr\"obner basis computed with threaded
│ │ │ │  Gr\"obner basis function _t_g_b in the expected Macaulay2 format, so that further
│ │ │ │  computation are one step easier to set up.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │ -                  ((1, 2), 0) => c
│ │ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize.html
│ │ │ @@ -76,18 +76,18 @@
│ │ │  
    		          
                  
    i2 : allowableThreads= 2;
    
│ │ │      		          
                  
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ +                                     2
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ -                                    2
│ │ │ -                  ((0, 2), 1) => a*c
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -99,16 +99,16 @@
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
    
│ │ │      		          
                  
    i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,18 +19,18 @@
│ │ │ │  minimal generators of the ideal generated by the leading terms of the values of
│ │ │ │  H. If the values of H constitute a Gr\"obner basis of the ideal they generate,
│ │ │ │  this method returns a minimal Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ +                                     2
│ │ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │                                     3
│ │ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │ -                                    2
│ │ │ │ -                  ((0, 2), 1) => a*c
│ │ │ │ +                  ((0, 2), 0) => -c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -40,16 +40,16 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │ -                  ((0, 2), 1) => null
│ │ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │ +                  ((0, 2), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -77,16 +77,18 @@
│ │ │  
    		          
                  
    i2 : allowableThreads= 2;
    
│ │ │      		          
                  
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                   3
│ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ +                                        3
│ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ +                                     2
│ │ │ +                  ((0, 1), 0) => -a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -98,15 +100,16 @@
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
    
│ │ │      		          
                  
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ +o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ +                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,16 +21,18 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gr\"obner basis of the ideal they generate, this
│ │ │ │  method returns a reduced Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                   3
│ │ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │ +                                        3
│ │ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ +                                     2
│ │ │ │ +                  ((0, 1), 0) => -a*c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -40,15 +42,16 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ +                  ((0, 1), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -92,44 +92,32 @@
│ │ │  
    		          
                  
    i3 : allowableThreads  = 4;
    
│ │ │      		          
                  
    i4 : H = tgb I
│ │ │  
│ │ │ -                                               2 5
│ │ │ -o4 = LineageTable{(((0, 3), 2), (0, 2)) => -22y z }
│ │ │ -                                              3 5
│ │ │ -                  (((0, 3), 2), (0, 3)) => 39y z
│ │ │ -                                        3 8
│ │ │ -                  (((0, 3), 2), 1) => 8y z
│ │ │ -                                        3 6
│ │ │ -                  (((0, 3), 2), 2) => 9y z
│ │ │ +                                        2 9
│ │ │ +o4 = LineageTable{(((0, 1), 3), 2) => 9y z           }
│ │ │                                     4 4     3 7
│ │ │                    ((0, 1), 2) => 9y z  - 6y z
│ │ │ -                                      3 9      3 8
│ │ │ -                  ((0, 1), 3) => - 14y z  - 38y z
│ │ │ -                                     3 9
│ │ │ -                  ((0, 2), 1) => -47y z
│ │ │ -                                      3 9     3 8
│ │ │ -                  ((0, 2), 3) => - 38y z  - 6y z
│ │ │ -                                    3 10      3 9
│ │ │ -                  ((0, 3), 1) => 43y z   - 42y z
│ │ │ -                                   4 4      3 6
│ │ │ -                  ((0, 3), 2) => 9y z  - 27y z
│ │ │ -                                               2 4
│ │ │ -                  ((2, 3), ((0, 1), 2)) => -47y z
│ │ │ -                                               2 4
│ │ │ -                  ((2, 3), ((0, 3), 2)) => -47y z
│ │ │ +                                     3 7      3 6
│ │ │ +                  ((0, 1), 3) => - 6y z  + 27y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((0, 3), (0, 1)) => 36y z  + 40y z
│ │ │ +                                          2 4
│ │ │ +                  ((1, 2), (0, 3)) => -10y z
│ │ │ +                                   2 8
│ │ │ +                  ((1, 2), 2) => 9y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                              5 3     2 4
│ │ │ -                  (0, 2) => 5y z  + 9y z
│ │ │ -                              5 2      5
│ │ │ -                  (0, 3) => 5y z  + 28y z
│ │ │ +                               5       3 4
│ │ │ +                  (0, 3) => 28y z - 24y z
│ │ │ +                                3 6
│ │ │ +                  (1, 2) => -19y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,44 +27,32 @@
│ │ │ │  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,
│ │ │ │  9*x*y^3*z + 10*x*y^3};
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : allowableThreads  = 4;
│ │ │ │  i4 : H = tgb I
│ │ │ │  
│ │ │ │ -                                               2 5
│ │ │ │ -o4 = LineageTable{(((0, 3), 2), (0, 2)) => -22y z }
│ │ │ │ -                                              3 5
│ │ │ │ -                  (((0, 3), 2), (0, 3)) => 39y z
│ │ │ │ -                                        3 8
│ │ │ │ -                  (((0, 3), 2), 1) => 8y z
│ │ │ │ -                                        3 6
│ │ │ │ -                  (((0, 3), 2), 2) => 9y z
│ │ │ │ +                                        2 9
│ │ │ │ +o4 = LineageTable{(((0, 1), 3), 2) => 9y z           }
│ │ │ │                                     4 4     3 7
│ │ │ │                    ((0, 1), 2) => 9y z  - 6y z
│ │ │ │ -                                      3 9      3 8
│ │ │ │ -                  ((0, 1), 3) => - 14y z  - 38y z
│ │ │ │ -                                     3 9
│ │ │ │ -                  ((0, 2), 1) => -47y z
│ │ │ │ -                                      3 9     3 8
│ │ │ │ -                  ((0, 2), 3) => - 38y z  - 6y z
│ │ │ │ -                                    3 10      3 9
│ │ │ │ -                  ((0, 3), 1) => 43y z   - 42y z
│ │ │ │ -                                   4 4      3 6
│ │ │ │ -                  ((0, 3), 2) => 9y z  - 27y z
│ │ │ │ -                                               2 4
│ │ │ │ -                  ((2, 3), ((0, 1), 2)) => -47y z
│ │ │ │ -                                               2 4
│ │ │ │ -                  ((2, 3), ((0, 3), 2)) => -47y z
│ │ │ │ +                                     3 7      3 6
│ │ │ │ +                  ((0, 1), 3) => - 6y z  + 27y z
│ │ │ │ +                                         2 5      2 4
│ │ │ │ +                  ((0, 3), (0, 1)) => 36y z  + 40y z
│ │ │ │ +                                          2 4
│ │ │ │ +                  ((1, 2), (0, 3)) => -10y z
│ │ │ │ +                                   2 8
│ │ │ │ +                  ((1, 2), 2) => 9y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                              5 3     2 4
│ │ │ │ -                  (0, 2) => 5y z  + 9y z
│ │ │ │ -                              5 2      5
│ │ │ │ -                  (0, 3) => 5y z  + 28y z
│ │ │ │ +                               5       3 4
│ │ │ │ +                  (0, 3) => 28y z - 24y z
│ │ │ │ +                                3 6
│ │ │ │ +                  (1, 2) => -19y z
│ │ │ │                                3 4     2 4
│ │ │ │                    (2, 3) => 7y z  - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html
│ │ │ @@ -75,46 +75,28 @@
│ │ │  
    		              
    i3 : allowableThreads  =  4
│ │ │  
│ │ │  o3 = 4
    
│ │ │      		          
                  
    i4 : g = tgb(rnc)
│ │ │  
│ │ │ -                                                 5 2    2 5
│ │ │ -o4 = LineageTable{(((((1, 2), 1), 1), 6), 6) => x x  - x x }
│ │ │ -                                                 0 5    0 2
│ │ │ -                                            4        2   3
│ │ │ -                  ((((1, 2), 1), 1), 6) => x x x  - x x x
│ │ │ -                                            0 5 4    0 3 2
│ │ │ -                                       3 2      4
│ │ │ -                  (((1, 2), 1), 1) => x x  - x x
│ │ │ -                                       0 4    0 2
│ │ │ -                                  2            2
│ │ │ -                  ((1, 2), 1) => x x x  - x x x
│ │ │ -                                  0 5 2    0 3 2
│ │ │ -                                      2      3
│ │ │ -                  ((1, 2), 8) => x x x  - x x
│ │ │ -                                  0 5 2    3 2
│ │ │ +                                  2
│ │ │ +o4 = LineageTable{((2, 3), 6) => x x  - x x x }
│ │ │ +                                  0 5    0 3 2
│ │ │                                          3
│ │ │                    (1, 2) => - x x x  + x
│ │ │                                 0 4 2    2
│ │ │                                            2
│ │ │                    (1, 4) => - x x x  + x x
│ │ │                                 0 5 2    3 2
│ │ │ -                                 2      2
│ │ │ -                  (1, 7) => - x x  + x x
│ │ │ -                               0 4    4 2
│ │ │                                      2
│ │ │                    (2, 3) => x x  - x
│ │ │                               0 4    2
│ │ │                    (4, 6) => x x  - x x
│ │ │                               0 5    3 2
│ │ │ -                                 2    2
│ │ │ -                  (6, 7) => - x x  + x x
│ │ │ -                               0 5    4 2
│ │ │                                 2      3
│ │ │                    (8, 9) => - x x  + x
│ │ │                                 5 2    4
│ │ │                            2
│ │ │                    0 => - x  + x x
│ │ │                            1    0 2
│ │ │                    1 => - x x  + x x
│ │ │ @@ -187,28 +169,22 @@
│ │ │            
    As the algorithm continues, keys are concatenated, so that for example the remainder of S(0,S(1,2)) will have lineage (0,(1,2)), and so on.   For more complicated lineage examples, see tgb.
    
│ │ │            
    Naturally, one can obtain a minimal basis or the reduced one as follows. In the output below, elements that are reduced are replaced by null, but their lineage keys are retained for informative purposes.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i8 : minimize g
│ │ │  
│ │ │ -o8 = LineageTable{(((((1, 2), 1), 1), 6), 6) => null}
│ │ │ -                  ((((1, 2), 1), 1), 6) => null
│ │ │ -                  (((1, 2), 1), 1) => null
│ │ │ -                  ((1, 2), 1) => null
│ │ │ -                  ((1, 2), 8) => null
│ │ │ +o8 = LineageTable{((2, 3), 6) => null  }
│ │ │                    (1, 2) => null
│ │ │                    (1, 4) => null
│ │ │ -                  (1, 7) => null
│ │ │                                      2
│ │ │                    (2, 3) => x x  - x
│ │ │                               0 4    2
│ │ │                    (4, 6) => x x  - x x
│ │ │                               0 5    3 2
│ │ │ -                  (6, 7) => null
│ │ │                               2      3
│ │ │                    (8, 9) => x x  - x
│ │ │                               5 2    4
│ │ │                          2
│ │ │                    0 => x  - x x
│ │ │                          1    0 2
│ │ │                    1 => x x  - x x
│ │ │ @@ -231,28 +207,22 @@
│ │ │                          3 5    4
│ │ │  
│ │ │  o8 : LineageTable
    
│ │ │      		          
                  
    i9 : gRed = reduce g
│ │ │  
│ │ │ -o9 = LineageTable{(((((1, 2), 1), 1), 6), 6) => null}
│ │ │ -                  ((((1, 2), 1), 1), 6) => null
│ │ │ -                  (((1, 2), 1), 1) => null
│ │ │ -                  ((1, 2), 1) => null
│ │ │ -                  ((1, 2), 8) => null
│ │ │ +o9 = LineageTable{((2, 3), 6) => null  }
│ │ │                    (1, 2) => null
│ │ │                    (1, 4) => null
│ │ │ -                  (1, 7) => null
│ │ │                                      2
│ │ │                    (2, 3) => x x  - x
│ │ │                               0 4    2
│ │ │                    (4, 6) => x x  - x x
│ │ │                               0 5    3 2
│ │ │ -                  (6, 7) => null
│ │ │                               2      3
│ │ │                    (8, 9) => x x  - x
│ │ │                               5 2    4
│ │ │                          2
│ │ │                    0 => x  - x x
│ │ │                          1    0 2
│ │ │                    1 => x x  - x x
│ │ │ @@ -320,25 +290,30 @@
│ │ │  
    		              
    i13 : allowableThreads =  2;
    
│ │ │      		          
                  
    i14 : T = tgb(I,Verbose=>true)
│ │ │  Scheduling a task for lineage (0,1)
│ │ │  Scheduling a task for lineage (0,2)
│ │ │  Scheduling a task for lineage (1,2)
│ │ │ -Scheduling task for lineage ((0,1),0)
│ │ │ -Scheduling task for lineage ((0,1),1)
│ │ │ -Scheduling task for lineage ((0,1),2)
│ │ │ -Adding the following remainder to GB: Adding the following remainder to GB: -1 from lineage (1,2)
│ │ │ --a^3*b-a*b^2*d+c^2 from lineage (0,1)
│ │ │  Scheduling task for lineage ((0,2),0)
│ │ │  Scheduling task for lineage ((0,2),1)
│ │ │ -Scheduling task for lineage ((0,2),(0,1))
│ │ │  Scheduling task for lineage ((0,2),2)
│ │ │ -Scheduling task for lineage ((0,2),(1,2))
│ │ │  Adding the following remainder to GB: -a^3*b-a*b^2*d from lineage (0,2)
│ │ │ +Scheduling task for lineage ((0,1),0)
│ │ │ +Scheduling task for lineage ((0,1),1)
│ │ │ +Scheduling task for lineage ((0,1),2)
│ │ │ +Scheduling task for lineage ((0,1),(0,2))
│ │ │ +Adding the following remainder to GB: c^2 from lineage (0,1)
│ │ │ +Adding the following remainder to GB: -1 from lineage (1,2)
│ │ │ +/usr/share/Macaulay2/Core/methods.m2:170:23:(1):[3]: error: no method found for applying leadTerm to:
│ │ │ +     argument 1 :  1 (of class ZZ)
│ │ │ +     argument 2 :  2 (of class ZZ)
│ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:308:46:(2):[2]: --back trace--
│ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:271:104:(2):[1]: --back trace--
│ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:270:10:(2): --back trace--
│ │ │  Found a unit in the Groebner basis; reducing now.
│ │ │  
│ │ │  o14 = LineageTable{(0, 1) => null}
│ │ │                     (0, 2) => null
│ │ │                     (1, 2) => 1
│ │ │                     0 => null
│ │ │                     1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,46 +39,28 @@
│ │ │ │  o2 : Ideal of QQ[x , x , x , x , x , x ]
│ │ │ │                    1   0   3   5   4   2
│ │ │ │  i3 : allowableThreads  =  4
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  i4 : g = tgb(rnc)
│ │ │ │  
│ │ │ │ -                                                 5 2    2 5
│ │ │ │ -o4 = LineageTable{(((((1, 2), 1), 1), 6), 6) => x x  - x x }
│ │ │ │ -                                                 0 5    0 2
│ │ │ │ -                                            4        2   3
│ │ │ │ -                  ((((1, 2), 1), 1), 6) => x x x  - x x x
│ │ │ │ -                                            0 5 4    0 3 2
│ │ │ │ -                                       3 2      4
│ │ │ │ -                  (((1, 2), 1), 1) => x x  - x x
│ │ │ │ -                                       0 4    0 2
│ │ │ │ -                                  2            2
│ │ │ │ -                  ((1, 2), 1) => x x x  - x x x
│ │ │ │ -                                  0 5 2    0 3 2
│ │ │ │ -                                      2      3
│ │ │ │ -                  ((1, 2), 8) => x x x  - x x
│ │ │ │ -                                  0 5 2    3 2
│ │ │ │ +                                  2
│ │ │ │ +o4 = LineageTable{((2, 3), 6) => x x  - x x x }
│ │ │ │ +                                  0 5    0 3 2
│ │ │ │                                          3
│ │ │ │                    (1, 2) => - x x x  + x
│ │ │ │                                 0 4 2    2
│ │ │ │                                            2
│ │ │ │                    (1, 4) => - x x x  + x x
│ │ │ │                                 0 5 2    3 2
│ │ │ │ -                                 2      2
│ │ │ │ -                  (1, 7) => - x x  + x x
│ │ │ │ -                               0 4    4 2
│ │ │ │                                      2
│ │ │ │                    (2, 3) => x x  - x
│ │ │ │                               0 4    2
│ │ │ │                    (4, 6) => x x  - x x
│ │ │ │                               0 5    3 2
│ │ │ │ -                                 2    2
│ │ │ │ -                  (6, 7) => - x x  + x x
│ │ │ │ -                               0 5    4 2
│ │ │ │                                 2      3
│ │ │ │                    (8, 9) => - x x  + x
│ │ │ │                                 5 2    4
│ │ │ │                            2
│ │ │ │                    0 => - x  + x x
│ │ │ │                            1    0 2
│ │ │ │                    1 => - x x  + x x
│ │ │ │ @@ -140,28 +122,22 @@
│ │ │ │  remainder of S(0,S(1,2)) will have lineage (0,(1,2)), and so on. For more
│ │ │ │  complicated lineage examples, see _t_g_b.
│ │ │ │  Naturally, one can obtain a minimal basis or the reduced one as follows. In the
│ │ │ │  output below, elements that are reduced are replaced by null, but their lineage
│ │ │ │  keys are retained for informative purposes.
│ │ │ │  i8 : minimize g
│ │ │ │  
│ │ │ │ -o8 = LineageTable{(((((1, 2), 1), 1), 6), 6) => null}
│ │ │ │ -                  ((((1, 2), 1), 1), 6) => null
│ │ │ │ -                  (((1, 2), 1), 1) => null
│ │ │ │ -                  ((1, 2), 1) => null
│ │ │ │ -                  ((1, 2), 8) => null
│ │ │ │ +o8 = LineageTable{((2, 3), 6) => null  }
│ │ │ │                    (1, 2) => null
│ │ │ │                    (1, 4) => null
│ │ │ │ -                  (1, 7) => null
│ │ │ │                                      2
│ │ │ │                    (2, 3) => x x  - x
│ │ │ │                               0 4    2
│ │ │ │                    (4, 6) => x x  - x x
│ │ │ │                               0 5    3 2
│ │ │ │ -                  (6, 7) => null
│ │ │ │                               2      3
│ │ │ │                    (8, 9) => x x  - x
│ │ │ │                               5 2    4
│ │ │ │                          2
│ │ │ │                    0 => x  - x x
│ │ │ │                          1    0 2
│ │ │ │                    1 => x x  - x x
│ │ │ │ @@ -182,28 +158,22 @@
│ │ │ │                                 2
│ │ │ │                    9 => x x  - x
│ │ │ │                          3 5    4
│ │ │ │  
│ │ │ │  o8 : LineageTable
│ │ │ │  i9 : gRed = reduce g
│ │ │ │  
│ │ │ │ -o9 = LineageTable{(((((1, 2), 1), 1), 6), 6) => null}
│ │ │ │ -                  ((((1, 2), 1), 1), 6) => null
│ │ │ │ -                  (((1, 2), 1), 1) => null
│ │ │ │ -                  ((1, 2), 1) => null
│ │ │ │ -                  ((1, 2), 8) => null
│ │ │ │ +o9 = LineageTable{((2, 3), 6) => null  }
│ │ │ │                    (1, 2) => null
│ │ │ │                    (1, 4) => null
│ │ │ │ -                  (1, 7) => null
│ │ │ │                                      2
│ │ │ │                    (2, 3) => x x  - x
│ │ │ │                               0 4    2
│ │ │ │                    (4, 6) => x x  - x x
│ │ │ │                               0 5    3 2
│ │ │ │ -                  (6, 7) => null
│ │ │ │                               2      3
│ │ │ │                    (8, 9) => x x  - x
│ │ │ │                               5 2    4
│ │ │ │                          2
│ │ │ │                    0 => x  - x x
│ │ │ │                          1    0 2
│ │ │ │                    1 => x x  - x x
│ │ │ │ @@ -259,26 +229,31 @@
│ │ │ │  
│ │ │ │  o12 : Ideal of QQ[a..d]
│ │ │ │  i13 : allowableThreads =  2;
│ │ │ │  i14 : T = tgb(I,Verbose=>true)
│ │ │ │  Scheduling a task for lineage (0,1)
│ │ │ │  Scheduling a task for lineage (0,2)
│ │ │ │  Scheduling a task for lineage (1,2)
│ │ │ │ -Scheduling task for lineage ((0,1),0)
│ │ │ │ -Scheduling task for lineage ((0,1),1)
│ │ │ │ -Scheduling task for lineage ((0,1),2)
│ │ │ │ -Adding the following remainder to GB: Adding the following remainder to GB: -
│ │ │ │ -1 from lineage (1,2)
│ │ │ │ --a^3*b-a*b^2*d+c^2 from lineage (0,1)
│ │ │ │  Scheduling task for lineage ((0,2),0)
│ │ │ │  Scheduling task for lineage ((0,2),1)
│ │ │ │ -Scheduling task for lineage ((0,2),(0,1))
│ │ │ │  Scheduling task for lineage ((0,2),2)
│ │ │ │ -Scheduling task for lineage ((0,2),(1,2))
│ │ │ │  Adding the following remainder to GB: -a^3*b-a*b^2*d from lineage (0,2)
│ │ │ │ +Scheduling task for lineage ((0,1),0)
│ │ │ │ +Scheduling task for lineage ((0,1),1)
│ │ │ │ +Scheduling task for lineage ((0,1),2)
│ │ │ │ +Scheduling task for lineage ((0,1),(0,2))
│ │ │ │ +Adding the following remainder to GB: c^2 from lineage (0,1)
│ │ │ │ +Adding the following remainder to GB: -1 from lineage (1,2)
│ │ │ │ +/usr/share/Macaulay2/Core/methods.m2:170:23:(1):[3]: error: no method found for
│ │ │ │ +applying leadTerm to:
│ │ │ │ +     argument 1 :  1 (of class ZZ)
│ │ │ │ +     argument 2 :  2 (of class ZZ)
│ │ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:308:46:(2):[2]: --back trace--
│ │ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:271:104:(2):[1]: --back trace--
│ │ │ │ +/usr/share/Macaulay2/ThreadedGB.m2:270:10:(2): --back trace--
│ │ │ │  Found a unit in the Groebner basis; reducing now.
│ │ │ │  
│ │ │ │  o14 = LineageTable{(0, 1) => null}
│ │ │ │                     (0, 2) => null
│ │ │ │                     (1, 2) => 1
│ │ │ │                     0 => null
│ │ │ │                     1 => null
│ │ ├── ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y2hpcm90b3BlU3RyaW5n
│ │ │  #:len=253
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJjaGlyb3RvcGVTdHJpbmciLCJjaGlyb3RvcGVTdHJp
│ │ ├── ./usr/share/doc/Macaulay2/TorAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  aXNHb3JlbnN0ZWlu
│ │ │  #:len=1402
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAid2hldGhlciB0aGUgcmluZyBpcyBHb3Jl
│ │ │  bnN0ZWluIiwgImxpbmVudW0iID0+IDEyNDMsIElucHV0cyA9PiB7U1BBTntUVHsiUiJ9LCIsICIs
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  ZWREZWc=
│ │ │  #:len=2501
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIChnZW5lcmljKSBF
│ │ │  dWNsaWRlYW4gZGlzdGFuY2UgZGVncmVlIG9mIGEgcHJvamVjdGl2ZSB0b3JpYyB2YXJpZXR5Iiwg
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │       | 0 1 2 0 2 0 |
│ │ │       | 1 1 1 1 1 1 |
│ │ │  
│ │ │                4       6
│ │ │  o3 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i4 : time edDeg(A)
│ │ │ - -- used 1.20987s (cpu); 0.750962s (thread); 0s (gc)
│ │ │ + -- used 1.3227s (cpu); 0.932002s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │ @@ -47,15 +47,15 @@
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ - -- used 4.36073s (cpu); 2.65103s (thread); 0s (gc)
│ │ │ + -- used 5.16812s (cpu); 3.43873s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │       | 1 1 1 1 1 1 |
│ │ │  
│ │ │                4       6
│ │ │  o3 : Matrix ZZ  <-- ZZ
    
│ │ │      		          
                  
    i4 : time edDeg(A)
│ │ │ - -- used 1.20987s (cpu); 0.750962s (thread); 0s (gc)
│ │ │ + -- used 1.3227s (cpu); 0.932002s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    
│ │ │      		          
                  
    i5 : time edDeg(A,ForceAmat=>true)
│ │ │ - -- used 4.36073s (cpu); 2.65103s (thread); 0s (gc)
│ │ │ + -- used 5.16812s (cpu); 3.43873s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,30 +58,30 @@
│ │ │ │       | 3 5 0 2 1 3 |
│ │ │ │       | 0 1 2 0 2 0 |
│ │ │ │       | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │                4       6
│ │ │ │  o3 : Matrix ZZ  <-- ZZ
│ │ │ │  i4 : time edDeg(A)
│ │ │ │ - -- used 1.20987s (cpu); 0.750962s (thread); 0s (gc)
│ │ │ │ + -- used 1.3227s (cpu); 0.932002s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │ - -- used 4.36073s (cpu); 2.65103s (thread); 0s (gc)
│ │ │ │ + -- used 5.16812s (cpu); 3.43873s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ ├── ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  aGVzc2VuYmVyZ1ZhcmlldHk=
│ │ │  #:len=736
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiSGVzc2VuYmVyZyB2YXJpZXR5IGFzc2Nv
│ │ │  aWF0ZWQgdG8gdGhlIG4tcGVybXV0YWhlZHJvbiIsICJsaW5lbnVtIiA9PiA2MzksIElucHV0cyA9
│ │ ├── ./usr/share/doc/Macaulay2/ToricVectorBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  ZHVhbChUb3JpY1ZlY3RvckJ1bmRsZSk=
│ │ │  #:len=1504
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiIHRoZSBkdWFsIGJ1bmRsZSBvZiBhIHRv
│ │ │  cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjgyMywgSW5wdXRzID0+IHtTUEFOe1RU
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Y2hlY2tJbnRlcmZhY2U=
│ │ │  #:len=792
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAid2hldGhlciB0aGUgTWFwbGUgaW50ZXJm
│ │ │  YWNlIGlzIHdvcmtpbmcgKGZvciBkZXZlbG9wZXJzKSIsICJsaW5lbnVtIiA9PiAzMzUsIElucHV0
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/example-output/___Triangular__Sets.out
│ │ │ @@ -4,16 +4,16 @@
│ │ │  
│ │ │  i2 : I = ideal {a*d - b*c, c*f - d*e, e*h - f*g};
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : triangularize I
│ │ │  
│ │ │ -o3 = {{c, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f - d*e, g, h} / {d,
│ │ │ +o3 = {{c, d, f, h}, {b, d, e, f}, {c, d, e, f}, {a*d - b*c, e, f} / d, {a*d -
│ │ │       ------------------------------------------------------------------------
│ │ │ -     f}, {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h}, {b, d, f, h}, {c, d,
│ │ │ +     b*c, c*f - d*e, g, h} / {d, f}, {a*d - b*c, c*f - d*e, e*h - f*g} / {d,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     e*h - f*g} / h, {c, d, g, h}, {c, d, f, h}, {b, d, e, f}}
│ │ │ +     f, h}, {b, d, f, h}, {c, d, e*h - f*g} / h, {c, d, g, h}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/html/index.html
│ │ │ @@ -52,19 +52,19 @@
│ │ │  
    		              
    i2 : I = ideal {a*d - b*c, c*f - d*e, e*h - f*g};
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │      		          
                  
    i3 : triangularize I
│ │ │  
│ │ │ -o3 = {{c, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f - d*e, g, h} / {d,
│ │ │ +o3 = {{c, d, f, h}, {b, d, e, f}, {c, d, e, f}, {a*d - b*c, e, f} / d, {a*d -
│ │ │       ------------------------------------------------------------------------
│ │ │ -     f}, {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h}, {b, d, f, h}, {c, d,
│ │ │ +     b*c, c*f - d*e, g, h} / {d, f}, {a*d - b*c, c*f - d*e, e*h - f*g} / {d,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     e*h - f*g} / h, {c, d, g, h}, {c, d, f, h}, {b, d, e, f}}
│ │ │ +     f, h}, {b, d, f, h}, {c, d, e*h - f*g} / h, {c, d, g, h}}
│ │ │  
│ │ │  o3 : List
    
│ │ │      		          
    
│ │ │  
    The method triangularize is implemented in M2 only for monomial and binomial ideals. For the general case we interface to Maple.

    This package also provides methods for manipulating triangular sets:         
      
│ │ │            
    • 
│ │ │  dim(TriaSystem) -- dimension of a triangular set          
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,19 +8,19 @@
│ │ │ │  This package allows to decompose polynomial ideals into _t_r_i_a_n_g_u_l_a_r_ _s_e_t_s
│ │ │ │  i1 : R = QQ[a..h, MonomialOrder=>Lex];
│ │ │ │  i2 : I = ideal {a*d - b*c, c*f - d*e, e*h - f*g};
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : triangularize I
│ │ │ │  
│ │ │ │ -o3 = {{c, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f - d*e, g, h} / {d,
│ │ │ │ +o3 = {{c, d, f, h}, {b, d, e, f}, {c, d, e, f}, {a*d - b*c, e, f} / d, {a*d -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     f}, {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h}, {b, d, f, h}, {c, d,
│ │ │ │ +     b*c, c*f - d*e, g, h} / {d, f}, {a*d - b*c, c*f - d*e, e*h - f*g} / {d,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     e*h - f*g} / h, {c, d, g, h}, {c, d, f, h}, {b, d, e, f}}
│ │ │ │ +     f, h}, {b, d, f, h}, {c, d, e*h - f*g} / h, {c, d, g, h}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  
│ │ │ │  The method _t_r_i_a_n_g_u_l_a_r_i_z_e is implemented in M2 only for monomial and binomial
│ │ │ │  ideals. For the general case we interface to Maple.
│ │ │ │  
│ │ │ │  This package also provides methods for manipulating triangular sets:
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  YWxsVHJpYW5ndWxhdGlvbnM=
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJhbGxUcmlhbmd1bGF0aW9ucyIsImFsbFRyaWFuZ3Vs
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │  
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .108538s elapsed
│ │ │ + -- .156496s elapsed
│ │ │  
│ │ │  i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -50,14 +50,14 @@
│ │ │  i7 : T = regularFineTriangulation A
│ │ │  
│ │ │  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}}
│ │ │  
│ │ │  o7 : Triangulation
│ │ │  
│ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.20095s elapsed
│ │ │ + -- 1.09493s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out
│ │ │ @@ -21,57 +21,15 @@
│ │ │  
│ │ │  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}}
│ │ │  
│ │ │  o3 : Triangulation
│ │ │  
│ │ │  i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -227,57 +185,63 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
│ │ │ -
│ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 5, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
│ │ │ +
│ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -409,63 +373,57 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
│ │ │ -
│ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
│ │ │ +
│ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -621,63 +579,63 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
│ │ │ -
│ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 5, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
│ │ │ +
│ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -833,15 +791,57 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}}
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 5, 6, 7}}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : all(Ts4, isFine)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │ @@ -858,133 +858,133 @@
│ │ │  o11 = Tally{false => 66}
│ │ │              true => 8
│ │ │  
│ │ │  o11 : Tally
│ │ │  
│ │ │  i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ -o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ -         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ -      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ -         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ -      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ -       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ -      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ -      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ -      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ -         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ -      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ -      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ -      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ -         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ -      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ -              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ -      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ -               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ -      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ -           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ -      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ -       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ -      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ -       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ -      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ -      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ -      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ -      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ -       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ -      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ -          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ -      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ -         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ -      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ -         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ -      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ -       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ -      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ -      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ -      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ -           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ -      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ -      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ -      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ -        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ -      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ -      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ -      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ -      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4  20              20  4    4        20  20        4    4 
│ │ │ -      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ -            3    3   3               3  3    3         3   3        3    3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ -      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ -            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ -      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ -      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ -      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ -      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  20
│ │ │ -      -, --}}
│ │ │ -      3   3
│ │ │ +        20        4  8  8  8          8  8  8  4        20    16  16     4 
│ │ │ +o12 = {{--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -,
│ │ │ +         3        3  3  3  3          3  3  3  3         3     3   3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8          8  4     8  8     20       20     4  4     20     
│ │ │ +      4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4},
│ │ │ +         3  3          3  3     3  3      3        3     3  3      3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       20  4              4  20       4  20        20  4       8     8  20 
│ │ │ +      {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --,
│ │ │ +        3  3              3   3       3   3         3  3       3     3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8          16  4  16  8        4    4     16  16     8  4     
│ │ │ +      8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4},
│ │ │ +         3  3           3  3   3  3        3    3      3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          4  8     16  16     4    4     20        20     4    8  8  8       
│ │ │ +      {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8,
│ │ │ +          3  3      3   3     3    3      3         3     3    3  3  3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8  8    8  20  8     8        4       16  8     4  16     4    4    
│ │ │ +      -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8,
│ │ │ +      3  3  3    3   3  3     3        3        3  3     3   3     3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  16  4  16       16  4     16  4     16  4    4     16  4  16  4 
│ │ │ +      4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -,
│ │ │ +         3   3  3   3        3  3      3  3      3  3    3      3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         16       8  8  8  8  8  8       20  8     8     8  4          4  8 
│ │ │ +      4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -,
│ │ │ +          3       3  3  3  3  3  3        3  3     3     3  3          3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16     16    20     4        4     20       4  4  16  4  16  16 
│ │ │ +      4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --,
│ │ │ +         3   3      3     3     3        3      3       3  3   3  3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  16     4     16  4    8  8     20     8  4          4  8    
│ │ │ +      4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +              3   3     3      3  3    3  3      3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20     8  8       8  4     16     16  4    8  8  20     8        4  
│ │ │ +      --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -},
│ │ │ +       3     3  3       3  3      3      3  3    3  3   3     3        3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       4        8     20  8  8    8     8  8  8  8     8    4  16  16       
│ │ │ +      {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8,
│ │ │ +       3        3      3  3  3    3     3  3  3  3     3    3   3   3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  4  16       20  4     8  8     8    8        4  8  20  8       16 
│ │ │ +      -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--,
│ │ │ +      3  3   3        3  3     3  3     3    3        3  3   3  3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         16  4     8  4       20     8  8     4  8          4  8     8     8 
│ │ │ +      4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -,
│ │ │ +          3  3     3  3        3     3  3     3  3          3  3     3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20       16  16  4  16  4  4       8        4     16  4  16         
│ │ │ +      --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4,
│ │ │ +       3        3   3  3   3  3  3       3        3      3  3   3         
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ +      --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ +       3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ +      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ +       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ +      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ +       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ +      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ +      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  8        4    4  20              20  4    4        20  20       
│ │ │ +      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ +       3  3        3    3   3               3  3    3         3   3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ +      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ +      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ +      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ +       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ +      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ +           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ +      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ +       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ +      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ +       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ +      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ +          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ +      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ +      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ +      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ +          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ +      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ +      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4        8
│ │ │ +      -, -, 4, 8, -}}
│ │ │ +      3  3        3
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : volume convexHull A -- 8
│ │ │  
│ │ │  o13 = 8
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html
│ │ │ @@ -116,57 +116,15 @@
│ │ │  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}}
│ │ │  
│ │ │  o3 : Triangulation
    
│ │ │      		          
                  
    i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -322,58 +280,64 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
    
│ │ │ -    		          
                  
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 5, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
    
│ │ │ +    		          
                  
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -505,64 +469,58 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
    
│ │ │ -    		          
                  
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
    
│ │ │ +    		          
                  
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -718,64 +676,64 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
    
│ │ │ -    		          
                  
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 5, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
    
│ │ │ +    		          
                  
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -931,15 +889,57 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}}
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 5, 6, 7}}}
│ │ │  
│ │ │  o7 : List
    
│ │ │      		          
                  
    i8 : all(Ts4, isFine)
│ │ │  
│ │ │  o8 = true
    
│ │ │ @@ -961,133 +961,133 @@
│ │ │              true => 8
│ │ │  
│ │ │  o11 : Tally
│ │ │      		          
                  
    i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ -o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ -         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ -      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ -         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ -      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ -       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ -      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ -      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ -      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ -         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ -      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ -      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ -      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ -         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ -      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ -              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ -      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ -               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ -      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ -           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ -      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ -       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ -      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ -       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ -      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ -      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ -      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ -      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ -       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ -      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ -          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ -      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ -         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ -      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ -         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ -      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ -       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ -      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ -      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ -      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ -           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ -      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ -      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ -      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ -        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ -      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ -      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ -      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ -      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4  20              20  4    4        20  20        4    4 
│ │ │ -      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ -            3    3   3               3  3    3         3   3        3    3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ -      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ -            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ -      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ -      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ -      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ -      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  20
│ │ │ -      -, --}}
│ │ │ -      3   3
│ │ │ +        20        4  8  8  8          8  8  8  4        20    16  16     4 
│ │ │ +o12 = {{--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -,
│ │ │ +         3        3  3  3  3          3  3  3  3         3     3   3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8          8  4     8  8     20       20     4  4     20     
│ │ │ +      4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4},
│ │ │ +         3  3          3  3     3  3      3        3     3  3      3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       20  4              4  20       4  20        20  4       8     8  20 
│ │ │ +      {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --,
│ │ │ +        3  3              3   3       3   3         3  3       3     3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8          16  4  16  8        4    4     16  16     8  4     
│ │ │ +      8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4},
│ │ │ +         3  3           3  3   3  3        3    3      3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          4  8     16  16     4    4     20        20     4    8  8  8       
│ │ │ +      {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8,
│ │ │ +          3  3      3   3     3    3      3         3     3    3  3  3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8  8    8  20  8     8        4       16  8     4  16     4    4    
│ │ │ +      -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8,
│ │ │ +      3  3  3    3   3  3     3        3        3  3     3   3     3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  16  4  16       16  4     16  4     16  4    4     16  4  16  4 
│ │ │ +      4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -,
│ │ │ +         3   3  3   3        3  3      3  3      3  3    3      3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         16       8  8  8  8  8  8       20  8     8     8  4          4  8 
│ │ │ +      4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -,
│ │ │ +          3       3  3  3  3  3  3        3  3     3     3  3          3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16     16    20     4        4     20       4  4  16  4  16  16 
│ │ │ +      4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --,
│ │ │ +         3   3      3     3     3        3      3       3  3   3  3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  16     4     16  4    8  8     20     8  4          4  8    
│ │ │ +      4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +              3   3     3      3  3    3  3      3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20     8  8       8  4     16     16  4    8  8  20     8        4  
│ │ │ +      --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -},
│ │ │ +       3     3  3       3  3      3      3  3    3  3   3     3        3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       4        8     20  8  8    8     8  8  8  8     8    4  16  16       
│ │ │ +      {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8,
│ │ │ +       3        3      3  3  3    3     3  3  3  3     3    3   3   3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  4  16       20  4     8  8     8    8        4  8  20  8       16 
│ │ │ +      -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--,
│ │ │ +      3  3   3        3  3     3  3     3    3        3  3   3  3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         16  4     8  4       20     8  8     4  8          4  8     8     8 
│ │ │ +      4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -,
│ │ │ +          3  3     3  3        3     3  3     3  3          3  3     3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20       16  16  4  16  4  4       8        4     16  4  16         
│ │ │ +      --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4,
│ │ │ +       3        3   3  3   3  3  3       3        3      3  3   3         
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ +      --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ +       3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ +      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ +       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ +      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ +       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ +      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ +      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  8        4    4  20              20  4    4        20  20       
│ │ │ +      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ +       3  3        3    3   3               3  3    3         3   3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ +      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ +      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ +      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ +       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ +      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ +           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ +      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ +       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ +      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ +       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ +      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ +          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ +      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ +      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ +      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ +          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ +      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ +      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4        8
│ │ │ +      -, -, 4, 8, -}}
│ │ │ +      3  3        3
│ │ │  
│ │ │  o12 : List
    
│ │ │      		          
                  
    i13 : volume convexHull A -- 8
│ │ │  
│ │ │  o13 = 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,57 +58,15 @@
│ │ │ │  
│ │ │ │  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}}
│ │ │ │  
│ │ │ │  o3 : Triangulation
│ │ │ │  i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │ │  
│ │ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -264,56 +222,62 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o4 : List
│ │ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ │ -
│ │ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 5, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o4 : List
│ │ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ │ +
│ │ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -445,62 +409,56 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o5 : List
│ │ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o5 : List
│ │ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ │ +
│ │ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -656,62 +614,62 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o6 : List
│ │ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6},
│ │ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │ +     {1, 5, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o6 : List
│ │ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ +
│ │ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -867,15 +825,57 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}}
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 2, 6}, {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 5, 6, 7}}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : all(Ts4, isFine)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : all(Ts4, isStar)
│ │ │ │  
│ │ │ │ @@ -887,133 +887,133 @@
│ │ │ │  
│ │ │ │  o11 = Tally{false => 66}
│ │ │ │              true => 8
│ │ │ │  
│ │ │ │  o11 : Tally
│ │ │ │  i12 : Ts4/gkzVector
│ │ │ │  
│ │ │ │ -        20        4  4        20       8  4     4     16  16    8        4
│ │ │ │ -o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ │ -         3        3  3         3       3  3     3      3   3    3        3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         4  16  16    4  16     16     4  8       4        20     8  8  8
│ │ │ │ -      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ │ -         3   3   3    3   3      3     3  3       3         3     3  3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -       8  8  8     20        4    4        8     8  20  8    4     16  4
│ │ │ │ -      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ │ -       3  3  3      3        3    3        3     3   3  3    3      3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8  16          4  20     8     8  8    16  4  4     16        8    16
│ │ │ │ -      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ │ -      3   3          3   3     3     3  3     3  3  3      3        3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         4  16  4  16     4       8  20  8  4        8    8  8     8     20
│ │ │ │ -      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ │ -         3   3  3   3     3       3   3  3  3        3    3  3     3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4       4  16     4  16     4  16       20  8  8  4        8    20
│ │ │ │ -      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ │ -      3       3   3     3   3     3   3        3  3  3  3        3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         4  8  8  8          8  8  8  4        20    16  16     4     4  8
│ │ │ │ -      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ │ -         3  3  3  3          3  3  3  3         3     3   3     3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -              8  4     8  8     20       20     4  4     20       20  4
│ │ │ │ -      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ │ -              3  3     3  3      3        3     3  3      3        3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -               4  20       4  20        20  4       8     8  20     4  8
│ │ │ │ -      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ │ -               3   3       3   3         3  3       3     3   3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          16  4  16  8        4    4     16  16     8  4          4  8
│ │ │ │ -      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ │ -           3  3   3  3        3    3      3   3     3  3          3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16  16     4    4     20        20     4    8  8  8        8  8  8
│ │ │ │ -      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ │ -       3   3     3    3      3         3     3    3  3  3        3  3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -       8  20  8     8        4       16  8     4  16     4    4        8  16
│ │ │ │ -      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ │ -       3   3  3     3        3        3  3     3   3     3    3        3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16       16  4     16  4     16  4    4     16  4  16  4     16
│ │ │ │ -      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ │ -      3   3        3  3      3  3      3  3    3      3  3   3  3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          8  8  8  8  8  8       20  8     8     8  4          4  8     4
│ │ │ │ -      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ │ -          3  3  3  3  3  3        3  3     3     3  3          3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16     16    20     4        4     20       4  4  16  4  16  16
│ │ │ │ -      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ │ -       3      3     3     3        3      3       3  3   3  3   3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          8  16     4     16  4    8  8     20     8  4          4  8     20
│ │ │ │ -      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ │ -          3   3     3      3  3    3  3      3     3  3          3  3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  8       8  4     16     16  4    8  8  20     8        4    4
│ │ │ │ -      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ │ -         3  3       3  3      3      3  3    3  3   3     3        3    3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4
│ │ │ │ -      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ │ -         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16       20  4     8  8     8    8        4  8  20  8       16     16
│ │ │ │ -      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ │ -       3        3  3     3  3     3    3        3  3   3  3        3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4     8  4       20     8  8     4  8          4  8     8     8  20
│ │ │ │ -      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ │ -      3     3  3        3     3  3     3  3          3  3     3     3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          16  16  4  16  4  4       8        4     16  4  16          20  4
│ │ │ │ -      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ │ -           3   3  3   3  3  3       3        3      3  3   3           3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  20          16        8  16  4  4          20  4        4  20
│ │ │ │ -      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ │ -      3   3           3        3   3  3  3           3  3        3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -       16  16  4     4        8       4  4  16  8        16    16  4  16
│ │ │ │ -      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ │ -        3   3  3     3        3       3  3   3  3         3     3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4        8    4  16     4     16  8       8  8     8  8     8  8
│ │ │ │ -      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ │ -      3        3    3   3     3      3  3       3  3     3  3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8
│ │ │ │ -      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ │ -      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -            4    4  20              20  4    4        20  20        4    4
│ │ │ │ -      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ │ -            3    3   3               3  3    3         3   3        3    3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -            8  16  16  4          4     20  20     4             4  20  20
│ │ │ │ -      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ │ -            3   3   3  3          3      3   3     3             3   3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4          8        16     4  4  16    8        4  8  8  20       8
│ │ │ │ -      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ │ -      3          3         3     3  3   3    3        3  3  3   3       3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4
│ │ │ │ -      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ │ -      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  20
│ │ │ │ -      -, --}}
│ │ │ │ -      3   3
│ │ │ │ +        20        4  8  8  8          8  8  8  4        20    16  16     4
│ │ │ │ +o12 = {{--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -,
│ │ │ │ +         3        3  3  3  3          3  3  3  3         3     3   3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  8          8  4     8  8     20       20     4  4     20
│ │ │ │ +      4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4},
│ │ │ │ +         3  3          3  3     3  3      3        3     3  3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       20  4              4  20       4  20        20  4       8     8  20
│ │ │ │ +      {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --,
│ │ │ │ +        3  3              3   3       3   3         3  3       3     3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  8          16  4  16  8        4    4     16  16     8  4
│ │ │ │ +      8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4},
│ │ │ │ +         3  3           3  3   3  3        3    3      3   3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          4  8     16  16     4    4     20        20     4    8  8  8
│ │ │ │ +      {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8,
│ │ │ │ +          3  3      3   3     3    3      3         3     3    3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  8  8    8  20  8     8        4       16  8     4  16     4    4
│ │ │ │ +      -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8,
│ │ │ │ +      3  3  3    3   3  3     3        3        3  3     3   3     3    3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         8  16  4  16       16  4     16  4     16  4    4     16  4  16  4
│ │ │ │ +      4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -,
│ │ │ │ +         3   3  3   3        3  3      3  3      3  3    3      3  3   3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         16       8  8  8  8  8  8       20  8     8     8  4          4  8
│ │ │ │ +      4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -,
│ │ │ │ +          3       3  3  3  3  3  3        3  3     3     3  3          3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  16     16    20     4        4     20       4  4  16  4  16  16
│ │ │ │ +      4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --,
│ │ │ │ +         3   3      3     3     3        3      3       3  3   3  3   3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +              8  16     4     16  4    8  8     20     8  4          4  8
│ │ │ │ +      4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ │ +              3   3     3      3  3    3  3      3     3  3          3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      20     8  8       8  4     16     16  4    8  8  20     8        4
│ │ │ │ +      --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -},
│ │ │ │ +       3     3  3       3  3      3      3  3    3  3   3     3        3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       4        8     20  8  8    8     8  8  8  8     8    4  16  16
│ │ │ │ +      {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8,
│ │ │ │ +       3        3      3  3  3    3     3  3  3  3     3    3   3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  4  16       20  4     8  8     8    8        4  8  20  8       16
│ │ │ │ +      -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--,
│ │ │ │ +      3  3   3        3  3     3  3     3    3        3  3   3  3        3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         16  4     8  4       20     8  8     4  8          4  8     8     8
│ │ │ │ +      4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -,
│ │ │ │ +          3  3     3  3        3     3  3     3  3          3  3     3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      20       16  16  4  16  4  4       8        4     16  4  16
│ │ │ │ +      --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4,
│ │ │ │ +       3        3   3  3   3  3  3       3        3      3  3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      20  4  4  20          16        8  16  4  4          20  4        4
│ │ │ │ +      --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ │ +       3  3  3   3           3        3   3  3  3           3  3        3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      20       16  16  4     4        8       4  4  16  8        16    16  4
│ │ │ │ +      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ │ +       3        3   3  3     3        3       3  3   3  3         3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16     4        8    4  16     4     16  8       8  8     8  8     8
│ │ │ │ +      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ │ +       3     3        3    3   3     3      3  3       3  3     3  3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8       8  4     20  8     8    16  4  4        16  16  4       4  16
│ │ │ │ +      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ │ +      3       3  3      3  3     3     3  3  3         3   3  3       3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16  8        4    4  20              20  4    4        20  20
│ │ │ │ +      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ │ +       3  3        3    3   3               3  3    3         3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4    4        8  16  16  4          4     20  20     4             4
│ │ │ │ +      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ │ +      3    3        3   3   3  3          3      3   3     3             3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      20  20  4          8        16     4  4  16    8        4  8  8  20
│ │ │ │ +      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ │ +       3   3  3          3         3     3  3   3    3        3  3  3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20
│ │ │ │ +      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ │ +           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      20  4  20  4  4  20    20        4  4        20       8  4     4
│ │ │ │ +      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ │ +       3  3   3  3  3   3     3        3  3         3       3  3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16  16    8        4     4  16  16    4  16     16     4  8       4
│ │ │ │ +      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ │ +       3   3    3        3     3   3   3    3   3      3     3  3       3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         20     8  8  8    8  8  8     20        4    4        8     8  20
│ │ │ │ +      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ │ +          3     3  3  3    3  3  3      3        3    3        3     3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8    4     16  4     8  16          4  20     8     8  8    16  4  4
│ │ │ │ +      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ │ +      3    3      3  3     3   3          3   3     3     3  3     3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         16        8    16     4  16  4  16     4       8  20  8  4
│ │ │ │ +      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ │ +          3        3     3     3   3  3   3     3       3   3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8    8  8     8     20  4       4  16     4  16     4  16       20  8
│ │ │ │ +      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ │ +      3    3  3     3      3  3       3   3     3   3     3   3        3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  4        8
│ │ │ │ +      -, -, 4, 8, -}}
│ │ │ │ +      3  3        3
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : volume convexHull A -- 8
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -158,15 +158,15 @@
│ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │  
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
    
│ │ │      		          
                  
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .108538s elapsed
    
│ │ │ + -- .156496s elapsed
│ │ │      		          
                  
    i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │ @@ -196,15 +196,15 @@
│ │ │  
│ │ │  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}}
│ │ │  
│ │ │  o7 : Triangulation
    
│ │ │      		          
                  
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.20095s elapsed
    
│ │ │ + -- 1.09493s elapsed
│ │ │      		          
                  
    i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │       | 0  0  0  1  0  0 -1 0 0 0 |
│ │ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │ │  
│ │ │ │                4       10
│ │ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ │ - -- .108538s elapsed
│ │ │ │ + -- .156496s elapsed
│ │ │ │  i4 : select(Ts, T -> isStar T)
│ │ │ │  
│ │ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6,
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │  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}}
│ │ │ │  
│ │ │ │  o7 : Triangulation
│ │ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ │ - -- 1.20095s elapsed
│ │ │ │ + -- 1.09493s elapsed
│ │ │ │  i9 : #Ts2 == #Ts
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_o_l_y_h_e_d_r_a -- for computations with convex polyhedra, cones, and fans
│ │ │ │      * _T_o_p_c_o_m -- interface to selected functions from topcom package
│ │ │ │      * _R_e_f_l_e_x_i_v_e_P_o_l_y_t_o_p_e_s_D_B -- simple access to Kreuzer-Skarke database of
│ │ ├── ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=7
│ │ │  cm90Rm9ydw==
│ │ │  #:len=229
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODI4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJyb3RGb3J3Iiwicm90Rm9ydyIsIlRyaXBsZXRzIn0s
│ │ ├── ./usr/share/doc/Macaulay2/Tropical/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  aXNCYWxhbmNlZA==
│ │ │  #:len=1097
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSB0cm9waWNh
│ │ │  bCBjeWNsZSBpcyBiYWxhbmNlZCIsICJsaW5lbnVtIiA9PiA5NTEsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/TropicalToric/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=43
│ │ │  Y2xhc3NGcm9tVHJvcGljYWwoTm9ybWFsVG9yaWNWYXJpZXR5LElkZWFsKQ==
│ │ │  #:len=335
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjbGFzc0Zyb21Ucm9waWNhbCxOb3JtYWxUb3JpY1Zh
│ │ ├── ./usr/share/doc/Macaulay2/Truncations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  dHJ1bmNhdGUoWlosTWF0cml4KQ==
│ │ │  #:len=276
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cnVuY2F0ZSxaWixNYXRyaXgpLCJ0cnVuY2F0ZSha
│ │ ├── ./usr/share/doc/Macaulay2/Units/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  VW5pdHM=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5pdHMgY29udmVyc2lvbiBhbmQgcGh5
│ │ │  c2ljYWwgY29uc3RhbnRzIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VEVYeyJUaGlzIHBh
│ │ ├── ./usr/share/doc/Macaulay2/VNumber/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  c3RhYmxlTWF4
│ │ │  #:len=1353
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgc2V0IG9mIHN0YWJs
│ │ │  ZSBwcmltZXMgb2YgYSBtb25vbWlhbCBpZGVhbCB0aGF0IGFyZSBtYXhpbWFsIHdpdGggcmVzcGVj
│ │ ├── ./usr/share/doc/Macaulay2/Valuations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  VmFsdWF0aW9u
│ │ │  #:len=1226
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHR5cGUgb2YgYWxsIHZhbHVhdGlv
│ │ │  bnMiLCAibGluZW51bSIgPT4gNzk5LCBTZWVBbHNvID0+IERJVntIRUFERVIyeyJTZWUgYWxzbyJ9
│ │ ├── ./usr/share/doc/Macaulay2/Varieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  PiBJbmZpbml0ZU51bWJlcg==
│ │ │  #:len=253
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPixJbmZpbml0ZU51bWJlciksIj4gSW5m
│ │ ├── ./usr/share/doc/Macaulay2/VectorFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Y29tbXV0YXRvcg==
│ │ │  #:len=3282
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNvbW11dGF0b3Igb2YgYSBjb2xs
│ │ │  ZWN0aW9uIG9mIHZlY3RvciBmaWVsZHMiLCAibGluZW51bSIgPT4gMjE1NSwgSW5wdXRzID0+IHtT
│ │ ├── ./usr/share/doc/Macaulay2/VectorGraphics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  dGV4KFNWRyk=
│ │ │  #:len=181
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4MywgInVuZG9jdW1lbnRlZCIgPT4g
│ │ │  dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4LFNW
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  bGlmdERlZm9ybWF0aW9uKC4uLixWZXJib3NlPT4uLi4p
│ │ │  #:len=588
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udHJvbCB0aGUgdmVyYm9zaXR5IG9m
│ │ │  IG91dHB1dCIsIERlc2NyaXB0aW9uID0+IDE6KFBBUkF7VFR7IlZlcmJvc2UifSwiIGlzIHRoZSBu
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out
│ │ │ @@ -6,30 +6,30 @@
│ │ │  
│ │ │  o2 = | xz yz z2 x3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix S  <-- S
│ │ │  
│ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 1.07659s (cpu); 0.666446s (thread); 0s (gc)
│ │ │ + -- used 1.50264s (cpu); 1.04548s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : T=ring first G;
│ │ │  
│ │ │  i5 : sum G
│ │ │  
│ │ │  o5 = | t_1t_16             |
│ │ │       | t_9t_16             |
│ │ │       | -t_4t_16            |
│ │ │       | -2t_14t_16+t_15t_16 |
│ │ │  
│ │ │               4      1
│ │ │  o5 : Matrix T  <-- T
│ │ │  
│ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.73081s (cpu); 0.444914s (thread); 0s (gc)
│ │ │ + -- used 0.80307s (cpu); 0.574068s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16                                                             
│ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │  o2 : Matrix S  <-- S
    │ │ │ │ │ │ │ │ │

    With the default setting SmartLift=>true we get very nice equations for the base space:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 1.07659s (cpu); 0.666446s (thread); 0s (gc)
    │ │ │ + -- used 1.50264s (cpu); 1.04548s (thread); 0s (gc) │ │ │ 
    i4 : T=ring first G;
    │ │ │ 
    i5 : sum G
│ │ │  
│ │ │ @@ -80,15 +80,15 @@
│ │ │  o5 : Matrix T  <-- T
    │ │ │
    │ │ │

    With the setting SmartLift=>false the calculation is faster, but the equations are no longer homogeneous:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.73081s (cpu); 0.444914s (thread); 0s (gc)
    │ │ │ + -- used 0.80307s (cpu); 0.574068s (thread); 0s (gc) │ │ │ 
    i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16                                                             
│ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,29 +18,29 @@
│ │ │ │  o2 = | xz yz z2 x3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix S  <-- S
│ │ │ │  With the default setting SmartLift=>true we get very nice equations for the
│ │ │ │  base space:
│ │ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ │ - -- used 1.07659s (cpu); 0.666446s (thread); 0s (gc)
│ │ │ │ + -- used 1.50264s (cpu); 1.04548s (thread); 0s (gc)
│ │ │ │  i4 : T=ring first G;
│ │ │ │  i5 : sum G
│ │ │ │  
│ │ │ │  o5 = | t_1t_16             |
│ │ │ │       | t_9t_16             |
│ │ │ │       | -t_4t_16            |
│ │ │ │       | -2t_14t_16+t_15t_16 |
│ │ │ │  
│ │ │ │               4      1
│ │ │ │  o5 : Matrix T  <-- T
│ │ │ │  With the setting SmartLift=>false the calculation is faster, but the equations
│ │ │ │  are no longer homogeneous:
│ │ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ │ - -- used 0.73081s (cpu); 0.444914s (thread); 0s (gc)
│ │ │ │ + -- used 0.80307s (cpu); 0.574068s (thread); 0s (gc)
│ │ │ │  i7 : sum G
│ │ │ │  
│ │ │ │  o7 = | t_1t_16
│ │ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/VirtualResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=42
│ │ │  bXVsdGlncmFkZWRSZWd1bGFyaXR5KC4uLixMb3dlckxpbWl0PT4uLi4p
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTgyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttdWx0aWdyYWRlZFJlZ3VsYXJpdHksTG93ZXJMaW1p
│ │ ├── ./usr/share/doc/Macaulay2/Visualize/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  dmlzdWFsaXplKEdyYXBoLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=1329
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gdG8gdmlldyBjb21tdW5pY2F0aW9u
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│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZXh0cmFjdFZhcnNBbGdlYnJh
│ │ │  #:len=1106
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5kZXJseWluZyBwb2x5bm9taWFsIHJp
│ │ │  bmcgaW4gdGhlIG9yZGluYXJ5IHZhcmlhYmxlcyBvZiBhIFdleWwgYWxnZWJyYSIsICJsaW5lbnVt
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  V2V5bEdyb3VwRWxlbWVudCAqIFJvb3Q=
│ │ │  #:len=994
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgYW4gZWxlbWVudCBvZiBhIFdl
│ │ │  eWwgZ3JvdXAgdG8gYSByb290IiwgImxpbmVudW0iID0+IDI5NjMsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        |  0 |       | -1 |                                              | -2 |        |  1 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1
│ │ │                                               |  3 |        |  1 |       |  1
│ │ │                                               | -2 |        | -1 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2,
│ │ │ +     |                                            | -1 |        | -1 |      
│ │ │ +     |                                            |  3 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ +     | -1 |}}}}
│ │ │ +     |  2 |
│ │ │       | -1 |
│ │ │ -     |  0 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_list__Weyl__Group__Elements_lp__Root__System_cm__Z__Z_rp.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1330776363660
│ │ │  
│ │ │  i1 : listWeylGroupElements(rootSystemG2,4)
│ │ │  
│ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, |  4 |},
│ │ │ -                                            | -3 |  
│ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |},
│ │ │ +                                            |  2 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     WeylGroupElement{RootSystem{...8...}, | -5 |}}
│ │ │ -                                           |  2 |
│ │ │ +     WeylGroupElement{RootSystem{...8...}, |  4 |}}
│ │ │ +                                           | -3 |
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_cm__Parabolic_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = set {1, 2}
│ │ │  
│ │ │  o2 : Parabolic
│ │ │  
│ │ │  i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ -          |  1 |  | -1 |  |  2 |
│ │ │ -          | -1 |  |  0 |  | -1 |
│ │ │ +o3 = set {| -1 |, |  1 |, |  2 |}
│ │ │ +          |  2 |  |  1 |  | -1 |
│ │ │ +          | -1 |  | -1 |  |  0 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.html
│ │ │ @@ -103,39 +103,39 @@
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
    │ │ │ 
    i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        |  0 |       | -1 |                                              | -2 |        |  1 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
    │ │ │
    │ │ │
    │ │ │

    Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1
│ │ │                                               |  3 |        |  1 |       |  1
│ │ │                                               | -2 |        | -1 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2,
│ │ │ +     |                                            | -1 |        | -1 |      
│ │ │ +     |                                            |  3 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ +     | -1 |}}}}
│ │ │ +     |  2 |
│ │ │       | -1 |
│ │ │ -     |  0 |
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -42,47 +42,47 @@ │ │ │ │ | -2 | │ │ │ │ | 1 | │ │ │ │ │ │ │ │ o4 : WeylGroupElement │ │ │ │ i5 : myInterval=intervalBruhat(w1 % P,w2 % P) │ │ │ │ │ │ │ │ o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 |}, │ │ │ │ -{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 |}, │ │ │ │ -{2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, │ │ │ │ -{2, | 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 │ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | -1 |}}}, │ │ │ │ -{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}}, { │ │ │ │ +{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, │ │ │ │ +{1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 2 |}, │ │ │ │ +{2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | - │ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}, │ │ │ │ +{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 |}}}}, { │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 2 |}, {}}}} │ │ │ │ | -3 | | 0 | │ │ │ │ | 1 | | 3 | | 1 | │ │ │ │ -| 1 | | -1 | | 2 | | │ │ │ │ --1 | | -2 | | -1 | │ │ │ │ -| 1 | | 1 | | 1 | | │ │ │ │ -1 | | -1 | │ │ │ │ +| 1 | | -1 | | -1 | | │ │ │ │ +2 | | 1 | | 1 | │ │ │ │ +| 1 | | 1 | | -2 | | │ │ │ │ +-1 | | -1 | │ │ │ │ | 1 | | 1 | │ │ │ │ | 1 | | -2 | | -1 | │ │ │ │ -| 1 | | 3 | | -1 | | │ │ │ │ -0 | | 3 | | 0 | │ │ │ │ -| -2 | | 1 | | 2 | | │ │ │ │ --1 | | 2 | │ │ │ │ +| 1 | | 3 | | 0 | | │ │ │ │ +-1 | | -2 | | 1 | │ │ │ │ +| 2 | | -1 | | 3 | | │ │ │ │ +0 | | 2 | │ │ │ │ │ │ │ │ o5 : HasseDiagram │ │ │ │ Each row of the Hasse diagram contains the elements of a certain length │ │ │ │ together with their links to the next row. │ │ │ │ i6 : myInterval#1 │ │ │ │ │ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 |}, {2, | -1 │ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 │ │ │ │ | 3 | | 1 | | 1 │ │ │ │ | -2 | | -1 | | 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, │ │ │ │ - | | -1 | | 2 | │ │ │ │ - | | 3 | | -1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 2 |}, {2, │ │ │ │ + | | -1 | | -1 | │ │ │ │ + | | 3 | | 0 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 2 |}}}} │ │ │ │ + | -1 |}}}} │ │ │ │ + | 2 | │ │ │ │ | -1 | │ │ │ │ - | 0 | │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_n_t_e_r_v_a_l_B_r_u_h_a_t_(_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_,_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_) -- elements between │ │ │ │ two given ones for the Bruhat order on a quotient of a Weyl group │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_list__Weyl__Group__Elements_lp__Root__System_cm__Z__Z_rp.html │ │ │ @@ -71,19 +71,19 @@ │ │ │
    │ │ │
    │ │ │

    Description

    │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : listWeylGroupElements(rootSystemG2,4)
│ │ │  
│ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, |  4 |},
│ │ │ -                                            | -3 |  
│ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |},
│ │ │ +                                            |  2 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     WeylGroupElement{RootSystem{...8...}, | -5 |}}
│ │ │ -                                           |  2 |
│ │ │ +     WeylGroupElement{RootSystem{...8...}, |  4 |}}
│ │ │ +                                           | -3 |
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -13,17 +13,17 @@ │ │ │ │ o R, an instance of the type _R_o_o_t_S_y_s_t_e_m, │ │ │ │ o k, an _i_n_t_e_g_e_r, │ │ │ │ * Outputs: │ │ │ │ o a _l_i_s_t, of all elements of length k in the Weyl group of R │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : listWeylGroupElements(rootSystemG2,4) │ │ │ │ │ │ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, | 4 |}, │ │ │ │ - | -3 | │ │ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |}, │ │ │ │ + | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - WeylGroupElement{RootSystem{...8...}, | -5 |}} │ │ │ │ - | 2 | │ │ │ │ + WeylGroupElement{RootSystem{...8...}, | 4 |}} │ │ │ │ + | -3 | │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _l_i_s_t_W_e_y_l_G_r_o_u_p_E_l_e_m_e_n_t_s_(_R_o_o_t_S_y_s_t_e_m_,_Z_Z_) -- list all elements of a given │ │ │ │ length in a Weyl group │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_positive__Roots_lp__Root__System_cm__Parabolic_rp.html │ │ │ @@ -85,17 +85,17 @@ │ │ │ o2 = set {1, 2} │ │ │ │ │ │ o2 : Parabolic │ │ │ │ │ │ │ │ │ 
    i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ -          |  1 |  | -1 |  |  2 |
│ │ │ -          | -1 |  |  0 |  | -1 |
│ │ │ +o3 = set {| -1 |, |  1 |, |  2 |}
│ │ │ +          |  2 |  |  1 |  | -1 |
│ │ │ +          | -1 |  | -1 |  |  0 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -24,16 +24,16 @@ │ │ │ │ i2 : P=parabolic(R,set{1,2}) │ │ │ │ │ │ │ │ o2 = set {1, 2} │ │ │ │ │ │ │ │ o2 : Parabolic │ │ │ │ i3 : positiveRoots(R,P) │ │ │ │ │ │ │ │ -o3 = set {| 1 |, | 2 |, | -1 |} │ │ │ │ - | 1 | | -1 | | 2 | │ │ │ │ - | -1 | | 0 | | -1 | │ │ │ │ +o3 = set {| -1 |, | 1 |, | 2 |} │ │ │ │ + | 2 | | 1 | | -1 | │ │ │ │ + | -1 | | -1 | | 0 | │ │ │ │ | 0 | | 0 | | 0 | │ │ │ │ │ │ │ │ o3 : Set │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_,_P_a_r_a_b_o_l_i_c_) -- the set of all positive roots in a │ │ │ │ parabolic sugroups │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ Y29ub3JtYWw= │ │ │ #:len=1037 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNvbm9ybWFsIHZh │ │ │ cmlldHkiLCAibGluZW51bSIgPT4gNjc2LCBJbnB1dHMgPT4ge1NQQU57VFR7IkkifSwiLCAiLFNQ │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out │ │ │ @@ -90,41 +90,41 @@ │ │ │ i22 : peek last ms │ │ │ │ │ │ o22 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ i23 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly") │ │ │ - -- used 1.80747s (cpu); 0.99033s (thread); 0s (gc) │ │ │ + -- used 1.61258s (cpu); 0.838479s (thread); 0s (gc) │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o23 : List │ │ │ │ │ │ i24 : peek last ms │ │ │ │ │ │ o24 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ i25 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most") │ │ │ - -- used 4.58985s (cpu); 2.45092s (thread); 0s (gc) │ │ │ + -- used 7.92286s (cpu); 2.37432s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ i26 : peek last ms │ │ │ │ │ │ o26 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ i27 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all") │ │ │ - -- used 5.53629s (cpu); 2.97634s (thread); 0s (gc) │ │ │ + -- used 9.38742s (cpu); 3.00015s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ i28 : peek last ms │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html │ │ │ @@ -222,45 +222,45 @@ │ │ │ │ │ │
    │ │ │

    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=>"all", 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 "all" setting. The other possible values are StratsToFind=>"singularOnly", and StratsToFind=>"most". The option StratsToFind=>"singularOnly" 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=>"most" 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=>"all" should be considered complete; StratsToFind=>"all" is run when no option is given.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")
│ │ │ - -- used 1.80747s (cpu); 0.99033s (thread); 0s (gc)
│ │ │ + -- used 1.61258s (cpu); 0.838479s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
    │ │ │ 
    i24 : peek last ms
│ │ │  
│ │ │  o24 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │                         1 => {ideal P, ideal M1, ideal(4M1 - P)}
│ │ │                         2 => {ideal 0}
    │ │ │ 
    i25 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")
│ │ │ - -- used 4.58985s (cpu); 2.45092s (thread); 0s (gc)
│ │ │ + -- used 7.92286s (cpu); 2.37432s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
    │ │ │ 
    i26 : peek last ms
│ │ │  
│ │ │  o26 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │                         1 => {ideal P, ideal M1, ideal(4M1 - P)}
│ │ │                         2 => {ideal 0}
    │ │ │ 
    i27 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")
│ │ │ - -- used 5.53629s (cpu); 2.97634s (thread); 0s (gc)
│ │ │ + -- used 9.38742s (cpu); 3.00015s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
    │ │ │ 
    i28 : peek last ms
│ │ │ ├── html2text {}
│ │ │ │ @@ -149,37 +149,37 @@
│ │ │ │  this command should be treated as a partial answer only. The option
│ │ │ │  StratsToFind=>"most" 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=>"all" should be considered complete; StratsToFind=>"all" is run
│ │ │ │  when no option is given.
│ │ │ │  i23 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")
│ │ │ │ - -- used 1.80747s (cpu); 0.99033s (thread); 0s (gc)
│ │ │ │ + -- used 1.61258s (cpu); 0.838479s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : peek last ms
│ │ │ │  
│ │ │ │  o24 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │                         1 => {ideal P, ideal M1, ideal(4M1 - P)}
│ │ │ │                         2 => {ideal 0}
│ │ │ │  i25 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")
│ │ │ │ - -- used 4.58985s (cpu); 2.45092s (thread); 0s (gc)
│ │ │ │ + -- used 7.92286s (cpu); 2.37432s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o25 : List
│ │ │ │  i26 : peek last ms
│ │ │ │  
│ │ │ │  o26 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │                         1 => {ideal P, ideal M1, ideal(4M1 - P)}
│ │ │ │                         2 => {ideal 0}
│ │ │ │  i27 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")
│ │ │ │ - -- used 5.53629s (cpu); 2.97634s (thread); 0s (gc)
│ │ │ │ + -- used 9.38742s (cpu); 3.00015s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : peek last ms
│ │ │ │  
│ │ │ │  o28 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ ├── ./usr/share/doc/Macaulay2/XML/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  Z2V0QXR0cmlidXRlcyhMaWJ4bWxOb2RlKQ==
│ │ │  #:len=465
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSBsaXN0IG9mIGF0dHJpYnV0
│ │ │  ZXMgb2YgYW4gWE1MIG5vZGUiLCBEZXNjcmlwdGlvbiA9PiAxOihUQUJMRXsiY2xhc3MiID0+ICJl
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  Z2ZhblBvbHlub21pYWxTZXRVbmlvbihMaXN0LExpc3Qp
│ │ │  #:len=309
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0Miwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2ZhblBvbHlub21pYWxTZXRVbmlvbixMaXN0LExp
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │  
│ │ │  i4 : prefixDirectory | currentLayout#"programs"
│ │ │  
│ │ │  o4 = /usr/x86_64-Linux-
│ │ │       Debian-trixie/libexec/Macaulay2/bin/
│ │ │  
│ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-37469-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-59381-0/172
│ │ │  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.
│ │ │  Options:
│ │ │  -g:
│ │ │   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.
│ │ │  
│ │ │  --symmetry:
│ │ │   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.
│ │ │ @@ -36,16 +36,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   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.
│ │ │  
│ │ │  --parameters value:
│ │ │   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.
│ │ │  --interrupt value:
│ │ │   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).
│ │ │ -using temporary file /tmp/M2-37469-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-37469-0/174
│ │ │ +using temporary file /tmp/M2-59381-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-59381-0/174
│ │ │  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,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   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.
│ │ │  
│ │ │  -r:
│ │ │   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.
│ │ │ @@ -54,69 +54,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   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.
│ │ │  
│ │ │  --parameters value:
│ │ │   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.
│ │ │ -using temporary file /tmp/M2-37469-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-37469-0/176
│ │ │ +using temporary file /tmp/M2-59381-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-59381-0/176
│ │ │  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.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-37469-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-37469-0/178
│ │ │ +using temporary file /tmp/M2-59381-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-59381-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-37469-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-37469-0/180
│ │ │ +using temporary file /tmp/M2-59381-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-59381-0/180
│ │ │  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.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-37469-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-37469-0/182
│ │ │ +using temporary file /tmp/M2-59381-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-59381-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-37469-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-37469-0/184
│ │ │ +using temporary file /tmp/M2-59381-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-59381-0/184
│ │ │  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.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   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.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   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.
│ │ │ -using temporary file /tmp/M2-37469-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-37469-0/186
│ │ │ +using temporary file /tmp/M2-59381-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-59381-0/186
│ │ │  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.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-37469-0/188
│ │ │ +using temporary file /tmp/M2-59381-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-59381-0/188
│ │ │  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.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -124,30 +124,30 @@
│ │ │  -i:
│ │ │   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.
│ │ │  -w:
│ │ │   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.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-37469-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-37469-0/190
│ │ │ +using temporary file /tmp/M2-59381-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-59381-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   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.
│ │ │  
│ │ │  --pair:
│ │ │   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.
│ │ │  
│ │ │  --mark:
│ │ │   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.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-37469-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-37469-0/192
│ │ │ +using temporary file /tmp/M2-59381-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-59381-0/192
│ │ │  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.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -162,57 +162,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-37469-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-37469-0/194
│ │ │ +using temporary file /tmp/M2-59381-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-59381-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-37469-0/196
│ │ │ +using temporary file /tmp/M2-59381-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-59381-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-37469-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-37469-0/198
│ │ │ +using temporary file /tmp/M2-59381-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-59381-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-37469-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-37469-0/200
│ │ │ +using temporary file /tmp/M2-59381-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-59381-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-37469-0/202
│ │ │ +using temporary file /tmp/M2-59381-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-59381-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-37469-0/204
│ │ │ +using temporary file /tmp/M2-59381-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-59381-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   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.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   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.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-37469-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-37469-0/206
│ │ │ +using temporary file /tmp/M2-59381-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-59381-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -227,16 +227,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-37469-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-37469-0/208
│ │ │ +using temporary file /tmp/M2-59381-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-59381-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -247,44 +247,44 @@
│ │ │   Use Katsura-n example instead of reading input.
│ │ │  --gaukwa value:
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │ -using temporary file /tmp/M2-37469-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-37469-0/210
│ │ │ +using temporary file /tmp/M2-59381-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-59381-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-37469-0/212
│ │ │ +using temporary file /tmp/M2-59381-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-59381-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-37469-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-37469-0/214
│ │ │ +using temporary file /tmp/M2-59381-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-59381-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-37469-0/216
│ │ │ +using temporary file /tmp/M2-59381-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-59381-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -297,25 +297,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-37469-0/218
│ │ │ +using temporary file /tmp/M2-59381-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-59381-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-37469-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-37469-0/220
│ │ │ +using temporary file /tmp/M2-59381-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-59381-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -324,70 +324,70 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   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).
│ │ │ -using temporary file /tmp/M2-37469-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-37469-0/222
│ │ │ +using temporary file /tmp/M2-59381-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-59381-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-37469-0/224
│ │ │ +using temporary file /tmp/M2-59381-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-59381-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-37469-0/226
│ │ │ +using temporary file /tmp/M2-59381-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-59381-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-37469-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-37469-0/228
│ │ │ +using temporary file /tmp/M2-59381-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-59381-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-37469-0/230
│ │ │ +using temporary file /tmp/M2-59381-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-59381-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-37469-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-37469-0/232
│ │ │ +using temporary file /tmp/M2-59381-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-59381-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-37469-0/234
│ │ │ +using temporary file /tmp/M2-59381-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-59381-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-37469-0/236
│ │ │ +using temporary file /tmp/M2-59381-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-59381-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-37469-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-37469-0/238
│ │ │ +using temporary file /tmp/M2-59381-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-59381-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-37469-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-37469-0/240
│ │ │ +using temporary file /tmp/M2-59381-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-59381-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -399,16 +399,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   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.
│ │ │ -using temporary file /tmp/M2-37469-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-37469-0/242
│ │ │ +using temporary file /tmp/M2-59381-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-59381-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -433,48 +433,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-37469-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-37469-0/244
│ │ │ +using temporary file /tmp/M2-59381-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-59381-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-37469-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-37469-0/246
│ │ │ +using temporary file /tmp/M2-59381-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-59381-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-37469-0/248
│ │ │ +using temporary file /tmp/M2-59381-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-59381-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-37469-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-37469-0/250
│ │ │ +using temporary file /tmp/M2-59381-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-59381-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-37469-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-37469-0/252
│ │ │ +using temporary file /tmp/M2-59381-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-59381-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -482,24 +482,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   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.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   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).
│ │ │ -using temporary file /tmp/M2-37469-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-37469-0/254
│ │ │ +using temporary file /tmp/M2-59381-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-59381-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-37469-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-37469-0/256
│ │ │ +using temporary file /tmp/M2-59381-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-59381-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -519,21 +519,21 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   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 the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/256
│ │ │ +using temporary file /tmp/M2-59381-0/256
│ │ │  
│ │ │  i6 : QQ[x,y];
│ │ │  
│ │ │  i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37469-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-59381-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-37469-0/258
│ │ │ +using temporary file /tmp/M2-59381-0/258
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │          
    
│ │ │            
    If you would like to see the input and output files used to communicate with gfan you can set the "keepfiles" configuration option to true. If "verbose" is set to true, gfanInterface will output the names of the temporary files used.
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-37469-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-59381-0/172
│ │ │  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.
│ │ │  Options:
│ │ │  -g:
│ │ │   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.
│ │ │  
│ │ │  --symmetry:
│ │ │   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.
│ │ │ @@ -110,16 +110,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   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.
│ │ │  
│ │ │  --parameters value:
│ │ │   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.
│ │ │  --interrupt value:
│ │ │   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).
│ │ │ -using temporary file /tmp/M2-37469-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-37469-0/174
│ │ │ +using temporary file /tmp/M2-59381-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-59381-0/174
│ │ │  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,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   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.
│ │ │  
│ │ │  -r:
│ │ │   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.
│ │ │ @@ -128,69 +128,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   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.
│ │ │  
│ │ │  --parameters value:
│ │ │   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.
│ │ │ -using temporary file /tmp/M2-37469-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-37469-0/176
│ │ │ +using temporary file /tmp/M2-59381-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-59381-0/176
│ │ │  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.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-37469-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-37469-0/178
│ │ │ +using temporary file /tmp/M2-59381-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-59381-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-37469-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-37469-0/180
│ │ │ +using temporary file /tmp/M2-59381-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-59381-0/180
│ │ │  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.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-37469-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-37469-0/182
│ │ │ +using temporary file /tmp/M2-59381-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-59381-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-37469-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-37469-0/184
│ │ │ +using temporary file /tmp/M2-59381-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-59381-0/184
│ │ │  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.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   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.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   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.
│ │ │ -using temporary file /tmp/M2-37469-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-37469-0/186
│ │ │ +using temporary file /tmp/M2-59381-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-59381-0/186
│ │ │  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.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-37469-0/188
│ │ │ +using temporary file /tmp/M2-59381-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-59381-0/188
│ │ │  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.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -198,30 +198,30 @@
│ │ │  -i:
│ │ │   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.
│ │ │  -w:
│ │ │   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.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-37469-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-37469-0/190
│ │ │ +using temporary file /tmp/M2-59381-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-59381-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   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.
│ │ │  
│ │ │  --pair:
│ │ │   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.
│ │ │  
│ │ │  --mark:
│ │ │   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.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-37469-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-37469-0/192
│ │ │ +using temporary file /tmp/M2-59381-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-59381-0/192
│ │ │  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.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -236,57 +236,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-37469-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-37469-0/194
│ │ │ +using temporary file /tmp/M2-59381-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-59381-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-37469-0/196
│ │ │ +using temporary file /tmp/M2-59381-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-59381-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-37469-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-37469-0/198
│ │ │ +using temporary file /tmp/M2-59381-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-59381-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-37469-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-37469-0/200
│ │ │ +using temporary file /tmp/M2-59381-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-59381-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-37469-0/202
│ │ │ +using temporary file /tmp/M2-59381-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-59381-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-37469-0/204
│ │ │ +using temporary file /tmp/M2-59381-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-59381-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   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.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   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.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-37469-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-37469-0/206
│ │ │ +using temporary file /tmp/M2-59381-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-59381-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -301,16 +301,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-37469-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-37469-0/208
│ │ │ +using temporary file /tmp/M2-59381-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-59381-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -321,44 +321,44 @@
│ │ │   Use Katsura-n example instead of reading input.
│ │ │  --gaukwa value:
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │ -using temporary file /tmp/M2-37469-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-37469-0/210
│ │ │ +using temporary file /tmp/M2-59381-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-59381-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-37469-0/212
│ │ │ +using temporary file /tmp/M2-59381-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-59381-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-37469-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-37469-0/214
│ │ │ +using temporary file /tmp/M2-59381-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-59381-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-37469-0/216
│ │ │ +using temporary file /tmp/M2-59381-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-59381-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -371,25 +371,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-37469-0/218
│ │ │ +using temporary file /tmp/M2-59381-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-59381-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-37469-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-37469-0/220
│ │ │ +using temporary file /tmp/M2-59381-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-59381-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -398,70 +398,70 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   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).
│ │ │ -using temporary file /tmp/M2-37469-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-37469-0/222
│ │ │ +using temporary file /tmp/M2-59381-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-59381-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-37469-0/224
│ │ │ +using temporary file /tmp/M2-59381-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-59381-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-37469-0/226
│ │ │ +using temporary file /tmp/M2-59381-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-59381-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-37469-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-37469-0/228
│ │ │ +using temporary file /tmp/M2-59381-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-59381-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-37469-0/230
│ │ │ +using temporary file /tmp/M2-59381-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-59381-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-37469-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-37469-0/232
│ │ │ +using temporary file /tmp/M2-59381-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-59381-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-37469-0/234
│ │ │ +using temporary file /tmp/M2-59381-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-59381-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-37469-0/236
│ │ │ +using temporary file /tmp/M2-59381-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-59381-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-37469-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-37469-0/238
│ │ │ +using temporary file /tmp/M2-59381-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-59381-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-37469-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-37469-0/240
│ │ │ +using temporary file /tmp/M2-59381-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-59381-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -473,16 +473,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   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.
│ │ │ -using temporary file /tmp/M2-37469-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-37469-0/242
│ │ │ +using temporary file /tmp/M2-59381-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-59381-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -507,48 +507,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-37469-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-37469-0/244
│ │ │ +using temporary file /tmp/M2-59381-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-59381-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-37469-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-37469-0/246
│ │ │ +using temporary file /tmp/M2-59381-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-59381-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37469-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-37469-0/248
│ │ │ +using temporary file /tmp/M2-59381-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-59381-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-37469-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-37469-0/250
│ │ │ +using temporary file /tmp/M2-59381-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-59381-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-37469-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-37469-0/252
│ │ │ +using temporary file /tmp/M2-59381-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-59381-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -556,24 +556,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   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.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   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).
│ │ │ -using temporary file /tmp/M2-37469-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-37469-0/254
│ │ │ +using temporary file /tmp/M2-59381-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-59381-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-37469-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-37469-0/256
│ │ │ +using temporary file /tmp/M2-59381-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-59381-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -593,28 +593,28 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   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 the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-37469-0/256
    
│ │ │ +using temporary file /tmp/M2-59381-0/256
│ │ │      		          
                  
    i6 : QQ[x,y];
    
│ │ │      		          
                  
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37469-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-59381-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-37469-0/258
    
│ │ │ +using temporary file /tmp/M2-59381-0/258
│ │ │      		          
    
│ │ │          
    
│ │ │            
    Finally, if you want to be able to render Groebner fans and monomial staircases to .png files, you should install fig2dev.  If it is installed in a non-standard location, then you may specify its path using programPaths.
    
│ │ │          
    
│ │ │        
│ │ │      
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  o4 = /usr/x86_64-Linux-
│ │ │ │       Debian-trixie/libexec/Macaulay2/bin/
│ │ │ │  If you would like to see the input and output files used to communicate with
│ │ │ │  gfan you can set the "keepfiles" configuration option to true. If "verbose" is
│ │ │ │  set to true, gfanInterface will output the names of the temporary files used.
│ │ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true,
│ │ │ │  "verbose" => true}, Reload => true);
│ │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-37469-0/172
│ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-59381-0/172
│ │ │ │  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.
│ │ │ │ @@ -77,16 +77,16 @@
│ │ │ │   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.
│ │ │ │  --interrupt value:
│ │ │ │   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).
│ │ │ │ -using temporary file /tmp/M2-37469-0/172
│ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-37469-0/174
│ │ │ │ +using temporary file /tmp/M2-59381-0/172
│ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-59381-0/174
│ │ │ │  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,...]).
│ │ │ │  Options:
│ │ │ │  -w:
│ │ │ │   Compute a Groebner basis with respect to a degree lexicographic order with
│ │ │ │ @@ -107,63 +107,63 @@
│ │ │ │  minimal Groebner basis with respect to the reverse lexicographic term order.
│ │ │ │  The target term order is always lexicographic. The -W option must be used.
│ │ │ │  
│ │ │ │  --parameters value:
│ │ │ │   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.
│ │ │ │ -using temporary file /tmp/M2-37469-0/174
│ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-37469-0/176
│ │ │ │ +using temporary file /tmp/M2-59381-0/174
│ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-59381-0/176
│ │ │ │  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.
│ │ │ │  Options:
│ │ │ │  --remainder:
│ │ │ │   Tell the program to output the remainders of the divisions rather than
│ │ │ │  outputting 0 or 1.
│ │ │ │  --multiplier:
│ │ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided
│ │ │ │  before doing the division.
│ │ │ │ -using temporary file /tmp/M2-37469-0/176
│ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-37469-0/178
│ │ │ │ +using temporary file /tmp/M2-59381-0/176
│ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-59381-0/178
│ │ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the first input file.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the second input file.
│ │ │ │  --stable:
│ │ │ │   Compute the stable intersection.
│ │ │ │ -using temporary file /tmp/M2-37469-0/178
│ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-37469-0/180
│ │ │ │ +using temporary file /tmp/M2-59381-0/178
│ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-59381-0/180
│ │ │ │  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.
│ │ │ │  Options:
│ │ │ │  -i value:
│ │ │ │   Specify the name of the input file.
│ │ │ │  --symmetry:
│ │ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must
│ │ │ │  be given on the standard input.
│ │ │ │  
│ │ │ │  --star:
│ │ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan
│ │ │ │  containing all cones of the original fan containing the vector.
│ │ │ │ -using temporary file /tmp/M2-37469-0/180
│ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-37469-0/182
│ │ │ │ +using temporary file /tmp/M2-59381-0/180
│ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-59381-0/182
│ │ │ │  This program takes two polyhedral fans and computes their product.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the first input file.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the second input file.
│ │ │ │ -using temporary file /tmp/M2-37469-0/182
│ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-37469-0/184
│ │ │ │ +using temporary file /tmp/M2-59381-0/182
│ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-59381-0/184
│ │ │ │  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.
│ │ │ │ @@ -180,24 +180,24 @@
│ │ │ │  --asfan:
│ │ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way
│ │ │ │  the extreme rays of the cone are also computed.
│ │ │ │  --vectorinput:
│ │ │ │   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.
│ │ │ │ -using temporary file /tmp/M2-37469-0/184
│ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-37469-0/186
│ │ │ │ +using temporary file /tmp/M2-59381-0/184
│ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-59381-0/186
│ │ │ │  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.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/186
│ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-37469-0/188
│ │ │ │ +using temporary file /tmp/M2-59381-0/186
│ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-59381-0/188
│ │ │ │  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.
│ │ │ │  Example:
│ │ │ │  Input:
│ │ │ │  Q[x,y]{y-1}
│ │ │ │ @@ -213,16 +213,16 @@
│ │ │ │   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.
│ │ │ │  
│ │ │ │  -H:
│ │ │ │   Let the name of the new variable be H rather than reading in a name from the
│ │ │ │  input.
│ │ │ │ -using temporary file /tmp/M2-37469-0/188
│ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-37469-0/190
│ │ │ │ +using temporary file /tmp/M2-59381-0/188
│ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-59381-0/190
│ │ │ │  This program converts a list of polynomials to a list of their initial forms
│ │ │ │  with respect to the vector given after the list.
│ │ │ │  Options:
│ │ │ │  --ideal:
│ │ │ │   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
│ │ │ │ @@ -238,16 +238,16 @@
│ │ │ │  --mark:
│ │ │ │   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.
│ │ │ │  --list:
│ │ │ │   Read in a list of vectors instead of a single vector and produce a list of
│ │ │ │  polynomial sets as output.
│ │ │ │ -using temporary file /tmp/M2-37469-0/190
│ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-37469-0/192
│ │ │ │ +using temporary file /tmp/M2-59381-0/190
│ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-59381-0/192
│ │ │ │  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
│ │ │ │ @@ -277,54 +277,54 @@
│ │ │ │  -W:
│ │ │ │   Print weight vector. This will make the program print an interior vector of
│ │ │ │  the current Groebner cone and a relative interior point for each flippable
│ │ │ │  facet of the current Groebner cone.
│ │ │ │  
│ │ │ │  --tropical:
│ │ │ │   Traverse a tropical variety interactively.
│ │ │ │ -using temporary file /tmp/M2-37469-0/192
│ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-37469-0/194
│ │ │ │ +using temporary file /tmp/M2-59381-0/192
│ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-59381-0/194
│ │ │ │  This program checks if a set of marked polynomials is a Groebner basis with
│ │ │ │  respect to its marking. First it is checked if the markings are consistent with
│ │ │ │  respect to a positive vector. Then Buchberger's S-criterion is checked. The
│ │ │ │  output is boolean value.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/194
│ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-37469-0/196
│ │ │ │ +using temporary file /tmp/M2-59381-0/194
│ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-59381-0/196
│ │ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the
│ │ │ │  polynomial ring. This is done by first computing a Groebner basis.
│ │ │ │  Options:
│ │ │ │  -g:
│ │ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ │ -using temporary file /tmp/M2-37469-0/196
│ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-37469-0/198
│ │ │ │ +using temporary file /tmp/M2-59381-0/196
│ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-59381-0/198
│ │ │ │  This program computes the lattice ideal of a lattice. The input is a list of
│ │ │ │  generators for the lattice.
│ │ │ │  Options:
│ │ │ │  -t:
│ │ │ │   Compute the toric ideal of the matrix whose rows are given on the input
│ │ │ │  instead.
│ │ │ │  --convert:
│ │ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ │ -using temporary file /tmp/M2-37469-0/198
│ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-37469-0/200
│ │ │ │ +using temporary file /tmp/M2-59381-0/198
│ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-59381-0/200
│ │ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │ │  Options:
│ │ │ │  -m:
│ │ │ │   Do the same thing for a list of polynomial sets. That is, output the set of
│ │ │ │  sets of leading terms.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-37469-0/200
│ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-37469-0/202
│ │ │ │ +using temporary file /tmp/M2-59381-0/200
│ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-59381-0/202
│ │ │ │  This program marks a set of polynomials with respect to the vector given at the
│ │ │ │  end of the input, meaning that the largest terms are moved to the front. In
│ │ │ │  case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/202
│ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-37469-0/204
│ │ │ │ +using temporary file /tmp/M2-59381-0/202
│ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-59381-0/204
│ │ │ │  This is a program for computing the normal fan of the Minkowski sum of the
│ │ │ │  Newton polytopes of a list of polynomials.
│ │ │ │  Options:
│ │ │ │  --symmetry:
│ │ │ │   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
│ │ │ │ @@ -334,16 +334,16 @@
│ │ │ │  --disableSymmetryTest:
│ │ │ │   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.
│ │ │ │  
│ │ │ │  --nocones:
│ │ │ │   Tell the program to not list cones in the output.
│ │ │ │ -using temporary file /tmp/M2-37469-0/204
│ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-37469-0/206
│ │ │ │ +using temporary file /tmp/M2-59381-0/204
│ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-59381-0/206
│ │ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │ │  Options:
│ │ │ │  -r value:
│ │ │ │   Specify r.
│ │ │ │  -d value:
│ │ │ │   Specify d.
│ │ │ │  -n value:
│ │ │ │ @@ -361,16 +361,16 @@
│ │ │ │   Produces a list of generators for the group of symmetries keeping the set of
│ │ │ │  minors fixed. (Only without --names).
│ │ │ │  --parametrize:
│ │ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal
│ │ │ │  to r-1 by a list of tropical polynomials.
│ │ │ │  --ultrametric:
│ │ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ │ -using temporary file /tmp/M2-37469-0/206
│ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-37469-0/208
│ │ │ │ +using temporary file /tmp/M2-59381-0/206
│ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-59381-0/208
│ │ │ │  This program computes the mixed volume of the Newton polytopes of a list of
│ │ │ │  polynomials. The ring is specified on the input. After this follows the list of
│ │ │ │  polynomials.
│ │ │ │  Options:
│ │ │ │  --vectorinput:
│ │ │ │   Read in a list of point configurations instead of a polynomial ring and a list
│ │ │ │  of polynomials.
│ │ │ │ @@ -384,25 +384,25 @@
│ │ │ │   Use Katsura-n example instead of reading input.
│ │ │ │  --gaukwa value:
│ │ │ │   Use Gaukwa-n example instead of reading input.
│ │ │ │  --eco value:
│ │ │ │   Use Eco-n example instead of reading input.
│ │ │ │  -j value:
│ │ │ │   Number of threads
│ │ │ │ -using temporary file /tmp/M2-37469-0/208
│ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-37469-0/210
│ │ │ │ +using temporary file /tmp/M2-59381-0/208
│ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-59381-0/210
│ │ │ │  This program computes the union of a list of polynomial sets given as input.
│ │ │ │  The polynomials must all belong to the same ring. The ring is specified on the
│ │ │ │  input. After this follows the list of polynomial sets.
│ │ │ │  Options:
│ │ │ │  -s:
│ │ │ │   Sort output by degree.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-37469-0/210
│ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-37469-0/212
│ │ │ │ +using temporary file /tmp/M2-59381-0/210
│ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-59381-0/212
│ │ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the
│ │ │ │  input is the list of all reduced Groebner bases of an ideal. The output is a
│ │ │ │  drawing of the Groebner fan intersected with a triangle. The corners of the
│ │ │ │  triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top.
│ │ │ │  If there are more than three variables in the ring these coordinates are
│ │ │ │  extended with zeros. It is possible to shift the 1 entry cyclic with the option
│ │ │ │  --shiftVariables.
│ │ │ │ @@ -410,16 +410,16 @@
│ │ │ │  -L:
│ │ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │ │  --shiftVariables value:
│ │ │ │   Shift the positions of the variables in the drawing. For example with the
│ │ │ │  value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...)
│ │ │ │  and top: (0,0,0,1,...). The shifting is done modulo the number of variables in
│ │ │ │  the polynomial ring. The default value is 0.
│ │ │ │ -using temporary file /tmp/M2-37469-0/212
│ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-37469-0/214
│ │ │ │ +using temporary file /tmp/M2-59381-0/212
│ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-59381-0/214
│ │ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig
│ │ │ │  file. The input is a Groebner basis of a (not necessarily monomial) polynomial
│ │ │ │  ideal. The initial ideal is given by the leading terms in the Groebner basis.
│ │ │ │  Using the -m option it is possible to render more than one staircase diagram.
│ │ │ │  The program only works for ideals in a polynomial ring with three variables.
│ │ │ │  Options:
│ │ │ │  -m:
│ │ │ │ @@ -432,16 +432,16 @@
│ │ │ │  number is large enough to give a correct picture of the standard monomials. The
│ │ │ │  default value is 8.
│ │ │ │  
│ │ │ │  -w value:
│ │ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file.
│ │ │ │  The default value is 5.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-37469-0/214
│ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-37469-0/216
│ │ │ │ +using temporary file /tmp/M2-59381-0/214
│ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-59381-0/216
│ │ │ │  This program computes the resultant fan as defined in "Computing Tropical
│ │ │ │  Resultants" by Jensen and Yu. The input is a polynomial ring followed by
│ │ │ │  polynomials, whose coefficients are ignored. The output is the fan of
│ │ │ │  coefficients such that the input system has a tropical solution.
│ │ │ │  Options:
│ │ │ │  --codimension:
│ │ │ │   Compute only the codimension of the resultant fan and return.
│ │ │ │ @@ -469,28 +469,28 @@
│ │ │ │  of polynomials.
│ │ │ │  
│ │ │ │  --projection:
│ │ │ │   Use the projection method to compute the resultant fan. This works only if the
│ │ │ │  resultant fan is a hypersurface. If this option is combined with --special,
│ │ │ │  then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-37469-0/216
│ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-37469-0/218
│ │ │ │ +using temporary file /tmp/M2-59381-0/216
│ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-59381-0/218
│ │ │ │  This program computes the saturation of the input ideal with the product of the
│ │ │ │  variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous
│ │ │ │  generators).
│ │ │ │  
│ │ │ │  --noideal:
│ │ │ │   Do not treat input as an ideal but just factor out common monomial factors of
│ │ │ │  the input polynomials.
│ │ │ │ -using temporary file /tmp/M2-37469-0/218
│ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-37469-0/220
│ │ │ │ +using temporary file /tmp/M2-59381-0/218
│ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-59381-0/220
│ │ │ │  This program computes the secondary fan of a vector configuration. The
│ │ │ │  configuration is given as an ordered list of vectors. In order to compute the
│ │ │ │  secondary fan of a point configuration an additional coordinate of ones must be
│ │ │ │  added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │ │  Options:
│ │ │ │  --unimodular:
│ │ │ │   Use heuristics to search for unimodular triangulation rather than computing
│ │ │ │ @@ -519,103 +519,103 @@
│ │ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but
│ │ │ │  still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is
│ │ │ │  used.
│ │ │ │  --interrupt value:
│ │ │ │   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).
│ │ │ │ -using temporary file /tmp/M2-37469-0/220
│ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-37469-0/222
│ │ │ │ +using temporary file /tmp/M2-59381-0/220
│ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-59381-0/222
│ │ │ │  This program takes a list of reduced Groebner bases for the same ideal and
│ │ │ │  computes various statistics. The following information is listed: the number of
│ │ │ │  bases in the input, the number of variables, the dimension of the homogeneity
│ │ │ │  space, the maximal total degree of any polynomial in the input and the minimal
│ │ │ │  total degree of any basis in the input, the maximal number of polynomials and
│ │ │ │  terms in a basis in the input.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/222
│ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-37469-0/224
│ │ │ │ +using temporary file /tmp/M2-59381-0/222
│ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-59381-0/224
│ │ │ │  This program changes the variable names of a polynomial ring. The input is a
│ │ │ │  polynomial ring, a polynomial set in the ring and a new polynomial ring with
│ │ │ │  the same coefficient field but different variable names. The output is the
│ │ │ │  polynomial set written with the variable names of the second polynomial ring.
│ │ │ │  Example:
│ │ │ │  Input:
│ │ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │ │  Output:
│ │ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/224
│ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-37469-0/226
│ │ │ │ +using temporary file /tmp/M2-59381-0/224
│ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-59381-0/226
│ │ │ │  This program converts ASCII math to TeX math. The data-type is specified by the
│ │ │ │  options.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Add a header to the output. Using this option the output will be LaTeXable
│ │ │ │  right away.
│ │ │ │  --polynomialset_:
│ │ │ │   The data to be converted is a list of polynomials.
│ │ │ │  --polynomialsetlist_:
│ │ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ │ -using temporary file /tmp/M2-37469-0/226
│ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-37469-0/228
│ │ │ │ +using temporary file /tmp/M2-59381-0/226
│ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-59381-0/228
│ │ │ │  This program takes a list of reduced Groebner bases and produces the fan of all
│ │ │ │  faces of these. In this way by giving the complete list of reduced Groebner
│ │ │ │  bases, the Groebner fan can be computed as a polyhedral complex. The option --
│ │ │ │  restrict lets the user choose between computing the Groebner fan or the
│ │ │ │  restricted Groebner fan.
│ │ │ │  Options:
│ │ │ │  --restrict:
│ │ │ │   Add an inequality for each coordinate, so that the the cones are restricted to
│ │ │ │  the non-negative orthant.
│ │ │ │  --symmetry:
│ │ │ │   Tell the program to read in generators for a group of symmetries (subgroup of
│ │ │ │  $S_n$) after having read in the ring. The output is grouped according to these
│ │ │ │  symmetries. Only one representative for each orbit is needed on the input.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-37469-0/228
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-37469-0/230
│ │ │ │ +using temporary file /tmp/M2-59381-0/228
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-59381-0/230
│ │ │ │  This program computes a tropical basis for an ideal defining a tropical curve.
│ │ │ │  Defining a tropical curve means that the Krull dimension of R/I is at most 1 +
│ │ │ │  the dimension of the homogeneity space of I where R is the polynomial ring. The
│ │ │ │  input is a generating set for the ideal. If the input is not homogeneous option
│ │ │ │  -h must be used.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Homogenise the input before computing a tropical basis and dehomogenise the
│ │ │ │  output. This is needed if the input generators are not already homogeneous.
│ │ │ │ -using temporary file /tmp/M2-37469-0/230
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-37469-0/232
│ │ │ │ +using temporary file /tmp/M2-59381-0/230
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-59381-0/232
│ │ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and
│ │ │ │  computes the tropical variety of the ideal as a subfan of the Groebner fan. The
│ │ │ │  program is slow but works for any homogeneous ideal. If you know that your
│ │ │ │  ideal is prime over the complex numbers or you simply know that its tropical
│ │ │ │  variety is pure and connected in codimension one then use
│ │ │ │  gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/232
│ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-37469-0/234
│ │ │ │ +using temporary file /tmp/M2-59381-0/232
│ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-59381-0/234
│ │ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/234
│ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-37469-0/236
│ │ │ │ +using temporary file /tmp/M2-59381-0/234
│ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-59381-0/236
│ │ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise
│ │ │ │  linear function represented by a fan whose cones are the linear regions. Each
│ │ │ │  ray of the fan gets the value of the tropical function assigned to it. In other
│ │ │ │  words this program computes the normal fan of the Newton polytope of the input
│ │ │ │  polynomial with additional information.Options:
│ │ │ │  --exponents:
│ │ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ │ -using temporary file /tmp/M2-37469-0/236
│ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-37469-0/238
│ │ │ │ +using temporary file /tmp/M2-59381-0/236
│ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-59381-0/238
│ │ │ │  This program computes the tropical hypersurface defined by a principal ideal.
│ │ │ │  The input is the polynomial ring followed by a set containing just a generator
│ │ │ │  of the ideal.Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/238
│ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-37469-0/240
│ │ │ │ +using temporary file /tmp/M2-59381-0/238
│ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-59381-0/240
│ │ │ │  This program computes the set theoretical intersection of a set of tropical
│ │ │ │  hypersurfaces (or to be precise, their common refinement as a fan). The input
│ │ │ │  is a list of polynomials with each polynomial defining a hypersurface.
│ │ │ │  Considering tropical hypersurfaces as fans, the intersection can be computed as
│ │ │ │  the common refinement of these. Thus the output is a fan whose support is the
│ │ │ │  intersection of the tropical hypersurfaces.
│ │ │ │  Options:
│ │ │ │ @@ -652,16 +652,16 @@
│ │ │ │  --stable:
│ │ │ │   Find the stable intersection of the input polynomials using tropical
│ │ │ │  intersection theory. This can be slow. Most other options are ignored.
│ │ │ │  --parameters value:
│ │ │ │   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.
│ │ │ │ -using temporary file /tmp/M2-37469-0/240
│ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-37469-0/242
│ │ │ │ +using temporary file /tmp/M2-59381-0/240
│ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-59381-0/242
│ │ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and
│ │ │ │  Singular. The Singular part of the implementation can be found in:
│ │ │ │  
│ │ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry,
│ │ │ │  2007
│ │ │ │  
│ │ │ │ @@ -689,54 +689,54 @@
│ │ │ │  Options:
│ │ │ │  --noMult:
│ │ │ │   Disable the multiplicity computation.
│ │ │ │  -n value:
│ │ │ │   Number of variables that should have negative weight.
│ │ │ │  -c:
│ │ │ │   Only output a list of vectors being the possible choices.
│ │ │ │ -using temporary file /tmp/M2-37469-0/242
│ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-37469-0/244
│ │ │ │ +using temporary file /tmp/M2-59381-0/242
│ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-59381-0/244
│ │ │ │  This program generates tropical equations for a tropical linear space in the
│ │ │ │  Speyer sense given the tropical Pluecker coordinates as input.
│ │ │ │  Options:
│ │ │ │  -d value:
│ │ │ │   Specify d.
│ │ │ │  -n value:
│ │ │ │   Specify n.
│ │ │ │  --trees:
│ │ │ │   list the boundary trees (assumes d=3)
│ │ │ │ -using temporary file /tmp/M2-37469-0/244
│ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-37469-0/246
│ │ │ │ +using temporary file /tmp/M2-59381-0/244
│ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-59381-0/246
│ │ │ │  This program computes the multiplicity of a tropical cone given a marked
│ │ │ │  reduced Groebner basis for its initial ideal.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-37469-0/246
│ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-37469-0/248
│ │ │ │ +using temporary file /tmp/M2-59381-0/246
│ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-59381-0/248
│ │ │ │  This program will compute the tropical rank of matrix given as input. Tropical
│ │ │ │  addition is MAXIMUM.
│ │ │ │  Options:
│ │ │ │  --kapranov:
│ │ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │ │  --determinant:
│ │ │ │   Compute the tropical determinant instead.
│ │ │ │ -using temporary file /tmp/M2-37469-0/248
│ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-37469-0/250
│ │ │ │ +using temporary file /tmp/M2-59381-0/248
│ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-59381-0/250
│ │ │ │  This program computes a starting pair of marked reduced Groebner bases to be
│ │ │ │  used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose
│ │ │ │  tropical variety is a pure d-dimensional polyhedral complex.
│ │ │ │  Options:
│ │ │ │  -g:
│ │ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ │  -d:
│ │ │ │   Output dimension information to standard error.
│ │ │ │  --stable:
│ │ │ │   Find starting cone in the stable intersection or, equivalently, pretend that
│ │ │ │  the coefficients are genereric.
│ │ │ │ -using temporary file /tmp/M2-37469-0/250
│ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-37469-0/252
│ │ │ │ +using temporary file /tmp/M2-59381-0/250
│ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-59381-0/252
│ │ │ │  This program computes a polyhedral fan representation of the tropical variety
│ │ │ │  of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let
│ │ │ │  $\omega$ be a relative interior point of $d$-dimensional Groebner cone
│ │ │ │  contained in the tropical variety. The input for this program is a pair of
│ │ │ │  marked reduced Groebner bases with respect to the term order represented by
│ │ │ │  $\omega$, tie-broken in some way. The first one is for the initial ideal
│ │ │ │  $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point
│ │ │ │ @@ -766,27 +766,27 @@
│ │ │ │  --stable:
│ │ │ │   Traverse the stable intersection or, equivalently, pretend that the
│ │ │ │  coefficients are genereric.
│ │ │ │  --interrupt value:
│ │ │ │   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).
│ │ │ │ -using temporary file /tmp/M2-37469-0/252
│ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-37469-0/254
│ │ │ │ +using temporary file /tmp/M2-59381-0/252
│ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-59381-0/254
│ │ │ │  This program computes the tropical Weil divisor of piecewise linear (or
│ │ │ │  tropical rational) function on a tropical k-cycle. See the Gfan manual for more
│ │ │ │  information.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the Polymake input file containing the piecewise linear
│ │ │ │  function.
│ │ │ │ -using temporary file /tmp/M2-37469-0/254
│ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-37469-0/256
│ │ │ │ +using temporary file /tmp/M2-59381-0/254
│ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-59381-0/256
│ │ │ │  This program is an experimental implementation of Groebner bases for ideals in
│ │ │ │  Z[x_1,...,x_n].
│ │ │ │  Several operations are supported by specifying the appropriate option:
│ │ │ │   (1) computation of the reduced Groebner basis with respect to a given vector
│ │ │ │  (tiebroken lexicographically),
│ │ │ │   (2) computation of an initial ideal,
│ │ │ │   (3) computation of the Groebner fan,
│ │ │ │ @@ -821,20 +821,20 @@
│ │ │ │   For the operations taking a vector as input, read in a list of vectors
│ │ │ │  instead, and perform the operation for each vector in the list.
│ │ │ │  -g:
│ │ │ │   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 the
│ │ │ │  usual --groebnerFan is too slow.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-37469-0/256
│ │ │ │ +using temporary file /tmp/M2-59381-0/256
│ │ │ │  i6 : QQ[x,y];
│ │ │ │  i7 : gfan {x,y};
│ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37469-0/258
│ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-59381-0/258
│ │ │ │  Q[x1,x2]
│ │ │ │  {{
│ │ │ │  x2,
│ │ │ │  x1}
│ │ │ │  }
│ │ │ │ -using temporary file /tmp/M2-37469-0/258
│ │ │ │ +using temporary file /tmp/M2-59381-0/258
│ │ │ │  Finally, if you want to be able to render Groebner fans and monomial staircases
│ │ │ │  to .png files, you should install fig2dev. If it is installed in a non-standard
│ │ │ │  location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s.
│ │ ├── ./usr/share/info/AInfinity.info.gz
│ │ │ ├── AInfinity.info
│ │ │ │ @@ -6047,15 +6047,15 @@
│ │ │ │  000179e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000179f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00017a10: 3320 3a20 656c 6170 7365 6454 696d 6520  3 : elapsedTime 
│ │ │ │  00017a20: 6275 726b 6552 6573 6f6c 7574 696f 6e28  burkeResolution(
│ │ │ │  00017a30: 4d2c 2037 2c20 4368 6563 6b20 3d3e 2066  M, 7, Check => f
│ │ │ │  00017a40: 616c 7365 2920 2020 2020 2020 2020 2020  alse)           
│ │ │ │ -00017a50: 7c0a 7c20 2d2d 2031 2e36 3332 3132 7320  |.| -- 1.63212s 
│ │ │ │ +00017a50: 7c0a 7c20 2d2d 2032 2e37 3135 3537 7320  |.| -- 2.71557s 
│ │ │ │  00017a60: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00017a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00017aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6089,16 +6089,16 @@
│ │ │ │  00017c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017cb0: 2d2d 2d2d 2b0a 7c69 3420 3a20 656c 6170  ----+.|i4 : elap
│ │ │ │  00017cc0: 7365 6454 696d 6520 6275 726b 6552 6573  sedTime burkeRes
│ │ │ │  00017cd0: 6f6c 7574 696f 6e28 4d2c 2037 2c20 4368  olution(M, 7, Ch
│ │ │ │  00017ce0: 6563 6b20 3d3e 2074 7275 6529 2020 2020  eck => true)    
│ │ │ │ -00017cf0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2031          |.| -- 1
│ │ │ │ -00017d00: 2e39 3337 3832 7320 656c 6170 7365 6420  .93782s elapsed 
│ │ │ │ +00017cf0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2033          |.| -- 3
│ │ │ │ +00017d00: 2e33 3838 3936 7320 656c 6170 7365 6420  .38896s elapsed 
│ │ │ │  00017d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d70: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/AbstractSimplicialComplexes.info.gz
│ │ │ ├── AbstractSimplicialComplexes.info
│ │ │ │ @@ -1782,19 +1782,19 @@
│ │ │ │  00006f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006f70: 2020 2020 7c0a 7c6f 3220 3d20 4162 7374      |.|o2 = Abst
│ │ │ │  00006f80: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │  00006f90: 6d70 6c65 787b 2d31 203d 3e20 7b7b 7d7d  mplex{-1 => {{}}
│ │ │ │  00006fa0: 2020 2020 207d 7c0a 7c20 2020 2020 2020       }|.|       
│ │ │ │  00006fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006fc0: 2020 2020 2020 2020 3020 3d3e 207b 7b32          0 => {{2
│ │ │ │ -00006fd0: 7d2c 207b 337d 7d20 7c0a 7c20 2020 2020  }, {3}} |.|     
│ │ │ │ +00006fc0: 2020 2020 2020 2020 3020 3d3e 207b 7b33          0 => {{3
│ │ │ │ +00006fd0: 7d2c 207b 347d 7d20 7c0a 7c20 2020 2020  }, {4}} |.|     
│ │ │ │  00006fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006ff0: 2020 2020 2020 2020 2020 3120 3d3e 207b            1 => {
│ │ │ │ -00007000: 7b32 2c20 337d 7d20 2020 7c0a 7c20 2020  {2, 3}}   |.|   
│ │ │ │ +00007000: 7b33 2c20 347d 7d20 2020 7c0a 7c20 2020  {3, 4}}   |.|   
│ │ │ │  00007010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007030: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  00007040: 3220 3a20 4162 7374 7261 6374 5369 6d70  2 : AbstractSimp
│ │ │ │  00007050: 6c69 6369 616c 436f 6d70 6c65 7820 2020  licialComplex   
│ │ │ │  00007060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00007070: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ @@ -1832,3601 +1832,3633 @@
│ │ │ │  00007270: 6e64 6f6d 2073 696d 706c 6963 6961 6c20  ndom simplicial 
│ │ │ │  00007280: 636f 6d70 6c65 7820 6f6e 205b 6e5d 2077  complex on [n] w
│ │ │ │  00007290: 6974 6820 6469 6d65 6e73 696f 6e20 6174  ith dimension at
│ │ │ │  000072a0: 206d 6f73 7420 6571 7561 6c20 746f 2072   most equal to r
│ │ │ │  000072b0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  000072c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000072d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00007980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000079a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +000079b0: 3a20 4d20 3d20 7261 6e64 6f6d 4162 7374  : M = randomAbst
│ │ │ │ +000079c0: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ +000079d0: 6d70 6c65 7828 362c 332c 3229 2020 2020  mplex(6,3,2)    
│ │ │ │ +000079e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000079f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007a40: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00007a50: 3d20 4162 7374 7261 6374 5369 6d70 6c69  = AbstractSimpli
│ │ │ │ +00007a60: 6369 616c 436f 6d70 6c65 787b 2d31 203d  cialComplex{-1 =
│ │ │ │ +00007a70: 3e20 7b7b 7d7d 2020 2020 2020 2020 2020  > {{}}          
│ │ │ │ +00007a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007ab0: 2020 2020 2020 2020 2020 2020 3020 3d3e              0 =>
│ │ │ │ +00007ac0: 207b 7b31 7d2c 207b 327d 2c20 7b33 7d2c   {{1}, {2}, {3},
│ │ │ │ +00007ad0: 207b 347d 2c20 7b35 7d2c 207b 367d 7d20   {4}, {5}, {6}} 
│ │ │ │ +00007ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007b00: 2020 2020 2020 2020 2020 2020 3120 3d3e              1 =>
│ │ │ │ +00007b10: 207b 7b31 2c20 337d 2c20 7b31 2c20 347d   {{1, 3}, {1, 4}
│ │ │ │ +00007b20: 2c20 7b31 2c20 357d 2c20 7b32 2c20 357d  , {1, 5}, {2, 5}
│ │ │ │ +00007b30: 2c20 7b32 2c20 367d 2c20 7c0a 7c20 2020  , {2, 6}, |.|   
│ │ │ │ +00007b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007b50: 2020 2020 2020 2020 2020 2020 3220 3d3e              2 =>
│ │ │ │ +00007b60: 207b 7b31 2c20 332c 2034 7d2c 207b 312c   {{1, 3, 4}, {1,
│ │ │ │ +00007b70: 2034 2c20 357d 2c20 7b32 2c20 352c 2036   4, 5}, {2, 5, 6
│ │ │ │ +00007b80: 7d7d 2020 2020 2020 2020 7c0a 7c20 2020  }}        |.|   
│ │ │ │  00007b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007bc0: 2020 7c0a 7c7b 332c 2034 7d2c 207b 332c    |.|{3, 4}, {3,
│ │ │ │ -00007bd0: 2035 7d2c 207b 332c 2036 7d2c 207b 352c   5}, {3, 6}, {5,
│ │ │ │ -00007be0: 2036 7d7d 2020 2020 2020 2020 2020 2020   6}}            
│ │ │ │ -00007bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007bd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00007be0: 3a20 4162 7374 7261 6374 5369 6d70 6c69  : AbstractSimpli
│ │ │ │ +00007bf0: 6369 616c 436f 6d70 6c65 7820 2020 2020  cialComplex     
│ │ │ │  00007c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007c10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00007c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007c20: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  00007c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007c60: 2d2d 2b0a 7c69 3920 3a20 7072 756e 6520  --+.|i9 : prune 
│ │ │ │ -00007c70: 4848 2073 696d 706c 6963 6961 6c43 6861  HH simplicialCha
│ │ │ │ -00007c80: 696e 436f 6d70 6c65 7820 4d20 2020 2020  inComplex M     
│ │ │ │ -00007c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007c70: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ +00007c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007c90: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │  00007ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00007cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00007cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007d00: 2020 7c0a 7c20 2020 2020 2020 3120 2020    |.|       1   
│ │ │ │ -00007d10: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -00007d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007d10: 2020 2020 2020 2020 2020 7c0a 7c7b 332c            |.|{3,
│ │ │ │ +00007d20: 2034 7d2c 207b 342c 2035 7d2c 207b 352c   4}, {4, 5}, {5,
│ │ │ │ +00007d30: 2036 7d7d 2020 2020 2020 2020 2020 2020   6}}            
│ │ │ │  00007d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007d50: 2020 7c0a 7c6f 3920 3d20 5a5a 2020 3c2d    |.|o9 = ZZ  <-
│ │ │ │ -00007d60: 2d20 5a5a 2020 2020 2020 2020 2020 2020  - ZZ            
│ │ │ │ -00007d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007da0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00007db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007df0: 2020 7c0a 7c20 2020 2020 3020 2020 2020    |.|     0     
│ │ │ │ -00007e00: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00007d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007d60: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00007d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920  ----------+.|i9 
│ │ │ │ +00007dc0: 3a20 7072 756e 6520 4848 2073 696d 706c  : prune HH simpl
│ │ │ │ +00007dd0: 6963 6961 6c43 6861 696e 436f 6d70 6c65  icialChainComple
│ │ │ │ +00007de0: 7820 4d20 2020 2020 2020 2020 2020 2020  x M             
│ │ │ │ +00007df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007e00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00007e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00007e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007e50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007e60: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00007e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007e90: 2020 7c0a 7c6f 3920 3a20 436f 6d70 6c65    |.|o9 : Comple
│ │ │ │ -00007ea0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00007eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007ea0: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +00007eb0: 3d20 5a5a 2020 2020 2020 2020 2020 2020  = ZZ            
│ │ │ │  00007ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007ee0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00007ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007f30: 2d2d 2b0a 0a43 7265 6174 6573 2061 2072  --+..Creates a r
│ │ │ │ -00007f40: 616e 646f 6d20 7375 622d 7369 6d70 6c69  andom sub-simpli
│ │ │ │ -00007f50: 6369 616c 2063 6f6d 706c 6578 206f 6620  cial complex of 
│ │ │ │ -00007f60: 6120 6769 7665 6e20 7369 6d70 6c69 6369  a given simplici
│ │ │ │ -00007f70: 616c 2063 6f6d 706c 6578 2e0a 0a2b 2d2d  al complex...+--
│ │ │ │ -00007f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00007fb0: 7c69 3130 203a 2073 6574 5261 6e64 6f6d  |i10 : setRandom
│ │ │ │ -00007fc0: 5365 6564 2863 7572 7265 6e74 5469 6d65  Seed(currentTime
│ │ │ │ -00007fd0: 2829 293b 2020 2020 2020 2020 2020 2020  ());            
│ │ │ │ -00007fe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00007ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008010: 2d2d 2d2d 2b0a 7c69 3131 203a 204b 203d  ----+.|i11 : K =
│ │ │ │ -00008020: 2072 616e 646f 6d41 6273 7472 6163 7453   randomAbstractS
│ │ │ │ -00008030: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ -00008040: 2834 2920 2020 207c 0a7c 2020 2020 2020  (4)    |.|      
│ │ │ │ -00008050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008070: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -00008080: 203d 2041 6273 7472 6163 7453 696d 706c   = AbstractSimpl
│ │ │ │ -00008090: 6963 6961 6c43 6f6d 706c 6578 7b2d 3120  icialComplex{-1 
│ │ │ │ -000080a0: 3d3e 207b 7b7d 7d20 2020 2020 7d7c 0a7c  => {{}}     }|.|
│ │ │ │ -000080b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000080c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000080d0: 3020 3d3e 207b 7b32 7d2c 207b 337d 7d20  0 => {{2}, {3}} 
│ │ │ │ -000080e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000080f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008100: 2020 2031 203d 3e20 7b7b 322c 2033 7d7d     1 => {{2, 3}}
│ │ │ │ -00008110: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00008120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008140: 2020 2020 2020 7c0a 7c6f 3131 203a 2041        |.|o11 : A
│ │ │ │ -00008150: 6273 7472 6163 7453 696d 706c 6963 6961  bstractSimplicia
│ │ │ │ -00008160: 6c43 6f6d 706c 6578 2020 2020 2020 2020  lComplex        
│ │ │ │ -00008170: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00008180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000081a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000081b0: 3132 203a 204a 203d 2072 616e 646f 6d53  12 : J = randomS
│ │ │ │ -000081c0: 7562 5369 6d70 6c69 6369 616c 436f 6d70  ubSimplicialComp
│ │ │ │ -000081d0: 6c65 7828 4b29 2020 2020 2020 2020 207c  lex(K)         |
│ │ │ │ -000081e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000081f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008210: 2020 7c0a 7c6f 3132 203d 2041 6273 7472    |.|o12 = Abstr
│ │ │ │ -00008220: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ -00008230: 706c 6578 7b2d 3120 3d3e 207b 7b7d 7d7d  plex{-1 => {{}}}
│ │ │ │ -00008240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00008250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008260: 2020 2020 2020 2020 3020 3d3e 207b 7b32          0 => {{2
│ │ │ │ -00008270: 7d7d 2020 2020 2020 7c0a 7c20 2020 2020  }}      |.|     
│ │ │ │ +00007ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007ef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007f50: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00007f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007f90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00007fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007fe0: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +00007ff0: 3a20 436f 6d70 6c65 7820 2020 2020 2020  : Complex       
│ │ │ │ +00008000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008030: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00008040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008080: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 7265  ----------+..Cre
│ │ │ │ +00008090: 6174 6573 2061 2072 616e 646f 6d20 7375  ates a random su
│ │ │ │ +000080a0: 622d 7369 6d70 6c69 6369 616c 2063 6f6d  b-simplicial com
│ │ │ │ +000080b0: 706c 6578 206f 6620 6120 6769 7665 6e20  plex of a given 
│ │ │ │ +000080c0: 7369 6d70 6c69 6369 616c 2063 6f6d 706c  simplicial compl
│ │ │ │ +000080d0: 6578 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ex...+----------
│ │ │ │ +000080e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000080f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008100: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2073  ------+.|i10 : s
│ │ │ │ +00008110: 6574 5261 6e64 6f6d 5365 6564 2863 7572  etRandomSeed(cur
│ │ │ │ +00008120: 7265 6e74 5469 6d65 2829 293b 2020 2020  rentTime());    
│ │ │ │ +00008130: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00008140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00008170: 3131 203a 204b 203d 2072 616e 646f 6d41  11 : K = randomA
│ │ │ │ +00008180: 6273 7472 6163 7453 696d 706c 6963 6961  bstractSimplicia
│ │ │ │ +00008190: 6c43 6f6d 706c 6578 2834 2920 2020 207c  lComplex(4)    |
│ │ │ │ +000081a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000081b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000081c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000081d0: 2020 7c0a 7c6f 3131 203d 2041 6273 7472    |.|o11 = Abstr
│ │ │ │ +000081e0: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ +000081f0: 706c 6578 7b2d 3120 3d3e 207b 7b7d 7d20  plex{-1 => {{}} 
│ │ │ │ +00008200: 2020 2020 7d7c 0a7c 2020 2020 2020 2020      }|.|        
│ │ │ │ +00008210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008220: 2020 2020 2020 2020 3020 3d3e 207b 7b33          0 => {{3
│ │ │ │ +00008230: 7d2c 207b 347d 7d20 7c0a 7c20 2020 2020  }, {4}} |.|     
│ │ │ │ +00008240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008250: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
│ │ │ │ +00008260: 7b7b 332c 2034 7d7d 2020 207c 0a7c 2020  {{3, 4}}   |.|  
│ │ │ │ +00008270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000082a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000082b0: 3220 3a20 4162 7374 7261 6374 5369 6d70  2 : AbstractSimp
│ │ │ │ -000082c0: 6c69 6369 616c 436f 6d70 6c65 7820 2020  licialComplex   
│ │ │ │ -000082d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000082e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00008290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000082a0: 7c6f 3131 203a 2041 6273 7472 6163 7453  |o11 : AbstractS
│ │ │ │ +000082b0: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +000082c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000082d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000082e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000088c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00008a50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00008a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00008c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00008bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00008c20: 2020 2020 2020 2020 2020 2020 2020 7d7c                }|
│ │ │ │ +00008c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00008c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008c50: 2030 203d 3e20 7b7b 317d 2c20 7b32 7d2c   0 => {{1}, {2},
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│ │ │ │ +00008c70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00008c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008c90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00008ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008c90: 2031 203d 3e20 7b7b 312c 2032 7d2c 207b   1 => {{1, 2}, {
│ │ │ │ +00008ca0: 312c 2033 7d2c 207b 322c 2033 7d7d 207c  1, 3}, {2, 3}} |
│ │ │ │ +00008cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00008cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008cd0: 2020 2020 2020 207c 0a7c 6f32 203d 2032         |.|o2 = 2
│ │ │ │ -00008ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008cd0: 2032 203d 3e20 7b7b 312c 2032 2c20 337d   2 => {{1, 2, 3}
│ │ │ │ +00008ce0: 7d20 2020 2020 2020 2020 2020 2020 207c  }              |
│ │ │ │ +00008cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00008d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008d10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00008d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00008d80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00008d90: 202a 6e6f 7465 2064 696d 2841 6273 7472   *note dim(Abstr
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│ │ │ │ -00008e30: 2c20 4e6f 6465 3a20 486f 7720 746f 206d  , Node: How to m
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│ │ │ │ -00008e50: 706c 6963 6961 6c20 636f 6d70 6c65 7865  plicial complexe
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│ │ │ │ -00008e70: 6d61 6b65 2072 6564 7563 6564 2061 6e64  make reduced and
│ │ │ │ -00008e80: 206e 6f6e 2d72 6564 7563 6564 2073 696d   non-reduced sim
│ │ │ │ -00008e90: 706c 6963 6961 6c20 6368 6169 6e20 636f  plicial chain co
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│ │ │ │ -00008ec0: 706c 6963 6961 6c43 6f6d 706c 6578 5f72  plicialComplex_r
│ │ │ │ -00008ed0: 702c 2055 703a 2054 6f70 0a0a 486f 7720  p, Up: Top..How 
│ │ │ │ -00008ee0: 746f 206d 616b 6520 6162 7374 7261 6374  to make abstract
│ │ │ │ -00008ef0: 2073 696d 706c 6963 6961 6c20 636f 6d70   simplicial comp
│ │ │ │ -00008f00: 6c65 7865 7320 2d2d 2055 7369 6e67 2074  lexes -- Using t
│ │ │ │ -00008f10: 6865 2074 7970 6520 4162 7374 7261 6374  he type Abstract
│ │ │ │ -00008f20: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ -00008f30: 7873 2074 6f20 7265 7072 6573 656e 7420  xs to represent 
│ │ │ │ -00008f40: 6162 7374 7261 6374 2073 696d 706c 6963  abstract simplic
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│ │ │ │ -00008f60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008f80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008f90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008fb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00008fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00008fe0: 0a54 6865 2074 7970 6520 4162 7374 7261  .The type Abstra
│ │ │ │ -00008ff0: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ -00009000: 6c65 7820 6973 2061 2064 6174 6120 7479  lex is a data ty
│ │ │ │ -00009010: 7065 2066 6f72 2077 6f72 6b69 6e67 2077  pe for working w
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│ │ │ │ -00009030: 706c 6963 6961 6c20 636f 6d70 6c65 7865  plicial complexe
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│ │ │ │ -00009050: 7375 7070 6f72 7465 6420 6f6e 205b 6e5d  supported on [n]
│ │ │ │ -00009060: 203d 207b 312c 2e2e 2e2c 6e7d 2e20 4865   = {1,...,n}. He
│ │ │ │ -00009070: 7265 2077 650a 696c 6c75 7374 7261 7465  re we.illustrate
│ │ │ │ -00009080: 2073 6f6d 6520 6f66 2074 6865 206d 6f73   some of the mos
│ │ │ │ -00009090: 7420 6261 7369 6320 7761 7973 2074 6f20  t basic ways to 
│ │ │ │ -000090a0: 696e 7465 7261 6374 2077 6974 6820 7468  interact with th
│ │ │ │ -000090b0: 6973 2064 6174 6120 7479 7065 2e0a 0a54  is data type...T
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│ │ │ │ -00009110: 6f6e 7374 7275 6374 6564 2069 6e20 7468  onstructed in th
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│ │ │ │ -00009130: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ -00009140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009180: 2b0a 7c69 3120 3a20 4b20 3d20 6162 7374  +.|i1 : K = abst
│ │ │ │ -00009190: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -000091a0: 6d70 6c65 7828 7b7b 312c 322c 332c 347d  mplex({{1,2,3,4}
│ │ │ │ -000091b0: 2c20 7b32 2c33 2c35 7d2c 7b31 2c35 7d7d  , {2,3,5},{1,5}}
│ │ │ │ -000091c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -000091d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000091e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000091f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009220: 7c0a 7c6f 3120 3d20 4162 7374 7261 6374  |.|o1 = Abstract
│ │ │ │ -00009230: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ -00009240: 787b 2d31 203d 3e20 7b7b 7d7d 2020 2020  x{-1 => {{}}    
│ │ │ │ -00009250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009270: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00009280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009290: 2020 3020 3d3e 207b 7b31 7d2c 207b 327d    0 => {{1}, {2}
│ │ │ │ -000092a0: 2c20 7b33 7d2c 207b 347d 2c20 7b35 7d7d  , {3}, {4}, {5}}
│ │ │ │ -000092b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000092c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000092d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000092e0: 2020 3120 3d3e 207b 7b31 2c20 327d 2c20    1 => {{1, 2}, 
│ │ │ │ -000092f0: 7b31 2c20 337d 2c20 7b31 2c20 347d 2c20  {1, 3}, {1, 4}, 
│ │ │ │ -00009300: 7b31 2c20 357d 2c20 7b32 2c20 337d 2c20  {1, 5}, {2, 3}, 
│ │ │ │ -00009310: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00009320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009330: 2020 3220 3d3e 207b 7b31 2c20 322c 2033    2 => {{1, 2, 3
│ │ │ │ -00009340: 7d2c 207b 312c 2032 2c20 347d 2c20 7b31  }, {1, 2, 4}, {1
│ │ │ │ -00009350: 2c20 332c 2034 7d2c 207b 322c 2033 2c20  , 3, 4}, {2, 3, 
│ │ │ │ -00009360: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00009370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009380: 2020 3320 3d3e 207b 7b31 2c20 322c 2033    3 => {{1, 2, 3
│ │ │ │ -00009390: 2c20 347d 7d20 2020 2020 2020 2020 2020  , 4}}           
│ │ │ │ -000093a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000093b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000093c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00008d30: 0a7c 6f31 203a 2041 6273 7472 6163 7453  .|o1 : AbstractS
│ │ │ │ +00008d40: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +00008d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00008d70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00008db0: 0a7c 6932 203a 2064 696d 204b 2020 2020  .|i2 : dim K    
│ │ │ │ +00008dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00008df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00008e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00008e30: 0a7c 6f32 203d 2032 2020 2020 2020 2020  .|o2 = 2        
│ │ │ │ +00008e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00008e70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │ +00008ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00008eb0: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
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│ │ │ │ +000090c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000090d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000090e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000090f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00009100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00009110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00009120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00009130: 2a2a 2a2a 2a2a 2a0a 0a54 6865 2074 7970  *******..The typ
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│ │ │ │ -000095b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00009620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00009690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00009660: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00009700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00009790: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000097b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00009800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009810: 2020 2020 2020 2020 2020 2020 3120 3d3e              1 =>
│ │ │ │ -00009820: 207b 7b31 2c20 327d 2c20 7b31 2c20 337d   {{1, 2}, {1, 3}
│ │ │ │ -00009830: 2c20 7b31 2c20 347d 2c20 7b32 2c20 337d  , {1, 4}, {2, 3}
│ │ │ │ -00009840: 2c20 7b32 2c20 347d 2c20 7c0a 7c20 2020  , {2, 4}, |.|   
│ │ │ │ +00009730: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00009740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000097c0: 6173 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  as..+-----------
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│ │ │ │ +000097e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000097f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00009860: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00009870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000098a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000098b0: 2020 2020 2020 2020 2020 2020 3320 3d3e              3 =>
│ │ │ │ -000098c0: 207b 7b31 2c20 322c 2033 2c20 347d 7d20   {{1, 2, 3, 4}} 
│ │ │ │ -000098d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000098e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000098b0: 2020 7c0a 7c6f 3220 3d20 4162 7374 7261    |.|o2 = Abstra
│ │ │ │ +000098c0: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ +000098d0: 6c65 787b 2d31 203d 3e20 7b7b 7d7d 2020  lex{-1 => {{}}  
│ │ │ │ +000098e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00009900: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00009910: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00009930: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00009940: 3a20 4162 7374 7261 6374 5369 6d70 6c69  : AbstractSimpli
│ │ │ │ -00009950: 6369 616c 436f 6d70 6c65 7820 2020 2020  cialComplex     
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│ │ │ │ -00009990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000099d0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ -000099e0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ -000099f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009970: 2020 2020 3120 3d3e 207b 7b31 2c20 327d      1 => {{1, 2}
│ │ │ │ +00009980: 2c20 7b31 2c20 337d 2c20 7b31 2c20 347d  , {1, 3}, {1, 4}
│ │ │ │ +00009990: 2c20 7b32 2c20 337d 2c20 7b32 2c20 347d  , {2, 3}, {2, 4}
│ │ │ │ +000099a0: 2c20 7c0a 7c20 2020 2020 2020 2020 2020  , |.|           
│ │ │ │ +000099b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000099d0: 2033 7d2c 207b 312c 2032 2c20 347d 2c20   3}, {1, 2, 4}, 
│ │ │ │ +000099e0: 7b31 2c20 332c 2034 7d2c 207b 322c 2033  {1, 3, 4}, {2, 3
│ │ │ │ +000099f0: 2c20 7c0a 7c20 2020 2020 2020 2020 2020  , |.|           
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│ │ │ │ +00009a10: 2020 2020 3320 3d3e 207b 7b31 2c20 322c      3 => {{1, 2,
│ │ │ │ +00009a20: 2033 2c20 347d 7d20 2020 2020 2020 2020   3, 4}}         
│ │ │ │  00009a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00009a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009a70: 2020 2020 2020 2020 2020 7c0a 7c7b 332c            |.|{3,
│ │ │ │ -00009a80: 2034 7d7d 2020 2020 2020 2020 2020 2020   4}}            
│ │ │ │ -00009a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00009ac0: 2020 2020 2020 2020 2020 7c0a 7c34 7d7d            |.|4}}
│ │ │ │ +00009a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009a90: 2020 7c0a 7c6f 3220 3a20 4162 7374 7261    |.|o2 : Abstra
│ │ │ │ +00009aa0: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ +00009ab0: 6c65 7820 2020 2020 2020 2020 2020 2020  lex             
│ │ │ │ +00009ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009b10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00009ae0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +00009af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865  ----------+..The
│ │ │ │ -00009b70: 2066 6163 6573 2061 6e64 2066 6163 6574   faces and facet
│ │ │ │ -00009b80: 7320 6f66 2073 7563 6820 7369 6d70 6c69  s of such simpli
│ │ │ │ -00009b90: 6369 616c 2063 6f6d 706c 6578 6573 2063  cial complexes c
│ │ │ │ -00009ba0: 616e 2062 6520 6163 6365 7373 6564 2061  an be accessed a
│ │ │ │ -00009bb0: 730a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  s..+------------
│ │ │ │ -00009bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009c00: 2d2b 0a7c 6933 203a 204b 5f28 2d31 2920  -+.|i3 : K_(-1) 
│ │ │ │ +00009b30: 2d2d 7c0a 7c20 2020 2020 2020 7d20 2020  --|.|       }   
│ │ │ │ +00009b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00009b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009bd0: 2020 7c0a 7c7b 332c 2034 7d7d 2020 2020    |.|{3, 4}}    
│ │ │ │ +00009be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009c20: 2020 7c0a 7c34 7d7d 2020 2020 2020 2020    |.|4}}        
│ │ │ │  00009c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009c50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00009c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ca0: 207c 0a7c 6f33 203d 207b 7b7d 7d20 2020   |.|o3 = {{}}   
│ │ │ │ -00009cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009cf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00009d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d40: 207c 0a7c 6f33 203a 204c 6973 7420 2020   |.|o3 : List   
│ │ │ │ -00009d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00009c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009cc0: 2d2d 2b0a 0a54 6865 2066 6163 6573 2061  --+..The faces a
│ │ │ │ +00009cd0: 6e64 2066 6163 6574 7320 6f66 2073 7563  nd facets of suc
│ │ │ │ +00009ce0: 6820 7369 6d70 6c69 6369 616c 2063 6f6d  h simplicial com
│ │ │ │ +00009cf0: 706c 6578 6573 2063 616e 2062 6520 6163  plexes can be ac
│ │ │ │ +00009d00: 6365 7373 6564 2061 730a 0a2b 2d2d 2d2d  cessed as..+----
│ │ │ │ +00009d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009d50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00009d60: 204b 5f28 2d31 2920 2020 2020 2020 2020   K_(-1)         
│ │ │ │  00009d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00009da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009de0: 2d2b 0a7c 6934 203a 204b 5f30 2020 2020  -+.|i4 : K_0    
│ │ │ │ -00009df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009da0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00009db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009df0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +00009e00: 207b 7b7d 7d20 2020 2020 2020 2020 2020   {{}}           
│ │ │ │  00009e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009e30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00009e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009e40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00009e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009e80: 207c 0a7c 6f34 203d 207b 7b31 7d2c 207b   |.|o4 = {{1}, {
│ │ │ │ -00009e90: 327d 2c20 7b33 7d2c 207b 347d 2c20 7b35  2}, {3}, {4}, {5
│ │ │ │ -00009ea0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00009e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009e90: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00009ea0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  00009eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ed0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00009ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009f20: 207c 0a7c 6f34 203a 204c 6973 7420 2020   |.|o4 : List   
│ │ │ │ -00009f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009ee0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00009ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009f30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00009f40: 204b 5f30 2020 2020 2020 2020 2020 2020   K_0            
│ │ │ │  00009f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009f70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00009f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009fc0: 2d2b 0a7c 6935 203a 204b 5f31 2020 2020  -+.|i5 : K_1    
│ │ │ │ -00009fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009f80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00009f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009fd0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +00009fe0: 207b 7b31 7d2c 207b 327d 2c20 7b33 7d2c   {{1}, {2}, {3},
│ │ │ │ +00009ff0: 207b 347d 2c20 7b35 7d7d 2020 2020 2020   {4}, {5}}      
│ │ │ │  0000a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a010: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a060: 207c 0a7c 6f35 203d 207b 7b31 2c20 327d   |.|o5 = {{1, 2}
│ │ │ │ -0000a070: 2c20 7b31 2c20 337d 2c20 7b31 2c20 347d  , {1, 3}, {1, 4}
│ │ │ │ -0000a080: 2c20 7b31 2c20 357d 2c20 7b32 2c20 337d  , {1, 5}, {2, 3}
│ │ │ │ -0000a090: 2c20 7b32 2c20 347d 2c20 7b32 2c20 357d  , {2, 4}, {2, 5}
│ │ │ │ -0000a0a0: 2c20 7b33 2c20 347d 2c20 7b33 2c20 357d  , {3, 4}, {3, 5}
│ │ │ │ -0000a0b0: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ -0000a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a100: 207c 0a7c 6f35 203a 204c 6973 7420 2020   |.|o5 : List   
│ │ │ │ -0000a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a070: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0000a080: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +0000a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a0c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a110: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0000a120: 204b 5f31 2020 2020 2020 2020 2020 2020   K_1            
│ │ │ │  0000a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a1a0: 2d2b 0a7c 6936 203a 204b 5f32 2020 2020  -+.|i6 : K_2    
│ │ │ │ -0000a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a1f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a160: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a1b0: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +0000a1c0: 207b 7b31 2c20 327d 2c20 7b31 2c20 337d   {{1, 2}, {1, 3}
│ │ │ │ +0000a1d0: 2c20 7b31 2c20 347d 2c20 7b31 2c20 357d  , {1, 4}, {1, 5}
│ │ │ │ +0000a1e0: 2c20 7b32 2c20 337d 2c20 7b32 2c20 347d  , {2, 3}, {2, 4}
│ │ │ │ +0000a1f0: 2c20 7b32 2c20 357d 2c20 7b33 2c20 347d  , {2, 5}, {3, 4}
│ │ │ │ +0000a200: 2c20 7b33 2c20 357d 7d7c 0a7c 2020 2020  , {3, 5}}|.|    
│ │ │ │  0000a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a240: 207c 0a7c 6f36 203d 207b 7b31 2c20 322c   |.|o6 = {{1, 2,
│ │ │ │ -0000a250: 2033 7d2c 207b 312c 2032 2c20 347d 2c20   3}, {1, 2, 4}, 
│ │ │ │ -0000a260: 7b31 2c20 332c 2034 7d2c 207b 322c 2033  {1, 3, 4}, {2, 3
│ │ │ │ -0000a270: 2c20 347d 2c20 7b32 2c20 332c 2035 7d7d  , 4}, {2, 3, 5}}
│ │ │ │ +0000a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a250: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +0000a260: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +0000a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a290: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a2e0: 207c 0a7c 6f36 203a 204c 6973 7420 2020   |.|o6 : List   
│ │ │ │ -0000a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a2a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0000a300: 204b 5f32 2020 2020 2020 2020 2020 2020   K_2            
│ │ │ │  0000a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a330: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a380: 2d2b 0a7c 6937 203a 2061 6273 7472 6163  -+.|i7 : abstrac
│ │ │ │ -0000a390: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ -0000a3a0: 6578 4661 6365 7473 204b 2020 2020 2020  exFacets K      
│ │ │ │ -0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a3d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a340: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a390: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ +0000a3a0: 207b 7b31 2c20 322c 2033 7d2c 207b 312c   {{1, 2, 3}, {1,
│ │ │ │ +0000a3b0: 2032 2c20 347d 2c20 7b31 2c20 332c 2034   2, 4}, {1, 3, 4
│ │ │ │ +0000a3c0: 7d2c 207b 322c 2033 2c20 347d 2c20 7b32  }, {2, 3, 4}, {2
│ │ │ │ +0000a3d0: 2c20 332c 2035 7d7d 2020 2020 2020 2020  , 3, 5}}        
│ │ │ │ +0000a3e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a420: 207c 0a7c 6f37 203d 207b 7b31 2c20 357d   |.|o7 = {{1, 5}
│ │ │ │ -0000a430: 2c20 7b32 2c20 332c 2035 7d2c 207b 312c  , {2, 3, 5}, {1,
│ │ │ │ -0000a440: 2032 2c20 332c 2034 7d7d 2020 2020 2020   2, 3, 4}}      
│ │ │ │ +0000a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a430: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ +0000a440: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0000a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a4c0: 207c 0a7c 6f37 203a 204c 6973 7420 2020   |.|o7 : List   
│ │ │ │ -0000a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a560: 2d2b 0a7c 6938 203a 204c 5f28 2d31 2920  -+.|i8 : L_(-1) 
│ │ │ │ -0000a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a480: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000a490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a4d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ +0000a4e0: 2061 6273 7472 6163 7453 696d 706c 6963   abstractSimplic
│ │ │ │ +0000a4f0: 6961 6c43 6f6d 706c 6578 4661 6365 7473  ialComplexFacets
│ │ │ │ +0000a500: 204b 2020 2020 2020 2020 2020 2020 2020   K              
│ │ │ │ +0000a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a520: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a570: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +0000a580: 207b 7b31 2c20 357d 2c20 7b32 2c20 332c   {{1, 5}, {2, 3,
│ │ │ │ +0000a590: 2035 7d2c 207b 312c 2032 2c20 332c 2034   5}, {1, 2, 3, 4
│ │ │ │ +0000a5a0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +0000a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a5c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a600: 207c 0a7c 6f38 203d 207b 7b7d 7d20 2020   |.|o8 = {{}}   
│ │ │ │ -0000a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a610: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0000a620: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0000a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a6a0: 207c 0a7c 6f38 203a 204c 6973 7420 2020   |.|o8 : List   
│ │ │ │ -0000a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a660: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a6b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +0000a6c0: 204c 5f28 2d31 2920 2020 2020 2020 2020   L_(-1)         
│ │ │ │  0000a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a6f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a740: 2d2b 0a7c 6939 203a 204c 5f30 2020 2020  -+.|i9 : L_0    
│ │ │ │ -0000a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a700: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a750: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +0000a760: 207b 7b7d 7d20 2020 2020 2020 2020 2020   {{}}           
│ │ │ │  0000a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a7a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a7e0: 207c 0a7c 6f39 203d 207b 7b31 7d2c 207b   |.|o9 = {{1}, {
│ │ │ │ -0000a7f0: 327d 2c20 7b33 7d2c 207b 347d 7d20 2020  2}, {3}, {4}}   
│ │ │ │ -0000a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a7f0: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ +0000a800: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0000a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a880: 207c 0a7c 6f39 203a 204c 6973 7420 2020   |.|o9 : List   
│ │ │ │ -0000a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a840: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a890: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ +0000a8a0: 204c 5f30 2020 2020 2020 2020 2020 2020   L_0            
│ │ │ │  0000a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a8d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a920: 2d2b 0a7c 6931 3020 3a20 4c5f 3120 2020  -+.|i10 : L_1   
│ │ │ │ -0000a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a8e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a930: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +0000a940: 207b 7b31 7d2c 207b 327d 2c20 7b33 7d2c   {{1}, {2}, {3},
│ │ │ │ +0000a950: 207b 347d 7d20 2020 2020 2020 2020 2020   {4}}           
│ │ │ │  0000a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a970: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a980: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a9c0: 207c 0a7c 6f31 3020 3d20 7b7b 312c 2032   |.|o10 = {{1, 2
│ │ │ │ -0000a9d0: 7d2c 207b 312c 2033 7d2c 207b 312c 2034  }, {1, 3}, {1, 4
│ │ │ │ -0000a9e0: 7d2c 207b 322c 2033 7d2c 207b 322c 2034  }, {2, 3}, {2, 4
│ │ │ │ -0000a9f0: 7d2c 207b 332c 2034 7d7d 2020 2020 2020  }, {3, 4}}      
│ │ │ │ +0000a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a9d0: 2020 2020 2020 2020 207c 0a7c 6f39 203a           |.|o9 :
│ │ │ │ +0000a9e0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +0000a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa60: 207c 0a7c 6f31 3020 3a20 4c69 7374 2020   |.|o10 : List  
│ │ │ │ -0000aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aa20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000aa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000aa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000aa70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ +0000aa80: 3a20 4c5f 3120 2020 2020 2020 2020 2020  : L_1           
│ │ │ │  0000aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aab0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0000aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ab00: 2d2b 0a7c 6931 3120 3a20 4c5f 3220 2020  -+.|i11 : L_2   
│ │ │ │ -0000ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ab50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aac0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ab10: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ +0000ab20: 3d20 7b7b 312c 2032 7d2c 207b 312c 2033  = {{1, 2}, {1, 3
│ │ │ │ +0000ab30: 7d2c 207b 312c 2034 7d2c 207b 322c 2033  }, {1, 4}, {2, 3
│ │ │ │ +0000ab40: 7d2c 207b 322c 2034 7d2c 207b 332c 2034  }, {2, 4}, {3, 4
│ │ │ │ +0000ab50: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +0000ab60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aba0: 207c 0a7c 6f31 3120 3d20 7b7b 312c 2032   |.|o11 = {{1, 2
│ │ │ │ -0000abb0: 2c20 337d 2c20 7b31 2c20 322c 2034 7d2c  , 3}, {1, 2, 4},
│ │ │ │ -0000abc0: 207b 312c 2033 2c20 347d 2c20 7b32 2c20   {1, 3, 4}, {2, 
│ │ │ │ -0000abd0: 332c 2034 7d7d 2020 2020 2020 2020 2020  3, 4}}          
│ │ │ │ +0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000abb0: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ +0000abc0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +0000abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000abf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0000ac40: 207c 0a7c 6f31 3120 3a20 4c69 7374 2020   |.|o11 : List  
│ │ │ │ -0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000ac50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +0000ac60: 3a20 4c5f 3220 2020 2020 2020 2020 2020  : L_2           
│ │ │ │  0000ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ac90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000acb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000acc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000acd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ace0: 2d2b 0a7c 6931 3220 3a20 4c5f 3320 2020  -+.|i12 : L_3   
│ │ │ │ -0000acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ad30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000acf0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ +0000ad00: 3d20 7b7b 312c 2032 2c20 337d 2c20 7b31  = {{1, 2, 3}, {1
│ │ │ │ +0000ad10: 2c20 322c 2034 7d2c 207b 312c 2033 2c20  , 2, 4}, {1, 3, 
│ │ │ │ +0000ad20: 347d 2c20 7b32 2c20 332c 2034 7d7d 2020  4}, {2, 3, 4}}  
│ │ │ │ +0000ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ad40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ad80: 207c 0a7c 6f31 3220 3d20 7b7b 312c 2032   |.|o12 = {{1, 2
│ │ │ │ -0000ad90: 2c20 332c 2034 7d7d 2020 2020 2020 2020  , 3, 4}}        
│ │ │ │ -0000ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ad90: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ +0000ada0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │  0000adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000add0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0000ae20: 207c 0a7c 6f31 3220 3a20 4c69 7374 2020   |.|o12 : List  
│ │ │ │ -0000ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0000ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0000ae30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220  ---------+.|i12 
│ │ │ │ +0000ae40: 3a20 4c5f 3320 2020 2020 2020 2020 2020  : L_3           
│ │ │ │  0000ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ae70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000aec0: 2d2b 0a7c 6931 3320 3a20 6162 7374 7261  -+.|i13 : abstra
│ │ │ │ -0000aed0: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ -0000aee0: 6c65 7846 6163 6574 7320 4c20 2020 2020  lexFacets L     
│ │ │ │ +0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ae80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aed0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ +0000aee0: 3d20 7b7b 312c 2032 2c20 332c 2034 7d7d  = {{1, 2, 3, 4}}
│ │ │ │  0000aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000af10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000af20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000af60: 207c 0a7c 6f31 3320 3d20 7b7b 312c 2032   |.|o13 = {{1, 2
│ │ │ │ -0000af70: 2c20 332c 2034 7d7d 2020 2020 2020 2020  , 3, 4}}        
│ │ │ │ -0000af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000af70: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ +0000af80: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │  0000af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000afb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0000afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b000: 207c 0a7c 6f31 3320 3a20 4c69 7374 2020   |.|o13 : List  
│ │ │ │ -0000b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b050: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0000b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b0a0: 2d2b 0a1f 0a46 696c 653a 2041 6273 7472  -+...File: Abstr
│ │ │ │ -0000b0b0: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ -0000b0c0: 706c 6578 6573 2e69 6e66 6f2c 204e 6f64  plexes.info, Nod
│ │ │ │ -0000b0d0: 653a 2048 6f77 2074 6f20 6d61 6b65 2072  e: How to make r
│ │ │ │ -0000b0e0: 6564 7563 6564 2061 6e64 206e 6f6e 2d72  educed and non-r
│ │ │ │ -0000b0f0: 6564 7563 6564 2073 696d 706c 6963 6961  educed simplicia
│ │ │ │ -0000b100: 6c20 6368 6169 6e20 636f 6d70 6c65 7865  l chain complexe
│ │ │ │ -0000b110: 732c 204e 6578 743a 2048 6f77 2074 6f20  s, Next: How to 
│ │ │ │ -0000b120: 6d61 6b65 2073 7562 7369 6d70 6c69 6361  make subsimplica
│ │ │ │ -0000b130: 6c20 636f 6d70 6c65 7865 7320 616e 6420  l complexes and 
│ │ │ │ -0000b140: 696e 6475 6365 6420 7369 6d70 6c69 6369  induced simplici
│ │ │ │ -0000b150: 616c 2063 6861 696e 2063 6f6d 706c 6578  al chain complex
│ │ │ │ -0000b160: 206d 6170 732c 2050 7265 763a 2048 6f77   maps, Prev: How
│ │ │ │ -0000b170: 2074 6f20 6d61 6b65 2061 6273 7472 6163   to make abstrac
│ │ │ │ -0000b180: 7420 7369 6d70 6c69 6369 616c 2063 6f6d  t simplicial com
│ │ │ │ -0000b190: 706c 6578 6573 2c20 5570 3a20 546f 700a  plexes, Up: Top.
│ │ │ │ -0000b1a0: 0a48 6f77 2074 6f20 6d61 6b65 2072 6564  .How to make red
│ │ │ │ -0000b1b0: 7563 6564 2061 6e64 206e 6f6e 2d72 6564  uced and non-red
│ │ │ │ -0000b1c0: 7563 6564 2073 696d 706c 6963 6961 6c20  uced simplicial 
│ │ │ │ -0000b1d0: 6368 6169 6e20 636f 6d70 6c65 7865 7320  chain complexes 
│ │ │ │ -0000b1e0: 2d2d 2053 696d 706c 6963 6961 6c20 686f  -- Simplicial ho
│ │ │ │ -0000b1f0: 6d6f 6c6f 6769 6361 6c20 636f 6e73 7472  mological constr
│ │ │ │ -0000b200: 7563 746f 7273 0a2a 2a2a 2a2a 2a2a 2a2a  uctors.*********
│ │ │ │ -0000b210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000b220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000b230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000b240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000b250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000b260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4e6f  ************..No
│ │ │ │ -0000b270: 6e2d 7265 6475 6365 6420 616e 6420 7265  n-reduced and re
│ │ │ │ -0000b280: 6475 6365 6420 7369 6d70 6c69 6369 616c  duced simplicial
│ │ │ │ -0000b290: 2063 6861 696e 2063 6f6d 706c 6578 6573   chain complexes
│ │ │ │ -0000b2a0: 2063 616e 2062 6520 636f 6e73 7472 7563   can be construc
│ │ │ │ -0000b2b0: 7465 6420 696e 2074 6865 0a66 6f6c 6c6f  ted in the.follo
│ │ │ │ -0000b2c0: 7769 6e67 2077 6179 2e0a 0a2b 2d2d 2d2d  wing way...+----
│ │ │ │ -0000b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b310: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0000b320: 204b 203d 2061 6273 7472 6163 7453 696d   K = abstractSim
│ │ │ │ -0000b330: 706c 6963 6961 6c43 6f6d 706c 6578 287b  plicialComplex({
│ │ │ │ -0000b340: 7b31 2c32 2c33 2c34 7d2c 207b 322c 332c  {1,2,3,4}, {2,3,
│ │ │ │ -0000b350: 357d 2c7b 312c 357d 7d29 2020 2020 2020  5},{1,5}})      
│ │ │ │ -0000b360: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b3b0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -0000b3c0: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000b3d0: 6961 6c43 6f6d 706c 6578 7b2d 3120 3d3e  ialComplex{-1 =>
│ │ │ │ -0000b3e0: 207b 7b7d 7d20 2020 2020 2020 2020 2020   {{}}           
│ │ │ │ -0000b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b420: 2020 2020 2020 2020 2020 2030 203d 3e20             0 => 
│ │ │ │ -0000b430: 7b7b 317d 2c20 7b32 7d2c 207b 337d 2c20  {{1}, {2}, {3}, 
│ │ │ │ -0000b440: 7b34 7d2c 207b 357d 7d20 2020 2020 2020  {4}, {5}}       
│ │ │ │ -0000b450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b470: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
│ │ │ │ -0000b480: 7b7b 312c 2032 7d2c 207b 312c 2033 7d2c  {{1, 2}, {1, 3},
│ │ │ │ -0000b490: 207b 312c 2034 7d2c 207b 312c 2035 7d2c   {1, 4}, {1, 5},
│ │ │ │ -0000b4a0: 207b 322c 2033 7d2c 207c 0a7c 2020 2020   {2, 3}, |.|    
│ │ │ │ -0000b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b4c0: 2020 2020 2020 2020 2020 2032 203d 3e20             2 => 
│ │ │ │ -0000b4d0: 7b7b 312c 2032 2c20 337d 2c20 7b31 2c20  {{1, 2, 3}, {1, 
│ │ │ │ -0000b4e0: 322c 2034 7d2c 207b 312c 2033 2c20 347d  2, 4}, {1, 3, 4}
│ │ │ │ -0000b4f0: 2c20 7b32 2c20 332c 207c 0a7c 2020 2020  , {2, 3, |.|    
│ │ │ │ +0000afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000afc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b010: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ +0000b020: 3a20 6162 7374 7261 6374 5369 6d70 6c69  : abstractSimpli
│ │ │ │ +0000b030: 6369 616c 436f 6d70 6c65 7846 6163 6574  cialComplexFacet
│ │ │ │ +0000b040: 7320 4c20 2020 2020 2020 2020 2020 2020  s L             
│ │ │ │ +0000b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b0b0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +0000b0c0: 3d20 7b7b 312c 2032 2c20 332c 2034 7d7d  = {{1, 2, 3, 4}}
│ │ │ │ +0000b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b150: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +0000b160: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +0000b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b1a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b1f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a1f 0a46 696c  ---------+...Fil
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│ │ │ │ +0000b220: 6e66 6f2c 204e 6f64 653a 2048 6f77 2074  nfo, Node: How t
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│ │ │ │ +0000b240: 6e64 206e 6f6e 2d72 6564 7563 6564 2073  nd non-reduced s
│ │ │ │ +0000b250: 696d 706c 6963 6961 6c20 6368 6169 6e20  implicial chain 
│ │ │ │ +0000b260: 636f 6d70 6c65 7865 732c 204e 6578 743a  complexes, Next:
│ │ │ │ +0000b270: 2048 6f77 2074 6f20 6d61 6b65 2073 7562   How to make sub
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│ │ │ │ +0000b2b0: 2063 6f6d 706c 6578 206d 6170 732c 2050   complex maps, P
│ │ │ │ +0000b2c0: 7265 763a 2048 6f77 2074 6f20 6d61 6b65  rev: How to make
│ │ │ │ +0000b2d0: 2061 6273 7472 6163 7420 7369 6d70 6c69   abstract simpli
│ │ │ │ +0000b2e0: 6369 616c 2063 6f6d 706c 6578 6573 2c20  cial complexes, 
│ │ │ │ +0000b2f0: 5570 3a20 546f 700a 0a48 6f77 2074 6f20  Up: Top..How to 
│ │ │ │ +0000b300: 6d61 6b65 2072 6564 7563 6564 2061 6e64  make reduced and
│ │ │ │ +0000b310: 206e 6f6e 2d72 6564 7563 6564 2073 696d   non-reduced sim
│ │ │ │ +0000b320: 706c 6963 6961 6c20 6368 6169 6e20 636f  plicial chain co
│ │ │ │ +0000b330: 6d70 6c65 7865 7320 2d2d 2053 696d 706c  mplexes -- Simpl
│ │ │ │ +0000b340: 6963 6961 6c20 686f 6d6f 6c6f 6769 6361  icial homologica
│ │ │ │ +0000b350: 6c20 636f 6e73 7472 7563 746f 7273 0a2a  l constructors.*
│ │ │ │ +0000b360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000b370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000b380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000b390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000b3a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000b3b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000b3c0: 2a2a 2a2a 0a0a 4e6f 6e2d 7265 6475 6365  ****..Non-reduce
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│ │ │ │ +0000b3e0: 6d70 6c69 6369 616c 2063 6861 696e 2063  mplicial chain c
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│ │ │ │ +0000b420: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
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│ │ │ │ +0000b460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b470: 2d2b 0a7c 6931 203a 204b 203d 2061 6273  -+.|i1 : K = abs
│ │ │ │ +0000b480: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +0000b490: 6f6d 706c 6578 287b 7b31 2c32 2c33 2c34  omplex({{1,2,3,4
│ │ │ │ +0000b4a0: 7d2c 207b 322c 332c 357d 2c7b 312c 357d  }, {2,3,5},{1,5}
│ │ │ │ +0000b4b0: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │ +0000b4c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b510: 2020 2020 2020 2020 2020 2033 203d 3e20             3 => 
│ │ │ │ -0000b520: 7b7b 312c 2032 2c20 332c 2034 7d7d 2020  {{1, 2, 3, 4}}  
│ │ │ │ -0000b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000b510: 207c 0a7c 6f31 203d 2041 6273 7472 6163   |.|o1 = Abstrac
│ │ │ │ +0000b520: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000b530: 6578 7b2d 3120 3d3e 207b 7b7d 7d20 2020  ex{-1 => {{}}   
│ │ │ │ +0000b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000b5a0: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000b5b0: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
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│ │ │ │ +0000b590: 7d2c 207b 337d 2c20 7b34 7d2c 207b 357d  }, {3}, {4}, {5}
│ │ │ │ +0000b5a0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0000b5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000b5e0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ -0000b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0000b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000b5d0: 2020 2031 203d 3e20 7b7b 312c 2032 7d2c     1 => {{1, 2},
│ │ │ │ +0000b5e0: 207b 312c 2033 7d2c 207b 312c 2034 7d2c   {1, 3}, {1, 4},
│ │ │ │ +0000b5f0: 207b 312c 2035 7d2c 207b 322c 2033 7d2c   {1, 5}, {2, 3},
│ │ │ │ +0000b600: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b620: 2020 2032 203d 3e20 7b7b 312c 2032 2c20     2 => {{1, 2, 
│ │ │ │ +0000b630: 337d 2c20 7b31 2c20 322c 2034 7d2c 207b  3}, {1, 2, 4}, {
│ │ │ │ +0000b640: 312c 2033 2c20 347d 2c20 7b32 2c20 332c  1, 3, 4}, {2, 3,
│ │ │ │ +0000b650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b680: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000b670: 2020 2033 203d 3e20 7b7b 312c 2032 2c20     3 => {{1, 2, 
│ │ │ │ +0000b680: 332c 2034 7d7d 2020 2020 2020 2020 2020  3, 4}}          
│ │ │ │  0000b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b6d0: 2020 2020 2020 2020 207c 0a7c 7b32 2c20           |.|{2, 
│ │ │ │ -0000b6e0: 347d 2c20 7b32 2c20 357d 2c20 7b33 2c20  4}, {2, 5}, {3, 
│ │ │ │ -0000b6f0: 347d 2c20 7b33 2c20 357d 7d20 2020 2020  4}, {3, 5}}     
│ │ │ │ -0000b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000b730: 7b32 2c20 332c 2035 7d7d 2020 2020 2020  {2, 3, 5}}      
│ │ │ │ -0000b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000b6f0: 207c 0a7c 6f31 203a 2041 6273 7472 6163   |.|o1 : Abstrac
│ │ │ │ +0000b700: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000b710: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
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│ │ │ │ +0000b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000b7c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0000b7d0: 206b 203d 2073 696d 706c 6963 6961 6c43   k = simplicialC
│ │ │ │ -0000b7e0: 6861 696e 436f 6d70 6c65 7820 4b20 2020  hainComplex K   
│ │ │ │ +0000b790: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0000b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b7b0: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │ +0000b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b7e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ +0000b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000b870: 2020 2035 2020 2020 2020 2039 2020 2020     5       9    
│ │ │ │ -0000b880: 2020 2035 2020 2020 2020 2031 2020 2020     5       1    
│ │ │ │ -0000b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000b830: 207c 0a7c 7b32 2c20 347d 2c20 7b32 2c20   |.|{2, 4}, {2, 
│ │ │ │ +0000b840: 357d 2c20 7b33 2c20 347d 2c20 7b33 2c20  5}, {3, 4}, {3, 
│ │ │ │ +0000b850: 357d 7d20 2020 2020 2020 2020 2020 2020  5}}             
│ │ │ │ +0000b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000b880: 207c 0a7c 347d 2c20 7b32 2c20 332c 2035   |.|4}, {2, 3, 5
│ │ │ │ +0000b890: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │  0000b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b8b0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -0000b8c0: 205a 5a20 203c 2d2d 205a 5a20 203c 2d2d   ZZ  <-- ZZ  <--
│ │ │ │ -0000b8d0: 205a 5a20 203c 2d2d 205a 5a20 2020 2020   ZZ  <-- ZZ     
│ │ │ │ -0000b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000b960: 2030 2020 2020 2020 2031 2020 2020 2020   0       1      
│ │ │ │ -0000b970: 2032 2020 2020 2020 2033 2020 2020 2020   2       3      
│ │ │ │ +0000b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000b8d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000b920: 2d2b 0a7c 6932 203a 206b 203d 2073 696d  -+.|i2 : k = sim
│ │ │ │ +0000b930: 706c 6963 6961 6c43 6861 696e 436f 6d70  plicialChainComp
│ │ │ │ +0000b940: 6c65 7820 4b20 2020 2020 2020 2020 2020  lex K           
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│ │ │ │ +0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000b9f0: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
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│ │ │ │ -0000ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0000bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000b9c0: 207c 0a7c 2020 2020 2020 2035 2020 2020   |.|       5    
│ │ │ │ +0000b9d0: 2020 2039 2020 2020 2020 2035 2020 2020     9       5    
│ │ │ │ +0000b9e0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +0000b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000ba10: 207c 0a7c 6f32 203d 205a 5a20 203c 2d2d   |.|o2 = ZZ  <--
│ │ │ │ +0000ba20: 205a 5a20 203c 2d2d 205a 5a20 203c 2d2d   ZZ  <-- ZZ  <--
│ │ │ │ +0000ba30: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +0000ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000bab0: 207c 0a7c 2020 2020 2030 2020 2020 2020   |.|     0      
│ │ │ │ +0000bac0: 2031 2020 2020 2020 2032 2020 2020 2020   1       2      
│ │ │ │ +0000bad0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0000bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bb30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000bb40: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ -0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bb60: 2020 2020 2020 2020 2020 2020 2020 2039                 9
│ │ │ │ +0000bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000bb50: 207c 0a7c 6f32 203a 2043 6f6d 706c 6578   |.|o2 : Complex
│ │ │ │ +0000bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bb80: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ -0000bb90: 2030 203a 205a 5a20 203c 2d2d 2d2d 2d2d   0 : ZZ  <------
│ │ │ │ -0000bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 205a 5a20  ------------ ZZ 
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│ │ │ │ +0000d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000d140: 203c 2d2d 2d2d 2d2d 2d2d 2d2d 205a 5a20   <---------- ZZ 
│ │ │ │ +0000d150: 203a 2033 2020 2020 2020 2020 2020 2020   : 3            
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│ │ │ │ +0000d1e0: 2020 2020 7c20 3120 207c 2020 2020 2020      | 1  |      
│ │ │ │  0000d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d220: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000d230: 2020 2020 7c20 2d31 207c 2020 2020 2020      | -1 |      
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│ │ │ │ -0000d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0000d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d4b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d4c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4769 7665  **********..Give
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│ │ │ │ -0000d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d680: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000d710: 347d 2c20 7b32 2c20 337d 2c20 7b32 2c20  4}, {2, 3}, {2, 
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│ │ │ │ -0000d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d3b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000d3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0000d4b0: 5072 6576 3a20 486f 7720 746f 206d 616b  Prev: How to mak
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│ │ │ │ +0000d590: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0000d5a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d5b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d5c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d5e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d5f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000d620: 2a2a 0a0a 4769 7665 6e20 6120 7375 6273  **..Given a subs
│ │ │ │ +0000d630: 696d 706c 6963 6961 6c20 636f 6d70 6c65  implicial comple
│ │ │ │ +0000d640: 7820 7468 6572 6520 6172 6520 696e 6475  x there are indu
│ │ │ │ +0000d650: 6365 6420 7369 6d70 6c69 6369 616c 2063  ced simplicial c
│ │ │ │ +0000d660: 6861 696e 2063 6f6d 706c 6578 206d 6170  hain complex map
│ │ │ │ +0000d670: 732e 0a54 6869 7320 6973 2069 6c6c 7573  s..This is illus
│ │ │ │ +0000d680: 7472 6174 6564 2069 6e20 7468 6520 666f  trated in the fo
│ │ │ │ +0000d690: 6c6c 6f77 696e 6720 7761 792e 0a0a 2b2d  llowing way...+-
│ │ │ │ +0000d6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0000d6f0: 3120 3a20 4b20 3d20 6162 7374 7261 6374  1 : K = abstract
│ │ │ │ +0000d700: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +0000d710: 7828 342c 3329 2020 2020 2020 2020 2020  x(4,3)          
│ │ │ │ +0000d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d780: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0000d790: 3120 3d20 4162 7374 7261 6374 5369 6d70  1 = AbstractSimp
│ │ │ │ +0000d7a0: 6c69 6369 616c 436f 6d70 6c65 787b 2d31  licialComplex{-1
│ │ │ │ +0000d7b0: 203d 3e20 7b7b 7d7d 2020 2020 2020 2020   => {{}}        
│ │ │ │ +0000d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d7d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d7f0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0000d800: 3d3e 207b 7b31 7d2c 207b 327d 2c20 7b33  => {{1}, {2}, {3
│ │ │ │ +0000d810: 7d2c 207b 347d 7d20 2020 2020 2020 2020  }, {4}}         
│ │ │ │ +0000d820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d840: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0000d850: 3d3e 207b 7b31 2c20 327d 2c20 7b31 2c20  => {{1, 2}, {1, 
│ │ │ │ +0000d860: 337d 2c20 7b31 2c20 347d 2c20 7b32 2c20  3}, {1, 4}, {2, 
│ │ │ │ +0000d870: 337d 2c20 7b32 2c20 347d 2c20 7c0a 7c20  3}, {2, 4}, |.| 
│ │ │ │  0000d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d8b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d890: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0000d8a0: 3d3e 207b 7b31 2c20 322c 2033 7d2c 207b  => {{1, 2, 3}, {
│ │ │ │ +0000d8b0: 312c 2032 2c20 347d 2c20 7b31 2c20 332c  1, 2, 4}, {1, 3,
│ │ │ │ +0000d8c0: 2034 7d2c 207b 322c 2033 2c20 7c0a 7c20   4}, {2, 3, |.| 
│ │ │ │  0000d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d900: 2020 2020 7c0a 7c7b 332c 2034 7d7d 2020      |.|{3, 4}}  
│ │ │ │ -0000d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d910: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0000d920: 3120 3a20 4162 7374 7261 6374 5369 6d70  1 : AbstractSimp
│ │ │ │ +0000d930: 6c69 6369 616c 436f 6d70 6c65 7820 2020  licialComplex   
│ │ │ │  0000d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d950: 2020 2020 7c0a 7c34 7d7d 2020 2020 2020      |.|4}}      
│ │ │ │ -0000d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d9a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0000d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d9f0: 2d2d 2d2d 2b0a 7c69 3220 3a20 4c20 3d20  ----+.|i2 : L = 
│ │ │ │ -0000da00: 6162 7374 7261 6374 5369 6d70 6c69 6369  abstractSimplici
│ │ │ │ -0000da10: 616c 436f 6d70 6c65 7828 342c 3229 2020  alComplex(4,2)  
│ │ │ │ +0000d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d960: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0000d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0000d9c0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0000d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000da00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000da40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000da50: 2020 2020 2020 2020 2020 2020 7c0a 7c7b              |.|{
│ │ │ │ +0000da60: 332c 2034 7d7d 2020 2020 2020 2020 2020  3, 4}}          
│ │ │ │  0000da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000da90: 2020 2020 7c0a 7c6f 3220 3d20 4162 7374      |.|o2 = Abst
│ │ │ │ -0000daa0: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -0000dab0: 6d70 6c65 787b 2d31 203d 3e20 7b7b 7d7d  mplex{-1 => {{}}
│ │ │ │ +0000da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000daa0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0000dab0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │  0000dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000db00: 2020 2020 2020 3020 3d3e 207b 7b31 7d2c        0 => {{1},
│ │ │ │ -0000db10: 207b 327d 2c20 7b33 7d2c 207b 347d 7d20   {2}, {3}, {4}} 
│ │ │ │ -0000db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000db30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000db50: 2020 2020 2020 3120 3d3e 207b 7b31 2c20        1 => {{1, 
│ │ │ │ -0000db60: 327d 2c20 7b31 2c20 337d 2c20 7b31 2c20  2}, {1, 3}, {1, 
│ │ │ │ -0000db70: 347d 2c20 7b32 2c20 337d 2c20 7b32 2c20  4}, {2, 3}, {2, 
│ │ │ │ -0000db80: 347d 2c20 7c0a 7c20 2020 2020 2020 2020  4}, |.|         
│ │ │ │ -0000db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000daf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0000db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0000db50: 3220 3a20 4c20 3d20 6162 7374 7261 6374  2 : L = abstract
│ │ │ │ +0000db60: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +0000db70: 7828 342c 3229 2020 2020 2020 2020 2020  x(4,2)          
│ │ │ │ +0000db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000db90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0000dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dbd0: 2020 2020 7c0a 7c6f 3220 3a20 4162 7374      |.|o2 : Abst
│ │ │ │ -0000dbe0: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -0000dbf0: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ -0000dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dc20: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ -0000dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dc70: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 7d20  ----|.|       } 
│ │ │ │ -0000dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dbe0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0000dbf0: 3220 3d20 4162 7374 7261 6374 5369 6d70  2 = AbstractSimp
│ │ │ │ +0000dc00: 6c69 6369 616c 436f 6d70 6c65 787b 2d31  licialComplex{-1
│ │ │ │ +0000dc10: 203d 3e20 7b7b 7d7d 2020 2020 2020 2020   => {{}}        
│ │ │ │ +0000dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dc50: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0000dc60: 3d3e 207b 7b31 7d2c 207b 327d 2c20 7b33  => {{1}, {2}, {3
│ │ │ │ +0000dc70: 7d2c 207b 347d 7d20 2020 2020 2020 2020  }, {4}}         
│ │ │ │ +0000dc80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0000dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dcc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dca0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0000dcb0: 3d3e 207b 7b31 2c20 327d 2c20 7b31 2c20  => {{1, 2}, {1, 
│ │ │ │ +0000dcc0: 337d 2c20 7b31 2c20 347d 2c20 7b32 2c20  3}, {1, 4}, {2, 
│ │ │ │ +0000dcd0: 337d 2c20 7b32 2c20 347d 2c20 7c0a 7c20  3}, {2, 4}, |.| 
│ │ │ │  0000dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dd10: 2020 2020 7c0a 7c7b 332c 2034 7d7d 2020      |.|{3, 4}}  
│ │ │ │ -0000dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dd20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0000dd30: 3220 3a20 4162 7374 7261 6374 5369 6d70  2 : AbstractSimp
│ │ │ │ +0000dd40: 6c69 6369 616c 436f 6d70 6c65 7820 2020  licialComplex   
│ │ │ │  0000dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dd60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0000dd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dd70: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  0000dd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ddb0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6620 3d20  ----+.|i3 : f = 
│ │ │ │ -0000ddc0: 696e 6475 6365 6453 696d 706c 6963 6961  inducedSimplicia
│ │ │ │ -0000ddd0: 6c43 6861 696e 436f 6d70 6c65 784d 6170  lChainComplexMap
│ │ │ │ -0000dde0: 284b 2c4c 2920 2020 2020 2020 2020 2020  (K,L)           
│ │ │ │ +0000ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0000ddd0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0000dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000de00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000de10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0000de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000de50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000de60: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ -0000de70: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0000de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000de60: 2020 2020 2020 2020 2020 2020 7c0a 7c7b              |.|{
│ │ │ │ +0000de70: 332c 2034 7d7d 2020 2020 2020 2020 2020  3, 4}}          
│ │ │ │  0000de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dea0: 2020 2020 7c0a 7c6f 3320 3d20 3020 3a20      |.|o3 = 0 : 
│ │ │ │ -0000deb0: 5a5a 2020 3c2d 2d2d 2d2d 2d2d 2d2d 2d2d  ZZ  <-----------
│ │ │ │ -0000dec0: 2d2d 2d2d 205a 5a20 203a 2030 2020 2020  ---- ZZ  : 0    
│ │ │ │ -0000ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000def0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000df00: 2020 2020 2020 207c 2031 2030 2030 2030         | 1 0 0 0
│ │ │ │ -0000df10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0000df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000df40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000df50: 2020 2020 2020 207c 2030 2031 2030 2030         | 0 1 0 0
│ │ │ │ -0000df60: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0000dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000deb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0000dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0000df10: 3320 3a20 6620 3d20 696e 6475 6365 6453  3 : f = inducedS
│ │ │ │ +0000df20: 696d 706c 6963 6961 6c43 6861 696e 436f  implicialChainCo
│ │ │ │ +0000df30: 6d70 6c65 784d 6170 284b 2c4c 2920 2020  mplexMap(K,L)   
│ │ │ │ +0000df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000df50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000df90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000dfa0: 2020 2020 2020 207c 2030 2030 2031 2030         | 0 0 1 0
│ │ │ │ -0000dfb0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0000dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000dfa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0000dfb0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0000dfc0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │  0000dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dfe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0000dff0: 2020 2020 2020 207c 2030 2030 2030 2031         | 0 0 0 1
│ │ │ │ -0000e000: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0000e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0000f210: 6520 706f 7373 6962 6c65 0a62 7920 7468  e possible.by th
│ │ │ │ +0000f220: 6520 6d65 7468 6f64 2069 6e64 7563 6564  e method induced
│ │ │ │ +0000f230: 5265 6475 6365 6453 696d 706c 6963 6961  ReducedSimplicia
│ │ │ │ +0000f240: 6c43 6861 696e 436f 6d70 6c65 784d 6170  lChainComplexMap
│ │ │ │ +0000f250: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +0000f260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f290: 2d2b 0a7c 6931 203a 204b 203d 2061 6273  -+.|i1 : K = abs
│ │ │ │ +0000f2a0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +0000f2b0: 6f6d 706c 6578 287b 7b31 2c32 7d2c 7b33  omplex({{1,2},{3
│ │ │ │ +0000f2c0: 7d7d 2920 2020 2020 2020 2020 2020 2020  }})             
│ │ │ │ +0000f2d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f2f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0000f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f330: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0000f340: 204a 203d 2061 6d62 6965 6e74 4162 7374   J = ambientAbst
│ │ │ │ -0000f350: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -0000f360: 6d70 6c65 7828 4b29 2020 2020 2020 2020  mplex(K)        
│ │ │ │ -0000f370: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f310: 207c 0a7c 6f31 203d 2041 6273 7472 6163   |.|o1 = Abstrac
│ │ │ │ +0000f320: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000f330: 6578 7b2d 3120 3d3e 207b 7b7d 7d20 2020  ex{-1 => {{}}   
│ │ │ │ +0000f340: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +0000f350: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f370: 2020 2030 203d 3e20 7b7b 317d 2c20 7b32     0 => {{1}, {2
│ │ │ │ +0000f380: 7d2c 207b 337d 7d20 2020 2020 2020 2020  }, {3}}         
│ │ │ │ +0000f390: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f3b0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -0000f3c0: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000f3d0: 6961 6c43 6f6d 706c 6578 7b2d 3120 3d3e  ialComplex{-1 =>
│ │ │ │ -0000f3e0: 207b 7b7d 7d20 2020 2020 2020 2020 2020   {{}}           
│ │ │ │ -0000f3f0: 2020 2020 2020 2020 7d7c 0a7c 2020 2020          }|.|    
│ │ │ │ +0000f3b0: 2020 2031 203d 3e20 7b7b 312c 2032 7d7d     1 => {{1, 2}}
│ │ │ │ +0000f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f3d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f410: 2020 2020 2020 2020 2020 2030 203d 3e20             0 => 
│ │ │ │ -0000f420: 7b7b 317d 2c20 7b32 7d2c 207b 337d 7d20  {{1}, {2}, {3}} 
│ │ │ │ -0000f430: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000f410: 207c 0a7c 6f31 203a 2041 6273 7472 6163   |.|o1 : Abstrac
│ │ │ │ +0000f420: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000f430: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  0000f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f450: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
│ │ │ │ -0000f460: 7b7b 312c 2032 7d2c 207b 312c 2033 7d2c  {{1, 2}, {1, 3},
│ │ │ │ -0000f470: 207b 322c 2033 7d7d 207c 0a7c 2020 2020   {2, 3}} |.|    
│ │ │ │ -0000f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f490: 2020 2020 2020 2020 2020 2032 203d 3e20             2 => 
│ │ │ │ -0000f4a0: 7b7b 312c 2032 2c20 337d 7d20 2020 2020  {{1, 2, 3}}     
│ │ │ │ -0000f4b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000f450: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f490: 2d2b 0a7c 6932 203a 204a 203d 2061 6d62  -+.|i2 : J = amb
│ │ │ │ +0000f4a0: 6965 6e74 4162 7374 7261 6374 5369 6d70  ientAbstractSimp
│ │ │ │ +0000f4b0: 6c69 6369 616c 436f 6d70 6c65 7828 4b29  licialComplex(K)
│ │ │ │  0000f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f4d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f4f0: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ -0000f500: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000f510: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ -0000f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f530: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0000f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f570: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0000f580: 2069 6e64 7563 6564 5265 6475 6365 6453   inducedReducedS
│ │ │ │ -0000f590: 696d 706c 6963 6961 6c43 6861 696e 436f  implicialChainCo
│ │ │ │ -0000f5a0: 6d70 6c65 784d 6170 284a 2c4b 2920 2020  mplexMap(J,K)   
│ │ │ │ -0000f5b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f510: 207c 0a7c 6f32 203d 2041 6273 7472 6163   |.|o2 = Abstrac
│ │ │ │ +0000f520: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000f530: 6578 7b2d 3120 3d3e 207b 7b7d 7d20 2020  ex{-1 => {{}}   
│ │ │ │ +0000f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f550: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ +0000f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f570: 2020 2030 203d 3e20 7b7b 317d 2c20 7b32     0 => {{1}, {2
│ │ │ │ +0000f580: 7d2c 207b 337d 7d20 2020 2020 2020 2020  }, {3}}         
│ │ │ │ +0000f590: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f5b0: 2020 2031 203d 3e20 7b7b 312c 2032 7d2c     1 => {{1, 2},
│ │ │ │ +0000f5c0: 207b 312c 2033 7d2c 207b 322c 2033 7d7d   {1, 3}, {2, 3}}
│ │ │ │ +0000f5d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f5f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f600: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -0000f610: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0000f5f0: 2020 2032 203d 3e20 7b7b 312c 2032 2c20     2 => {{1, 2, 
│ │ │ │ +0000f600: 337d 7d20 2020 2020 2020 2020 2020 2020  3}}             
│ │ │ │ +0000f610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f630: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ -0000f640: 202d 3120 3a20 5a5a 2020 3c2d 2d2d 2d2d   -1 : ZZ  <-----
│ │ │ │ -0000f650: 2d2d 2d2d 205a 5a20 203a 202d 3120 2020  ---- ZZ  : -1   
│ │ │ │ -0000f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f670: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f680: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ -0000f690: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0000f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f6b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f6f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f700: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -0000f710: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +0000f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f650: 207c 0a7c 6f32 203a 2041 6273 7472 6163   |.|o2 : Abstrac
│ │ │ │ +0000f660: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000f670: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │ +0000f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f690: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000f6d0: 2d2b 0a7c 6933 203a 2069 6e64 7563 6564  -+.|i3 : induced
│ │ │ │ +0000f6e0: 5265 6475 6365 6453 696d 706c 6963 6961  ReducedSimplicia
│ │ │ │ +0000f6f0: 6c43 6861 696e 436f 6d70 6c65 784d 6170  lChainComplexMap
│ │ │ │ +0000f700: 284a 2c4b 2920 2020 2020 2020 2020 2020  (J,K)           
│ │ │ │ +0000f710: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f730: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f740: 2030 203a 205a 5a20 203c 2d2d 2d2d 2d2d   0 : ZZ  <------
│ │ │ │ -0000f750: 2d2d 2d2d 2d2d 2d20 5a5a 2020 3a20 3020  ------- ZZ  : 0 
│ │ │ │ -0000f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f780: 2020 2020 2020 2020 2020 2020 7c20 3120              | 1 
│ │ │ │ -0000f790: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │ -0000f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f7c0: 2020 2020 2020 2020 2020 2020 7c20 3020              | 0 
│ │ │ │ -0000f7d0: 3120 3020 7c20 2020 2020 2020 2020 2020  1 0 |           
│ │ │ │ -0000f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f7f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f800: 2020 2020 2020 2020 2020 2020 7c20 3020              | 0 
│ │ │ │ -0000f810: 3020 3120 7c20 2020 2020 2020 2020 2020  0 1 |           
│ │ │ │ +0000f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f760: 3120 2020 2020 2020 2020 2020 2020 2031  1              1
│ │ │ │ +0000f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f790: 207c 0a7c 6f33 203d 202d 3120 3a20 5a5a   |.|o3 = -1 : ZZ
│ │ │ │ +0000f7a0: 2020 3c2d 2d2d 2d2d 2d2d 2d2d 205a 5a20    <--------- ZZ 
│ │ │ │ +0000f7b0: 203a 202d 3120 2020 2020 2020 2020 2020   : -1           
│ │ │ │ +0000f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f7d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f7e0: 2020 2020 207c 2031 207c 2020 2020 2020       | 1 |      
│ │ │ │ +0000f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f810: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f830: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f850: 207c 0a7c 2020 2020 2020 2020 2020 2033   |.|           3
│ │ │ │  0000f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f870: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f880: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -0000f890: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -0000f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f8b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f8c0: 2031 203a 205a 5a20 203c 2d2d 2d2d 2d2d   1 : ZZ  <------
│ │ │ │ -0000f8d0: 2d2d 2d20 5a5a 2020 3a20 3120 2020 2020  --- ZZ  : 1     
│ │ │ │ -0000f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f8f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f900: 2020 2020 2020 2020 2020 2020 7c20 3120              | 1 
│ │ │ │ -0000f910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0000f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f930: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f940: 2020 2020 2020 2020 2020 2020 7c20 3020              | 0 
│ │ │ │ -0000f950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0000f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f970: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000f980: 2020 2020 2020 2020 2020 2020 7c20 3020              | 0 
│ │ │ │ -0000f990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000f870: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0000f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f890: 207c 0a7c 2020 2020 2030 203a 205a 5a20   |.|     0 : ZZ 
│ │ │ │ +0000f8a0: 203c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20   <------------- 
│ │ │ │ +0000f8b0: 5a5a 2020 3a20 3020 2020 2020 2020 2020  ZZ  : 0         
│ │ │ │ +0000f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f8e0: 2020 2020 7c20 3120 3020 3020 7c20 2020      | 1 0 0 |   
│ │ │ │ +0000f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f910: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f920: 2020 2020 7c20 3020 3120 3020 7c20 2020      | 0 1 0 |   
│ │ │ │ +0000f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f950: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000f960: 2020 2020 7c20 3020 3020 3120 7c20 2020      | 0 0 1 |   
│ │ │ │ +0000f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f990: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f9b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f9f0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -0000fa00: 2043 6f6d 706c 6578 4d61 7020 2020 2020   ComplexMap     
│ │ │ │ -0000fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fa30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0000fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fa70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0000fa80: 204c 203d 2061 6273 7472 6163 7453 696d   L = abstractSim
│ │ │ │ -0000fa90: 706c 6963 6961 6c43 6f6d 706c 6578 207b  plicialComplex {
│ │ │ │ -0000faa0: 7b7d 7d20 2020 2020 2020 2020 2020 2020  {}}             
│ │ │ │ -0000fab0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000f9d0: 207c 0a7c 2020 2020 2020 2020 2020 2033   |.|           3
│ │ │ │ +0000f9e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0000f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fa10: 207c 0a7c 2020 2020 2031 203a 205a 5a20   |.|     1 : ZZ 
│ │ │ │ +0000fa20: 203c 2d2d 2d2d 2d2d 2d2d 2d20 5a5a 2020   <--------- ZZ  
│ │ │ │ +0000fa30: 3a20 3120 2020 2020 2020 2020 2020 2020  : 1             
│ │ │ │ +0000fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fa50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000fa60: 2020 2020 7c20 3120 7c20 2020 2020 2020      | 1 |       
│ │ │ │ +0000fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fa90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000faa0: 2020 2020 7c20 3020 7c20 2020 2020 2020      | 0 |       
│ │ │ │ +0000fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000faf0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -0000fb00: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000fb10: 6961 6c43 6f6d 706c 6578 7b2d 3120 3d3e  ialComplex{-1 =>
│ │ │ │ -0000fb20: 207b 7b7d 7d7d 2020 2020 2020 2020 2020   {{}}}          
│ │ │ │ -0000fb30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000fad0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000fae0: 2020 2020 7c20 3020 7c20 2020 2020 2020      | 0 |       
│ │ │ │ +0000faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fb10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fb70: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -0000fb80: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000fb90: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ -0000fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fbb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000fb50: 207c 0a7c 6f33 203a 2043 6f6d 706c 6578   |.|o3 : Complex
│ │ │ │ +0000fb60: 4d61 7020 2020 2020 2020 2020 2020 2020  Map             
│ │ │ │ +0000fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fb90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fbf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -0000fc00: 2069 6e64 7563 6564 5265 6475 6365 6453   inducedReducedS
│ │ │ │ -0000fc10: 696d 706c 6963 6961 6c43 6861 696e 436f  implicialChainCo
│ │ │ │ -0000fc20: 6d70 6c65 784d 6170 284c 2c4c 2920 2020  mplexMap(L,L)   
│ │ │ │ -0000fc30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000fbd0: 2d2b 0a7c 6934 203a 204c 203d 2061 6273  -+.|i4 : L = abs
│ │ │ │ +0000fbe0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +0000fbf0: 6f6d 706c 6578 207b 7b7d 7d20 2020 2020  omplex {{}}     
│ │ │ │ +0000fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fc10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fc70: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ -0000fc80: 202d 3220 3a20 3020 3c2d 2d2d 2d2d 2030   -2 : 0 <----- 0
│ │ │ │ -0000fc90: 203a 202d 3220 2020 2020 2020 2020 2020   : -2           
│ │ │ │ +0000fc50: 207c 0a7c 6f34 203d 2041 6273 7472 6163   |.|o4 = Abstrac
│ │ │ │ +0000fc60: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000fc70: 6578 7b2d 3120 3d3e 207b 7b7d 7d7d 2020  ex{-1 => {{}}}  
│ │ │ │ +0000fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fcb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000fcc0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -0000fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fcf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fcd0: 207c 0a7c 6f34 203a 2041 6273 7472 6163   |.|o4 : Abstrac
│ │ │ │ +0000fce0: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +0000fcf0: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  0000fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000fd30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000fd40: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -0000fd50: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ -0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fd70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000fd80: 202d 3120 3a20 5a5a 2020 3c2d 2d2d 2d2d   -1 : ZZ  <-----
│ │ │ │ -0000fd90: 2d2d 2d2d 205a 5a20 203a 202d 3120 2020  ---- ZZ  : -1   
│ │ │ │ +0000fd10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000fd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0000fd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000fd50: 2d2b 0a7c 6935 203a 2069 6e64 7563 6564  -+.|i5 : induced
│ │ │ │ +0000fd60: 5265 6475 6365 6453 696d 706c 6963 6961  ReducedSimplicia
│ │ │ │ +0000fd70: 6c43 6861 696e 436f 6d70 6c65 784d 6170  lChainComplexMap
│ │ │ │ +0000fd80: 284c 2c4c 2920 2020 2020 2020 2020 2020  (L,L)           
│ │ │ │ +0000fd90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fdb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0000fdc0: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ -0000fdd0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0000fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fdf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fdd0: 207c 0a7c 6f35 203d 202d 3220 3a20 3020   |.|o5 = -2 : 0 
│ │ │ │ +0000fde0: 3c2d 2d2d 2d2d 2030 203a 202d 3220 2020  <----- 0 : -2   
│ │ │ │ +0000fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fe30: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -0000fe40: 2043 6f6d 706c 6578 4d61 7020 2020 2020   ComplexMap     
│ │ │ │ -0000fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fe10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000fe20: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +0000fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fe50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0000fe70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0000fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000feb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -0000fec0: 204d 203d 2061 6273 7472 6163 7453 696d   M = abstractSim
│ │ │ │ -0000fed0: 706c 6963 6961 6c43 6f6d 706c 6578 207b  plicialComplex {
│ │ │ │ -0000fee0: 7b31 7d7d 2020 2020 2020 2020 2020 2020  {1}}            
│ │ │ │ -0000fef0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fe90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000fea0: 3120 2020 2020 2020 2020 2020 2020 2031  1              1
│ │ │ │ +0000feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000fed0: 207c 0a7c 2020 2020 202d 3120 3a20 5a5a   |.|     -1 : ZZ
│ │ │ │ +0000fee0: 2020 3c2d 2d2d 2d2d 2d2d 2d2d 205a 5a20    <--------- ZZ 
│ │ │ │ +0000fef0: 203a 202d 3120 2020 2020 2020 2020 2020   : -1           
│ │ │ │  0000ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ff30: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ -0000ff40: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -0000ff50: 6961 6c43 6f6d 706c 6578 7b2d 3120 3d3e  ialComplex{-1 =>
│ │ │ │ -0000ff60: 207b 7b7d 7d7d 2020 2020 2020 2020 2020   {{}}}          
│ │ │ │ -0000ff70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000ff10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000ff20: 2020 2020 207c 2031 207c 2020 2020 2020       | 1 |      
│ │ │ │ +0000ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ff50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0000ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ff90: 2020 2020 2020 2020 2020 2030 203d 3e20             0 => 
│ │ │ │ -0000ffa0: 7b7b 317d 7d20 2020 2020 2020 2020 2020  {{1}}           
│ │ │ │ -0000ffb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000ff90: 207c 0a7c 6f35 203a 2043 6f6d 706c 6578   |.|o5 : Complex
│ │ │ │ +0000ffa0: 4d61 7020 2020 2020 2020 2020 2020 2020  Map             
│ │ │ │ +0000ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fff0: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ -00010000: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -00010010: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ -00010020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010030: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00010040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010070: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ -00010080: 204c 203d 2061 6273 7472 6163 7453 696d   L = abstractSim
│ │ │ │ -00010090: 706c 6963 6961 6c43 6f6d 706c 6578 207b  plicialComplex {
│ │ │ │ -000100a0: 7b7d 7d20 2020 2020 2020 2020 2020 2020  {}}             
│ │ │ │ -000100b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0000ffd0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000ffe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000fff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00010010: 2d2b 0a7c 6936 203a 204d 203d 2061 6273  -+.|i6 : M = abs
│ │ │ │ +00010020: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00010030: 6f6d 706c 6578 207b 7b31 7d7d 2020 2020  omplex {{1}}    
│ │ │ │ +00010040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010050: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00010060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010090: 207c 0a7c 6f36 203d 2041 6273 7472 6163   |.|o6 = Abstrac
│ │ │ │ +000100a0: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +000100b0: 6578 7b2d 3120 3d3e 207b 7b7d 7d7d 2020  ex{-1 => {{}}}  
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│ │ │ │  000100e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000100f0: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ -00010100: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -00010110: 6961 6c43 6f6d 706c 6578 7b2d 3120 3d3e  ialComplex{-1 =>
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│ │ │ │ -00010130: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000100f0: 2020 2030 203d 3e20 7b7b 317d 7d20 2020     0 => {{1}}   
│ │ │ │ +00010100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00010120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010170: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -00010180: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -00010190: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ -000101a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000101b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00010150: 207c 0a7c 6f36 203a 2041 6273 7472 6163   |.|o6 : Abstrac
│ │ │ │ +00010160: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00010170: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │ +00010180: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000101b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000101c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000101d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000101e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000101f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -00010200: 2069 6e64 7563 6564 5265 6475 6365 6453   inducedReducedS
│ │ │ │ -00010210: 696d 706c 6963 6961 6c43 6861 696e 436f  implicialChainCo
│ │ │ │ -00010220: 6d70 6c65 784d 6170 284d 2c4c 2920 2020  mplexMap(M,L)   
│ │ │ │ -00010230: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000101d0: 2d2b 0a7c 6937 203a 204c 203d 2061 6273  -+.|i7 : L = abs
│ │ │ │ +000101e0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +000101f0: 6f6d 706c 6578 207b 7b7d 7d20 2020 2020  omplex {{}}     
│ │ │ │ +00010200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010210: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00010220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010270: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ -00010280: 202d 3220 3a20 3020 3c2d 2d2d 2d2d 2030   -2 : 0 <----- 0
│ │ │ │ -00010290: 203a 202d 3220 2020 2020 2020 2020 2020   : -2           
│ │ │ │ +00010250: 207c 0a7c 6f37 203d 2041 6273 7472 6163   |.|o7 = Abstrac
│ │ │ │ +00010260: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00010270: 6578 7b2d 3120 3d3e 207b 7b7d 7d7d 2020  ex{-1 => {{}}}  
│ │ │ │ +00010280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010290: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000102a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000102b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000102c0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -000102d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000102e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000102f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000102b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000102c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000102d0: 207c 0a7c 6f37 203a 2041 6273 7472 6163   |.|o7 : Abstrac
│ │ │ │ +000102e0: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +000102f0: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │  00010300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010330: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00010340: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00010350: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ -00010360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010370: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00010380: 202d 3120 3a20 5a5a 2020 3c2d 2d2d 2d2d   -1 : ZZ  <-----
│ │ │ │ -00010390: 2d2d 2d2d 205a 5a20 203a 202d 3120 2020  ---- ZZ  : -1   
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│ │ │ │ +00010320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010350: 2d2b 0a7c 6938 203a 2069 6e64 7563 6564  -+.|i8 : induced
│ │ │ │ +00010360: 5265 6475 6365 6453 696d 706c 6963 6961  ReducedSimplicia
│ │ │ │ +00010370: 6c43 6861 696e 436f 6d70 6c65 784d 6170  lChainComplexMap
│ │ │ │ +00010380: 284d 2c4c 2920 2020 2020 2020 2020 2020  (M,L)           
│ │ │ │ +00010390: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000103a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000103b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000103c0: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ -000103d0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -000103e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000103f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000103b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000103c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000103d0: 207c 0a7c 6f38 203d 202d 3220 3a20 3020   |.|o8 = -2 : 0 
│ │ │ │ +000103e0: 3c2d 2d2d 2d2d 2030 203a 202d 3220 2020  <----- 0 : -2   
│ │ │ │ +000103f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00010700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00010710: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +000107b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
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│ │ │ │ +000107f0: 6d70 6c65 784d 6170 3a0a 696e 6475 6365  mplexMap:.induce
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│ │ │ │ +00010980: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00010990: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000109a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000109b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000109c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000109d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00010a60: 7369 6d70 6c69 6369 616c 2063 6f6d 706c  simplicial compl
│ │ │ │ +00010a70: 6578 2c20 7468 656e 2069 7420 6973 2075  ex, then it is u
│ │ │ │ +00010a80: 7365 6675 6c20 746f 2063 616c 6375 6c61  seful to calcula
│ │ │ │ +00010a90: 7465 2074 6865 2069 6e64 7563 6564 0a6d  te the induced.m
│ │ │ │ +00010aa0: 6170 2061 7420 7468 6520 6c65 7665 6c20  ap at the level 
│ │ │ │ +00010ab0: 6f66 2053 696d 706c 6963 6961 6c20 4368  of Simplicial Ch
│ │ │ │ +00010ac0: 6169 6e20 436f 6d70 6c65 7865 732e 2020  ain Complexes.  
│ │ │ │ +00010ad0: 5468 6973 2069 7320 6d61 6465 2070 6f73  This is made pos
│ │ │ │ +00010ae0: 7369 626c 6520 6279 2074 6865 0a6d 6574  sible by the.met
│ │ │ │ +00010af0: 686f 6420 696e 6475 6365 6453 696d 706c  hod inducedSimpl
│ │ │ │ +00010b00: 6963 6961 6c43 6861 696e 436f 6d70 6c65  icialChainComple
│ │ │ │ +00010b10: 784d 6170 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  xMap...+--------
│ │ │ │ +00010b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010b50: 2d2d 2d2d 2d2b 0a7c 6931 203a 204b 203d  -----+.|i1 : K =
│ │ │ │ +00010b60: 2061 6273 7472 6163 7453 696d 706c 6963   abstractSimplic
│ │ │ │ +00010b70: 6961 6c43 6f6d 706c 6578 287b 7b31 2c32  ialComplex({{1,2
│ │ │ │ +00010b80: 7d2c 7b33 7d7d 2920 2020 2020 2020 2020  },{3}})         
│ │ │ │ +00010b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010bb0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00010bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00010c00: 6932 203a 204a 203d 2061 6d62 6965 6e74  i2 : J = ambient
│ │ │ │ -00010c10: 4162 7374 7261 6374 5369 6d70 6c69 6369  AbstractSimplici
│ │ │ │ -00010c20: 616c 436f 6d70 6c65 7828 4b29 2020 2020  alComplex(K)    
│ │ │ │ -00010c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010bd0: 2020 2020 207c 0a7c 6f31 203d 2041 6273       |.|o1 = Abs
│ │ │ │ +00010be0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00010bf0: 6f6d 706c 6578 7b2d 3120 3d3e 207b 7b7d  omplex{-1 => {{}
│ │ │ │ +00010c00: 7d20 2020 2020 2020 2020 207d 2020 2020  }          }    
│ │ │ │ +00010c10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00010c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010c30: 2020 2020 2020 2030 203d 3e20 7b7b 317d         0 => {{1}
│ │ │ │ +00010c40: 2c20 7b32 7d2c 207b 337d 7d20 2020 2020  , {2}, {3}}     
│ │ │ │ +00010c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010c80: 6f32 203d 2041 6273 7472 6163 7453 696d  o2 = AbstractSim
│ │ │ │ -00010c90: 706c 6963 6961 6c43 6f6d 706c 6578 7b2d  plicialComplex{-
│ │ │ │ -00010ca0: 3120 3d3e 207b 7b7d 7d20 2020 2020 2020  1 => {{}}       
│ │ │ │ -00010cb0: 2020 2020 2020 2020 2020 2020 7d7c 0a7c              }|.|
│ │ │ │ +00010c70: 2020 2020 2020 2031 203d 3e20 7b7b 312c         1 => {{1,
│ │ │ │ +00010c80: 2032 7d7d 2020 2020 2020 2020 2020 2020   2}}            
│ │ │ │ +00010c90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010cd0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00010ce0: 203d 3e20 7b7b 317d 2c20 7b32 7d2c 207b   => {{1}, {2}, {
│ │ │ │ -00010cf0: 337d 7d20 2020 2020 2020 2020 207c 0a7c  3}}          |.|
│ │ │ │ +00010cd0: 2020 2020 207c 0a7c 6f31 203a 2041 6273       |.|o1 : Abs
│ │ │ │ +00010ce0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00010cf0: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │  00010d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d10: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00010d20: 203d 3e20 7b7b 312c 2032 7d2c 207b 312c   => {{1, 2}, {1,
│ │ │ │ -00010d30: 2033 7d2c 207b 322c 2033 7d7d 207c 0a7c   3}, {2, 3}} |.|
│ │ │ │ -00010d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d50: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00010d60: 203d 3e20 7b7b 312c 2032 2c20 337d 7d20   => {{1, 2, 3}} 
│ │ │ │ -00010d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010d10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00010d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010d50: 2d2d 2d2d 2d2b 0a7c 6932 203a 204a 203d  -----+.|i2 : J =
│ │ │ │ +00010d60: 2061 6d62 6965 6e74 4162 7374 7261 6374   ambientAbstract
│ │ │ │ +00010d70: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00010d80: 7828 4b29 2020 2020 2020 2020 2020 2020  x(K)            
│ │ │ │ +00010d90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010dc0: 6f32 203a 2041 6273 7472 6163 7453 696d  o2 : AbstractSim
│ │ │ │ -00010dd0: 706c 6963 6961 6c43 6f6d 706c 6578 2020  plicialComplex  
│ │ │ │ -00010de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010df0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00010e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00010e40: 6933 203a 2069 6e64 7563 6564 5369 6d70  i3 : inducedSimp
│ │ │ │ -00010e50: 6c69 6369 616c 4368 6169 6e43 6f6d 706c  licialChainCompl
│ │ │ │ -00010e60: 6578 4d61 7028 4a2c 4b29 2020 2020 2020  exMap(J,K)      
│ │ │ │ -00010e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00010dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010dd0: 2020 2020 207c 0a7c 6f32 203d 2041 6273       |.|o2 = Abs
│ │ │ │ +00010de0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00010df0: 6f6d 706c 6578 7b2d 3120 3d3e 207b 7b7d  omplex{-1 => {{}
│ │ │ │ +00010e00: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00010e10: 2020 2020 7d7c 0a7c 2020 2020 2020 2020      }|.|        
│ │ │ │ +00010e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e30: 2020 2020 2020 2030 203d 3e20 7b7b 317d         0 => {{1}
│ │ │ │ +00010e40: 2c20 7b32 7d2c 207b 337d 7d20 2020 2020  , {2}, {3}}     
│ │ │ │ +00010e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00010e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e70: 2020 2020 2020 2031 203d 3e20 7b7b 312c         1 => {{1,
│ │ │ │ +00010e80: 2032 7d2c 207b 312c 2033 7d2c 207b 322c   2}, {1, 3}, {2,
│ │ │ │ +00010e90: 2033 7d7d 207c 0a7c 2020 2020 2020 2020   3}} |.|        
│ │ │ │  00010ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010ec0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00010ed0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00010eb0: 2020 2020 2020 2032 203d 3e20 7b7b 312c         2 => {{1,
│ │ │ │ +00010ec0: 2032 2c20 337d 7d20 2020 2020 2020 2020   2, 3}}         
│ │ │ │ +00010ed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010ef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010f00: 6f33 203d 2030 203a 205a 5a20 203c 2d2d  o3 = 0 : ZZ  <--
│ │ │ │ -00010f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 5a5a 2020  ----------- ZZ  
│ │ │ │ -00010f20: 3a20 3020 2020 2020 2020 2020 2020 2020  : 0             
│ │ │ │ -00010f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00010ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f10: 2020 2020 207c 0a7c 6f32 203a 2041 6273       |.|o2 : Abs
│ │ │ │ +00010f20: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00010f30: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │  00010f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f50: 7c20 3120 3020 3020 7c20 2020 2020 2020  | 1 0 0 |       
│ │ │ │ -00010f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f90: 7c20 3020 3120 3020 7c20 2020 2020 2020  | 0 1 0 |       
│ │ │ │ -00010fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010fd0: 7c20 3020 3020 3120 7c20 2020 2020 2020  | 0 0 1 |       
│ │ │ │ +00010f50: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f90: 2d2d 2d2d 2d2b 0a7c 6933 203a 2069 6e64  -----+.|i3 : ind
│ │ │ │ +00010fa0: 7563 6564 5369 6d70 6c69 6369 616c 4368  ucedSimplicialCh
│ │ │ │ +00010fb0: 6169 6e43 6f6d 706c 6578 4d61 7028 4a2c  ainComplexMap(J,
│ │ │ │ +00010fc0: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +00010fd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00010ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011040: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00011050: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00011060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011080: 2020 2020 2031 203a 205a 5a20 203c 2d2d       1 : ZZ  <--
│ │ │ │ -00011090: 2d2d 2d2d 2d2d 2d20 5a5a 2020 3a20 3120  ------- ZZ  : 1 
│ │ │ │ -000110a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000110b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011020: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00011030: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00011040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011050: 2020 2020 207c 0a7c 6f33 203d 2030 203a       |.|o3 = 0 :
│ │ │ │ +00011060: 205a 5a20 203c 2d2d 2d2d 2d2d 2d2d 2d2d   ZZ  <----------
│ │ │ │ +00011070: 2d2d 2d20 5a5a 2020 3a20 3020 2020 2020  --- ZZ  : 0     
│ │ │ │ +00011080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000110a0: 2020 2020 2020 2020 7c20 3120 3020 3020          | 1 0 0 
│ │ │ │ +000110b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000110c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000110d0: 7c20 3120 7c20 2020 2020 2020 2020 2020  | 1 |           
│ │ │ │ -000110e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000110f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000110d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000110e0: 2020 2020 2020 2020 7c20 3020 3120 3020          | 0 1 0 
│ │ │ │ +000110f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00011100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011110: 7c20 3020 7c20 2020 2020 2020 2020 2020  | 0 |           
│ │ │ │ -00011120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011130: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011120: 2020 2020 2020 2020 7c20 3020 3020 3120          | 0 0 1 
│ │ │ │ +00011130: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00011140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011150: 7c20 3020 7c20 2020 2020 2020 2020 2020  | 0 |           
│ │ │ │ +00011150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000111a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000111b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000111c0: 6f33 203a 2043 6f6d 706c 6578 4d61 7020  o3 : ComplexMap 
│ │ │ │ -000111d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000111e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000111f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00011200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00011240: 6934 203a 204c 203d 2061 6273 7472 6163  i4 : L = abstrac
│ │ │ │ -00011250: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ -00011260: 6578 207b 7b7d 7d20 2020 2020 2020 2020  ex {{}}         
│ │ │ │ -00011270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000111a0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +000111b0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +000111c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000111d0: 2020 2020 207c 0a7c 2020 2020 2031 203a       |.|     1 :
│ │ │ │ +000111e0: 205a 5a20 203c 2d2d 2d2d 2d2d 2d2d 2d20   ZZ  <--------- 
│ │ │ │ +000111f0: 5a5a 2020 3a20 3120 2020 2020 2020 2020  ZZ  : 1         
│ │ │ │ +00011200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011220: 2020 2020 2020 2020 7c20 3120 7c20 2020          | 1 |   
│ │ │ │ +00011230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011260: 2020 2020 2020 2020 7c20 3020 7c20 2020          | 0 |   
│ │ │ │ +00011270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000112a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000112b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000112c0: 6f34 203d 2041 6273 7472 6163 7453 696d  o4 = AbstractSim
│ │ │ │ -000112d0: 706c 6963 6961 6c43 6f6d 706c 6578 7b2d  plicialComplex{-
│ │ │ │ -000112e0: 3120 3d3e 207b 7b7d 7d7d 2020 2020 2020  1 => {{}}}      
│ │ │ │ -000112f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000112a0: 2020 2020 2020 2020 7c20 3020 7c20 2020          | 0 |   
│ │ │ │ +000112b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000112c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000112d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000112e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000112f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011340: 6f34 203a 2041 6273 7472 6163 7453 696d  o4 : AbstractSim
│ │ │ │ -00011350: 706c 6963 6961 6c43 6f6d 706c 6578 2020  plicialComplex  
│ │ │ │ -00011360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011370: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00011310: 2020 2020 207c 0a7c 6f33 203a 2043 6f6d       |.|o3 : Com
│ │ │ │ +00011320: 706c 6578 4d61 7020 2020 2020 2020 2020  plexMap         
│ │ │ │ +00011330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011350: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00011360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000113a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000113b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000113c0: 6935 203a 2069 6e64 7563 6564 5369 6d70  i5 : inducedSimp
│ │ │ │ -000113d0: 6c69 6369 616c 4368 6169 6e43 6f6d 706c  licialChainCompl
│ │ │ │ -000113e0: 6578 4d61 7028 4c2c 4c29 2020 2020 2020  exMap(L,L)      
│ │ │ │ -000113f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011390: 2d2d 2d2d 2d2b 0a7c 6934 203a 204c 203d  -----+.|i4 : L =
│ │ │ │ +000113a0: 2061 6273 7472 6163 7453 696d 706c 6963   abstractSimplic
│ │ │ │ +000113b0: 6961 6c43 6f6d 706c 6578 207b 7b7d 7d20  ialComplex {{}} 
│ │ │ │ +000113c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000113d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000113e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000113f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011440: 6f35 203d 2030 2020 2020 2020 2020 2020  o5 = 0          
│ │ │ │ -00011450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011410: 2020 2020 207c 0a7c 6f34 203d 2041 6273       |.|o4 = Abs
│ │ │ │ +00011420: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00011430: 6f6d 706c 6578 7b2d 3120 3d3e 207b 7b7d  omplex{-1 => {{}
│ │ │ │ +00011440: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00011450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000114a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000114b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000114c0: 6f35 203a 2043 6f6d 706c 6578 4d61 7020  o5 : ComplexMap 
│ │ │ │ -000114d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000114e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000114f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00011490: 2020 2020 207c 0a7c 6f34 203a 2041 6273       |.|o4 : Abs
│ │ │ │ +000114a0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +000114b0: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +000114c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000114d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000114e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000114f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00011540: 6936 203a 204d 203d 2061 6273 7472 6163  i6 : M = abstrac
│ │ │ │ -00011550: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ -00011560: 6578 207b 7b31 7d7d 2020 2020 2020 2020  ex {{1}}        
│ │ │ │ -00011570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011510: 2d2d 2d2d 2d2b 0a7c 6935 203a 2069 6e64  -----+.|i5 : ind
│ │ │ │ +00011520: 7563 6564 5369 6d70 6c69 6369 616c 4368  ucedSimplicialCh
│ │ │ │ +00011530: 6169 6e43 6f6d 706c 6578 4d61 7028 4c2c  ainComplexMap(L,
│ │ │ │ +00011540: 4c29 2020 2020 2020 2020 2020 2020 2020  L)              
│ │ │ │ +00011550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011590: 2020 2020 207c 0a7c 6f35 203d 2030 2020       |.|o5 = 0  
│ │ │ │  000115a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000115b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000115c0: 6f36 203d 2041 6273 7472 6163 7453 696d  o6 = AbstractSim
│ │ │ │ -000115d0: 706c 6963 6961 6c43 6f6d 706c 6578 7b2d  plicialComplex{-
│ │ │ │ -000115e0: 3120 3d3e 207b 7b7d 7d7d 2020 2020 2020  1 => {{}}}      
│ │ │ │ -000115f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000115b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000115c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000115d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000115e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000115f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011610: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00011620: 203d 3e20 7b7b 317d 7d20 2020 2020 2020   => {{1}}       
│ │ │ │ -00011630: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011610: 2020 2020 207c 0a7c 6f35 203a 2043 6f6d       |.|o5 : Com
│ │ │ │ +00011620: 706c 6578 4d61 7020 2020 2020 2020 2020  plexMap         
│ │ │ │ +00011630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011640: 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                  
│ │ │ │ -00011670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011680: 6f36 203a 2041 6273 7472 6163 7453 696d  o6 : AbstractSim
│ │ │ │ -00011690: 706c 6963 6961 6c43 6f6d 706c 6578 2020  plicialComplex  
│ │ │ │ -000116a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000116b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000116c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000116d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000116e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000116f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00011700: 6937 203a 204c 203d 2061 6273 7472 6163  i7 : L = abstrac
│ │ │ │ -00011710: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ -00011720: 6578 207b 7b7d 7d20 2020 2020 2020 2020  ex {{}}         
│ │ │ │ -00011730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011650: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00011660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011690: 2d2d 2d2d 2d2b 0a7c 6936 203a 204d 203d  -----+.|i6 : M =
│ │ │ │ +000116a0: 2061 6273 7472 6163 7453 696d 706c 6963   abstractSimplic
│ │ │ │ +000116b0: 6961 6c43 6f6d 706c 6578 207b 7b31 7d7d  ialComplex {{1}}
│ │ │ │ +000116c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000116d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000116e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000116f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011710: 2020 2020 207c 0a7c 6f36 203d 2041 6273       |.|o6 = Abs
│ │ │ │ +00011720: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00011730: 6f6d 706c 6578 7b2d 3120 3d3e 207b 7b7d  omplex{-1 => {{}
│ │ │ │ +00011740: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00011750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011770: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011780: 6f37 203d 2041 6273 7472 6163 7453 696d  o7 = AbstractSim
│ │ │ │ -00011790: 706c 6963 6961 6c43 6f6d 706c 6578 7b2d  plicialComplex{-
│ │ │ │ -000117a0: 3120 3d3e 207b 7b7d 7d7d 2020 2020 2020  1 => {{}}}      
│ │ │ │ -000117b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011770: 2020 2020 2020 2030 203d 3e20 7b7b 317d         0 => {{1}
│ │ │ │ +00011780: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00011790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000117a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000117b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000117c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000117d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000117e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000117f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011800: 6f37 203a 2041 6273 7472 6163 7453 696d  o7 : AbstractSim
│ │ │ │ -00011810: 706c 6963 6961 6c43 6f6d 706c 6578 2020  plicialComplex  
│ │ │ │ -00011820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011830: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000117d0: 2020 2020 207c 0a7c 6f36 203a 2041 6273       |.|o6 : Abs
│ │ │ │ +000117e0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +000117f0: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00011800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011810: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00011820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00011880: 6938 203a 2069 6e64 7563 6564 5369 6d70  i8 : inducedSimp
│ │ │ │ -00011890: 6c69 6369 616c 4368 6169 6e43 6f6d 706c  licialChainCompl
│ │ │ │ -000118a0: 6578 4d61 7028 4d2c 4c29 2020 2020 2020  exMap(M,L)      
│ │ │ │ -000118b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011850: 2d2d 2d2d 2d2b 0a7c 6937 203a 204c 203d  -----+.|i7 : L =
│ │ │ │ +00011860: 2061 6273 7472 6163 7453 696d 706c 6963   abstractSimplic
│ │ │ │ +00011870: 6961 6c43 6f6d 706c 6578 207b 7b7d 7d20  ialComplex {{}} 
│ │ │ │ +00011880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000118a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000118b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000118c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000118d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000118e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000118f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011900: 6f38 203d 2030 2020 2020 2020 2020 2020  o8 = 0          
│ │ │ │ -00011910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000118d0: 2020 2020 207c 0a7c 6f37 203d 2041 6273       |.|o7 = Abs
│ │ │ │ +000118e0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +000118f0: 6f6d 706c 6578 7b2d 3120 3d3e 207b 7b7d  omplex{-1 => {{}
│ │ │ │ +00011900: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00011910: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00011930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00011980: 6f38 203a 2043 6f6d 706c 6578 4d61 7020  o8 : ComplexMap 
│ │ │ │ -00011990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000119a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000119b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00011950: 2020 2020 207c 0a7c 6f37 203a 2041 6273       |.|o7 : Abs
│ │ │ │ +00011960: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00011970: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00011980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011990: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000119a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000119b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000119c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000119d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000119e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000119f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00011a00: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -00011a10: 3d0a 0a20 202a 202a 6e6f 7465 2069 6e64  =..  * *note ind
│ │ │ │ -00011a20: 7563 6564 5265 6475 6365 6453 696d 706c  ucedReducedSimpl
│ │ │ │ -00011a30: 6963 6961 6c43 6861 696e 436f 6d70 6c65  icialChainComple
│ │ │ │ -00011a40: 784d 6170 3a0a 2020 2020 696e 6475 6365  xMap:.    induce
│ │ │ │ -00011a50: 6452 6564 7563 6564 5369 6d70 6c69 6369  dReducedSimplici
│ │ │ │ -00011a60: 616c 4368 6169 6e43 6f6d 706c 6578 4d61  alChainComplexMa
│ │ │ │ -00011a70: 702c 202d 2d20 496e 6475 6365 6420 6d61  p, -- Induced ma
│ │ │ │ -00011a80: 7073 2074 6861 7420 6172 6973 6520 7669  ps that arise vi
│ │ │ │ -00011a90: 610a 2020 2020 696e 636c 7573 696f 6e73  a.    inclusions
│ │ │ │ -00011aa0: 206f 6620 6162 7374 7261 6374 2073 696d   of abstract sim
│ │ │ │ -00011ab0: 706c 6963 6961 6c20 636f 6d70 6c65 7865  plicial complexe
│ │ │ │ -00011ac0: 730a 0a57 6179 7320 746f 2075 7365 2069  s..Ways to use i
│ │ │ │ -00011ad0: 6e64 7563 6564 5369 6d70 6c69 6369 616c  nducedSimplicial
│ │ │ │ -00011ae0: 4368 6169 6e43 6f6d 706c 6578 4d61 703a  ChainComplexMap:
│ │ │ │ -00011af0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00011b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00011b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00011b20: 2020 2a20 2269 6e64 7563 6564 5369 6d70    * "inducedSimp
│ │ │ │ -00011b30: 6c69 6369 616c 4368 6169 6e43 6f6d 706c  licialChainCompl
│ │ │ │ -00011b40: 6578 4d61 7028 4162 7374 7261 6374 5369  exMap(AbstractSi
│ │ │ │ -00011b50: 6d70 6c69 6369 616c 436f 6d70 6c65 782c  mplicialComplex,
│ │ │ │ -00011b60: 0a20 2020 2041 6273 7472 6163 7453 696d  .    AbstractSim
│ │ │ │ -00011b70: 706c 6963 6961 6c43 6f6d 706c 6578 2922  plicialComplex)"
│ │ │ │ -00011b80: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -00011b90: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -00011ba0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -00011bb0: 6563 7420 2a6e 6f74 6520 696e 6475 6365  ect *note induce
│ │ │ │ -00011bc0: 6453 696d 706c 6963 6961 6c43 6861 696e  dSimplicialChain
│ │ │ │ -00011bd0: 436f 6d70 6c65 784d 6170 3a0a 696e 6475  ComplexMap:.indu
│ │ │ │ -00011be0: 6365 6453 696d 706c 6963 6961 6c43 6861  cedSimplicialCha
│ │ │ │ -00011bf0: 696e 436f 6d70 6c65 784d 6170 2c20 6973  inComplexMap, is
│ │ │ │ -00011c00: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -00011c10: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ -00011c20: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -00011c30: 6e63 7469 6f6e 2c2e 0a1f 0a46 696c 653a  nction,....File:
│ │ │ │ -00011c40: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -00011c50: 6961 6c43 6f6d 706c 6578 6573 2e69 6e66  ialComplexes.inf
│ │ │ │ -00011c60: 6f2c 204e 6f64 653a 206e 6577 2041 6273  o, Node: new Abs
│ │ │ │ -00011c70: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ -00011c80: 6f6d 706c 6578 2c20 4e65 7874 3a20 7261  omplex, Next: ra
│ │ │ │ -00011c90: 6e64 6f6d 4162 7374 7261 6374 5369 6d70  ndomAbstractSimp
│ │ │ │ -00011ca0: 6c69 6369 616c 436f 6d70 6c65 782c 2050  licialComplex, P
│ │ │ │ -00011cb0: 7265 763a 2069 6e64 7563 6564 5369 6d70  rev: inducedSimp
│ │ │ │ -00011cc0: 6c69 6369 616c 4368 6169 6e43 6f6d 706c  licialChainCompl
│ │ │ │ -00011cd0: 6578 4d61 702c 2055 703a 2054 6f70 0a0a  exMap, Up: Top..
│ │ │ │ -00011ce0: 6e65 7720 4162 7374 7261 6374 5369 6d70  new AbstractSimp
│ │ │ │ -00011cf0: 6c69 6369 616c 436f 6d70 6c65 780a 2a2a  licialComplex.**
│ │ │ │ -00011d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a57 6179  ***********..Way
│ │ │ │ -00011d20: 7320 746f 2075 7365 2074 6869 7320 6d65  s to use this me
│ │ │ │ -00011d30: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  thod:.==========
│ │ │ │ -00011d40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00011d50: 2020 2a20 2a6e 6f74 6520 6e65 7720 4162    * *note new Ab
│ │ │ │ -00011d60: 7374 7261 6374 5369 6d70 6c69 6369 616c  stractSimplicial
│ │ │ │ -00011d70: 436f 6d70 6c65 783a 206e 6577 2041 6273  Complex: new Abs
│ │ │ │ -00011d80: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ -00011d90: 6f6d 706c 6578 2c0a 1f0a 4669 6c65 3a20  omplex,...File: 
│ │ │ │ -00011da0: 4162 7374 7261 6374 5369 6d70 6c69 6369  AbstractSimplici
│ │ │ │ -00011db0: 616c 436f 6d70 6c65 7865 732e 696e 666f  alComplexes.info
│ │ │ │ -00011dc0: 2c20 4e6f 6465 3a20 7261 6e64 6f6d 4162  , Node: randomAb
│ │ │ │ -00011dd0: 7374 7261 6374 5369 6d70 6c69 6369 616c  stractSimplicial
│ │ │ │ -00011de0: 436f 6d70 6c65 782c 204e 6578 743a 2072  Complex, Next: r
│ │ │ │ -00011df0: 616e 646f 6d53 7562 5369 6d70 6c69 6369  andomSubSimplici
│ │ │ │ -00011e00: 616c 436f 6d70 6c65 782c 2050 7265 763a  alComplex, Prev:
│ │ │ │ -00011e10: 206e 6577 2041 6273 7472 6163 7453 696d   new AbstractSim
│ │ │ │ -00011e20: 706c 6963 6961 6c43 6f6d 706c 6578 2c20  plicialComplex, 
│ │ │ │ -00011e30: 5570 3a20 546f 700a 0a72 616e 646f 6d41  Up: Top..randomA
│ │ │ │ -00011e40: 6273 7472 6163 7453 696d 706c 6963 6961  bstractSimplicia
│ │ │ │ -00011e50: 6c43 6f6d 706c 6578 202d 2d20 4372 6561  lComplex -- Crea
│ │ │ │ -00011e60: 7465 2061 2072 616e 646f 6d20 7369 6d70  te a random simp
│ │ │ │ -00011e70: 6c69 6369 616c 2073 6574 0a2a 2a2a 2a2a  licial set.*****
│ │ │ │ -00011e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465  ************..De
│ │ │ │ -00011ec0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -00011ed0: 3d3d 3d3d 3d0a 0a43 7265 6174 6520 6120  =====..Create a 
│ │ │ │ -00011ee0: 7261 6e64 6f6d 2061 6273 7472 6163 7420  random abstract 
│ │ │ │ -00011ef0: 7369 6d70 6c69 6369 616c 2063 6f6d 706c  simplicial compl
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│ │ │ │ -00011f10: 2073 7570 706f 7274 6564 206f 6e20 6120   supported on a 
│ │ │ │ -00011f20: 7375 6273 6574 0a6f 6620 5b6e 5d20 3d20  subset.of [n] = 
│ │ │ │ -00011f30: 7b31 2c2e 2e2e 2c6e 7d2e 0a0a 2b2d 2d2d  {1,...,n}...+---
│ │ │ │ -00011f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -00011f80: 3a20 7365 7452 616e 646f 6d53 6565 6428  : setRandomSeed(
│ │ │ │ -00011f90: 6375 7272 656e 7454 696d 6528 2929 3b20  currentTime()); 
│ │ │ │ -00011fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011fb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00011fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00012000: 3a20 4b20 3d20 7261 6e64 6f6d 4162 7374  : K = randomAbst
│ │ │ │ -00012010: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -00012020: 6d70 6c65 7828 3429 2020 2020 2020 2020  mplex(4)        
│ │ │ │ -00012030: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00012040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012070: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00012080: 3d20 4162 7374 7261 6374 5369 6d70 6c69  = AbstractSimpli
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│ │ │ │ -000120c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000120d0: 2020 2020 2020 2020 2020 2020 3020 3d3e              0 =>
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│ │ │ │ -000120f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00012100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012110: 2020 2020 2020 2020 2020 2020 3120 3d3e              1 =>
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│ │ │ │ -00012140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012150: 2020 2020 2020 2020 2020 2020 3220 3d3e              2 =>
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│ │ │ │ -00012170: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00012180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000121a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000121b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -000121c0: 3a20 4162 7374 7261 6374 5369 6d70 6c69  : AbstractSimpli
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│ │ │ │ -000121f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00012200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012230: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 7265  ----------+..Cre
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│ │ │ │ -00012260: 6f6e 205b 6e5d 2077 6974 6820 6469 6d65  on [n] with dime
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│ │ │ │ +00011a00: 4c29 2020 2020 2020 2020 2020 2020 2020  L)              
│ │ │ │ +00011a10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a50: 2020 2020 207c 0a7c 6f38 203d 2030 2020       |.|o8 = 0  
│ │ │ │ +00011a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00011aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ad0: 2020 2020 207c 0a7c 6f38 203a 2043 6f6d       |.|o8 : Com
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│ │ │ │ +00011b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00011b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011b50: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ +00011b60: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00011b70: 6e6f 7465 2069 6e64 7563 6564 5265 6475  note inducedRedu
│ │ │ │ +00011b80: 6365 6453 696d 706c 6963 6961 6c43 6861  cedSimplicialCha
│ │ │ │ +00011b90: 696e 436f 6d70 6c65 784d 6170 3a0a 2020  inComplexMap:.  
│ │ │ │ +00011ba0: 2020 696e 6475 6365 6452 6564 7563 6564    inducedReduced
│ │ │ │ +00011bb0: 5369 6d70 6c69 6369 616c 4368 6169 6e43  SimplicialChainC
│ │ │ │ +00011bc0: 6f6d 706c 6578 4d61 702c 202d 2d20 496e  omplexMap, -- In
│ │ │ │ +00011bd0: 6475 6365 6420 6d61 7073 2074 6861 7420  duced maps that 
│ │ │ │ +00011be0: 6172 6973 6520 7669 610a 2020 2020 696e  arise via.    in
│ │ │ │ +00011bf0: 636c 7573 696f 6e73 206f 6620 6162 7374  clusions of abst
│ │ │ │ +00011c00: 7261 6374 2073 696d 706c 6963 6961 6c20  ract simplicial 
│ │ │ │ +00011c10: 636f 6d70 6c65 7865 730a 0a57 6179 7320  complexes..Ways 
│ │ │ │ +00011c20: 746f 2075 7365 2069 6e64 7563 6564 5369  to use inducedSi
│ │ │ │ +00011c30: 6d70 6c69 6369 616c 4368 6169 6e43 6f6d  mplicialChainCom
│ │ │ │ +00011c40: 706c 6578 4d61 703a 0a3d 3d3d 3d3d 3d3d  plexMap:.=======
│ │ │ │ +00011c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00011c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00011c70: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 6e64  ======..  * "ind
│ │ │ │ +00011c80: 7563 6564 5369 6d70 6c69 6369 616c 4368  ucedSimplicialCh
│ │ │ │ +00011c90: 6169 6e43 6f6d 706c 6578 4d61 7028 4162  ainComplexMap(Ab
│ │ │ │ +00011ca0: 7374 7261 6374 5369 6d70 6c69 6369 616c  stractSimplicial
│ │ │ │ +00011cb0: 436f 6d70 6c65 782c 0a20 2020 2041 6273  Complex,.    Abs
│ │ │ │ +00011cc0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00011cd0: 6f6d 706c 6578 2922 0a0a 466f 7220 7468  omplex)"..For th
│ │ │ │ +00011ce0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +00011cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00011d00: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00011d10: 6520 696e 6475 6365 6453 696d 706c 6963  e inducedSimplic
│ │ │ │ +00011d20: 6961 6c43 6861 696e 436f 6d70 6c65 784d  ialChainComplexM
│ │ │ │ +00011d30: 6170 3a0a 696e 6475 6365 6453 696d 706c  ap:.inducedSimpl
│ │ │ │ +00011d40: 6963 6961 6c43 6861 696e 436f 6d70 6c65  icialChainComple
│ │ │ │ +00011d50: 784d 6170 2c20 6973 2061 202a 6e6f 7465  xMap, is a *note
│ │ │ │ +00011d60: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00011d70: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00011d80: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +00011d90: 0a1f 0a46 696c 653a 2041 6273 7472 6163  ...File: Abstrac
│ │ │ │ +00011da0: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00011db0: 6578 6573 2e69 6e66 6f2c 204e 6f64 653a  exes.info, Node:
│ │ │ │ +00011dc0: 206e 6577 2041 6273 7472 6163 7453 696d   new AbstractSim
│ │ │ │ +00011dd0: 706c 6963 6961 6c43 6f6d 706c 6578 2c20  plicialComplex, 
│ │ │ │ +00011de0: 4e65 7874 3a20 7261 6e64 6f6d 4162 7374  Next: randomAbst
│ │ │ │ +00011df0: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ +00011e00: 6d70 6c65 782c 2050 7265 763a 2069 6e64  mplex, Prev: ind
│ │ │ │ +00011e10: 7563 6564 5369 6d70 6c69 6369 616c 4368  ucedSimplicialCh
│ │ │ │ +00011e20: 6169 6e43 6f6d 706c 6578 4d61 702c 2055  ainComplexMap, U
│ │ │ │ +00011e30: 703a 2054 6f70 0a0a 6e65 7720 4162 7374  p: Top..new Abst
│ │ │ │ +00011e40: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ +00011e50: 6d70 6c65 780a 2a2a 2a2a 2a2a 2a2a 2a2a  mplex.**********
│ │ │ │ +00011e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011e70: 2a2a 2a0a 0a57 6179 7320 746f 2075 7365  ***..Ways to use
│ │ │ │ +00011e80: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │ +00011e90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00011ea0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00011eb0: 6520 6e65 7720 4162 7374 7261 6374 5369  e new AbstractSi
│ │ │ │ +00011ec0: 6d70 6c69 6369 616c 436f 6d70 6c65 783a  mplicialComplex:
│ │ │ │ +00011ed0: 206e 6577 2041 6273 7472 6163 7453 696d   new AbstractSim
│ │ │ │ +00011ee0: 706c 6963 6961 6c43 6f6d 706c 6578 2c0a  plicialComplex,.
│ │ │ │ +00011ef0: 1f0a 4669 6c65 3a20 4162 7374 7261 6374  ..File: Abstract
│ │ │ │ +00011f00: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00011f10: 7865 732e 696e 666f 2c20 4e6f 6465 3a20  xes.info, Node: 
│ │ │ │ +00011f20: 7261 6e64 6f6d 4162 7374 7261 6374 5369  randomAbstractSi
│ │ │ │ +00011f30: 6d70 6c69 6369 616c 436f 6d70 6c65 782c  mplicialComplex,
│ │ │ │ +00011f40: 204e 6578 743a 2072 616e 646f 6d53 7562   Next: randomSub
│ │ │ │ +00011f50: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00011f60: 782c 2050 7265 763a 206e 6577 2041 6273  x, Prev: new Abs
│ │ │ │ +00011f70: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00011f80: 6f6d 706c 6578 2c20 5570 3a20 546f 700a  omplex, Up: Top.
│ │ │ │ +00011f90: 0a72 616e 646f 6d41 6273 7472 6163 7453  .randomAbstractS
│ │ │ │ +00011fa0: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +00011fb0: 202d 2d20 4372 6561 7465 2061 2072 616e   -- Create a ran
│ │ │ │ +00011fc0: 646f 6d20 7369 6d70 6c69 6369 616c 2073  dom simplicial s
│ │ │ │ +00011fd0: 6574 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  et.*************
│ │ │ │ +00011fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012010: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f  ****..Descriptio
│ │ │ │ +00012020: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43  n.===========..C
│ │ │ │ +00012030: 7265 6174 6520 6120 7261 6e64 6f6d 2061  reate a random a
│ │ │ │ +00012040: 6273 7472 6163 7420 7369 6d70 6c69 6369  bstract simplici
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│ │ │ │ +00012060: 7665 7274 6963 6573 2073 7570 706f 7274  vertices support
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│ │ │ │ +00012090: 7d2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  }...+-----------
│ │ │ │ +000120a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120c0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7365 7452  ----+.|i1 : setR
│ │ │ │ +000120d0: 616e 646f 6d53 6565 6428 6375 7272 656e  andomSeed(curren
│ │ │ │ +000120e0: 7454 696d 6528 2929 3b20 2020 2020 2020  tTime());       
│ │ │ │ +000120f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00012100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012120: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00012130: 4b20 3d20 7261 6e64 6f6d 4162 7374 7261  K = randomAbstra
│ │ │ │ +00012140: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ +00012150: 6c65 7828 3429 2020 2020 7c0a 7c20 2020  lex(4)    |.|   
│ │ │ │ +00012160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012180: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00012190: 3220 3d20 4162 7374 7261 6374 5369 6d70  2 = AbstractSimp
│ │ │ │ +000121a0: 6c69 6369 616c 436f 6d70 6c65 787b 2d31  licialComplex{-1
│ │ │ │ +000121b0: 203d 3e20 7b7b 7d7d 2020 2020 207d 7c0a   => {{}}     }|.
│ │ │ │ +000121c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000121d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000121e0: 3020 3d3e 207b 7b31 7d2c 207b 337d 7d20  0 => {{1}, {3}} 
│ │ │ │ +000121f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00012200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012210: 2020 3120 3d3e 207b 7b31 2c20 337d 7d20    1 => {{1, 3}} 
│ │ │ │ +00012220: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00012230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012250: 2020 2020 7c0a 7c6f 3220 3a20 4162 7374      |.|o2 : Abst
│ │ │ │ +00012260: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ +00012270: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ +00012280: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00012290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000122a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -000122d0: 2073 6574 5261 6e64 6f6d 5365 6564 2863   setRandomSeed(c
│ │ │ │ -000122e0: 7572 7265 6e74 5469 6d65 2829 293b 2020  urrentTime());  
│ │ │ │ -000122f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012300: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000122b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 7265 6174  --------+..Creat
│ │ │ │ +000122c0: 6520 6120 7261 6e64 6f6d 2073 696d 706c  e a random simpl
│ │ │ │ +000122d0: 6963 6961 6c20 636f 6d70 6c65 7820 6f6e  icial complex on
│ │ │ │ +000122e0: 205b 6e5d 2077 6974 6820 6469 6d65 6e73   [n] with dimens
│ │ │ │ +000122f0: 696f 6e20 6174 206d 6f73 7420 6571 7561  ion at most equa
│ │ │ │ +00012300: 6c20 746f 2072 2e0a 0a2b 2d2d 2d2d 2d2d  l to r...+------
│ │ │ │  00012310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012340: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -00012350: 204c 203d 2072 616e 646f 6d41 6273 7472   L = randomAbstr
│ │ │ │ -00012360: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ -00012370: 706c 6578 2836 2c33 2920 2020 2020 2020  plex(6,3)       
│ │ │ │ -00012380: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00012390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000123a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000123b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000123c0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -000123d0: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -000123e0: 6961 6c43 6f6d 706c 6578 7b2d 3120 3d3e  ialComplex{-1 =>
│ │ │ │ -000123f0: 207b 7b7d 7d20 2020 2020 2020 2020 2020   {{}}           
│ │ │ │ -00012400: 2020 2020 2020 2020 7d7c 0a7c 2020 2020          }|.|    
│ │ │ │ -00012410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012420: 2020 2020 2020 2020 2020 2030 203d 3e20             0 => 
│ │ │ │ -00012430: 7b7b 317d 2c20 7b34 7d2c 207b 367d 7d20  {{1}, {4}, {6}} 
│ │ │ │ -00012440: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00012450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012460: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
│ │ │ │ -00012470: 7b7b 312c 2034 7d2c 207b 312c 2036 7d2c  {{1, 4}, {1, 6},
│ │ │ │ -00012480: 207b 342c 2036 7d7d 207c 0a7c 2020 2020   {4, 6}} |.|    
│ │ │ │ -00012490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000124a0: 2020 2020 2020 2020 2020 2032 203d 3e20             2 => 
│ │ │ │ -000124b0: 7b7b 312c 2034 2c20 367d 7d20 2020 2020  {{1, 4, 6}}     
│ │ │ │ -000124c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000124d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000124e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000124f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012500: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -00012510: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
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│ │ │ │ -00012530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012540: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012580: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4372 6561  ---------+..Crea
│ │ │ │ -00012590: 7465 2074 6865 2072 616e 646f 6d20 636f  te the random co
│ │ │ │ -000125a0: 6d70 6c65 7820 595f 6428 6e2c 6d29 2077  mplex Y_d(n,m) w
│ │ │ │ -000125b0: 6869 6368 2068 6173 2076 6572 7465 7820  hich has vertex 
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│ │ │ │ -000125e0: 6b65 6c65 746f 6e2c 2061 6e64 2068 6173  keleton, and has
│ │ │ │ -000125f0: 2065 7861 6374 6c79 206d 2064 2d64 696d   exactly m d-dim
│ │ │ │ -00012600: 656e 7369 6f6e 616c 2066 6163 6573 2c20  ensional faces, 
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│ │ │ │ -00012620: 2066 726f 6d20 616c 6c0a 6269 6e6f 6d69   from all.binomi
│ │ │ │ -00012630: 616c 2862 696e 6f6d 6961 6c28 6e2c 642b  al(binomial(n,d+
│ │ │ │ -00012640: 3129 2c6d 2920 706f 7373 6962 696c 6974  1),m) possibilit
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│ │ │ │ -00012660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000126a0: 2d2d 2d2d 2b0a 7c69 3520 3a20 7365 7452  ----+.|i5 : setR
│ │ │ │ -000126b0: 616e 646f 6d53 6565 6428 6375 7272 656e  andomSeed(curren
│ │ │ │ -000126c0: 7454 696d 6528 2929 3b20 2020 2020 2020  tTime());       
│ │ │ │ -000126d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000126e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000126f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00012700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012740: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d20  ----+.|i6 : M = 
│ │ │ │ -00012750: 7261 6e64 6f6d 4162 7374 7261 6374 5369  randomAbstractSi
│ │ │ │ -00012760: 6d70 6c69 6369 616c 436f 6d70 6c65 7828  mplicialComplex(
│ │ │ │ -00012770: 362c 332c 3229 2020 2020 2020 2020 2020  6,3,2)          
│ │ │ │ -00012780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000127a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000127b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012330: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
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│ │ │ │ +00012350: 7572 7265 6e74 5469 6d65 2829 293b 2020  urrentTime());  
│ │ │ │ +00012360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00012370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000123a0: 6934 203a 204c 203d 2072 616e 646f 6d41  i4 : L = randomA
│ │ │ │ +000123b0: 6273 7472 6163 7453 696d 706c 6963 6961  bstractSimplicia
│ │ │ │ +000123c0: 6c43 6f6d 706c 6578 2836 2c33 2920 207c  lComplex(6,3)  |
│ │ │ │ +000123d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000123e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000123f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012400: 207c 0a7c 6f34 203d 2041 6273 7472 6163   |.|o4 = Abstrac
│ │ │ │ +00012410: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00012420: 6578 7b2d 3120 3d3e 207b 7b7d 7d20 2020  ex{-1 => {{}}   
│ │ │ │ +00012430: 2020 7d7c 0a7c 2020 2020 2020 2020 2020    }|.|          
│ │ │ │ +00012440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012450: 2020 2020 2030 203d 3e20 7b7b 317d 2c20       0 => {{1}, 
│ │ │ │ +00012460: 7b33 7d7d 207c 0a7c 2020 2020 2020 2020  {3}} |.|        
│ │ │ │ +00012470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012480: 2020 2020 2020 2031 203d 3e20 7b7b 312c         1 => {{1,
│ │ │ │ +00012490: 2033 7d7d 2020 207c 0a7c 2020 2020 2020   3}}   |.|      
│ │ │ │ +000124a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000124b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000124c0: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +000124d0: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ +000124e0: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ +000124f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00012500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00012530: 4372 6561 7465 2074 6865 2072 616e 646f  Create the rando
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│ │ │ │ +000125c0: 6e64 6f6d 2066 726f 6d20 616c 6c0a 6269  ndom from all.bi
│ │ │ │ +000125d0: 6e6f 6d69 616c 2862 696e 6f6d 6961 6c28  nomial(binomial(
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│ │ │ │ +00012610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012640: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
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│ │ │ │ +00012670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012690: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000126a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +000126f0: 4d20 3d20 7261 6e64 6f6d 4162 7374 7261  M = randomAbstra
│ │ │ │ +00012700: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ +00012710: 6c65 7828 362c 332c 3229 2020 2020 2020  lex(6,3,2)      
│ │ │ │ +00012720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00012740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00012760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012780: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +00012790: 4162 7374 7261 6374 5369 6d70 6c69 6369  AbstractSimplici
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│ │ │ │ +000127b0: 7b7b 7d7d 2020 2020 2020 2020 2020 2020  {{}}            
│ │ │ │  000127c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000127d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000127e0: 2020 2020 7c0a 7c6f 3620 3d20 4162 7374      |.|o6 = Abst
│ │ │ │ -000127f0: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -00012800: 6d70 6c65 787b 2d31 203d 3e20 7b7b 7d7d  mplex{-1 => {{}}
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│ │ │ │ -00012820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012850: 2020 2020 2020 3020 3d3e 207b 7b31 7d2c        0 => {{1},
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│ │ │ │ -00012880: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000128b0: 337d 2c20 7b31 2c20 357d 2c20 7b32 2c20  3}, {1, 5}, {2, 
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│ │ │ │ -000128d0: 357d 2c20 7c0a 7c20 2020 2020 2020 2020  5}, |.|         
│ │ │ │ +000127d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000127e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00012830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00012850: 7b31 2c20 337d 2c20 7b31 2c20 347d 2c20  {1, 3}, {1, 4}, 
│ │ │ │ +00012860: 7b31 2c20 357d 2c20 7b32 2c20 337d 2c20  {1, 5}, {2, 3}, 
│ │ │ │ +00012870: 7b32 2c20 357d 2c20 7c0a 7c20 2020 2020  {2, 5}, |.|     
│ │ │ │ +00012880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012890: 2020 2020 2020 2020 2020 3220 3d3e 207b            2 => {
│ │ │ │ +000128a0: 7b31 2c20 332c 2034 7d2c 207b 312c 2033  {1, 3, 4}, {1, 3
│ │ │ │ +000128b0: 2c20 357d 2c20 7b32 2c20 332c 2035 7d7d  , 5}, {2, 3, 5}}
│ │ │ │ +000128c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000128d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00012900: 332c 2035 7d2c 207b 322c 2034 2c20 357d  3, 5}, {2, 4, 5}
│ │ │ │ -00012910: 2c20 7b34 2c20 352c 2036 7d7d 2020 2020  , {4, 5, 6}}    
│ │ │ │ -00012920: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000128f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012910: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +00012920: 4162 7374 7261 6374 5369 6d70 6c69 6369  AbstractSimplici
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│ │ │ │  00012940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012970: 2020 2020 7c0a 7c6f 3620 3a20 4162 7374      |.|o6 : Abst
│ │ │ │ -00012980: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -00012990: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
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│ │ │ │ -00012a20: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
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│ │ │ │ +00012990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000129a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000129b0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +000129c0: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │ +000129d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000129e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000129f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012a00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00012a50: 2020 2020 2020 2020 7c0a 7c7b 332c 2034          |.|{3, 4
│ │ │ │ +00012a60: 7d2c 207b 332c 2035 7d7d 2020 2020 2020  }, {3, 5}}      
│ │ │ │  00012a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012ab0: 2020 2020 7c0a 7c7b 342c 2035 7d2c 207b      |.|{4, 5}, {
│ │ │ │ -00012ac0: 342c 2036 7d2c 207b 352c 2036 7d7d 2020  4, 6}, {5, 6}}  
│ │ │ │ -00012ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012b00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00012b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012b50: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ -00012b60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00012b70: 6f74 6520 7261 6e64 6f6d 3a20 284d 6163  ote random: (Mac
│ │ │ │ -00012b80: 6175 6c61 7932 446f 6329 7261 6e64 6f6d  aulay2Doc)random
│ │ │ │ -00012b90: 2c20 2d2d 2067 6574 2061 2072 616e 646f  , -- get a rando
│ │ │ │ -00012ba0: 6d20 6f62 6a65 6374 0a20 202a 2072 616e  m object.  * ran
│ │ │ │ -00012bb0: 646f 6d53 7175 6172 6546 7265 654d 6f6e  domSquareFreeMon
│ │ │ │ -00012bc0: 6f6d 6961 6c49 6465 616c 2028 6d69 7373  omialIdeal (miss
│ │ │ │ -00012bd0: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -00012be0: 6e29 0a0a 5761 7973 2074 6f20 7573 6520  n)..Ways to use 
│ │ │ │ -00012bf0: 7261 6e64 6f6d 4162 7374 7261 6374 5369  randomAbstractSi
│ │ │ │ -00012c00: 6d70 6c69 6369 616c 436f 6d70 6c65 783a  mplicialComplex:
│ │ │ │ -00012c10: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00012c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00012c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00012c40: 202a 2022 7261 6e64 6f6d 4162 7374 7261   * "randomAbstra
│ │ │ │ -00012c50: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ -00012c60: 6c65 7828 5a5a 2922 0a20 202a 2022 7261  lex(ZZ)".  * "ra
│ │ │ │ -00012c70: 6e64 6f6d 4162 7374 7261 6374 5369 6d70  ndomAbstractSimp
│ │ │ │ -00012c80: 6c69 6369 616c 436f 6d70 6c65 7828 5a5a  licialComplex(ZZ
│ │ │ │ -00012c90: 2c5a 5a29 220a 2020 2a20 2272 616e 646f  ,ZZ)".  * "rando
│ │ │ │ -00012ca0: 6d41 6273 7472 6163 7453 696d 706c 6963  mAbstractSimplic
│ │ │ │ -00012cb0: 6961 6c43 6f6d 706c 6578 285a 5a2c 5a5a  ialComplex(ZZ,ZZ
│ │ │ │ -00012cc0: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070  ,ZZ)"..For the p
│ │ │ │ -00012cd0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00012ce0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00012cf0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ -00012d00: 616e 646f 6d41 6273 7472 6163 7453 696d  andomAbstractSim
│ │ │ │ -00012d10: 706c 6963 6961 6c43 6f6d 706c 6578 3a0a  plicialComplex:.
│ │ │ │ -00012d20: 7261 6e64 6f6d 4162 7374 7261 6374 5369  randomAbstractSi
│ │ │ │ -00012d30: 6d70 6c69 6369 616c 436f 6d70 6c65 782c  mplicialComplex,
│ │ │ │ -00012d40: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -00012d50: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ -00012d60: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00012d70: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ -00012d80: 6c65 3a20 4162 7374 7261 6374 5369 6d70  le: AbstractSimp
│ │ │ │ -00012d90: 6c69 6369 616c 436f 6d70 6c65 7865 732e  licialComplexes.
│ │ │ │ -00012da0: 696e 666f 2c20 4e6f 6465 3a20 7261 6e64  info, Node: rand
│ │ │ │ -00012db0: 6f6d 5375 6253 696d 706c 6963 6961 6c43  omSubSimplicialC
│ │ │ │ -00012dc0: 6f6d 706c 6578 2c20 4e65 7874 3a20 7265  omplex, Next: re
│ │ │ │ -00012dd0: 6475 6365 6453 696d 706c 6963 6961 6c43  ducedSimplicialC
│ │ │ │ -00012de0: 6861 696e 436f 6d70 6c65 782c 2050 7265  hainComplex, Pre
│ │ │ │ -00012df0: 763a 2072 616e 646f 6d41 6273 7472 6163  v: randomAbstrac
│ │ │ │ -00012e00: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ -00012e10: 6578 2c20 5570 3a20 546f 700a 0a72 616e  ex, Up: Top..ran
│ │ │ │ -00012e20: 646f 6d53 7562 5369 6d70 6c69 6369 616c  domSubSimplicial
│ │ │ │ -00012e30: 436f 6d70 6c65 7820 2d2d 2043 7265 6174  Complex -- Creat
│ │ │ │ -00012e40: 6520 6120 7261 6e64 6f6d 2073 7562 2d73  e a random sub-s
│ │ │ │ -00012e50: 696d 706c 6963 6961 6c20 636f 6d70 6c65  implicial comple
│ │ │ │ -00012e60: 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  x.**************
│ │ │ │ -00012e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012ea0: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ -00012eb0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00012ec0: 0a43 7265 6174 6573 2061 2072 616e 646f  .Creates a rando
│ │ │ │ -00012ed0: 6d20 7375 622d 7369 6d70 6c69 6369 616c  m sub-simplicial
│ │ │ │ -00012ee0: 2063 6f6d 706c 6578 206f 6620 6120 6769   complex of a gi
│ │ │ │ -00012ef0: 7665 6e20 7369 6d70 6c69 6369 616c 2063  ven simplicial c
│ │ │ │ -00012f00: 6f6d 706c 6578 2e0a 0a2b 2d2d 2d2d 2d2d  omplex...+------
│ │ │ │ -00012f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f30: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574  -----+.|i1 : set
│ │ │ │ -00012f40: 5261 6e64 6f6d 5365 6564 2863 7572 7265  RandomSeed(curre
│ │ │ │ -00012f50: 6e74 5469 6d65 2829 293b 2020 2020 2020  ntTime());      
│ │ │ │ -00012f60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00012f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f90: 2d2b 0a7c 6932 203a 204b 203d 2072 616e  -+.|i2 : K = ran
│ │ │ │ -00012fa0: 646f 6d41 6273 7472 6163 7453 696d 706c  domAbstractSimpl
│ │ │ │ -00012fb0: 6963 6961 6c43 6f6d 706c 6578 2834 297c  icialComplex(4)|
│ │ │ │ -00012fc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012aa0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00012ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012af0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +00012b00: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +00012b10: 2a20 2a6e 6f74 6520 7261 6e64 6f6d 3a20  * *note random: 
│ │ │ │ +00012b20: 284d 6163 6175 6c61 7932 446f 6329 7261  (Macaulay2Doc)ra
│ │ │ │ +00012b30: 6e64 6f6d 2c20 2d2d 2067 6574 2061 2072  ndom, -- get a r
│ │ │ │ +00012b40: 616e 646f 6d20 6f62 6a65 6374 0a20 202a  andom object.  *
│ │ │ │ +00012b50: 2072 616e 646f 6d53 7175 6172 6546 7265   randomSquareFre
│ │ │ │ +00012b60: 654d 6f6e 6f6d 6961 6c49 6465 616c 2028  eMonomialIdeal (
│ │ │ │ +00012b70: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ +00012b80: 6174 696f 6e29 0a0a 5761 7973 2074 6f20  ation)..Ways to 
│ │ │ │ +00012b90: 7573 6520 7261 6e64 6f6d 4162 7374 7261  use randomAbstra
│ │ │ │ +00012ba0: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ +00012bb0: 6c65 783a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  lex:.===========
│ │ │ │ +00012bc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00012bd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00012be0: 3d0a 0a20 202a 2022 7261 6e64 6f6d 4162  =..  * "randomAb
│ │ │ │ +00012bf0: 7374 7261 6374 5369 6d70 6c69 6369 616c  stractSimplicial
│ │ │ │ +00012c00: 436f 6d70 6c65 7828 5a5a 2922 0a20 202a  Complex(ZZ)".  *
│ │ │ │ +00012c10: 2022 7261 6e64 6f6d 4162 7374 7261 6374   "randomAbstract
│ │ │ │ +00012c20: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00012c30: 7828 5a5a 2c5a 5a29 220a 2020 2a20 2272  x(ZZ,ZZ)".  * "r
│ │ │ │ +00012c40: 616e 646f 6d41 6273 7472 6163 7453 696d  andomAbstractSim
│ │ │ │ +00012c50: 706c 6963 6961 6c43 6f6d 706c 6578 285a  plicialComplex(Z
│ │ │ │ +00012c60: 5a2c 5a5a 2c5a 5a29 220a 0a46 6f72 2074  Z,ZZ,ZZ)"..For t
│ │ │ │ +00012c70: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +00012c80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00012c90: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +00012ca0: 7465 2072 616e 646f 6d41 6273 7472 6163  te randomAbstrac
│ │ │ │ +00012cb0: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00012cc0: 6578 3a0a 7261 6e64 6f6d 4162 7374 7261  ex:.randomAbstra
│ │ │ │ +00012cd0: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ +00012ce0: 6c65 782c 2069 7320 6120 2a6e 6f74 6520  lex, is a *note 
│ │ │ │ +00012cf0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +00012d00: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00012d10: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +00012d20: 1f0a 4669 6c65 3a20 4162 7374 7261 6374  ..File: Abstract
│ │ │ │ +00012d30: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00012d40: 7865 732e 696e 666f 2c20 4e6f 6465 3a20  xes.info, Node: 
│ │ │ │ +00012d50: 7261 6e64 6f6d 5375 6253 696d 706c 6963  randomSubSimplic
│ │ │ │ +00012d60: 6961 6c43 6f6d 706c 6578 2c20 4e65 7874  ialComplex, Next
│ │ │ │ +00012d70: 3a20 7265 6475 6365 6453 696d 706c 6963  : reducedSimplic
│ │ │ │ +00012d80: 6961 6c43 6861 696e 436f 6d70 6c65 782c  ialChainComplex,
│ │ │ │ +00012d90: 2050 7265 763a 2072 616e 646f 6d41 6273   Prev: randomAbs
│ │ │ │ +00012da0: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00012db0: 6f6d 706c 6578 2c20 5570 3a20 546f 700a  omplex, Up: Top.
│ │ │ │ +00012dc0: 0a72 616e 646f 6d53 7562 5369 6d70 6c69  .randomSubSimpli
│ │ │ │ +00012dd0: 6369 616c 436f 6d70 6c65 7820 2d2d 2043  cialComplex -- C
│ │ │ │ +00012de0: 7265 6174 6520 6120 7261 6e64 6f6d 2073  reate a random s
│ │ │ │ +00012df0: 7562 2d73 696d 706c 6963 6961 6c20 636f  ub-simplicial co
│ │ │ │ +00012e00: 6d70 6c65 780a 2a2a 2a2a 2a2a 2a2a 2a2a  mplex.**********
│ │ │ │ +00012e10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012e20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012e40: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ +00012e50: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00012e60: 3d3d 3d0a 0a43 7265 6174 6573 2061 2072  ===..Creates a r
│ │ │ │ +00012e70: 616e 646f 6d20 7375 622d 7369 6d70 6c69  andom sub-simpli
│ │ │ │ +00012e80: 6369 616c 2063 6f6d 706c 6578 206f 6620  cial complex of 
│ │ │ │ +00012e90: 6120 6769 7665 6e20 7369 6d70 6c69 6369  a given simplici
│ │ │ │ +00012ea0: 616c 2063 6f6d 706c 6578 2e0a 0a2b 2d2d  al complex...+--
│ │ │ │ +00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00012ef0: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ +00012f00: 2863 7572 7265 6e74 5469 6d65 2829 293b  (currentTime());
│ │ │ │ +00012f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012f20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00012f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00012f70: 203a 204b 203d 2072 616e 646f 6d41 6273   : K = randomAbs
│ │ │ │ +00012f80: 7472 6163 7453 696d 706c 6963 6961 6c43  tractSimplicialC
│ │ │ │ +00012f90: 6f6d 706c 6578 2834 2920 2020 2020 2020  omplex(4)       
│ │ │ │ +00012fa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00012fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00012ff0: 6f32 203d 2041 6273 7472 6163 7453 696d  o2 = AbstractSim
│ │ │ │ -00013000: 706c 6963 6961 6c43 6f6d 706c 6578 7b2d  plicialComplex{-
│ │ │ │ -00013010: 3120 3d3e 207b 7b7d 7d7d 207c 0a7c 2020  1 => {{}}} |.|  
│ │ │ │ -00013020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013030: 2020 2020 2020 2020 2020 2020 2030 203d               0 =
│ │ │ │ -00013040: 3e20 7b7b 317d 7d20 207c 0a7c 2020 2020  > {{1}}  |.|    
│ │ │ │ -00013050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013070: 2020 2020 2020 207c 0a7c 6f32 203a 2041         |.|o2 : A
│ │ │ │ -00013080: 6273 7472 6163 7453 696d 706c 6963 6961  bstractSimplicia
│ │ │ │ -00013090: 6c43 6f6d 706c 6578 2020 2020 2020 2020  lComplex        
│ │ │ │ -000130a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000130b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130d0: 2d2d 2d2b 0a7c 6933 203a 204a 203d 2072  ---+.|i3 : J = r
│ │ │ │ -000130e0: 616e 646f 6d53 7562 5369 6d70 6c69 6369  andomSubSimplici
│ │ │ │ -000130f0: 616c 436f 6d70 6c65 7828 4b29 2020 2020  alComplex(K)    
│ │ │ │ -00013100: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00012fe0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00012ff0: 203d 2041 6273 7472 6163 7453 696d 706c   = AbstractSimpl
│ │ │ │ +00013000: 6963 6961 6c43 6f6d 706c 6578 7b2d 3120  icialComplex{-1 
│ │ │ │ +00013010: 3d3e 207b 7b7d 7d20 2020 2020 2020 2020  => {{}}         
│ │ │ │ +00013020: 2020 2020 2020 2020 2020 7d7c 0a7c 2020            }|.|  
│ │ │ │ +00013030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013040: 2020 2020 2020 2020 2020 2020 2030 203d               0 =
│ │ │ │ +00013050: 3e20 7b7b 327d 2c20 7b33 7d2c 207b 347d  > {{2}, {3}, {4}
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│ │ │ │ -00013170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013180: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
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│ │ │ │ -00013220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00013440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000134a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -000135e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000135f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000136a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000137c0: 2020 2020 2020 2020 2020 2032 203d 3e20             2 => 
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│ │ │ │ -00013800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013810: 2020 2020 2020 2020 2020 2033 203d 3e20             3 => 
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│ │ │ │ -00013850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013860: 2020 2020 2020 2020 2020 2034 203d 3e20             4 => 
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│ │ │ │ +00013120: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
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│ │ │ │ +00013160: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ +00013180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000131d0: 7828 4b29 2020 2020 2020 2020 2020 2020  x(K)            
│ │ │ │ +000131e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013220: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
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│ │ │ │ +00013260: 2020 2020 2020 2020 2020 7d7c 0a7c 2020            }|.|  
│ │ │ │ +00013270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013280: 2020 2020 2020 2020 2020 2020 2030 203d               0 =
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│ │ │ │ +000132b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000132c0: 2020 2020 2020 2020 2020 2020 2031 203d               1 =
│ │ │ │ +000132d0: 3e20 7b7b 322c 2033 7d2c 207b 322c 2034  > {{2, 3}, {2, 4
│ │ │ │ +000132e0: 7d2c 207b 332c 2034 7d7d 207c 0a7c 2020  }, {3, 4}} |.|  
│ │ │ │ +000132f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013300: 2020 2020 2020 2020 2020 2020 2032 203d               2 =
│ │ │ │ +00013310: 3e20 7b7b 322c 2033 2c20 347d 7d20 2020  > {{2, 3, 4}}   
│ │ │ │ +00013320: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00013330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013360: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00013370: 203a 2041 6273 7472 6163 7453 696d 706c   : AbstractSimpl
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│ │ │ │ +00013390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000133a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000133b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00013430: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
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│ │ │ │ +00013490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00013540: 6c43 6f6d 706c 6578 6573 2e69 6e66 6f2c  lComplexes.info,
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│ │ │ │ +000135c0: 6564 7563 6564 5369 6d70 6c69 6369 616c  educedSimplicial
│ │ │ │ +000135d0: 4368 6169 6e43 6f6d 706c 6578 202d 2d20  ChainComplex -- 
│ │ │ │ +000135e0: 5468 6520 7265 6475 6365 6420 686f 6d6f  The reduced homo
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│ │ │ │ +00013620: 6273 7472 6163 7420 7369 6d70 6c69 6369  bstract simplici
│ │ │ │ +00013630: 616c 2063 6f6d 706c 6578 0a2a 2a2a 2a2a  al complex.*****
│ │ │ │ +00013640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00013650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00013660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00013670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00013680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00013690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000136a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000136b0: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ +000136c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +000136d0: 0a54 6869 7320 6d65 7468 6f64 2072 6574  .This method ret
│ │ │ │ +000136e0: 7572 6e73 2074 6865 2072 6564 7563 6564  urns the reduced
│ │ │ │ +000136f0: 2068 6f6d 6f6c 6f67 6963 616c 2063 6861   homological cha
│ │ │ │ +00013700: 696e 2063 6f6d 706c 6578 2028 692e 652e  in complex (i.e.
│ │ │ │ +00013710: 2c20 7468 6572 6520 6973 2061 0a6e 6f6e  , there is a.non
│ │ │ │ +00013720: 7a65 726f 2074 6572 6d20 696e 2068 6f6d  zero term in hom
│ │ │ │ +00013730: 6f6c 6f67 6963 616c 2064 6567 7265 6520  ological degree 
│ │ │ │ +00013740: 2d31 2074 6861 7420 636f 7272 6573 706f  -1 that correspo
│ │ │ │ +00013750: 6e64 7320 746f 2074 6865 2065 6d70 7479  nds to the empty
│ │ │ │ +00013760: 2066 6163 6529 2074 6861 740a 6973 2061   face) that.is a
│ │ │ │ +00013770: 7373 6f63 6961 7465 6420 746f 2061 6e20  ssociated to an 
│ │ │ │ +00013780: 6162 7374 7261 6374 2073 696d 706c 6963  abstract simplic
│ │ │ │ +00013790: 6961 6c20 636f 6d70 6c65 782e 2020 5468  ial complex.  Th
│ │ │ │ +000137a0: 6520 6368 6169 6e20 636f 6d70 6c65 7820  e chain complex 
│ │ │ │ +000137b0: 6973 2064 6566 696e 6564 0a6f 7665 7220  is defined.over 
│ │ │ │ +000137c0: 7468 6520 696e 7465 6765 7273 2e0a 0a2b  the integers...+
│ │ │ │ +000137d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000137e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000137f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00013820: 6931 203a 204b 203d 2061 6273 7472 6163  i1 : K = abstrac
│ │ │ │ +00013830: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00013840: 6578 287b 7b31 2c32 2c33 7d2c 7b32 2c34  ex({{1,2,3},{2,4
│ │ │ │ +00013850: 2c39 7d2c 7b31 2c32 2c33 2c35 2c37 2c38  ,9},{1,2,3,5,7,8
│ │ │ │ +00013860: 7d2c 7b33 2c34 7d7d 2920 2020 207c 0a7c  },{3,4}})    |.|
│ │ │ │ +00013870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000138a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000138b0: 2020 2020 2020 2020 2020 2035 203d 3e20             5 => 
│ │ │ │ -000138c0: 7b7b 312c 2032 2c20 332c 2035 2c20 372c  {{1, 2, 3, 5, 7,
│ │ │ │ -000138d0: 2038 7d7d 2020 2020 2020 2020 2020 2020   8}}            
│ │ │ │ -000138e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000138b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000138c0: 6f31 203d 2041 6273 7472 6163 7453 696d  o1 = AbstractSim
│ │ │ │ +000138d0: 706c 6963 6961 6c43 6f6d 706c 6578 7b2d  plicialComplex{-
│ │ │ │ +000138e0: 3120 3d3e 207b 7b7d 7d20 2020 2020 2020  1 => {{}}       
│ │ │ │  000138f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013900: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00013910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013930: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -00013940: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ -00013950: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ +00013920: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00013930: 203d 3e20 7b7b 317d 2c20 7b32 7d2c 207b   => {{1}, {2}, {
│ │ │ │ +00013940: 337d 2c20 7b34 7d2c 207b 357d 2c20 7b37  3}, {4}, {5}, {7
│ │ │ │ +00013950: 7d2c 207b 387d 2c20 7b39 7d7d 207c 0a7c  }, {8}, {9}} |.|
│ │ │ │  00013960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013980: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ -00013990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139d0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -000139e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000139f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013970: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00013980: 203d 3e20 7b7b 312c 2032 7d2c 207b 312c   => {{1, 2}, {1,
│ │ │ │ +00013990: 2033 7d2c 207b 312c 2035 7d2c 207b 312c   3}, {1, 5}, {1,
│ │ │ │ +000139a0: 2037 7d2c 207b 312c 2038 7d2c 207c 0a7c   7}, {1, 8}, |.|
│ │ │ │ +000139b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000139c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000139d0: 203d 3e20 7b7b 312c 2032 2c20 337d 2c20   => {{1, 2, 3}, 
│ │ │ │ +000139e0: 7b31 2c20 322c 2035 7d2c 207b 312c 2032  {1, 2, 5}, {1, 2
│ │ │ │ +000139f0: 2c20 377d 2c20 7b31 2c20 322c 207c 0a7c  , 7}, {1, 2, |.|
│ │ │ │  00013a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00013a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013a10: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00013a20: 203d 3e20 7b7b 312c 2032 2c20 332c 2035   => {{1, 2, 3, 5
│ │ │ │ +00013a30: 7d2c 207b 312c 2032 2c20 332c 2037 7d2c  }, {1, 2, 3, 7},
│ │ │ │ +00013a40: 207b 312c 2032 2c20 332c 2038 207c 0a7c   {1, 2, 3, 8 |.|
│ │ │ │  00013a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a70: 2020 2020 2020 2020 207c 0a7c 207b 322c           |.| {2,
│ │ │ │ -00013a80: 2033 7d2c 207b 322c 2034 7d2c 207b 322c   3}, {2, 4}, {2,
│ │ │ │ -00013a90: 2035 7d2c 207b 322c 2037 7d2c 207b 322c   5}, {2, 7}, {2,
│ │ │ │ -00013aa0: 2038 7d2c 207b 322c 2039 7d2c 207b 332c   8}, {2, 9}, {3,
│ │ │ │ -00013ab0: 2034 7d2c 207b 332c 2035 7d2c 207b 332c   4}, {3, 5}, {3,
│ │ │ │ -00013ac0: 2037 7d2c 207b 332c 207c 0a7c 2038 7d2c   7}, {3, |.| 8},
│ │ │ │ -00013ad0: 207b 312c 2033 2c20 357d 2c20 7b31 2c20   {1, 3, 5}, {1, 
│ │ │ │ -00013ae0: 332c 2037 7d2c 207b 312c 2033 2c20 387d  3, 7}, {1, 3, 8}
│ │ │ │ -00013af0: 2c20 7b31 2c20 352c 2037 7d2c 207b 312c  , {1, 5, 7}, {1,
│ │ │ │ -00013b00: 2035 2c20 387d 2c20 7b31 2c20 372c 2038   5, 8}, {1, 7, 8
│ │ │ │ -00013b10: 7d2c 207b 322c 2033 2c7c 0a7c 7d2c 207b  }, {2, 3,|.|}, {
│ │ │ │ -00013b20: 312c 2032 2c20 352c 2037 7d2c 207b 312c  1, 2, 5, 7}, {1,
│ │ │ │ -00013b30: 2032 2c20 352c 2038 7d2c 207b 312c 2032   2, 5, 8}, {1, 2
│ │ │ │ -00013b40: 2c20 372c 2038 7d2c 207b 312c 2033 2c20  , 7, 8}, {1, 3, 
│ │ │ │ -00013b50: 352c 2037 7d2c 207b 312c 2033 2c20 352c  5, 7}, {1, 3, 5,
│ │ │ │ -00013b60: 2038 7d2c 207b 312c 207c 0a7c 2c20 332c   8}, {1, |.|, 3,
│ │ │ │ -00013b70: 2037 2c20 387d 2c20 7b31 2c20 322c 2035   7, 8}, {1, 2, 5
│ │ │ │ -00013b80: 2c20 372c 2038 7d2c 207b 312c 2033 2c20  , 7, 8}, {1, 3, 
│ │ │ │ -00013b90: 352c 2037 2c20 387d 2c20 7b32 2c20 332c  5, 7, 8}, {2, 3,
│ │ │ │ -00013ba0: 2035 2c20 372c 2038 7d7d 2020 2020 2020   5, 7, 8}}      
│ │ │ │ -00013bb0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +00013a60: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +00013a70: 203d 3e20 7b7b 312c 2032 2c20 332c 2035   => {{1, 2, 3, 5
│ │ │ │ +00013a80: 2c20 377d 2c20 7b31 2c20 322c 2033 2c20  , 7}, {1, 2, 3, 
│ │ │ │ +00013a90: 352c 2038 7d2c 207b 312c 2032 207c 0a7c  5, 8}, {1, 2 |.|
│ │ │ │ +00013aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ab0: 2020 2020 2020 2020 2020 2020 2020 2035                 5
│ │ │ │ +00013ac0: 203d 3e20 7b7b 312c 2032 2c20 332c 2035   => {{1, 2, 3, 5
│ │ │ │ +00013ad0: 2c20 372c 2038 7d7d 2020 2020 2020 2020  , 7, 8}}        
│ │ │ │ +00013ae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013b40: 6f31 203a 2041 6273 7472 6163 7453 696d  o1 : AbstractSim
│ │ │ │ +00013b50: 706c 6963 6961 6c43 6f6d 706c 6578 2020  plicialComplex  
│ │ │ │ +00013b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013c00: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ +00013bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00013be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00013c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ca0: 2020 2020 2020 2020 207c 0a7c 387d 2c20           |.|8}, 
│ │ │ │ -00013cb0: 7b34 2c20 397d 2c20 7b35 2c20 377d 2c20  {4, 9}, {5, 7}, 
│ │ │ │ -00013cc0: 7b35 2c20 387d 2c20 7b37 2c20 387d 7d20  {5, 8}, {7, 8}} 
│ │ │ │ -00013cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013cf0: 2020 2020 2020 2020 207c 0a7c 2035 7d2c           |.| 5},
│ │ │ │ -00013d00: 207b 322c 2033 2c20 377d 2c20 7b32 2c20   {2, 3, 7}, {2, 
│ │ │ │ -00013d10: 332c 2038 7d2c 207b 322c 2034 2c20 397d  3, 8}, {2, 4, 9}
│ │ │ │ -00013d20: 2c20 7b32 2c20 352c 2037 7d2c 207b 322c  , {2, 5, 7}, {2,
│ │ │ │ -00013d30: 2035 2c20 2020 2020 2020 2020 2020 2020   5,             
│ │ │ │ -00013d40: 2020 2020 2020 2020 207c 0a7c 332c 2037           |.|3, 7
│ │ │ │ -00013d50: 2c20 387d 2c20 7b31 2c20 352c 2037 2c20  , 8}, {1, 5, 7, 
│ │ │ │ -00013d60: 387d 2c20 7b32 2c20 332c 2035 2c20 377d  8}, {2, 3, 5, 7}
│ │ │ │ -00013d70: 2c20 7b32 2c20 332c 2035 2c20 387d 2c20  , {2, 3, 5, 8}, 
│ │ │ │ -00013d80: 7b32 2c20 2020 2020 2020 2020 2020 2020  {2,             
│ │ │ │ -00013d90: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ -00013da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013c80: 207b 322c 2033 7d2c 207b 322c 2034 7d2c   {2, 3}, {2, 4},
│ │ │ │ +00013c90: 207b 322c 2035 7d2c 207b 322c 2037 7d2c   {2, 5}, {2, 7},
│ │ │ │ +00013ca0: 207b 322c 2038 7d2c 207b 322c 2039 7d2c   {2, 8}, {2, 9},
│ │ │ │ +00013cb0: 207b 332c 2034 7d2c 207b 332c 2035 7d2c   {3, 4}, {3, 5},
│ │ │ │ +00013cc0: 207b 332c 2037 7d2c 207b 332c 207c 0a7c   {3, 7}, {3, |.|
│ │ │ │ +00013cd0: 2038 7d2c 207b 312c 2033 2c20 357d 2c20   8}, {1, 3, 5}, 
│ │ │ │ +00013ce0: 7b31 2c20 332c 2037 7d2c 207b 312c 2033  {1, 3, 7}, {1, 3
│ │ │ │ +00013cf0: 2c20 387d 2c20 7b31 2c20 352c 2037 7d2c  , 8}, {1, 5, 7},
│ │ │ │ +00013d00: 207b 312c 2035 2c20 387d 2c20 7b31 2c20   {1, 5, 8}, {1, 
│ │ │ │ +00013d10: 372c 2038 7d2c 207b 322c 2033 2c7c 0a7c  7, 8}, {2, 3,|.|
│ │ │ │ +00013d20: 7d2c 207b 312c 2032 2c20 352c 2037 7d2c  }, {1, 2, 5, 7},
│ │ │ │ +00013d30: 207b 312c 2032 2c20 352c 2038 7d2c 207b   {1, 2, 5, 8}, {
│ │ │ │ +00013d40: 312c 2032 2c20 372c 2038 7d2c 207b 312c  1, 2, 7, 8}, {1,
│ │ │ │ +00013d50: 2033 2c20 352c 2037 7d2c 207b 312c 2033   3, 5, 7}, {1, 3
│ │ │ │ +00013d60: 2c20 352c 2038 7d2c 207b 312c 207c 0a7c  , 5, 8}, {1, |.|
│ │ │ │ +00013d70: 2c20 332c 2037 2c20 387d 2c20 7b31 2c20  , 3, 7, 8}, {1, 
│ │ │ │ +00013d80: 322c 2035 2c20 372c 2038 7d2c 207b 312c  2, 5, 7, 8}, {1,
│ │ │ │ +00013d90: 2033 2c20 352c 2037 2c20 387d 2c20 7b32   3, 5, 7, 8}, {2
│ │ │ │ +00013da0: 2c20 332c 2035 2c20 372c 2038 7d7d 2020  , 3, 5, 7, 8}}  
│ │ │ │ +00013db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00013dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013de0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -00013df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │  00013e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e20: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ -00013e30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013e50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00013e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ed0: 2020 2020 2020 2020 207c 0a7c 387d 2c20           |.|8}, 
│ │ │ │ -00013ee0: 7b32 2c20 372c 2038 7d2c 207b 332c 2035  {2, 7, 8}, {3, 5
│ │ │ │ -00013ef0: 2c20 377d 2c20 7b33 2c20 352c 2038 7d2c  , 7}, {3, 5, 8},
│ │ │ │ -00013f00: 207b 332c 2037 2c20 387d 2c20 7b35 2c20   {3, 7, 8}, {5, 
│ │ │ │ -00013f10: 372c 2038 7d7d 2020 2020 2020 2020 2020  7, 8}}          
│ │ │ │ -00013f20: 2020 2020 2020 2020 207c 0a7c 332c 2037           |.|3, 7
│ │ │ │ -00013f30: 2c20 387d 2c20 7b32 2c20 352c 2037 2c20  , 8}, {2, 5, 7, 
│ │ │ │ -00013f40: 387d 2c20 7b33 2c20 352c 2037 2c20 387d  8}, {3, 5, 7, 8}
│ │ │ │ -00013f50: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -00013f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00013f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013eb0: 387d 2c20 7b34 2c20 397d 2c20 7b35 2c20  8}, {4, 9}, {5, 
│ │ │ │ +00013ec0: 377d 2c20 7b35 2c20 387d 2c20 7b37 2c20  7}, {5, 8}, {7, 
│ │ │ │ +00013ed0: 387d 7d20 2020 2020 2020 2020 2020 2020  8}}             
│ │ │ │ +00013ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -000140d0: 205a 5a20 2020 3c2d 2d20 5a5a 2020 203c   ZZ   <-- ZZ   <
│ │ │ │ -000140e0: 2d2d 205a 5a20 2020 3c2d 2d20 5a5a 2020  -- ZZ   <-- ZZ  
│ │ │ │ -000140f0: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020  <-- ZZ          
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│ │ │ │ -00014160: 202d 3120 2020 2020 2030 2020 2020 2020   -1      0      
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│ │ │ │ -00014180: 2020 2033 2020 2020 2020 2020 3420 2020     3        4   
│ │ │ │ -00014190: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ -000141a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000141b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000141c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000141d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000141e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000140e0: 387d 2c20 7b32 2c20 372c 2038 7d2c 207b  8}, {2, 7, 8}, {
│ │ │ │ +000140f0: 332c 2035 2c20 377d 2c20 7b33 2c20 352c  3, 5, 7}, {3, 5,
│ │ │ │ +00014100: 2038 7d2c 207b 332c 2037 2c20 387d 2c20   8}, {3, 7, 8}, 
│ │ │ │ +00014110: 7b35 2c20 372c 2038 7d7d 2020 2020 2020  {5, 7, 8}}      
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│ │ │ │ +00014140: 2037 2c20 387d 2c20 7b33 2c20 352c 2037   7, 8}, {3, 5, 7
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│ │ │ │ +00014180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000141a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000141b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000141c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
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│ │ │ │ +000141f0: 6578 284b 2920 2020 2020 2020 2020 2020  ex(K)           
│ │ │ │ +00014200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00014220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000142b0: 696d 706c 6963 6961 6c43 6861 696e 436f  implicialChainCo
│ │ │ │ -000142c0: 6d70 6c65 783a 0a3d 3d3d 3d3d 3d3d 3d3d  mplex:.=========
│ │ │ │ -000142d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000142e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000142f0: 3d0a 0a20 202a 2022 7265 6475 6365 6453  =..  * "reducedS
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│ │ │ │ -000143a0: 436f 6d70 6c65 782c 0a69 7320 6120 2a6e  Complex,.is a *n
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│ │ │ │ -00014400: 6d70 6c65 7865 732e 696e 666f 2c20 4e6f  mplexes.info, No
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│ │ │ │ -00014470: 202d 2d20 5468 6520 6e6f 6e2d 7265 6475   -- The non-redu
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│ │ │ │ -000144c0: 7369 6d70 6c69 6369 616c 2063 6f6d 706c  simplicial compl
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│ │ │ │ -000144e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000144f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00014500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00014510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00014520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00014530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00014540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
│ │ │ │ -00014550: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00014560: 3d3d 3d3d 0a0a 5468 6973 206d 6574 686f  ====..This metho
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│ │ │ │ -00014680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
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│ │ │ │ -000147b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147c0: 2020 2020 2020 2020 2020 2030 203d 3e20             0 => 
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│ │ │ │ -000147f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014810: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
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│ │ │ │ -00014890: 2c20 7b32 2c20 342c 207c 0a7c 2020 2020  , {2, 4, |.|    
│ │ │ │ -000148a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000148b0: 2020 2020 2020 2020 2020 2033 203d 3e20             3 => 
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│ │ │ │ -000148d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000148f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00014270: 2020 2020 2020 2031 2020 2020 2020 2038         1       8
│ │ │ │ +00014280: 2020 2020 2020 2031 3920 2020 2020 2020         19       
│ │ │ │ +00014290: 3231 2020 2020 2020 2031 3520 2020 2020  21       15     
│ │ │ │ +000142a0: 2020 3620 2020 2020 2020 3120 2020 2020    6       1     
│ │ │ │ +000142b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000142c0: 6f32 203d 205a 5a20 203c 2d2d 205a 5a20  o2 = ZZ  <-- ZZ 
│ │ │ │ +000142d0: 203c 2d2d 205a 5a20 2020 3c2d 2d20 5a5a   <-- ZZ   <-- ZZ
│ │ │ │ +000142e0: 2020 203c 2d2d 205a 5a20 2020 3c2d 2d20     <-- ZZ   <-- 
│ │ │ │ +000142f0: 5a5a 2020 3c2d 2d20 5a5a 2020 2020 2020  ZZ  <-- ZZ      
│ │ │ │ +00014300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00014360: 2020 2020 202d 3120 2020 2020 2030 2020       -1      0  
│ │ │ │ +00014370: 2020 2020 2031 2020 2020 2020 2020 3220       1        2 
│ │ │ │ +00014380: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00014390: 3420 2020 2020 2020 3520 2020 2020 2020  4       5       
│ │ │ │ +000143a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000143b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000143c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000143d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000143e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000143f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00014420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014440: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00014450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +000144a0: 5761 7973 2074 6f20 7573 6520 7265 6475  Ways to use redu
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│ │ │ │ +00014710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00014720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00014740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
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│ │ │ │ +00014790: 686f 6d6f 6c6f 6769 6361 6c20 6368 6169  homological chai
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│ │ │ │  00014920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000149e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ +000149c0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
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│ │ │ │ +00014a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2031                 1
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│ │ │ │ +00014a30: 2033 7d2c 207b 312c 2034 7d2c 207b 312c   3}, {1, 4}, {1,
│ │ │ │ +00014a40: 2035 7d2c 207b 322c 2033 7d2c 207c 0a7c   5}, {2, 3}, |.|
│ │ │ │  00014a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a70: 2020 2020 2020 2020 207c 0a7c 7b32 2c20           |.|{2, 
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│ │ │ │ -00014ad0: 7b32 2c20 352c 2037 7d2c 207b 342c 2035  {2, 5, 7}, {4, 5
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│ │ │ │ +00014a60: 2020 2020 2020 2020 2020 2020 2020 2032                 2
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│ │ │ │ +00014a90: 2c20 357d 2c20 7b32 2c20 342c 207c 0a7c  , 5}, {2, 4, |.|
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│ │ │ │ +00014ab0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00014ac0: 203d 3e20 7b7b 322c 2034 2c20 352c 2037   => {{2, 4, 5, 7
│ │ │ │ +00014ad0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00014ae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00014b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014b60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
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│ │ │ │ -00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00014b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00014b40: 6f31 203a 2041 6273 7472 6163 7453 696d  o1 : AbstractSim
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│ │ │ │ +00014b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00014b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00014bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │  00014be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014c10: 2020 2036 2020 2020 2020 2031 3120 2020     6       11   
│ │ │ │ -00014c20: 2020 2020 3620 2020 2020 2020 3120 2020      6       1   
│ │ │ │ +00014c00: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ +00014c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00014c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c50: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -00014c60: 205a 5a20 203c 2d2d 205a 5a20 2020 3c2d   ZZ  <-- ZZ   <-
│ │ │ │ -00014c70: 2d20 5a5a 2020 3c2d 2d20 5a5a 2020 2020  - ZZ  <-- ZZ    
│ │ │ │ -00014c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00014c80: 7b32 2c20 347d 2c20 7b32 2c20 357d 2c20  {2, 4}, {2, 5}, 
│ │ │ │ +00014c90: 7b32 2c20 377d 2c20 7b34 2c20 357d 2c20  {2, 7}, {4, 5}, 
│ │ │ │ +00014ca0: 7b34 2c20 377d 2c20 7b35 2c20 377d 7d20  {4, 7}, {5, 7}} 
│ │ │ │  00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014d00: 2030 2020 2020 2020 2031 2020 2020 2020   0       1      
│ │ │ │ -00014d10: 2020 3220 2020 2020 2020 3320 2020 2020    2       3     
│ │ │ │ -00014d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d90: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ -00014da0: 2043 6f6d 706c 6578 2020 2020 2020 2020   Complex        
│ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00014cd0: 377d 2c20 7b32 2c20 352c 2037 7d2c 207b  7}, {2, 5, 7}, {
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│ │ │ │ +00014cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014d10: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00014d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00014d70: 6932 203a 2043 203d 2073 696d 706c 6963  i2 : C = simplic
│ │ │ │ +00014d80: 6961 6c43 6861 696e 436f 6d70 6c65 7828  ialChainComplex(
│ │ │ │ +00014d90: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +00014da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00014dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014de0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014e30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973  ---------+..Ways
│ │ │ │ -00014e40: 2074 6f20 7573 6520 7369 6d70 6c69 6369   to use simplici
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│ │ │ │ -00014e60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00014e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00014ec0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00014ed0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
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│ │ │ │ -00014ef0: 2a6e 6f74 6520 7369 6d70 6c69 6369 616c  *note simplicial
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│ │ │ │ -00014f10: 6d70 6c69 6369 616c 4368 6169 6e43 6f6d  mplicialChainCom
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│ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz
│ │ │ ├── AdjunctionForSurfaces.info
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│ │ │ │  00006040: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ @@ -1582,16 +1582,16 @@
│ │ │ │  000062d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000062e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000062f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006300: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │  00006310: 203a 2065 6c61 7073 6564 5469 6d65 2062   : elapsedTime b
│ │ │ │  00006320: 6173 6550 7473 3d70 7269 6d61 7279 4465  asePts=primaryDe
│ │ │ │  00006330: 636f 6d70 6f73 6974 696f 6e20 6964 6561  composition idea
│ │ │ │ -00006340: 6c20 483b 207c 0a7c 202d 2d20 372e 3237  l H; |.| -- 7.27
│ │ │ │ -00006350: 3731 3373 2065 6c61 7073 6564 2020 2020  713s elapsed    
│ │ │ │ +00006340: 6c20 483b 207c 0a7c 202d 2d20 3131 2e36  l H; |.| -- 11.6
│ │ │ │ +00006350: 3039 3573 2065 6c61 7073 6564 2020 2020  095s elapsed    
│ │ │ │  00006360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006380: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00006390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  000063c0: 3820 3a20 7461 6c6c 7920 6170 706c 7928  8 : tally apply(
│ │ │ │ @@ -2502,15 +2502,15 @@
│ │ │ │  00009c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009c80: 2d2b 0a7c 6931 3420 3a20 656c 6170 7365  -+.|i14 : elapse
│ │ │ │  00009c90: 6454 696d 6520 7375 6228 492c 4829 2020  dTime sub(I,H)  
│ │ │ │  00009ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00009cc0: 2d2d 202e 3035 3437 3832 3973 2065 6c61  -- .0547829s ela
│ │ │ │ +00009cc0: 2d2d 202e 3037 3030 3636 3473 2065 6c61  -- .0700664s ela
│ │ │ │  00009cd0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00009ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009cf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00009d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009d30: 2020 7c0a 7c6f 3134 203d 2069 6465 616c    |.|o14 = ideal
│ │ │ │ @@ -2542,16 +2542,16 @@
│ │ │ │  00009ed0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00009ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │  00009f10: 203a 2065 6c61 7073 6564 5469 6d65 2062   : elapsedTime b
│ │ │ │  00009f20: 6574 7469 2849 273d 7472 696d 206b 6572  etti(I'=trim ker
│ │ │ │  00009f30: 2070 6869 2920 2020 2020 2020 2020 2020   phi)           
│ │ │ │ -00009f40: 2020 2020 207c 0a7c 202d 2d20 2e31 3932       |.| -- .192
│ │ │ │ -00009f50: 3735 3673 2065 6c61 7073 6564 2020 2020  756s elapsed    
│ │ │ │ +00009f40: 2020 2020 207c 0a7c 202d 2d20 2e31 3034       |.| -- .104
│ │ │ │ +00009f50: 3632 3173 2065 6c61 7073 6564 2020 2020  621s elapsed    
│ │ │ │  00009f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009fb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00009fc0: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ @@ -2594,15 +2594,15 @@
│ │ │ │  0000a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a240: 2d2d 2d2d 2b0a 7c69 3138 203a 2065 6c61  ----+.|i18 : ela
│ │ │ │  0000a250: 7073 6564 5469 6d65 2062 6173 6550 7473  psedTime basePts
│ │ │ │  0000a260: 3d70 7269 6d61 7279 4465 636f 6d70 6f73  =primaryDecompos
│ │ │ │  0000a270: 6974 696f 6e20 6964 6561 6c20 483b 207c  ition ideal H; |
│ │ │ │ -0000a280: 0a7c 202d 2d20 322e 3230 3436 3473 2065  .| -- 2.20464s e
│ │ │ │ +0000a280: 0a7c 202d 2d20 332e 3430 3032 3173 2065  .| -- 3.40021s e
│ │ │ │  0000a290: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0000a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a2b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0000a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a2f0: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 7461  -----+.|i19 : ta
│ │ ├── ./usr/share/info/BGG.info.gz
│ │ │ ├── BGG.info
│ │ │ │ @@ -4214,17 +4214,17 @@
│ │ │ │  00010750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010780: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 7469  -----+.|i14 : ti
│ │ │ │  00010790: 6d65 2062 6574 7469 2028 4620 3d20 7075  me betti (F = pu
│ │ │ │  000107a0: 7265 5265 736f 6c75 7469 6f6e 284d 2c7b  reResolution(M,{
│ │ │ │  000107b0: 302c 322c 347d 2929 2020 2020 207c 0a7c  0,2,4}))     |.|
│ │ │ │ -000107c0: 202d 2d20 7573 6564 2030 2e36 3931 3935   -- used 0.69195
│ │ │ │ -000107d0: 3273 2028 6370 7529 3b20 302e 3437 3633  2s (cpu); 0.4763
│ │ │ │ -000107e0: 3732 7320 2874 6872 6561 6429 3b20 3073  72s (thread); 0s
│ │ │ │ +000107c0: 202d 2d20 7573 6564 2030 2e38 3439 3438   -- used 0.84948
│ │ │ │ +000107d0: 3273 2028 6370 7529 3b20 302e 3532 3530  2s (cpu); 0.5250
│ │ │ │ +000107e0: 3132 7320 2874 6872 6561 6429 3b20 3073  12s (thread); 0s
│ │ │ │  000107f0: 2028 6763 297c 0a7c 2020 2020 2020 2020   (gc)|.|        
│ │ │ │  00010800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00010830: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │  00010840: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00010850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4275,894 +4275,896 @@
│ │ │ │  00010b20: 6961 626c 6573 2069 6e0a 4120 7468 6973  iables in.A this
│ │ │ │  00010b30: 2072 756e 7320 6d75 6368 2066 6173 7465   runs much faste
│ │ │ │  00010b40: 7220 7468 616e 2074 616b 696e 6720 6120  r than taking a 
│ │ │ │  00010b50: 7261 6e64 6f6d 206d 6174 7269 7820 4d2e  random matrix M.
│ │ │ │  00010b60: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00010b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010b90: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20  -------+.|i15 : 
│ │ │ │ -00010ba0: 7469 6d65 2062 6574 7469 2028 4620 3d20  time betti (F = 
│ │ │ │ -00010bb0: 7075 7265 5265 736f 6c75 7469 6f6e 2831  pureResolution(1
│ │ │ │ -00010bc0: 312c 342c 7b30 2c32 2c34 7d29 2920 7c0a  1,4,{0,2,4})) |.
│ │ │ │ -00010bd0: 7c20 2d2d 2075 7365 6420 302e 3739 3633  | -- used 0.7963
│ │ │ │ -00010be0: 3738 7320 2863 7075 293b 2030 2e35 3735  78s (cpu); 0.575
│ │ │ │ -00010bf0: 3334 7320 2874 6872 6561 6429 3b20 3073  34s (thread); 0s
│ │ │ │ -00010c00: 2028 6763 297c 0a7c 2020 2020 2020 2020   (gc)|.|        
│ │ │ │ +00010b90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a  --------+.|i15 :
│ │ │ │ +00010ba0: 2074 696d 6520 6265 7474 6920 2846 203d   time betti (F =
│ │ │ │ +00010bb0: 2070 7572 6552 6573 6f6c 7574 696f 6e28   pureResolution(
│ │ │ │ +00010bc0: 3131 2c34 2c7b 302c 322c 347d 2929 2020  11,4,{0,2,4}))  
│ │ │ │ +00010bd0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3730  |.| -- used 0.70
│ │ │ │ +00010be0: 3637 3931 7320 2863 7075 293b 2030 2e35  6791s (cpu); 0.5
│ │ │ │ +00010bf0: 3435 3237 3473 2028 7468 7265 6164 293b  45274s (thread);
│ │ │ │ +00010c00: 2030 7320 2867 6329 7c0a 7c20 2020 2020   0s (gc)|.|     
│ │ │ │  00010c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00010c40: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -00010c50: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00010c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010c40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00010c50: 3020 3120 3220 2020 2020 2020 2020 2020  0 1 2           
│ │ │ │  00010c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c70: 2020 207c 0a7c 6f31 3520 3d20 746f 7461     |.|o15 = tota
│ │ │ │ -00010c80: 6c3a 2033 2036 2033 2020 2020 2020 2020  l: 3 6 3        
│ │ │ │ +00010c70: 2020 2020 2020 2020 7c0a 7c6f 3135 203d          |.|o15 =
│ │ │ │ +00010c80: 2074 6f74 616c 3a20 3320 3620 3320 2020   total: 3 6 3   
│ │ │ │  00010c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00010cb0: 2020 2020 2020 2030 3a20 3320 2e20 2e20         0: 3 . . 
│ │ │ │ -00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010cb0: 7c0a 7c20 2020 2020 2020 2020 2030 3a20  |.|          0: 
│ │ │ │ +00010cc0: 3320 2e20 2e20 2020 2020 2020 2020 2020  3 . .           
│ │ │ │  00010cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010ce0: 207c 0a7c 2020 2020 2020 2020 2020 313a   |.|          1:
│ │ │ │ -00010cf0: 202e 2036 202e 2020 2020 2020 2020 2020   . 6 .          
│ │ │ │ +00010ce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00010cf0: 2020 2020 2031 3a20 2e20 3620 2e20 2020       1: . 6 .   
│ │ │ │  00010d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00010d20: 2020 2020 2032 3a20 2e20 2e20 3320 2020       2: . . 3   
│ │ │ │ -00010d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00010d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010d20: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ +00010d30: 2e20 2e20 3320 2020 2020 2020 2020 2020  . . 3           
│ │ │ │ +00010d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00010d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d80: 2020 2020 2020 7c0a 7c6f 3135 203a 2042        |.|o15 : B
│ │ │ │ -00010d90: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ -00010da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010db0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00010dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010d90: 7c0a 7c6f 3135 203a 2042 6574 7469 5461  |.|o15 : BettiTa
│ │ │ │ +00010da0: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ +00010db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010dc0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00010dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010df0: 2d2d 2d2d 2b0a 7c69 3136 203a 2072 696e  ----+.|i16 : rin
│ │ │ │ -00010e00: 6720 4620 2020 2020 2020 2020 2020 2020  g F             
│ │ │ │ +00010df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010e00: 2b0a 7c69 3136 203a 2072 696e 6720 4620  +.|i16 : ring F 
│ │ │ │  00010e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010e20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00010e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00010e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010e60: 2020 7c0a 7c20 2020 2020 205a 5a20 2020    |.|      ZZ   
│ │ │ │ -00010e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e70: 7c0a 7c20 2020 2020 205a 5a20 2020 2020  |.|      ZZ     
│ │ │ │  00010e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010e90: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ -00010ea0: 3d20 2d2d 5b61 202e 2e61 2020 5d20 2020  = --[a ..a  ]   
│ │ │ │ -00010eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010ea0: 2020 2020 2020 2020 7c0a 7c6f 3136 203d          |.|o16 =
│ │ │ │ +00010eb0: 202d 2d5b 6120 2e2e 6120 205d 2020 2020   --[a ..a  ]    
│ │ │ │  00010ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010ed0: 7c0a 7c20 2020 2020 2031 3120 2030 2020  |.|      11  0  
│ │ │ │ -00010ee0: 2031 3520 2020 2020 2020 2020 2020 2020   15             
│ │ │ │ -00010ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00010f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010ee0: 7c0a 7c20 2020 2020 2031 3120 2030 2020  |.|      11  0  
│ │ │ │ +00010ef0: 2031 3520 2020 2020 2020 2020 2020 2020   15             
│ │ │ │ +00010f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00010f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00010f40: 7c6f 3136 203a 2050 6f6c 796e 6f6d 6961  |o16 : Polynomia
│ │ │ │ -00010f50: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ -00010f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00010f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f50: 7c0a 7c6f 3136 203a 2050 6f6c 796e 6f6d  |.|o16 : Polynom
│ │ │ │ +00010f60: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +00010f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f80: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00010f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -00010fb0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -00010fc0: 0a0a 2020 2a20 2a6e 6f74 6520 6469 7265  ..  * *note dire
│ │ │ │ -00010fd0: 6374 496d 6167 6543 6f6d 706c 6578 3a20  ctImageComplex: 
│ │ │ │ +00010fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010fc0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ +00010fd0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │  00010fe0: 6469 7265 6374 496d 6167 6543 6f6d 706c  directImageCompl
│ │ │ │ -00010ff0: 6578 2c20 2d2d 2064 6972 6563 7420 696d  ex, -- direct im
│ │ │ │ -00011000: 6167 6520 636f 6d70 6c65 780a 2020 2a20  age complex.  * 
│ │ │ │ -00011010: 2a6e 6f74 6520 756e 6976 6572 7361 6c45  *note universalE
│ │ │ │ -00011020: 7874 656e 7369 6f6e 3a20 756e 6976 6572  xtension: univer
│ │ │ │ -00011030: 7361 6c45 7874 656e 7369 6f6e 2c20 2d2d  salExtension, --
│ │ │ │ -00011040: 2055 6e69 7665 7273 616c 2065 7874 656e   Universal exten
│ │ │ │ -00011050: 7369 6f6e 206f 660a 2020 2020 7665 6374  sion of.    vect
│ │ │ │ -00011060: 6f72 2062 756e 646c 6573 206f 6e20 505e  or bundles on P^
│ │ │ │ -00011070: 310a 0a57 6179 7320 746f 2075 7365 2070  1..Ways to use p
│ │ │ │ -00011080: 7572 6552 6573 6f6c 7574 696f 6e3a 0a3d  ureResolution:.=
│ │ │ │ -00011090: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000110a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -000110b0: 2270 7572 6552 6573 6f6c 7574 696f 6e28  "pureResolution(
│ │ │ │ -000110c0: 4d61 7472 6978 2c4c 6973 7429 220a 2020  Matrix,List)".  
│ │ │ │ -000110d0: 2a20 2270 7572 6552 6573 6f6c 7574 696f  * "pureResolutio
│ │ │ │ -000110e0: 6e28 5269 6e67 2c4c 6973 7429 220a 2020  n(Ring,List)".  
│ │ │ │ -000110f0: 2a20 2270 7572 6552 6573 6f6c 7574 696f  * "pureResolutio
│ │ │ │ -00011100: 6e28 5a5a 2c4c 6973 7429 220a 2020 2a20  n(ZZ,List)".  * 
│ │ │ │ -00011110: 2270 7572 6552 6573 6f6c 7574 696f 6e28  "pureResolution(
│ │ │ │ -00011120: 5a5a 2c5a 5a2c 4c69 7374 2922 0a0a 466f  ZZ,ZZ,List)"..Fo
│ │ │ │ -00011130: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00011140: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00011150: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00011160: 2a6e 6f74 6520 7075 7265 5265 736f 6c75  *note pureResolu
│ │ │ │ -00011170: 7469 6f6e 3a20 7075 7265 5265 736f 6c75  tion: pureResolu
│ │ │ │ -00011180: 7469 6f6e 2c20 6973 2061 202a 6e6f 7465  tion, is a *note
│ │ │ │ -00011190: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ -000111a0: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ -000111b0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ -000111c0: 0a1f 0a46 696c 653a 2042 4747 2e69 6e66  ...File: BGG.inf
│ │ │ │ -000111d0: 6f2c 204e 6f64 653a 2052 6567 756c 6172  o, Node: Regular
│ │ │ │ -000111e0: 6974 792c 204e 6578 743a 2073 796d 4578  ity, Next: symEx
│ │ │ │ -000111f0: 742c 2050 7265 763a 2070 7572 6552 6573  t, Prev: pureRes
│ │ │ │ -00011200: 6f6c 7574 696f 6e2c 2055 703a 2054 6f70  olution, Up: Top
│ │ │ │ -00011210: 0a0a 5265 6775 6c61 7269 7479 202d 2d20  ..Regularity -- 
│ │ │ │ -00011220: 4f70 7469 6f6e 2066 6f72 2064 6972 6563  Option for direc
│ │ │ │ -00011230: 7449 6d61 6765 436f 6d70 6c65 780a 2a2a  tImageComplex.**
│ │ │ │ -00011240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011260: 2a2a 2a2a 2a2a 2a2a 2a0a 0a43 6176 6561  *********..Cavea
│ │ │ │ -00011270: 740a 3d3d 3d3d 3d3d 0a0a 4375 7272 656e  t.======..Curren
│ │ │ │ -00011280: 746c 7920 6e6f 7420 7375 7070 6f72 7465  tly not supporte
│ │ │ │ -00011290: 640a 0a46 756e 6374 696f 6e73 2077 6974  d..Functions wit
│ │ │ │ -000112a0: 6820 6f70 7469 6f6e 616c 2061 7267 756d  h optional argum
│ │ │ │ -000112b0: 656e 7420 6e61 6d65 6420 5265 6775 6c61  ent named Regula
│ │ │ │ -000112c0: 7269 7479 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  rity:.==========
│ │ │ │ -000112d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00010ff0: 6578 3a20 6469 7265 6374 496d 6167 6543  ex: directImageC
│ │ │ │ +00011000: 6f6d 706c 6578 2c20 2d2d 2064 6972 6563  omplex, -- direc
│ │ │ │ +00011010: 7420 696d 6167 6520 636f 6d70 6c65 780a  t image complex.
│ │ │ │ +00011020: 2020 2a20 2a6e 6f74 6520 756e 6976 6572    * *note univer
│ │ │ │ +00011030: 7361 6c45 7874 656e 7369 6f6e 3a20 756e  salExtension: un
│ │ │ │ +00011040: 6976 6572 7361 6c45 7874 656e 7369 6f6e  iversalExtension
│ │ │ │ +00011050: 2c20 2d2d 2055 6e69 7665 7273 616c 2065  , -- Universal e
│ │ │ │ +00011060: 7874 656e 7369 6f6e 206f 660a 2020 2020  xtension of.    
│ │ │ │ +00011070: 7665 6374 6f72 2062 756e 646c 6573 206f  vector bundles o
│ │ │ │ +00011080: 6e20 505e 310a 0a57 6179 7320 746f 2075  n P^1..Ways to u
│ │ │ │ +00011090: 7365 2070 7572 6552 6573 6f6c 7574 696f  se pureResolutio
│ │ │ │ +000110a0: 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n:.=============
│ │ │ │ +000110b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +000110c0: 2020 2a20 2270 7572 6552 6573 6f6c 7574    * "pureResolut
│ │ │ │ +000110d0: 696f 6e28 4d61 7472 6978 2c4c 6973 7429  ion(Matrix,List)
│ │ │ │ +000110e0: 220a 2020 2a20 2270 7572 6552 6573 6f6c  ".  * "pureResol
│ │ │ │ +000110f0: 7574 696f 6e28 5269 6e67 2c4c 6973 7429  ution(Ring,List)
│ │ │ │ +00011100: 220a 2020 2a20 2270 7572 6552 6573 6f6c  ".  * "pureResol
│ │ │ │ +00011110: 7574 696f 6e28 5a5a 2c4c 6973 7429 220a  ution(ZZ,List)".
│ │ │ │ +00011120: 2020 2a20 2270 7572 6552 6573 6f6c 7574    * "pureResolut
│ │ │ │ +00011130: 696f 6e28 5a5a 2c5a 5a2c 4c69 7374 2922  ion(ZZ,ZZ,List)"
│ │ │ │ +00011140: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00011150: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00011160: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00011170: 6563 7420 2a6e 6f74 6520 7075 7265 5265  ect *note pureRe
│ │ │ │ +00011180: 736f 6c75 7469 6f6e 3a20 7075 7265 5265  solution: pureRe
│ │ │ │ +00011190: 736f 6c75 7469 6f6e 2c20 6973 2061 202a  solution, is a *
│ │ │ │ +000111a0: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +000111b0: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +000111c0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +000111d0: 6f6e 2c2e 0a1f 0a46 696c 653a 2042 4747  on,....File: BGG
│ │ │ │ +000111e0: 2e69 6e66 6f2c 204e 6f64 653a 2052 6567  .info, Node: Reg
│ │ │ │ +000111f0: 756c 6172 6974 792c 204e 6578 743a 2073  ularity, Next: s
│ │ │ │ +00011200: 796d 4578 742c 2050 7265 763a 2070 7572  ymExt, Prev: pur
│ │ │ │ +00011210: 6552 6573 6f6c 7574 696f 6e2c 2055 703a  eResolution, Up:
│ │ │ │ +00011220: 2054 6f70 0a0a 5265 6775 6c61 7269 7479   Top..Regularity
│ │ │ │ +00011230: 202d 2d20 4f70 7469 6f6e 2066 6f72 2064   -- Option for d
│ │ │ │ +00011240: 6972 6563 7449 6d61 6765 436f 6d70 6c65  irectImageComple
│ │ │ │ +00011250: 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  x.**************
│ │ │ │ +00011260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011270: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a43  *************..C
│ │ │ │ +00011280: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4375  aveat.======..Cu
│ │ │ │ +00011290: 7272 656e 746c 7920 6e6f 7420 7375 7070  rrently not supp
│ │ │ │ +000112a0: 6f72 7465 640a 0a46 756e 6374 696f 6e73  orted..Functions
│ │ │ │ +000112b0: 2077 6974 6820 6f70 7469 6f6e 616c 2061   with optional a
│ │ │ │ +000112c0: 7267 756d 656e 7420 6e61 6d65 6420 5265  rgument named Re
│ │ │ │ +000112d0: 6775 6c61 7269 7479 3a0a 3d3d 3d3d 3d3d  gularity:.======
│ │ │ │  000112e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000112f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2264  ========..  * "d
│ │ │ │ -00011300: 6972 6563 7449 6d61 6765 436f 6d70 6c65  irectImageComple
│ │ │ │ -00011310: 7828 2e2e 2e2c 5265 6775 6c61 7269 7479  x(...,Regularity
│ │ │ │ -00011320: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ -00011330: 6e6f 7465 2064 6972 6563 7449 6d61 6765  note directImage
│ │ │ │ -00011340: 436f 6d70 6c65 783a 0a20 2020 2064 6972  Complex:.    dir
│ │ │ │ -00011350: 6563 7449 6d61 6765 436f 6d70 6c65 782c  ectImageComplex,
│ │ │ │ -00011360: 202d 2d20 6469 7265 6374 2069 6d61 6765   -- direct image
│ │ │ │ -00011370: 2063 6f6d 706c 6578 0a0a 466f 7220 7468   complex..For th
│ │ │ │ -00011380: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00011390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -000113a0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -000113b0: 6520 5265 6775 6c61 7269 7479 3a20 2852  e Regularity: (R
│ │ │ │ -000113c0: 6567 756c 6172 6974 7929 546f 702c 2069  egularity)Top, i
│ │ │ │ -000113d0: 7320 6120 2a6e 6f74 6520 7379 6d62 6f6c  s a *note symbol
│ │ │ │ -000113e0: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ -000113f0: 5379 6d62 6f6c 2c2e 0a1f 0a46 696c 653a  Symbol,....File:
│ │ │ │ -00011400: 2042 4747 2e69 6e66 6f2c 204e 6f64 653a   BGG.info, Node:
│ │ │ │ -00011410: 2073 796d 4578 742c 204e 6578 743a 2074   symExt, Next: t
│ │ │ │ -00011420: 6174 6552 6573 6f6c 7574 696f 6e2c 2050  ateResolution, P
│ │ │ │ -00011430: 7265 763a 2052 6567 756c 6172 6974 792c  rev: Regularity,
│ │ │ │ -00011440: 2055 703a 2054 6f70 0a0a 7379 6d45 7874   Up: Top..symExt
│ │ │ │ -00011450: 202d 2d20 7468 6520 6669 7273 7420 6469   -- the first di
│ │ │ │ -00011460: 6666 6572 656e 7469 616c 206f 6620 7468  fferential of th
│ │ │ │ -00011470: 6520 636f 6d70 6c65 7820 5228 4d29 0a2a  e complex R(M).*
│ │ │ │ -00011480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000112f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00011300: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00011310: 2a20 2264 6972 6563 7449 6d61 6765 436f  * "directImageCo
│ │ │ │ +00011320: 6d70 6c65 7828 2e2e 2e2c 5265 6775 6c61  mplex(...,Regula
│ │ │ │ +00011330: 7269 7479 3d3e 2e2e 2e29 2220 2d2d 2073  rity=>...)" -- s
│ │ │ │ +00011340: 6565 202a 6e6f 7465 2064 6972 6563 7449  ee *note directI
│ │ │ │ +00011350: 6d61 6765 436f 6d70 6c65 783a 0a20 2020  mageComplex:.   
│ │ │ │ +00011360: 2064 6972 6563 7449 6d61 6765 436f 6d70   directImageComp
│ │ │ │ +00011370: 6c65 782c 202d 2d20 6469 7265 6374 2069  lex, -- direct i
│ │ │ │ +00011380: 6d61 6765 2063 6f6d 706c 6578 0a0a 466f  mage complex..Fo
│ │ │ │ +00011390: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +000113a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +000113b0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +000113c0: 2a6e 6f74 6520 5265 6775 6c61 7269 7479  *note Regularity
│ │ │ │ +000113d0: 3a20 2852 6567 756c 6172 6974 7929 546f  : (Regularity)To
│ │ │ │ +000113e0: 702c 2069 7320 6120 2a6e 6f74 6520 7379  p, is a *note sy
│ │ │ │ +000113f0: 6d62 6f6c 3a0a 284d 6163 6175 6c61 7932  mbol:.(Macaulay2
│ │ │ │ +00011400: 446f 6329 5379 6d62 6f6c 2c2e 0a1f 0a46  Doc)Symbol,....F
│ │ │ │ +00011410: 696c 653a 2042 4747 2e69 6e66 6f2c 204e  ile: BGG.info, N
│ │ │ │ +00011420: 6f64 653a 2073 796d 4578 742c 204e 6578  ode: symExt, Nex
│ │ │ │ +00011430: 743a 2074 6174 6552 6573 6f6c 7574 696f  t: tateResolutio
│ │ │ │ +00011440: 6e2c 2050 7265 763a 2052 6567 756c 6172  n, Prev: Regular
│ │ │ │ +00011450: 6974 792c 2055 703a 2054 6f70 0a0a 7379  ity, Up: Top..sy
│ │ │ │ +00011460: 6d45 7874 202d 2d20 7468 6520 6669 7273  mExt -- the firs
│ │ │ │ +00011470: 7420 6469 6666 6572 656e 7469 616c 206f  t differential o
│ │ │ │ +00011480: 6620 7468 6520 636f 6d70 6c65 7820 5228  f the complex R(
│ │ │ │ +00011490: 4d29 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  M).*************
│ │ │ │  000114a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000114b0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -000114c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ -000114d0: 653a 200a 2020 2020 2020 2020 7379 6d45  e: .        symE
│ │ │ │ -000114e0: 7874 286d 2c45 290a 2020 2a20 496e 7075  xt(m,E).  * Inpu
│ │ │ │ -000114f0: 7473 3a0a 2020 2020 2020 2a20 6d2c 2061  ts:.      * m, a
│ │ │ │ -00011500: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ -00011510: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ -00011520: 7269 782c 2c20 6120 7072 6573 656e 7461  rix,, a presenta
│ │ │ │ -00011530: 7469 6f6e 206d 6174 7269 7820 666f 7220  tion matrix for 
│ │ │ │ -00011540: 610a 2020 2020 2020 2020 706f 7369 7469  a.        positi
│ │ │ │ -00011550: 7665 6c79 2067 7261 6465 6420 6d6f 6475  vely graded modu
│ │ │ │ -00011560: 6c65 204d 206f 7665 7220 6120 706f 6c79  le M over a poly
│ │ │ │ -00011570: 6e6f 6d69 616c 2072 696e 670a 2020 2020  nomial ring.    
│ │ │ │ -00011580: 2020 2a20 452c 2061 202a 6e6f 7465 2070    * E, a *note p
│ │ │ │ -00011590: 6f6c 796e 6f6d 6961 6c20 7269 6e67 3a20  olynomial ring: 
│ │ │ │ -000115a0: 284d 6163 6175 6c61 7932 446f 6329 506f  (Macaulay2Doc)Po
│ │ │ │ -000115b0: 6c79 6e6f 6d69 616c 5269 6e67 2c2c 2065  lynomialRing,, e
│ │ │ │ -000115c0: 7874 6572 696f 720a 2020 2020 2020 2020  xterior.        
│ │ │ │ -000115d0: 616c 6765 6272 610a 2020 2a20 4f75 7470  algebra.  * Outp
│ │ │ │ -000115e0: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ -000115f0: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -00011600: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -00011610: 782c 2c20 6120 6d61 7472 6978 2072 6570  x,, a matrix rep
│ │ │ │ -00011620: 7265 7365 6e74 696e 6720 7468 6520 6d61  resenting the ma
│ │ │ │ -00011630: 700a 2020 2020 2020 2020 4d5f 3120 2a2a  p.        M_1 **
│ │ │ │ -00011640: 206f 6d65 6761 5f45 203c 2d2d 204d 5f30   omega_E <-- M_0
│ │ │ │ -00011650: 202a 2a20 6f6d 6567 615f 450a 0a44 6573   ** omega_E..Des
│ │ │ │ -00011660: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00011670: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374  ====..This funct
│ │ │ │ -00011680: 696f 6e20 7461 6b65 7320 6173 2069 6e70  ion takes as inp
│ │ │ │ -00011690: 7574 2061 206d 6174 7269 7820 6d20 7769  ut a matrix m wi
│ │ │ │ -000116a0: 7468 206c 696e 6561 7220 656e 7472 6965  th linear entrie
│ │ │ │ -000116b0: 732c 2077 6869 6368 2077 6520 7468 696e  s, which we thin
│ │ │ │ -000116c0: 6b20 6f66 0a61 7320 6120 7072 6573 656e  k of.as a presen
│ │ │ │ -000116d0: 7461 7469 6f6e 206d 6174 7269 7820 666f  tation matrix fo
│ │ │ │ -000116e0: 7220 6120 706f 7369 7469 7665 6c79 2067  r a positively g
│ │ │ │ -000116f0: 7261 6465 6420 532d 6d6f 6475 6c65 204d  raded S-module M
│ │ │ │ -00011700: 206d 6174 7269 7820 7265 7072 6573 656e   matrix represen
│ │ │ │ -00011710: 7469 6e67 0a74 6865 206d 6170 204d 5f31  ting.the map M_1
│ │ │ │ -00011720: 202a 2a20 6f6d 6567 615f 4520 3c2d 2d20   ** omega_E <-- 
│ │ │ │ -00011730: 4d5f 3020 2a2a 206f 6d65 6761 5f45 2077  M_0 ** omega_E w
│ │ │ │ -00011740: 6869 6368 2069 7320 7468 6520 6669 7273  hich is the firs
│ │ │ │ -00011750: 7420 6469 6666 6572 656e 7469 616c 206f  t differential o
│ │ │ │ -00011760: 660a 7468 6520 636f 6d70 6c65 7820 5228  f.the complex R(
│ │ │ │ -00011770: 4d29 2e0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  M)..+-----------
│ │ │ │ -00011780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000114b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000114c0: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +000114d0: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +000114e0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000114f0: 7379 6d45 7874 286d 2c45 290a 2020 2a20  symExt(m,E).  * 
│ │ │ │ +00011500: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00011510: 6d2c 2061 202a 6e6f 7465 206d 6174 7269  m, a *note matri
│ │ │ │ +00011520: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +00011530: 294d 6174 7269 782c 2c20 6120 7072 6573  )Matrix,, a pres
│ │ │ │ +00011540: 656e 7461 7469 6f6e 206d 6174 7269 7820  entation matrix 
│ │ │ │ +00011550: 666f 7220 610a 2020 2020 2020 2020 706f  for a.        po
│ │ │ │ +00011560: 7369 7469 7665 6c79 2067 7261 6465 6420  sitively graded 
│ │ │ │ +00011570: 6d6f 6475 6c65 204d 206f 7665 7220 6120  module M over a 
│ │ │ │ +00011580: 706f 6c79 6e6f 6d69 616c 2072 696e 670a  polynomial ring.
│ │ │ │ +00011590: 2020 2020 2020 2a20 452c 2061 202a 6e6f        * E, a *no
│ │ │ │ +000115a0: 7465 2070 6f6c 796e 6f6d 6961 6c20 7269  te polynomial ri
│ │ │ │ +000115b0: 6e67 3a20 284d 6163 6175 6c61 7932 446f  ng: (Macaulay2Do
│ │ │ │ +000115c0: 6329 506f 6c79 6e6f 6d69 616c 5269 6e67  c)PolynomialRing
│ │ │ │ +000115d0: 2c2c 2065 7874 6572 696f 720a 2020 2020  ,, exterior.    
│ │ │ │ +000115e0: 2020 2020 616c 6765 6272 610a 2020 2a20      algebra.  * 
│ │ │ │ +000115f0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00011600: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +00011610: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00011620: 6174 7269 782c 2c20 6120 6d61 7472 6978  atrix,, a matrix
│ │ │ │ +00011630: 2072 6570 7265 7365 6e74 696e 6720 7468   representing th
│ │ │ │ +00011640: 6520 6d61 700a 2020 2020 2020 2020 4d5f  e map.        M_
│ │ │ │ +00011650: 3120 2a2a 206f 6d65 6761 5f45 203c 2d2d  1 ** omega_E <--
│ │ │ │ +00011660: 204d 5f30 202a 2a20 6f6d 6567 615f 450a   M_0 ** omega_E.
│ │ │ │ +00011670: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00011680: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2066  ========..This f
│ │ │ │ +00011690: 756e 6374 696f 6e20 7461 6b65 7320 6173  unction takes as
│ │ │ │ +000116a0: 2069 6e70 7574 2061 206d 6174 7269 7820   input a matrix 
│ │ │ │ +000116b0: 6d20 7769 7468 206c 696e 6561 7220 656e  m with linear en
│ │ │ │ +000116c0: 7472 6965 732c 2077 6869 6368 2077 6520  tries, which we 
│ │ │ │ +000116d0: 7468 696e 6b20 6f66 0a61 7320 6120 7072  think of.as a pr
│ │ │ │ +000116e0: 6573 656e 7461 7469 6f6e 206d 6174 7269  esentation matri
│ │ │ │ +000116f0: 7820 666f 7220 6120 706f 7369 7469 7665  x for a positive
│ │ │ │ +00011700: 6c79 2067 7261 6465 6420 532d 6d6f 6475  ly graded S-modu
│ │ │ │ +00011710: 6c65 204d 206d 6174 7269 7820 7265 7072  le M matrix repr
│ │ │ │ +00011720: 6573 656e 7469 6e67 0a74 6865 206d 6170  esenting.the map
│ │ │ │ +00011730: 204d 5f31 202a 2a20 6f6d 6567 615f 4520   M_1 ** omega_E 
│ │ │ │ +00011740: 3c2d 2d20 4d5f 3020 2a2a 206f 6d65 6761  <-- M_0 ** omega
│ │ │ │ +00011750: 5f45 2077 6869 6368 2069 7320 7468 6520  _E which is the 
│ │ │ │ +00011760: 6669 7273 7420 6469 6666 6572 656e 7469  first differenti
│ │ │ │ +00011770: 616c 206f 660a 7468 6520 636f 6d70 6c65  al of.the comple
│ │ │ │ +00011780: 7820 5228 4d29 2e0a 2b2d 2d2d 2d2d 2d2d  x R(M)..+-------
│ │ │ │  00011790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000117a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -000117b0: 5320 3d20 5a5a 2f33 3230 3033 5b78 5f30  S = ZZ/32003[x_0
│ │ │ │ -000117c0: 2e2e 785f 325d 3b20 2020 2020 2020 2020  ..x_2];         
│ │ │ │ -000117d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000117e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000117f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000117a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000117b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000117c0: 3120 3a20 5320 3d20 5a5a 2f33 3230 3033  1 : S = ZZ/32003
│ │ │ │ +000117d0: 5b78 5f30 2e2e 785f 325d 3b20 2020 2020  [x_0..x_2];     
│ │ │ │ +000117e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000117f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00011800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011810: 2d2d 2d2d 2b0a 7c69 3220 3a20 4520 3d20  ----+.|i2 : E = 
│ │ │ │ -00011820: 5a5a 2f33 3230 3033 5b65 5f30 2e2e 655f  ZZ/32003[e_0..e_
│ │ │ │ -00011830: 322c 2053 6b65 7743 6f6d 6d75 7461 7469  2, SkewCommutati
│ │ │ │ -00011840: 7665 3d3e 7472 7565 5d3b 7c0a 2b2d 2d2d  ve=>true];|.+---
│ │ │ │ -00011850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011820: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00011830: 4520 3d20 5a5a 2f33 3230 3033 5b65 5f30  E = ZZ/32003[e_0
│ │ │ │ +00011840: 2e2e 655f 322c 2053 6b65 7743 6f6d 6d75  ..e_2, SkewCommu
│ │ │ │ +00011850: 7461 7469 7665 3d3e 7472 7565 5d3b 7c0a  tative=>true];|.
│ │ │ │ +00011860: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011880: 2b0a 7c69 3320 3a20 4d20 3d20 636f 6b65  +.|i3 : M = coke
│ │ │ │ -00011890: 7220 6d61 7472 6978 207b 7b78 5f30 5e32  r matrix {{x_0^2
│ │ │ │ -000118a0: 2c20 785f 315e 327d 7d3b 2020 2020 2020  , x_1^2}};      
│ │ │ │ -000118b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -000118c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011890: 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20 3d20  ----+.|i3 : M = 
│ │ │ │ +000118a0: 636f 6b65 7220 6d61 7472 6978 207b 7b78  coker matrix {{x
│ │ │ │ +000118b0: 5f30 5e32 2c20 785f 315e 327d 7d3b 2020  _0^2, x_1^2}};  
│ │ │ │ +000118c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000118d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000118e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000118f0: 3420 3a20 6d20 3d20 7072 6573 656e 7461  4 : m = presenta
│ │ │ │ -00011900: 7469 6f6e 2074 7275 6e63 6174 6528 7265  tion truncate(re
│ │ │ │ -00011910: 6775 6c61 7269 7479 204d 2c4d 293b 2020  gularity M,M);  
│ │ │ │ -00011920: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00011930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000118e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000118f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011900: 2b0a 7c69 3420 3a20 6d20 3d20 7072 6573  +.|i4 : m = pres
│ │ │ │ +00011910: 656e 7461 7469 6f6e 2074 7275 6e63 6174  entation truncat
│ │ │ │ +00011920: 6528 7265 6775 6c61 7269 7479 204d 2c4d  e(regularity M,M
│ │ │ │ +00011930: 293b 2020 2020 7c0a 7c20 2020 2020 2020  );    |.|       
│ │ │ │  00011940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011950: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00011960: 2020 2020 2020 2020 3420 2020 2020 2038          4      8
│ │ │ │ -00011970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00011990: 7c6f 3420 3a20 4d61 7472 6978 2053 2020  |o4 : Matrix S  
│ │ │ │ -000119a0: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -000119b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000119c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000119d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00011970: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00011980: 2020 2038 2020 2020 2020 2020 2020 2020     8            
│ │ │ │ +00011990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000119a0: 2020 7c0a 7c6f 3420 3a20 4d61 7472 6978    |.|o4 : Matrix
│ │ │ │ +000119b0: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ +000119c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000119d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  000119e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000119f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -00011a00: 3a20 7379 6d45 7874 286d 2c45 2920 2020  : symExt(m,E)   
│ │ │ │ -00011a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011a30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00011a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000119f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00011a10: 7c69 3520 3a20 7379 6d45 7874 286d 2c45  |i5 : symExt(m,E
│ │ │ │ +00011a20: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00011a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00011a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011a60: 2020 2020 2020 7c0a 7c6f 3520 3d20 7b2d        |.|o5 = {-
│ │ │ │ -00011a70: 317d 207c 2065 5f32 2030 2020 2030 2020  1} | e_2 0   0  
│ │ │ │ -00011a80: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ -00011a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00011aa0: 2020 2020 7b2d 317d 207c 2065 5f31 2065      {-1} | e_1 e
│ │ │ │ -00011ab0: 5f32 2030 2020 2030 2020 207c 2020 2020  _2 0   0   |    
│ │ │ │ -00011ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011ad0: 2020 7c0a 7c20 2020 2020 7b2d 317d 207c    |.|     {-1} |
│ │ │ │ -00011ae0: 2065 5f30 2030 2020 2065 5f32 2030 2020   e_0 0   e_2 0  
│ │ │ │ -00011af0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00011b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00011b10: 7b2d 317d 207c 2030 2020 2065 5f30 2065  {-1} | 0   e_0 e
│ │ │ │ -00011b20: 5f31 2065 5f32 207c 2020 2020 2020 2020  _1 e_2 |        
│ │ │ │ -00011b30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00011b40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00011b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011a70: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00011a80: 3d20 7b2d 317d 207c 2065 5f32 2030 2020  = {-1} | e_2 0  
│ │ │ │ +00011a90: 2030 2020 2030 2020 207c 2020 2020 2020   0   0   |      
│ │ │ │ +00011aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ab0: 7c0a 7c20 2020 2020 7b2d 317d 207c 2065  |.|     {-1} | e
│ │ │ │ +00011ac0: 5f31 2065 5f32 2030 2020 2030 2020 207c  _1 e_2 0   0   |
│ │ │ │ +00011ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ae0: 2020 2020 2020 7c0a 7c20 2020 2020 7b2d        |.|     {-
│ │ │ │ +00011af0: 317d 207c 2065 5f30 2030 2020 2065 5f32  1} | e_0 0   e_2
│ │ │ │ +00011b00: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +00011b10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00011b20: 2020 2020 7b2d 317d 207c 2030 2020 2065      {-1} | 0   e
│ │ │ │ +00011b30: 5f30 2065 5f31 2065 5f32 207c 2020 2020  _0 e_1 e_2 |    
│ │ │ │ +00011b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011b50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00011b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011b70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00011b80: 2020 2020 3420 2020 2020 2034 2020 2020      4      4    
│ │ │ │ -00011b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011ba0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00011bb0: 3a20 4d61 7472 6978 2045 2020 3c2d 2d20  : Matrix E  <-- 
│ │ │ │ -00011bc0: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │ -00011bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011be0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00011bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011b80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00011b90: 2020 2020 2020 2020 3420 2020 2020 2034          4      4
│ │ │ │ +00011ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00011bc0: 7c6f 3520 3a20 4d61 7472 6978 2045 2020  |o5 : Matrix E  
│ │ │ │ +00011bd0: 3c2d 2d20 4520 2020 2020 2020 2020 2020  <-- E           
│ │ │ │ +00011be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011bf0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00011c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011c10: 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561 740a  ------+..Caveat.
│ │ │ │ -00011c20: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e  ======..This fun
│ │ │ │ -00011c30: 6374 696f 6e20 6973 2061 2071 7569 636b  ction is a quick
│ │ │ │ -00011c40: 2d61 6e64 2d64 6972 7479 2074 6f6f 6c20  -and-dirty tool 
│ │ │ │ -00011c50: 7768 6963 6820 7265 7175 6972 6573 206c  which requires l
│ │ │ │ -00011c60: 6974 746c 6520 636f 6d70 7574 6174 696f  ittle computatio
│ │ │ │ -00011c70: 6e2e 0a48 6f77 6576 6572 2069 6620 6974  n..However if it
│ │ │ │ -00011c80: 2069 7320 6361 6c6c 6564 206f 6e20 7477   is called on tw
│ │ │ │ -00011c90: 6f20 7375 6363 6573 7369 7665 2074 7275  o successive tru
│ │ │ │ -00011ca0: 6e63 6174 696f 6e73 206f 6620 6120 6d6f  ncations of a mo
│ │ │ │ -00011cb0: 6475 6c65 2c20 7468 656e 2074 6865 0a6d  dule, then the.m
│ │ │ │ -00011cc0: 6170 7320 6974 2070 726f 6475 6365 7320  aps it produces 
│ │ │ │ -00011cd0: 6d61 7920 4e4f 5420 636f 6d70 6f73 6520  may NOT compose 
│ │ │ │ -00011ce0: 746f 207a 6572 6f20 6265 6361 7573 6520  to zero because 
│ │ │ │ -00011cf0: 7468 6520 6368 6f69 6365 206f 6620 6261  the choice of ba
│ │ │ │ -00011d00: 7365 7320 6973 206e 6f74 0a63 6f6e 7369  ses is not.consi
│ │ │ │ -00011d10: 7374 656e 742e 0a0a 5365 6520 616c 736f  stent...See also
│ │ │ │ -00011d20: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00011d30: 6e6f 7465 2062 6767 3a20 6267 672c 202d  note bgg: bgg, -
│ │ │ │ -00011d40: 2d20 7468 6520 6974 6820 6469 6666 6572  - the ith differ
│ │ │ │ -00011d50: 656e 7469 616c 206f 6620 7468 6520 636f  ential of the co
│ │ │ │ -00011d60: 6d70 6c65 7820 5228 4d29 0a0a 5761 7973  mplex R(M)..Ways
│ │ │ │ -00011d70: 2074 6f20 7573 6520 7379 6d45 7874 3a0a   to use symExt:.
│ │ │ │ -00011d80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00011d90: 3d3d 3d0a 0a20 202a 2022 7379 6d45 7874  ===..  * "symExt
│ │ │ │ -00011da0: 284d 6174 7269 782c 506f 6c79 6e6f 6d69  (Matrix,Polynomi
│ │ │ │ -00011db0: 616c 5269 6e67 2922 0a0a 466f 7220 7468  alRing)"..For th
│ │ │ │ -00011dc0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00011dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00011de0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00011df0: 6520 7379 6d45 7874 3a20 7379 6d45 7874  e symExt: symExt
│ │ │ │ -00011e00: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -00011e10: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ -00011e20: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00011e30: 6f64 4675 6e63 7469 6f6e 2c2e 0a1f 0a46  odFunction,....F
│ │ │ │ -00011e40: 696c 653a 2042 4747 2e69 6e66 6f2c 204e  ile: BGG.info, N
│ │ │ │ -00011e50: 6f64 653a 2074 6174 6552 6573 6f6c 7574  ode: tateResolut
│ │ │ │ -00011e60: 696f 6e2c 204e 6578 743a 2075 6e69 7665  ion, Next: unive
│ │ │ │ -00011e70: 7273 616c 4578 7465 6e73 696f 6e2c 2050  rsalExtension, P
│ │ │ │ -00011e80: 7265 763a 2073 796d 4578 742c 2055 703a  rev: symExt, Up:
│ │ │ │ -00011e90: 2054 6f70 0a0a 7461 7465 5265 736f 6c75   Top..tateResolu
│ │ │ │ -00011ea0: 7469 6f6e 202d 2d20 6669 6e69 7465 2070  tion -- finite p
│ │ │ │ -00011eb0: 6965 6365 206f 6620 7468 6520 5461 7465  iece of the Tate
│ │ │ │ -00011ec0: 2072 6573 6f6c 7574 696f 6e0a 2a2a 2a2a   resolution.****
│ │ │ │ -00011ed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176  ----------+..Cav
│ │ │ │ +00011c30: 6561 740a 3d3d 3d3d 3d3d 0a0a 5468 6973  eat.======..This
│ │ │ │ +00011c40: 2066 756e 6374 696f 6e20 6973 2061 2071   function is a q
│ │ │ │ +00011c50: 7569 636b 2d61 6e64 2d64 6972 7479 2074  uick-and-dirty t
│ │ │ │ +00011c60: 6f6f 6c20 7768 6963 6820 7265 7175 6972  ool which requir
│ │ │ │ +00011c70: 6573 206c 6974 746c 6520 636f 6d70 7574  es little comput
│ │ │ │ +00011c80: 6174 696f 6e2e 0a48 6f77 6576 6572 2069  ation..However i
│ │ │ │ +00011c90: 6620 6974 2069 7320 6361 6c6c 6564 206f  f it is called o
│ │ │ │ +00011ca0: 6e20 7477 6f20 7375 6363 6573 7369 7665  n two successive
│ │ │ │ +00011cb0: 2074 7275 6e63 6174 696f 6e73 206f 6620   truncations of 
│ │ │ │ +00011cc0: 6120 6d6f 6475 6c65 2c20 7468 656e 2074  a module, then t
│ │ │ │ +00011cd0: 6865 0a6d 6170 7320 6974 2070 726f 6475  he.maps it produ
│ │ │ │ +00011ce0: 6365 7320 6d61 7920 4e4f 5420 636f 6d70  ces may NOT comp
│ │ │ │ +00011cf0: 6f73 6520 746f 207a 6572 6f20 6265 6361  ose to zero beca
│ │ │ │ +00011d00: 7573 6520 7468 6520 6368 6f69 6365 206f  use the choice o
│ │ │ │ +00011d10: 6620 6261 7365 7320 6973 206e 6f74 0a63  f bases is not.c
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│ │ │ │ +00011d70: 6520 636f 6d70 6c65 7820 5228 4d29 0a0a  e complex R(M)..
│ │ │ │ +00011d80: 5761 7973 2074 6f20 7573 6520 7379 6d45  Ways to use symE
│ │ │ │ +00011d90: 7874 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  xt:.============
│ │ │ │ +00011da0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7379  =======..  * "sy
│ │ │ │ +00011db0: 6d45 7874 284d 6174 7269 782c 506f 6c79  mExt(Matrix,Poly
│ │ │ │ +00011dc0: 6e6f 6d69 616c 5269 6e67 2922 0a0a 466f  nomialRing)"..Fo
│ │ │ │ +00011dd0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00011de0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00011df0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00011e00: 2a6e 6f74 6520 7379 6d45 7874 3a20 7379  *note symExt: sy
│ │ │ │ +00011e10: 6d45 7874 2c20 6973 2061 202a 6e6f 7465  mExt, is a *note
│ │ │ │ +00011e20: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00011e30: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00011e40: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +00011e50: 0a1f 0a46 696c 653a 2042 4747 2e69 6e66  ...File: BGG.inf
│ │ │ │ +00011e60: 6f2c 204e 6f64 653a 2074 6174 6552 6573  o, Node: tateRes
│ │ │ │ +00011e70: 6f6c 7574 696f 6e2c 204e 6578 743a 2075  olution, Next: u
│ │ │ │ +00011e80: 6e69 7665 7273 616c 4578 7465 6e73 696f  niversalExtensio
│ │ │ │ +00011e90: 6e2c 2050 7265 763a 2073 796d 4578 742c  n, Prev: symExt,
│ │ │ │ +00011ea0: 2055 703a 2054 6f70 0a0a 7461 7465 5265   Up: Top..tateRe
│ │ │ │ +00011eb0: 736f 6c75 7469 6f6e 202d 2d20 6669 6e69  solution -- fini
│ │ │ │ +00011ec0: 7465 2070 6965 6365 206f 6620 7468 6520  te piece of the 
│ │ │ │ +00011ed0: 5461 7465 2072 6573 6f6c 7574 696f 6e0a  Tate resolution.
│ │ │ │  00011ee0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00011ef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011f00: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -00011f10: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -00011f20: 200a 2020 2020 2020 2020 7461 7465 5265   .        tateRe
│ │ │ │ -00011f30: 736f 6c75 7469 6f6e 286d 2c45 2c6c 2c68  solution(m,E,l,h
│ │ │ │ -00011f40: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -00011f50: 2020 2020 2a20 6d2c 2061 202a 6e6f 7465      * m, a *note
│ │ │ │ -00011f60: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ -00011f70: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ -00011f80: 6120 7072 6573 656e 7461 7469 6f6e 206d  a presentation m
│ │ │ │ -00011f90: 6174 7269 7820 666f 7220 610a 2020 2020  atrix for a.    
│ │ │ │ -00011fa0: 2020 2020 6d6f 6475 6c65 0a20 2020 2020      module.     
│ │ │ │ -00011fb0: 202a 2045 2c20 6120 2a6e 6f74 6520 706f   * E, a *note po
│ │ │ │ -00011fc0: 6c79 6e6f 6d69 616c 2072 696e 673a 2028  lynomial ring: (
│ │ │ │ -00011fd0: 4d61 6361 756c 6179 3244 6f63 2950 6f6c  Macaulay2Doc)Pol
│ │ │ │ -00011fe0: 796e 6f6d 6961 6c52 696e 672c 2c20 6578  ynomialRing,, ex
│ │ │ │ -00011ff0: 7465 7269 6f72 0a20 2020 2020 2020 2061  terior.        a
│ │ │ │ -00012000: 6c67 6562 7261 0a20 2020 2020 202a 206c  lgebra.      * l
│ │ │ │ -00012010: 2c20 616e 202a 6e6f 7465 2069 6e74 6567  , an *note integ
│ │ │ │ -00012020: 6572 3a20 284d 6163 6175 6c61 7932 446f  er: (Macaulay2Do
│ │ │ │ -00012030: 6329 5a5a 2c2c 206c 6f77 6572 2063 6f68  c)ZZ,, lower coh
│ │ │ │ -00012040: 6f6d 6f6c 6f67 6963 616c 2064 6567 7265  omological degre
│ │ │ │ -00012050: 650a 2020 2020 2020 2a20 682c 2061 6e20  e.      * h, an 
│ │ │ │ -00012060: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ -00012070: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ -00012080: 2c20 7570 7065 7220 626f 756e 6420 6f6e  , upper bound on
│ │ │ │ -00012090: 2074 6865 0a20 2020 2020 2020 2063 6f68   the.        coh
│ │ │ │ -000120a0: 6f6d 6f6c 6f67 6963 616c 2064 6567 7265  omological degre
│ │ │ │ -000120b0: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ -000120c0: 2020 2020 202a 2061 202a 6e6f 7465 2063       * a *note c
│ │ │ │ -000120d0: 6861 696e 2063 6f6d 706c 6578 3a20 284d  hain complex: (M
│ │ │ │ -000120e0: 6163 6175 6c61 7932 446f 6329 4368 6169  acaulay2Doc)Chai
│ │ │ │ -000120f0: 6e43 6f6d 706c 6578 2c2c 2061 2066 696e  nComplex,, a fin
│ │ │ │ -00012100: 6974 6520 7069 6563 6520 6f66 0a20 2020  ite piece of.   
│ │ │ │ -00012110: 2020 2020 2074 6865 2054 6174 6520 7265       the Tate re
│ │ │ │ -00012120: 736f 6c75 7469 6f6e 0a0a 4465 7363 7269  solution..Descri
│ │ │ │ -00012130: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -00012140: 3d0a 0a54 6869 7320 6675 6e63 7469 6f6e  =..This function
│ │ │ │ -00012150: 2074 616b 6573 2061 7320 696e 7075 7420   takes as input 
│ │ │ │ -00012160: 6120 7072 6573 656e 7461 7469 6f6e 206d  a presentation m
│ │ │ │ -00012170: 6174 7269 7820 6d20 6f66 2061 2066 696e  atrix m of a fin
│ │ │ │ -00012180: 6974 656c 7920 6765 6e65 7261 7465 640a  itely generated.
│ │ │ │ -00012190: 6772 6164 6564 2053 2d6d 6f64 756c 6520  graded S-module 
│ │ │ │ -000121a0: 4d20 616e 2065 7874 6572 696f 7220 616c  M an exterior al
│ │ │ │ -000121b0: 6765 6272 6120 4520 616e 6420 7477 6f20  gebra E and two 
│ │ │ │ -000121c0: 696e 7465 6765 7273 206c 2061 6e64 2068  integers l and h
│ │ │ │ -000121d0: 2e20 4966 2072 2069 7320 7468 650a 7265  . If r is the.re
│ │ │ │ -000121e0: 6775 6c61 7269 7479 206f 6620 4d2c 2074  gularity of M, t
│ │ │ │ -000121f0: 6865 6e20 7468 6973 2066 756e 6374 696f  hen this functio
│ │ │ │ -00012200: 6e20 636f 6d70 7574 6573 2074 6865 2070  n computes the p
│ │ │ │ -00012210: 6965 6365 206f 6620 7468 6520 5461 7465  iece of the Tate
│ │ │ │ -00012220: 2072 6573 6f6c 7574 696f 6e0a 6672 6f6d   resolution.from
│ │ │ │ -00012230: 2063 6f68 6f6d 6f6c 6f67 6963 616c 2064   cohomological d
│ │ │ │ -00012240: 6567 7265 6520 6c20 746f 2063 6f68 6f6d  egree l to cohom
│ │ │ │ -00012250: 6f6c 6f67 6963 616c 2064 6567 7265 6520  ological degree 
│ │ │ │ -00012260: 6d61 7828 722b 322c 6829 2e20 466f 7220  max(r+2,h). For 
│ │ │ │ -00012270: 696e 7374 616e 6365 2c0a 666f 7220 7468  instance,.for th
│ │ │ │ -00012280: 6520 686f 6d6f 6765 6e65 6f75 7320 636f  e homogeneous co
│ │ │ │ -00012290: 6f72 6469 6e61 7465 2072 696e 6720 6f66  ordinate ring of
│ │ │ │ -000122a0: 2061 2070 6f69 6e74 2069 6e20 7468 6520   a point in the 
│ │ │ │ -000122b0: 7072 6f6a 6563 7469 7665 2070 6c61 6e65  projective plane
│ │ │ │ -000122c0: 3a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :.+-------------
│ │ │ │ -000122d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011f10: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +00011f20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +00011f30: 6167 653a 200a 2020 2020 2020 2020 7461  age: .        ta
│ │ │ │ +00011f40: 7465 5265 736f 6c75 7469 6f6e 286d 2c45  teResolution(m,E
│ │ │ │ +00011f50: 2c6c 2c68 290a 2020 2a20 496e 7075 7473  ,l,h).  * Inputs
│ │ │ │ +00011f60: 3a0a 2020 2020 2020 2a20 6d2c 2061 202a  :.      * m, a *
│ │ │ │ +00011f70: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ +00011f80: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ +00011f90: 782c 2c20 6120 7072 6573 656e 7461 7469  x,, a presentati
│ │ │ │ +00011fa0: 6f6e 206d 6174 7269 7820 666f 7220 610a  on matrix for a.
│ │ │ │ +00011fb0: 2020 2020 2020 2020 6d6f 6475 6c65 0a20          module. 
│ │ │ │ +00011fc0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74       * E, a *not
│ │ │ │ +00011fd0: 6520 706f 6c79 6e6f 6d69 616c 2072 696e  e polynomial rin
│ │ │ │ +00011fe0: 673a 2028 4d61 6361 756c 6179 3244 6f63  g: (Macaulay2Doc
│ │ │ │ +00011ff0: 2950 6f6c 796e 6f6d 6961 6c52 696e 672c  )PolynomialRing,
│ │ │ │ +00012000: 2c20 6578 7465 7269 6f72 0a20 2020 2020  , exterior.     
│ │ │ │ +00012010: 2020 2061 6c67 6562 7261 0a20 2020 2020     algebra.     
│ │ │ │ +00012020: 202a 206c 2c20 616e 202a 6e6f 7465 2069   * l, an *note i
│ │ │ │ +00012030: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +00012040: 7932 446f 6329 5a5a 2c2c 206c 6f77 6572  y2Doc)ZZ,, lower
│ │ │ │ +00012050: 2063 6f68 6f6d 6f6c 6f67 6963 616c 2064   cohomological d
│ │ │ │ +00012060: 6567 7265 650a 2020 2020 2020 2a20 682c  egree.      * h,
│ │ │ │ +00012070: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ +00012080: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00012090: 295a 5a2c 2c20 7570 7065 7220 626f 756e  )ZZ,, upper boun
│ │ │ │ +000120a0: 6420 6f6e 2074 6865 0a20 2020 2020 2020  d on the.       
│ │ │ │ +000120b0: 2063 6f68 6f6d 6f6c 6f67 6963 616c 2064   cohomological d
│ │ │ │ +000120c0: 6567 7265 650a 2020 2a20 4f75 7470 7574  egree.  * Output
│ │ │ │ +000120d0: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ +000120e0: 7465 2063 6861 696e 2063 6f6d 706c 6578  te chain complex
│ │ │ │ +000120f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00012100: 4368 6169 6e43 6f6d 706c 6578 2c2c 2061  ChainComplex,, a
│ │ │ │ +00012110: 2066 696e 6974 6520 7069 6563 6520 6f66   finite piece of
│ │ │ │ +00012120: 0a20 2020 2020 2020 2074 6865 2054 6174  .        the Tat
│ │ │ │ +00012130: 6520 7265 736f 6c75 7469 6f6e 0a0a 4465  e resolution..De
│ │ │ │ +00012140: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +00012150: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63  =====..This func
│ │ │ │ +00012160: 7469 6f6e 2074 616b 6573 2061 7320 696e  tion takes as in
│ │ │ │ +00012170: 7075 7420 6120 7072 6573 656e 7461 7469  put a presentati
│ │ │ │ +00012180: 6f6e 206d 6174 7269 7820 6d20 6f66 2061  on matrix m of a
│ │ │ │ +00012190: 2066 696e 6974 656c 7920 6765 6e65 7261   finitely genera
│ │ │ │ +000121a0: 7465 640a 6772 6164 6564 2053 2d6d 6f64  ted.graded S-mod
│ │ │ │ +000121b0: 756c 6520 4d20 616e 2065 7874 6572 696f  ule M an exterio
│ │ │ │ +000121c0: 7220 616c 6765 6272 6120 4520 616e 6420  r algebra E and 
│ │ │ │ +000121d0: 7477 6f20 696e 7465 6765 7273 206c 2061  two integers l a
│ │ │ │ +000121e0: 6e64 2068 2e20 4966 2072 2069 7320 7468  nd h. If r is th
│ │ │ │ +000121f0: 650a 7265 6775 6c61 7269 7479 206f 6620  e.regularity of 
│ │ │ │ +00012200: 4d2c 2074 6865 6e20 7468 6973 2066 756e  M, then this fun
│ │ │ │ +00012210: 6374 696f 6e20 636f 6d70 7574 6573 2074  ction computes t
│ │ │ │ +00012220: 6865 2070 6965 6365 206f 6620 7468 6520  he piece of the 
│ │ │ │ +00012230: 5461 7465 2072 6573 6f6c 7574 696f 6e0a  Tate resolution.
│ │ │ │ +00012240: 6672 6f6d 2063 6f68 6f6d 6f6c 6f67 6963  from cohomologic
│ │ │ │ +00012250: 616c 2064 6567 7265 6520 6c20 746f 2063  al degree l to c
│ │ │ │ +00012260: 6f68 6f6d 6f6c 6f67 6963 616c 2064 6567  ohomological deg
│ │ │ │ +00012270: 7265 6520 6d61 7828 722b 322c 6829 2e20  ree max(r+2,h). 
│ │ │ │ +00012280: 466f 7220 696e 7374 616e 6365 2c0a 666f  For instance,.fo
│ │ │ │ +00012290: 7220 7468 6520 686f 6d6f 6765 6e65 6f75  r the homogeneou
│ │ │ │ +000122a0: 7320 636f 6f72 6469 6e61 7465 2072 696e  s coordinate rin
│ │ │ │ +000122b0: 6720 6f66 2061 2070 6f69 6e74 2069 6e20  g of a point in 
│ │ │ │ +000122c0: 7468 6520 7072 6f6a 6563 7469 7665 2070  the projective p
│ │ │ │ +000122d0: 6c61 6e65 3a0a 2b2d 2d2d 2d2d 2d2d 2d2d  lane:.+---------
│ │ │ │  000122e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122f0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5320  ------+.|i1 : S 
│ │ │ │ -00012300: 3d20 5a5a 2f33 3230 3033 5b78 5f30 2e2e  = ZZ/32003[x_0..
│ │ │ │ -00012310: 785f 325d 3b20 2020 2020 2020 2020 2020  x_2];           
│ │ │ │ -00012320: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000122f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012300: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00012310: 3a20 5320 3d20 5a5a 2f33 3230 3033 5b78  : S = ZZ/32003[x
│ │ │ │ +00012320: 5f30 2e2e 785f 325d 3b20 2020 2020 2020  _0..x_2];       
│ │ │ │ +00012330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012340: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00012350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012360: 2d2d 2b0a 7c69 3220 3a20 4520 3d20 5a5a  --+.|i2 : E = ZZ
│ │ │ │ -00012370: 2f33 3230 3033 5b65 5f30 2e2e 655f 322c  /32003[e_0..e_2,
│ │ │ │ -00012380: 2053 6b65 7743 6f6d 6d75 7461 7469 7665   SkewCommutative
│ │ │ │ -00012390: 3d3e 7472 7565 5d3b 7c0a 2b2d 2d2d 2d2d  =>true];|.+-----
│ │ │ │ -000123a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012370: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 4520  ------+.|i2 : E 
│ │ │ │ +00012380: 3d20 5a5a 2f33 3230 3033 5b65 5f30 2e2e  = ZZ/32003[e_0..
│ │ │ │ +00012390: 655f 322c 2053 6b65 7743 6f6d 6d75 7461  e_2, SkewCommuta
│ │ │ │ +000123a0: 7469 7665 3d3e 7472 7565 5d3b 7c0a 2b2d  tive=>true];|.+-
│ │ │ │  000123b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000123c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000123d0: 7c69 3320 3a20 6d20 3d20 6d61 7472 6978  |i3 : m = matrix
│ │ │ │ -000123e0: 7b7b 785f 302c 785f 317d 7d3b 2020 2020  {{x_0,x_1}};    
│ │ │ │ -000123f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012400: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000123c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000123d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000123e0: 2d2d 2b0a 7c69 3320 3a20 6d20 3d20 6d61  --+.|i3 : m = ma
│ │ │ │ +000123f0: 7472 6978 7b7b 785f 302c 785f 317d 7d3b  trix{{x_0,x_1}};
│ │ │ │ +00012400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00012420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012430: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00012440: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00012450: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00012460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012470: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -00012480: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -00012490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000124a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -000124b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00012450: 7c20 2020 2020 2020 2020 2020 2020 3120  |             1 
│ │ │ │ +00012460: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00012470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012480: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472      |.|o3 : Matr
│ │ │ │ +00012490: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +000124a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000124b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000124c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000124d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000124e0: 3420 3a20 7265 6775 6c61 7269 7479 2063  4 : regularity c
│ │ │ │ -000124f0: 6f6b 6572 206d 2020 2020 2020 2020 2020  oker m          
│ │ │ │ -00012500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012510: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00012520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000124d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000124e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000124f0: 2b0a 7c69 3420 3a20 7265 6775 6c61 7269  +.|i4 : regulari
│ │ │ │ +00012500: 7479 2063 6f6b 6572 206d 2020 2020 2020  ty coker m      
│ │ │ │ +00012510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012520: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00012530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012540: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -00012550: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00012560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00012580: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00012590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00012560: 3420 3d20 3020 2020 2020 2020 2020 2020  4 = 0           
│ │ │ │ +00012570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012590: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000125a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000125b0: 2d2d 2d2d 2b0a 7c69 3520 3a20 5420 3d20  ----+.|i5 : T = 
│ │ │ │ -000125c0: 7461 7465 5265 736f 6c75 7469 6f6e 286d  tateResolution(m
│ │ │ │ -000125d0: 2c45 2c2d 322c 3429 2020 2020 2020 2020  ,E,-2,4)        
│ │ │ │ -000125e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000125f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000125b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000125c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +000125d0: 5420 3d20 7461 7465 5265 736f 6c75 7469  T = tateResoluti
│ │ │ │ +000125e0: 6f6e 286d 2c45 2c2d 322c 3429 2020 2020  on(m,E,-2,4)    
│ │ │ │ +000125f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00012600: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00012610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012620: 7c0a 7c20 2020 2020 2031 2020 2020 2020  |.|      1      
│ │ │ │ -00012630: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00012640: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00012650: 2020 2031 2020 7c0a 7c6f 3520 3d20 4520     1  |.|o5 = E 
│ │ │ │ -00012660: 203c 2d2d 2045 2020 3c2d 2d20 4520 203c   <-- E  <-- E  <
│ │ │ │ -00012670: 2d2d 2045 2020 3c2d 2d20 4520 203c 2d2d  -- E  <-- E  <--
│ │ │ │ -00012680: 2045 2020 3c2d 2d20 4520 2020 7c0a 7c20   E  <-- E   |.| 
│ │ │ │ -00012690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000126a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012630: 2020 2020 7c0a 7c20 2020 2020 2031 2020      |.|      1  
│ │ │ │ +00012640: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00012650: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00012660: 3120 2020 2020 2031 2020 7c0a 7c6f 3520  1      1  |.|o5 
│ │ │ │ +00012670: 3d20 4520 203c 2d2d 2045 2020 3c2d 2d20  = E  <-- E  <-- 
│ │ │ │ +00012680: 4520 203c 2d2d 2045 2020 3c2d 2d20 4520  E  <-- E  <-- E 
│ │ │ │ +00012690: 203c 2d2d 2045 2020 3c2d 2d20 4520 2020   <-- E  <-- E   
│ │ │ │ +000126a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000126b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000126c0: 2020 7c0a 7c20 2020 2020 3020 2020 2020    |.|     0     
│ │ │ │ -000126d0: 2031 2020 2020 2020 3220 2020 2020 2033   1      2      3
│ │ │ │ -000126e0: 2020 2020 2020 3420 2020 2020 2035 2020        4      5  
│ │ │ │ -000126f0: 2020 2020 3620 2020 7c0a 7c20 2020 2020      6   |.|     
│ │ │ │ -00012700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000126c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000126d0: 2020 2020 2020 7c0a 7c20 2020 2020 3020        |.|     0 
│ │ │ │ +000126e0: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +000126f0: 2020 2033 2020 2020 2020 3420 2020 2020     3      4     
│ │ │ │ +00012700: 2035 2020 2020 2020 3620 2020 7c0a 7c20   5      6   |.| 
│ │ │ │  00012710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00012730: 7c6f 3520 3a20 4368 6169 6e43 6f6d 706c  |o5 : ChainCompl
│ │ │ │ -00012740: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │ -00012750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012760: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00012770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012740: 2020 7c0a 7c6f 3520 3a20 4368 6169 6e43    |.|o5 : ChainC
│ │ │ │ +00012750: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00012760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012770: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00012780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012790: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -000127a0: 3a20 6265 7474 6920 5420 2020 2020 2020  : betti T       
│ │ │ │ -000127b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000127a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000127b0: 7c69 3620 3a20 6265 7474 6920 5420 2020  |i6 : betti T   
│ │ │ │  000127c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000127d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000127e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000127d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000127e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000127f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012800: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00012810: 2020 2020 2030 2031 2032 2033 2034 2035       0 1 2 3 4 5
│ │ │ │ -00012820: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ -00012830: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00012840: 3620 3d20 746f 7461 6c3a 2031 2031 2031  6 = total: 1 1 1
│ │ │ │ -00012850: 2031 2031 2031 2031 2020 2020 2020 2020   1 1 1 1        
│ │ │ │ -00012860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012870: 2020 7c0a 7c20 2020 2020 2020 202d 343a    |.|        -4:
│ │ │ │ -00012880: 2031 2031 2031 2031 2031 2031 2031 2020   1 1 1 1 1 1 1  
│ │ │ │ -00012890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000128a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000128b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00012820: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ +00012830: 2034 2035 2036 2020 2020 2020 2020 2020   4 5 6          
│ │ │ │ +00012840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012850: 7c0a 7c6f 3620 3d20 746f 7461 6c3a 2031  |.|o6 = total: 1
│ │ │ │ +00012860: 2031 2031 2031 2031 2031 2031 2020 2020   1 1 1 1 1 1    
│ │ │ │ +00012870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012890: 202d 343a 2031 2031 2031 2031 2031 2031   -4: 1 1 1 1 1 1
│ │ │ │ +000128a0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +000128b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000128c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000128d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000128e0: 7c6f 3620 3a20 4265 7474 6954 616c 6c79  |o6 : BettiTally
│ │ │ │ -000128f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012910: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00012920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000128d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000128e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000128f0: 2020 7c0a 7c6f 3620 3a20 4265 7474 6954    |.|o6 : BettiT
│ │ │ │ +00012900: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │ +00012910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012920: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00012930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012940: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -00012950: 3a20 542e 6464 5f31 2020 2020 2020 2020  : T.dd_1        
│ │ │ │ -00012960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00012960: 7c69 3720 3a20 542e 6464 5f31 2020 2020  |i7 : T.dd_1    
│ │ │ │  00012970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012980: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00012990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000129a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000129b0: 2020 2020 2020 7c0a 7c6f 3720 3d20 7b2d        |.|o7 = {-
│ │ │ │ -000129c0: 347d 207c 2065 5f32 207c 2020 2020 2020  4} | e_2 |      
│ │ │ │ -000129d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000129e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000129b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000129c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +000129d0: 3d20 7b2d 347d 207c 2065 5f32 207c 2020  = {-4} | e_2 |  
│ │ │ │ +000129e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012a00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00012a30: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00012a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a50: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -00012a60: 4d61 7472 6978 2045 2020 3c2d 2d20 4520  Matrix E  <-- E 
│ │ │ │ -00012a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00012a90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00012aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012a30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012a40: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00012a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012a60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00012a70: 3720 3a20 4d61 7472 6978 2045 2020 3c2d  7 : Matrix E  <-
│ │ │ │ +00012a80: 2d20 4520 2020 2020 2020 2020 2020 2020  - E             
│ │ │ │ +00012a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012aa0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00012ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012ac0: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ -00012ad0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00012ae0: 6f74 6520 7379 6d45 7874 3a20 7379 6d45  ote symExt: symE
│ │ │ │ -00012af0: 7874 2c20 2d2d 2074 6865 2066 6972 7374  xt, -- the first
│ │ │ │ -00012b00: 2064 6966 6665 7265 6e74 6961 6c20 6f66   differential of
│ │ │ │ -00012b10: 2074 6865 2063 6f6d 706c 6578 2052 284d   the complex R(M
│ │ │ │ -00012b20: 290a 0a57 6179 7320 746f 2075 7365 2074  )..Ways to use t
│ │ │ │ -00012b30: 6174 6552 6573 6f6c 7574 696f 6e3a 0a3d  ateResolution:.=
│ │ │ │ -00012b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00012b50: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00012b60: 2274 6174 6552 6573 6f6c 7574 696f 6e28  "tateResolution(
│ │ │ │ -00012b70: 4d61 7472 6978 2c50 6f6c 796e 6f6d 6961  Matrix,Polynomia
│ │ │ │ -00012b80: 6c52 696e 672c 5a5a 2c5a 5a29 220a 0a46  lRing,ZZ,ZZ)"..F
│ │ │ │ -00012b90: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00012ba0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00012bb0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00012bc0: 202a 6e6f 7465 2074 6174 6552 6573 6f6c   *note tateResol
│ │ │ │ -00012bd0: 7574 696f 6e3a 2074 6174 6552 6573 6f6c  ution: tateResol
│ │ │ │ -00012be0: 7574 696f 6e2c 2069 7320 6120 2a6e 6f74  ution, is a *not
│ │ │ │ -00012bf0: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ -00012c00: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ -00012c10: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ -00012c20: 2e0a 1f0a 4669 6c65 3a20 4247 472e 696e  ....File: BGG.in
│ │ │ │ -00012c30: 666f 2c20 4e6f 6465 3a20 756e 6976 6572  fo, Node: univer
│ │ │ │ -00012c40: 7361 6c45 7874 656e 7369 6f6e 2c20 5072  salExtension, Pr
│ │ │ │ -00012c50: 6576 3a20 7461 7465 5265 736f 6c75 7469  ev: tateResoluti
│ │ │ │ -00012c60: 6f6e 2c20 5570 3a20 546f 700a 0a75 6e69  on, Up: Top..uni
│ │ │ │ -00012c70: 7665 7273 616c 4578 7465 6e73 696f 6e20  versalExtension 
│ │ │ │ -00012c80: 2d2d 2055 6e69 7665 7273 616c 2065 7874  -- Universal ext
│ │ │ │ -00012c90: 656e 7369 6f6e 206f 6620 7665 6374 6f72  ension of vector
│ │ │ │ -00012ca0: 2062 756e 646c 6573 206f 6e20 505e 310a   bundles on P^1.
│ │ │ │ -00012cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ad0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +00012ae0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +00012af0: 2a20 2a6e 6f74 6520 7379 6d45 7874 3a20  * *note symExt: 
│ │ │ │ +00012b00: 7379 6d45 7874 2c20 2d2d 2074 6865 2066  symExt, -- the f
│ │ │ │ +00012b10: 6972 7374 2064 6966 6665 7265 6e74 6961  irst differentia
│ │ │ │ +00012b20: 6c20 6f66 2074 6865 2063 6f6d 706c 6578  l of the complex
│ │ │ │ +00012b30: 2052 284d 290a 0a57 6179 7320 746f 2075   R(M)..Ways to u
│ │ │ │ +00012b40: 7365 2074 6174 6552 6573 6f6c 7574 696f  se tateResolutio
│ │ │ │ +00012b50: 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n:.=============
│ │ │ │ +00012b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00012b70: 2020 2a20 2274 6174 6552 6573 6f6c 7574    * "tateResolut
│ │ │ │ +00012b80: 696f 6e28 4d61 7472 6978 2c50 6f6c 796e  ion(Matrix,Polyn
│ │ │ │ +00012b90: 6f6d 6961 6c52 696e 672c 5a5a 2c5a 5a29  omialRing,ZZ,ZZ)
│ │ │ │ +00012ba0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00012bb0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00012bc0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00012bd0: 6a65 6374 202a 6e6f 7465 2074 6174 6552  ject *note tateR
│ │ │ │ +00012be0: 6573 6f6c 7574 696f 6e3a 2074 6174 6552  esolution: tateR
│ │ │ │ +00012bf0: 6573 6f6c 7574 696f 6e2c 2069 7320 6120  esolution, is a 
│ │ │ │ +00012c00: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +00012c10: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ +00012c20: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +00012c30: 696f 6e2c 2e0a 1f0a 4669 6c65 3a20 4247  ion,....File: BG
│ │ │ │ +00012c40: 472e 696e 666f 2c20 4e6f 6465 3a20 756e  G.info, Node: un
│ │ │ │ +00012c50: 6976 6572 7361 6c45 7874 656e 7369 6f6e  iversalExtension
│ │ │ │ +00012c60: 2c20 5072 6576 3a20 7461 7465 5265 736f  , Prev: tateReso
│ │ │ │ +00012c70: 6c75 7469 6f6e 2c20 5570 3a20 546f 700a  lution, Up: Top.
│ │ │ │ +00012c80: 0a75 6e69 7665 7273 616c 4578 7465 6e73  .universalExtens
│ │ │ │ +00012c90: 696f 6e20 2d2d 2055 6e69 7665 7273 616c  ion -- Universal
│ │ │ │ +00012ca0: 2065 7874 656e 7369 6f6e 206f 6620 7665   extension of ve
│ │ │ │ +00012cb0: 6374 6f72 2062 756e 646c 6573 206f 6e20  ctor bundles on 
│ │ │ │ +00012cc0: 505e 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  P^1.************
│ │ │ │  00012cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00012ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012cf0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ -00012d00: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ -00012d10: 3a20 0a20 2020 2020 2020 2045 203d 2075  : .        E = u
│ │ │ │ -00012d20: 6e69 7665 7273 616c 4578 7465 6e73 696f  niversalExtensio
│ │ │ │ -00012d30: 6e28 4c61 2c20 4c62 290a 2020 2a20 496e  n(La, Lb).  * In
│ │ │ │ -00012d40: 7075 7473 3a0a 2020 2020 2020 2a20 4c61  puts:.      * La
│ │ │ │ -00012d50: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -00012d60: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -00012d70: 7374 2c2c 206f 6620 696e 7465 6765 7273  st,, of integers
│ │ │ │ -00012d80: 0a20 2020 2020 202a 204c 622c 2061 202a  .      * Lb, a *
│ │ │ │ -00012d90: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -00012da0: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -00012db0: 6f66 2069 6e74 6567 6572 730a 2020 2a20  of integers.  * 
│ │ │ │ -00012dc0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00012dd0: 2045 2c20 6120 2a6e 6f74 6520 6d6f 6475   E, a *note modu
│ │ │ │ -00012de0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00012df0: 6329 4d6f 6475 6c65 2c2c 2072 6570 7265  c)Module,, repre
│ │ │ │ -00012e00: 7365 6e74 696e 6720 7468 6520 6578 7465  senting the exte
│ │ │ │ -00012e10: 6e73 696f 6e0a 0a44 6573 6372 6970 7469  nsion..Descripti
│ │ │ │ -00012e20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00012e30: 4576 6572 7920 7665 6374 6f72 2062 756e  Every vector bun
│ │ │ │ -00012e40: 646c 6520 4520 6f6e 2024 5c50 505e 3124  dle E on $\PP^1$
│ │ │ │ -00012e50: 2073 706c 6974 7320 6173 2061 2073 756d   splits as a sum
│ │ │ │ -00012e60: 206f 6620 6c69 6e65 2062 756e 646c 6573   of line bundles
│ │ │ │ -00012e70: 204f 4f28 615f 6929 2e20 4966 204c 610a   OO(a_i). If La.
│ │ │ │ -00012e80: 6973 2061 206c 6973 7420 6f66 2069 6e74  is a list of int
│ │ │ │ -00012e90: 6567 6572 732c 2077 6520 7772 6974 6520  egers, we write 
│ │ │ │ -00012ea0: 4528 4c61 2920 666f 7220 7468 6520 6469  E(La) for the di
│ │ │ │ -00012eb0: 7265 6374 2073 756d 206f 6620 7468 6520  rect sum of the 
│ │ │ │ -00012ec0: 6c69 6e65 2062 756e 646c 650a 4f4f 284c  line bundle.OO(L
│ │ │ │ -00012ed0: 615f 6929 2e20 2047 6976 656e 2074 776f  a_i).  Given two
│ │ │ │ -00012ee0: 2073 7563 6820 6275 6e64 6c65 7320 7370   such bundles sp
│ │ │ │ -00012ef0: 6563 6966 6965 6420 6279 2074 6865 206c  ecified by the l
│ │ │ │ -00012f00: 6973 7473 204c 6120 616e 6420 4c62 2074  ists La and Lb t
│ │ │ │ -00012f10: 6869 7320 7363 7269 7074 0a63 6f6e 7374  his script.const
│ │ │ │ -00012f20: 7275 6374 7320 6120 6d6f 6475 6c65 2072  ructs a module r
│ │ │ │ -00012f30: 6570 7265 7365 6e74 696e 6720 7468 6520  epresenting the 
│ │ │ │ -00012f40: 756e 6976 6572 7361 6c20 6578 7465 6e73  universal extens
│ │ │ │ -00012f50: 696f 6e20 6f66 2045 284c 6229 2062 7920  ion of E(Lb) by 
│ │ │ │ -00012f60: 4528 4c61 292e 2049 740a 6973 2064 6566  E(La). It.is def
│ │ │ │ -00012f70: 696e 6564 206f 6e20 7468 6520 7072 6f64  ined on the prod
│ │ │ │ -00012f80: 7563 7420 7661 7269 6574 7920 4578 745e  uct variety Ext^
│ │ │ │ -00012f90: 3128 4528 4c61 292c 2045 284c 6229 2920  1(E(La), E(Lb)) 
│ │ │ │ -00012fa0: 7820 245c 5050 5e31 242c 2061 6e64 0a72  x $\PP^1$, and.r
│ │ │ │ -00012fb0: 6570 7265 7365 6e74 6564 2068 6572 6520  epresented here 
│ │ │ │ -00012fc0: 6279 2061 2067 7261 6465 6420 6d6f 6475  by a graded modu
│ │ │ │ -00012fd0: 6c65 206f 7665 7220 7468 6520 636f 6f72  le over the coor
│ │ │ │ -00012fe0: 6469 6e61 7465 2072 696e 6720 5320 3d20  dinate ring S = 
│ │ │ │ -00012ff0: 415b 795f 302c 795f 315d 206f 660a 7468  A[y_0,y_1] of.th
│ │ │ │ -00013000: 6973 2076 6172 6965 7479 3b20 6865 7265  is variety; here
│ │ │ │ -00013010: 2041 2069 7320 7468 6520 636f 6f72 6469   A is the coordi
│ │ │ │ -00013020: 6e61 7465 2072 696e 6720 6f66 2045 7874  nate ring of Ext
│ │ │ │ -00013030: 5e31 2845 284c 6129 2c20 4528 4c62 2929  ^1(E(La), E(Lb))
│ │ │ │ -00013040: 2c20 7768 6963 6820 6973 2061 0a70 6f6c  , which is a.pol
│ │ │ │ -00013050: 796e 6f6d 6961 6c20 7269 6e67 2e0a 0a2b  ynomial ring...+
│ │ │ │ -00013060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012d00: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00012d10: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +00012d20: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ +00012d30: 203d 2075 6e69 7665 7273 616c 4578 7465   = universalExte
│ │ │ │ +00012d40: 6e73 696f 6e28 4c61 2c20 4c62 290a 2020  nsion(La, Lb).  
│ │ │ │ +00012d50: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +00012d60: 2a20 4c61 2c20 6120 2a6e 6f74 6520 6c69  * La, a *note li
│ │ │ │ +00012d70: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +00012d80: 6329 4c69 7374 2c2c 206f 6620 696e 7465  c)List,, of inte
│ │ │ │ +00012d90: 6765 7273 0a20 2020 2020 202a 204c 622c  gers.      * Lb,
│ │ │ │ +00012da0: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +00012db0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +00012dc0: 742c 2c20 6f66 2069 6e74 6567 6572 730a  t,, of integers.
│ │ │ │ +00012dd0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00012de0: 2020 202a 2045 2c20 6120 2a6e 6f74 6520     * E, a *note 
│ │ │ │ +00012df0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +00012e00: 7932 446f 6329 4d6f 6475 6c65 2c2c 2072  y2Doc)Module,, r
│ │ │ │ +00012e10: 6570 7265 7365 6e74 696e 6720 7468 6520  epresenting the 
│ │ │ │ +00012e20: 6578 7465 6e73 696f 6e0a 0a44 6573 6372  extension..Descr
│ │ │ │ +00012e30: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00012e40: 3d3d 0a0a 4576 6572 7920 7665 6374 6f72  ==..Every vector
│ │ │ │ +00012e50: 2062 756e 646c 6520 4520 6f6e 2024 5c50   bundle E on $\P
│ │ │ │ +00012e60: 505e 3124 2073 706c 6974 7320 6173 2061  P^1$ splits as a
│ │ │ │ +00012e70: 2073 756d 206f 6620 6c69 6e65 2062 756e   sum of line bun
│ │ │ │ +00012e80: 646c 6573 204f 4f28 615f 6929 2e20 4966  dles OO(a_i). If
│ │ │ │ +00012e90: 204c 610a 6973 2061 206c 6973 7420 6f66   La.is a list of
│ │ │ │ +00012ea0: 2069 6e74 6567 6572 732c 2077 6520 7772   integers, we wr
│ │ │ │ +00012eb0: 6974 6520 4528 4c61 2920 666f 7220 7468  ite E(La) for th
│ │ │ │ +00012ec0: 6520 6469 7265 6374 2073 756d 206f 6620  e direct sum of 
│ │ │ │ +00012ed0: 7468 6520 6c69 6e65 2062 756e 646c 650a  the line bundle.
│ │ │ │ +00012ee0: 4f4f 284c 615f 6929 2e20 2047 6976 656e  OO(La_i).  Given
│ │ │ │ +00012ef0: 2074 776f 2073 7563 6820 6275 6e64 6c65   two such bundle
│ │ │ │ +00012f00: 7320 7370 6563 6966 6965 6420 6279 2074  s specified by t
│ │ │ │ +00012f10: 6865 206c 6973 7473 204c 6120 616e 6420  he lists La and 
│ │ │ │ +00012f20: 4c62 2074 6869 7320 7363 7269 7074 0a63  Lb this script.c
│ │ │ │ +00012f30: 6f6e 7374 7275 6374 7320 6120 6d6f 6475  onstructs a modu
│ │ │ │ +00012f40: 6c65 2072 6570 7265 7365 6e74 696e 6720  le representing 
│ │ │ │ +00012f50: 7468 6520 756e 6976 6572 7361 6c20 6578  the universal ex
│ │ │ │ +00012f60: 7465 6e73 696f 6e20 6f66 2045 284c 6229  tension of E(Lb)
│ │ │ │ +00012f70: 2062 7920 4528 4c61 292e 2049 740a 6973   by E(La). It.is
│ │ │ │ +00012f80: 2064 6566 696e 6564 206f 6e20 7468 6520   defined on the 
│ │ │ │ +00012f90: 7072 6f64 7563 7420 7661 7269 6574 7920  product variety 
│ │ │ │ +00012fa0: 4578 745e 3128 4528 4c61 292c 2045 284c  Ext^1(E(La), E(L
│ │ │ │ +00012fb0: 6229 2920 7820 245c 5050 5e31 242c 2061  b)) x $\PP^1$, a
│ │ │ │ +00012fc0: 6e64 0a72 6570 7265 7365 6e74 6564 2068  nd.represented h
│ │ │ │ +00012fd0: 6572 6520 6279 2061 2067 7261 6465 6420  ere by a graded 
│ │ │ │ +00012fe0: 6d6f 6475 6c65 206f 7665 7220 7468 6520  module over the 
│ │ │ │ +00012ff0: 636f 6f72 6469 6e61 7465 2072 696e 6720  coordinate ring 
│ │ │ │ +00013000: 5320 3d20 415b 795f 302c 795f 315d 206f  S = A[y_0,y_1] o
│ │ │ │ +00013010: 660a 7468 6973 2076 6172 6965 7479 3b20  f.this variety; 
│ │ │ │ +00013020: 6865 7265 2041 2069 7320 7468 6520 636f  here A is the co
│ │ │ │ +00013030: 6f72 6469 6e61 7465 2072 696e 6720 6f66  ordinate ring of
│ │ │ │ +00013040: 2045 7874 5e31 2845 284c 6129 2c20 4528   Ext^1(E(La), E(
│ │ │ │ +00013050: 4c62 2929 2c20 7768 6963 6820 6973 2061  Lb)), which is a
│ │ │ │ +00013060: 0a70 6f6c 796e 6f6d 6961 6c20 7269 6e67  .polynomial ring
│ │ │ │ +00013070: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  00013080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000130b0: 6931 203a 204d 203d 2075 6e69 7665 7273  i1 : M = univers
│ │ │ │ -000130c0: 616c 4578 7465 6e73 696f 6e28 7b2d 327d  alExtension({-2}
│ │ │ │ -000130d0: 2c20 7b32 7d29 2020 2020 2020 2020 2020  , {2})          
│ │ │ │ -000130e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000130f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000130a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000130b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000130c0: 2d2b 0a7c 6931 203a 204d 203d 2075 6e69  -+.|i1 : M = uni
│ │ │ │ +000130d0: 7665 7273 616c 4578 7465 6e73 696f 6e28  versalExtension(
│ │ │ │ +000130e0: 7b2d 327d 2c20 7b32 7d29 2020 2020 2020  {-2}, {2})      
│ │ │ │ +000130f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013110: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00013120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013150: 6f31 203d 2063 6f6b 6572 6e65 6c20 7b32  o1 = cokernel {2
│ │ │ │ -00013160: 2c20 307d 207c 2078 5f30 2078 5f31 2078  , 0} | x_0 x_1 x
│ │ │ │ -00013170: 5f32 207c 2020 2020 2020 2020 2020 2020  _2 |            
│ │ │ │ -00013180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000131a0: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -000131b0: 2c20 317d 207c 2079 5f30 2030 2020 2030  , 1} | y_0 0   0
│ │ │ │ -000131c0: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ -000131d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000131e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000131f0: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -00013200: 2c20 317d 207c 2079 5f31 2079 5f30 2030  , 1} | y_1 y_0 0
│ │ │ │ -00013210: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ -00013220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013240: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -00013250: 2c20 317d 207c 2030 2020 2079 5f31 2079  , 1} | 0   y_1 y
│ │ │ │ -00013260: 5f30 207c 2020 2020 2020 2020 2020 2020  _0 |            
│ │ │ │ -00013270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013290: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -000132a0: 2c20 317d 207c 2030 2020 2030 2020 2079  , 1} | 0   0   y
│ │ │ │ -000132b0: 5f31 207c 2020 2020 2020 2020 2020 2020  _1 |            
│ │ │ │ -000132c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013160: 207c 0a7c 6f31 203d 2063 6f6b 6572 6e65   |.|o1 = cokerne
│ │ │ │ +00013170: 6c20 7b32 2c20 307d 207c 2078 5f30 2078  l {2, 0} | x_0 x
│ │ │ │ +00013180: 5f31 2078 5f32 207c 2020 2020 2020 2020  _1 x_2 |        
│ │ │ │ +00013190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000131a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000131b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000131c0: 2020 7b31 2c20 317d 207c 2079 5f30 2030    {1, 1} | y_0 0
│ │ │ │ +000131d0: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +000131e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013200: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013210: 2020 7b31 2c20 317d 207c 2079 5f31 2079    {1, 1} | y_1 y
│ │ │ │ +00013220: 5f30 2030 2020 207c 2020 2020 2020 2020  _0 0   |        
│ │ │ │ +00013230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013250: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013260: 2020 7b31 2c20 317d 207c 2030 2020 2079    {1, 1} | 0   y
│ │ │ │ +00013270: 5f31 2079 5f30 207c 2020 2020 2020 2020  _1 y_0 |        
│ │ │ │ +00013280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000132a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000132b0: 2020 7b31 2c20 317d 207c 2030 2020 2030    {1, 1} | 0   0
│ │ │ │ +000132c0: 2020 2079 5f31 207c 2020 2020 2020 2020     y_1 |        
│ │ │ │ +000132d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000132e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000132f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00013300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013330: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ -00013340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013350: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ -00013360: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │ -00013370: 2020 3520 2020 2020 2020 2020 207c 0a7c    5          |.|
│ │ │ │ -00013380: 6f31 203a 202d 2d2d 5b78 202e 2e78 205d  o1 : ---[x ..x ]
│ │ │ │ -00013390: 5b79 202e 2e79 205d 2d6d 6f64 756c 652c  [y ..y ]-module,
│ │ │ │ -000133a0: 2071 756f 7469 656e 7420 6f66 2028 2d2d   quotient of (--
│ │ │ │ -000133b0: 2d5b 7820 2e2e 7820 5d5b 7920 2e2e 7920  -[x ..x ][y ..y 
│ │ │ │ -000133c0: 5d29 2020 2020 2020 2020 2020 207c 0a7c  ])           |.|
│ │ │ │ -000133d0: 2020 2020 2031 3031 2020 3020 2020 3220       101  0   2 
│ │ │ │ -000133e0: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ -000133f0: 2020 2020 2020 2020 2020 2020 2020 3130                10
│ │ │ │ -00013400: 3120 2030 2020 2032 2020 2030 2020 2031  1  0   2   0   1
│ │ │ │ -00013410: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00013420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013340: 207c 0a7c 2020 2020 2020 5a5a 2020 2020   |.|      ZZ    
│ │ │ │ +00013350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013370: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +00013380: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +00013390: 207c 0a7c 6f31 203a 202d 2d2d 5b78 202e   |.|o1 : ---[x .
│ │ │ │ +000133a0: 2e78 205d 5b79 202e 2e79 205d 2d6d 6f64  .x ][y ..y ]-mod
│ │ │ │ +000133b0: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ +000133c0: 2028 2d2d 2d5b 7820 2e2e 7820 5d5b 7920   (---[x ..x ][y 
│ │ │ │ +000133d0: 2e2e 7920 5d29 2020 2020 2020 2020 2020  ..y ])          
│ │ │ │ +000133e0: 207c 0a7c 2020 2020 2031 3031 2020 3020   |.|     101  0 
│ │ │ │ +000133f0: 2020 3220 2020 3020 2020 3120 2020 2020    2   0   1     
│ │ │ │ +00013400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013410: 2020 3130 3120 2030 2020 2032 2020 2030    101  0   2   0
│ │ │ │ +00013420: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00013430: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00013440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00013470: 6932 203a 204d 203d 2075 6e69 7665 7273  i2 : M = univers
│ │ │ │ -00013480: 616c 4578 7465 6e73 696f 6e28 7b2d 322c  alExtension({-2,
│ │ │ │ -00013490: 2d33 7d2c 207b 322c 337d 2920 2020 2020  -3}, {2,3})     
│ │ │ │ -000134a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000134b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013480: 2d2b 0a7c 6932 203a 204d 203d 2075 6e69  -+.|i2 : M = uni
│ │ │ │ +00013490: 7665 7273 616c 4578 7465 6e73 696f 6e28  versalExtension(
│ │ │ │ +000134a0: 7b2d 322c 2d33 7d2c 207b 322c 337d 2920  {-2,-3}, {2,3}) 
│ │ │ │ +000134b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000134d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000134d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000134e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013510: 6f32 203d 2063 6f6b 6572 6e65 6c20 7b32  o2 = cokernel {2
│ │ │ │ -00013520: 2c20 307d 207c 2078 5f30 795f 3120 785f  , 0} | x_0y_1 x_
│ │ │ │ -00013530: 3179 5f31 2078 5f32 795f 3120 785f 3379  1y_1 x_2y_1 x_3y
│ │ │ │ -00013540: 5f31 2078 5f34 795f 3120 785f 3579 5f31  _1 x_4y_1 x_5y_1
│ │ │ │ -00013550: 2078 5f36 795f 3120 2020 2020 207c 0a7c   x_6y_1      |.|
│ │ │ │ -00013560: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ -00013570: 2c20 307d 207c 2078 5f39 2020 2020 785f  , 0} | x_9    x_
│ │ │ │ -00013580: 3130 2020 2078 5f31 3120 2020 785f 3132  10   x_11   x_12
│ │ │ │ -00013590: 2020 2078 5f31 3320 2020 785f 3134 2020     x_13   x_14  
│ │ │ │ -000135a0: 2078 5f31 3520 2020 2020 2020 207c 0a7c   x_15        |.|
│ │ │ │ -000135b0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -000135c0: 2c20 317d 207c 2079 5f30 2020 2020 3020  , 1} | y_0    0 
│ │ │ │ -000135d0: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -000135e0: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -000135f0: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -00013600: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -00013610: 2c20 317d 207c 2079 5f31 2020 2020 795f  , 1} | y_1    y_
│ │ │ │ -00013620: 3020 2020 2030 2020 2020 2020 3020 2020  0    0      0   
│ │ │ │ -00013630: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -00013640: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -00013650: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -00013660: 2c20 317d 207c 2030 2020 2020 2020 795f  , 1} | 0      y_
│ │ │ │ -00013670: 3120 2020 2079 5f30 2020 2020 3020 2020  1    y_0    0   
│ │ │ │ -00013680: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -00013690: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -000136a0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -000136b0: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -000136c0: 2020 2020 2079 5f31 2020 2020 795f 3020       y_1    y_0 
│ │ │ │ -000136d0: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -000136e0: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -000136f0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -00013700: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -00013710: 2020 2020 2030 2020 2020 2020 795f 3120       0      y_1 
│ │ │ │ -00013720: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -00013730: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -00013740: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -00013750: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -00013760: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -00013770: 2020 2079 5f30 2020 2020 3020 2020 2020     y_0    0     
│ │ │ │ -00013780: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -00013790: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -000137a0: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -000137b0: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -000137c0: 2020 2079 5f31 2020 2020 795f 3020 2020     y_1    y_0   
│ │ │ │ -000137d0: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -000137e0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -000137f0: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -00013800: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -00013810: 2020 2030 2020 2020 2020 795f 3120 2020     0      y_1   
│ │ │ │ -00013820: 2079 5f30 2020 2020 2020 2020 207c 0a7c   y_0         |.|
│ │ │ │ -00013830: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -00013840: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -00013850: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -00013860: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -00013870: 2079 5f31 2020 2020 2020 2020 207c 0a7c   y_1         |.|
│ │ │ │ -00013880: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -00013890: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -000138a0: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -000138b0: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -000138c0: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -000138d0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -000138e0: 2c20 317d 207c 2030 2020 2020 2020 3020  , 1} | 0      0 
│ │ │ │ -000138f0: 2020 2020 2030 2020 2020 2020 3020 2020       0      0   
│ │ │ │ -00013900: 2020 2030 2020 2020 2020 3020 2020 2020     0      0     
│ │ │ │ -00013910: 2030 2020 2020 2020 2020 2020 207c 0a7c   0           |.|
│ │ │ │ -00013920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013930: 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                  
│ │ │ │ +00013520: 207c 0a7c 6f32 203d 2063 6f6b 6572 6e65   |.|o2 = cokerne
│ │ │ │ +00013530: 6c20 7b32 2c20 307d 207c 2078 5f30 795f  l {2, 0} | x_0y_
│ │ │ │ +00013540: 3120 785f 3179 5f31 2078 5f32 795f 3120  1 x_1y_1 x_2y_1 
│ │ │ │ +00013550: 785f 3379 5f31 2078 5f34 795f 3120 785f  x_3y_1 x_4y_1 x_
│ │ │ │ +00013560: 3579 5f31 2078 5f36 795f 3120 2020 2020  5y_1 x_6y_1     
│ │ │ │ +00013570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013580: 2020 7b33 2c20 307d 207c 2078 5f39 2020    {3, 0} | x_9  
│ │ │ │ +00013590: 2020 785f 3130 2020 2078 5f31 3120 2020    x_10   x_11   
│ │ │ │ +000135a0: 785f 3132 2020 2078 5f31 3320 2020 785f  x_12   x_13   x_
│ │ │ │ +000135b0: 3134 2020 2078 5f31 3520 2020 2020 2020  14   x_15       
│ │ │ │ +000135c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000135d0: 2020 7b32 2c20 317d 207c 2079 5f30 2020    {2, 1} | y_0  
│ │ │ │ +000135e0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +000135f0: 3020 2020 2020 2030 2020 2020 2020 3020  0      0      0 
│ │ │ │ +00013600: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00013610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013620: 2020 7b32 2c20 317d 207c 2079 5f31 2020    {2, 1} | y_1  
│ │ │ │ +00013630: 2020 795f 3020 2020 2030 2020 2020 2020    y_0    0      
│ │ │ │ +00013640: 3020 2020 2020 2030 2020 2020 2020 3020  0      0      0 
│ │ │ │ +00013650: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00013660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013670: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +00013680: 2020 795f 3120 2020 2079 5f30 2020 2020    y_1    y_0    
│ │ │ │ +00013690: 3020 2020 2020 2030 2020 2020 2020 3020  0      0      0 
│ │ │ │ +000136a0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +000136b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000136c0: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +000136d0: 2020 3020 2020 2020 2079 5f31 2020 2020    0      y_1    
│ │ │ │ +000136e0: 795f 3020 2020 2030 2020 2020 2020 3020  y_0    0      0 
│ │ │ │ +000136f0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00013700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013710: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +00013720: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013730: 795f 3120 2020 2030 2020 2020 2020 3020  y_1    0      0 
│ │ │ │ +00013740: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00013750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013760: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +00013770: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013780: 3020 2020 2020 2079 5f30 2020 2020 3020  0      y_0    0 
│ │ │ │ +00013790: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +000137a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000137b0: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +000137c0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +000137d0: 3020 2020 2020 2079 5f31 2020 2020 795f  0      y_1    y_
│ │ │ │ +000137e0: 3020 2020 2030 2020 2020 2020 2020 2020  0    0          
│ │ │ │ +000137f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013800: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +00013810: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013820: 3020 2020 2020 2030 2020 2020 2020 795f  0      0      y_
│ │ │ │ +00013830: 3120 2020 2079 5f30 2020 2020 2020 2020  1    y_0        
│ │ │ │ +00013840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013850: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +00013860: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013870: 3020 2020 2020 2030 2020 2020 2020 3020  0      0      0 
│ │ │ │ +00013880: 2020 2020 2079 5f31 2020 2020 2020 2020       y_1        
│ │ │ │ +00013890: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000138a0: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +000138b0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +000138c0: 3020 2020 2020 2030 2020 2020 2020 3020  0      0      0 
│ │ │ │ +000138d0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +000138e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000138f0: 2020 7b32 2c20 317d 207c 2030 2020 2020    {2, 1} | 0    
│ │ │ │ +00013900: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013910: 3020 2020 2020 2030 2020 2020 2020 3020  0      0      0 
│ │ │ │ +00013920: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00013930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00013940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013970: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ -00013980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013980: 207c 0a7c 2020 2020 2020 5a5a 2020 2020   |.|      ZZ    
│ │ │ │  00013990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000139a0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ -000139b0: 2020 2020 3133 2020 2020 2020 207c 0a7c      13       |.|
│ │ │ │ -000139c0: 6f32 203a 202d 2d2d 5b78 202e 2e78 2020  o2 : ---[x ..x  
│ │ │ │ -000139d0: 5d5b 7920 2e2e 7920 5d2d 6d6f 6475 6c65  ][y ..y ]-module
│ │ │ │ -000139e0: 2c20 7175 6f74 6965 6e74 206f 6620 282d  , quotient of (-
│ │ │ │ -000139f0: 2d2d 5b78 202e 2e78 2020 5d5b 7920 2e2e  --[x ..x  ][y ..
│ │ │ │ -00013a00: 7920 5d29 2020 2020 2020 2020 207c 0a7c  y ])         |.|
│ │ │ │ -00013a10: 2020 2020 2031 3031 2020 3020 2020 3137       101  0   17
│ │ │ │ -00013a20: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ -00013a30: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00013a40: 3031 2020 3020 2020 3137 2020 2030 2020  01  0   17   0  
│ │ │ │ -00013a50: 2031 2020 2020 2020 2020 2020 207c 0a7c   1           |.|
│ │ │ │ -00013a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000139a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000139b0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +000139c0: 2020 2020 2020 2020 3133 2020 2020 2020          13      
│ │ │ │ +000139d0: 207c 0a7c 6f32 203a 202d 2d2d 5b78 202e   |.|o2 : ---[x .
│ │ │ │ +000139e0: 2e78 2020 5d5b 7920 2e2e 7920 5d2d 6d6f  .x  ][y ..y ]-mo
│ │ │ │ +000139f0: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +00013a00: 6620 282d 2d2d 5b78 202e 2e78 2020 5d5b  f (---[x ..x  ][
│ │ │ │ +00013a10: 7920 2e2e 7920 5d29 2020 2020 2020 2020  y ..y ])        
│ │ │ │ +00013a20: 207c 0a7c 2020 2020 2031 3031 2020 3020   |.|     101  0 
│ │ │ │ +00013a30: 2020 3137 2020 2030 2020 2031 2020 2020    17   0   1    
│ │ │ │ +00013a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013a50: 2020 2031 3031 2020 3020 2020 3137 2020     101  0   17  
│ │ │ │ +00013a60: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │ +00013a70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00013a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00013ab0: 785f 3779 5f31 2078 5f38 795f 3120 7c20  x_7y_1 x_8y_1 | 
│ │ │ │ -00013ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ac0: 2d7c 0a7c 785f 3779 5f31 2078 5f38 795f  -|.|x_7y_1 x_8y_
│ │ │ │ +00013ad0: 3120 7c20 2020 2020 2020 2020 2020 2020  1 |             
│ │ │ │  00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013b00: 785f 3136 2020 2078 5f31 3720 2020 7c20  x_16   x_17   | 
│ │ │ │ -00013b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b10: 207c 0a7c 785f 3136 2020 2078 5f31 3720   |.|x_16   x_17 
│ │ │ │ +00013b20: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013b50: 3020 2020 2020 2030 2020 2020 2020 7c20  0      0      | 
│ │ │ │ -00013b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b60: 207c 0a7c 3020 2020 2020 2030 2020 2020   |.|0      0    
│ │ │ │ +00013b70: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013ba0: 3020 2020 2020 2030 2020 2020 2020 7c20  0      0      | 
│ │ │ │ -00013bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013bb0: 207c 0a7c 3020 2020 2020 2030 2020 2020   |.|0      0    
│ │ │ │ +00013bc0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013be0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013bf0: 3020 2020 2020 2030 2020 2020 2020 7c20  0      0      | 
│ │ │ │ -00013c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c00: 207c 0a7c 3020 2020 2020 2030 2020 2020   |.|0      0    
│ │ │ │ +00013c10: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013c40: 3020 2020 2020 2030 2020 2020 2020 7c20  0      0      | 
│ │ │ │ -00013c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c50: 207c 0a7c 3020 2020 2020 2030 2020 2020   |.|0      0    
│ │ │ │ +00013c60: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013c90: 3020 2020 2020 2030 2020 2020 2020 7c20  0      0      | 
│ │ │ │ -00013ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ca0: 207c 0a7c 3020 2020 2020 2030 2020 2020   |.|0      0    
│ │ │ │ +00013cb0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013cd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00013ce0: 3020 2020 2020 2030 2020 2020 2020 7c20  0      0      | 
│ │ │ │ -00013cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013cf0: 207c 0a7c 3020 2020 2020 2030 2020 2020   |.|0      0    
│ │ │ │ +00013d00: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00013d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013d20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ ├── ./usr/share/info/Benchmark.info.gz
│ │ │ ├── Benchmark.info
│ │ │ │ @@ -146,46 +146,46 @@
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│ │ │ │ -00000ae0: 5a2f 3130 313a 202e 3131 3335 3837 2073  Z/101: .113587 s
│ │ │ │ +00000ae0: 5a2f 3130 313a 202e 3133 3034 3832 2073  Z/101: .130482 s
│ │ │ │  00000af0: 6563 6f6e 6473 2020 2020 2020 2020 7c0a  econds        |.
│ │ │ │  00000b00: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00000b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00000b50: 7c28 3230 3235 2d30 322d 3037 2920 7838  |(2025-02-07) x8
│ │ │ │ -00000b60: 365f 3634 2047 4e55 2f4c 696e 7578 2020  6_64 GNU/Linux  
│ │ │ │ -00000b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00000b50: 7c36 2e31 322e 3335 2d31 2028 3230 3235  |6.12.35-1 (2025
│ │ │ │ +00000b60: 2d30 372d 3033 2920 7838 365f 3634 2047  -07-03) x86_64 G
│ │ │ │ +00000b70: 4e55 2f4c 696e 7578 2020 2020 2020 2020  NU/Linux        
│ │ │ │  00000b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000b90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00000ba0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00000bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ ├── ./usr/share/info/Bertini.info.gz
│ │ │ ├── Bertini.info
│ │ │ │ @@ -2113,7845 +2113,7845 @@
│ │ │ │  00008400: 6e75 6d62 6572 0a20 2020 2020 2020 206f  number.        o
│ │ │ │  00008410: 7220 7261 6e64 6f6d 2063 6f6d 706c 6578  r random complex
│ │ │ │  00008420: 206e 756d 6265 720a 2020 2020 2020 2a20   number.      * 
│ │ │ │  00008430: 2a6e 6f74 6520 546f 7044 6972 6563 746f  *note TopDirecto
│ │ │ │  00008440: 7279 3a20 546f 7044 6972 6563 746f 7279  ry: TopDirectory
│ │ │ │  00008450: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │  00008460: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ -00008470: 222f 746d 702f 4d32 2d37 3134 3831 2d30  "/tmp/M2-71481-0
│ │ │ │ -00008480: 2f30 222c 204f 7074 696f 6e20 746f 2063  /0", Option to c
│ │ │ │ -00008490: 6861 6e67 6520 6469 7265 6374 6f72 7920  hange directory 
│ │ │ │ -000084a0: 666f 7220 6669 6c65 2073 746f 7261 6765  for file storage
│ │ │ │ -000084b0: 2e0a 2020 2020 2020 2a20 2a6e 6f74 6520  ..      * *note 
│ │ │ │ -000084c0: 5665 7262 6f73 653a 2062 6572 7469 6e69  Verbose: bertini
│ │ │ │ -000084d0: 5472 6163 6b48 6f6d 6f74 6f70 795f 6c70  TrackHomotopy_lp
│ │ │ │ -000084e0: 5f70 645f 7064 5f70 645f 636d 5665 7262  _pd_pd_pd_cmVerb
│ │ │ │ -000084f0: 6f73 653d 3e5f 7064 5f70 645f 7064 5f72  ose=>_pd_pd_pd_r
│ │ │ │ -00008500: 700a 2020 2020 2020 2020 2c20 3d3e 202e  p.        , => .
│ │ │ │ -00008510: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -00008520: 6520 6661 6c73 652c 204f 7074 696f 6e20  e false, Option 
│ │ │ │ -00008530: 746f 2073 696c 656e 6365 2061 6464 6974  to silence addit
│ │ │ │ -00008540: 696f 6e61 6c20 6f75 7470 7574 0a20 202a  ional output.  *
│ │ │ │ -00008550: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -00008560: 2a20 532c 2061 202a 6e6f 7465 206c 6973  * S, a *note lis
│ │ │ │ -00008570: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -00008580: 294c 6973 742c 2c20 6120 6c69 7374 2077  )List,, a list w
│ │ │ │ -00008590: 686f 7365 2065 6e74 7269 6573 2061 7265  hose entries are
│ │ │ │ -000085a0: 206c 6973 7473 206f 660a 2020 2020 2020   lists of.      
│ │ │ │ -000085b0: 2020 736f 6c75 7469 6f6e 7320 666f 7220    solutions for 
│ │ │ │ -000085c0: 6561 6368 2074 6172 6765 7420 7379 7374  each target syst
│ │ │ │ -000085d0: 656d 0a0a 4465 7363 7269 7074 696f 6e0a  em..Description.
│ │ │ │ -000085e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -000085f0: 7320 6d65 7468 6f64 206e 756d 6572 6963  s method numeric
│ │ │ │ -00008600: 616c 6c79 2073 6f6c 7665 7320 7365 7665  ally solves seve
│ │ │ │ -00008610: 7261 6c20 706f 6c79 6e6f 6d69 616c 2073  ral polynomial s
│ │ │ │ -00008620: 7973 7465 6d73 2066 726f 6d20 6120 7061  ystems from a pa
│ │ │ │ -00008630: 7261 6d65 7465 7269 7a65 640a 6661 6d69  rameterized.fami
│ │ │ │ -00008640: 6c79 2061 7420 6f6e 6365 2e20 2054 6865  ly at once.  The
│ │ │ │ -00008650: 206c 6973 7420 4620 6973 2061 2073 7973   list F is a sys
│ │ │ │ -00008660: 7465 6d20 6f66 2070 6f6c 796e 6f6d 6961  tem of polynomia
│ │ │ │ -00008670: 6c73 2069 6e20 7269 6e67 2076 6172 6961  ls in ring varia
│ │ │ │ -00008680: 626c 6573 2061 6e64 0a74 6865 2070 6172  bles and.the par
│ │ │ │ -00008690: 616d 6574 6572 7320 6c69 7374 6564 2069  ameters listed i
│ │ │ │ -000086a0: 6e20 502e 2020 5468 6520 6c69 7374 2054  n P.  The list T
│ │ │ │ -000086b0: 2069 7320 7468 6520 7365 7420 6f66 2070   is the set of p
│ │ │ │ -000086c0: 6172 616d 6574 6572 2076 616c 7565 7320  arameter values 
│ │ │ │ -000086d0: 666f 720a 7768 6963 6820 736f 6c75 7469  for.which soluti
│ │ │ │ -000086e0: 6f6e 7320 746f 2046 2061 7265 2064 6573  ons to F are des
│ │ │ │ -000086f0: 6972 6564 2e20 2042 6f74 6820 7374 6167  ired.  Both stag
│ │ │ │ -00008700: 6573 206f 6620 4265 7274 696e 6927 7320  es of Bertini's 
│ │ │ │ -00008710: 7061 7261 6d65 7465 7220 686f 6d6f 746f  parameter homoto
│ │ │ │ -00008720: 7079 0a6d 6574 686f 6420 6172 6520 6361  py.method are ca
│ │ │ │ -00008730: 6c6c 6564 2077 6974 6820 6265 7274 696e  lled with bertin
│ │ │ │ -00008740: 6950 6172 616d 6574 6572 486f 6d6f 746f  iParameterHomoto
│ │ │ │ -00008750: 7079 2e20 4669 7273 742c 2042 6572 7469  py. First, Berti
│ │ │ │ -00008760: 6e69 2061 7373 6967 6e73 2061 0a72 616e  ni assigns a.ran
│ │ │ │ -00008770: 646f 6d20 636f 6d70 6c65 7820 6e75 6d62  dom complex numb
│ │ │ │ -00008780: 6572 2074 6f20 6561 6368 2070 6172 616d  er to each param
│ │ │ │ -00008790: 6574 6572 2061 6e64 2073 6f6c 7665 7320  eter and solves 
│ │ │ │ -000087a0: 7468 6520 7265 7375 6c74 696e 6720 7379  the resulting sy
│ │ │ │ -000087b0: 7374 656d 2c20 7468 656e 2c0a 6166 7465  stem, then,.afte
│ │ │ │ -000087c0: 7220 7468 6973 2069 6e69 7469 616c 2070  r this initial p
│ │ │ │ -000087d0: 6861 7365 2c20 4265 7274 696e 6920 636f  hase, Bertini co
│ │ │ │ -000087e0: 6d70 7574 6573 2073 6f6c 7574 696f 6e73  mputes solutions
│ │ │ │ -000087f0: 2066 6f72 2065 7665 7279 2067 6976 656e   for every given
│ │ │ │ -00008800: 2063 686f 6963 6520 6f66 0a70 6172 616d   choice of.param
│ │ │ │ -00008810: 6574 6572 7320 7573 696e 6720 6120 6e75  eters using a nu
│ │ │ │ -00008820: 6d62 6572 206f 6620 7061 7468 7320 6571  mber of paths eq
│ │ │ │ -00008830: 7561 6c20 746f 2074 6865 2065 7861 6374  ual to the exact
│ │ │ │ -00008840: 2072 6f6f 7420 636f 756e 7420 696e 2074   root count in t
│ │ │ │ -00008850: 6865 2066 6972 7374 0a73 7461 6765 2e0a  he first.stage..
│ │ │ │ -00008860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008470: 222f 746d 702f 4d32 2d31 3234 3033 342d  "/tmp/M2-124034-
│ │ │ │ +00008480: 302f 3022 2c20 4f70 7469 6f6e 2074 6f20  0/0", Option to 
│ │ │ │ +00008490: 6368 616e 6765 2064 6972 6563 746f 7279  change directory
│ │ │ │ +000084a0: 2066 6f72 2066 696c 6520 7374 6f72 6167   for file storag
│ │ │ │ +000084b0: 652e 0a20 2020 2020 202a 202a 6e6f 7465  e..      * *note
│ │ │ │ +000084c0: 2056 6572 626f 7365 3a20 6265 7274 696e   Verbose: bertin
│ │ │ │ +000084d0: 6954 7261 636b 486f 6d6f 746f 7079 5f6c  iTrackHomotopy_l
│ │ │ │ +000084e0: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ +000084f0: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ +00008500: 7270 0a20 2020 2020 2020 202c 203d 3e20  rp.        , => 
│ │ │ │ +00008510: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00008520: 7565 2066 616c 7365 2c20 4f70 7469 6f6e  ue false, Option
│ │ │ │ +00008530: 2074 6f20 7369 6c65 6e63 6520 6164 6469   to silence addi
│ │ │ │ +00008540: 7469 6f6e 616c 206f 7574 7075 740a 2020  tional output.  
│ │ │ │ +00008550: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00008560: 202a 2053 2c20 6120 2a6e 6f74 6520 6c69   * S, a *note li
│ │ │ │ +00008570: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +00008580: 6329 4c69 7374 2c2c 2061 206c 6973 7420  c)List,, a list 
│ │ │ │ +00008590: 7768 6f73 6520 656e 7472 6965 7320 6172  whose entries ar
│ │ │ │ +000085a0: 6520 6c69 7374 7320 6f66 0a20 2020 2020  e lists of.     
│ │ │ │ +000085b0: 2020 2073 6f6c 7574 696f 6e73 2066 6f72     solutions for
│ │ │ │ +000085c0: 2065 6163 6820 7461 7267 6574 2073 7973   each target sys
│ │ │ │ +000085d0: 7465 6d0a 0a44 6573 6372 6970 7469 6f6e  tem..Description
│ │ │ │ +000085e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ +000085f0: 6973 206d 6574 686f 6420 6e75 6d65 7269  is method numeri
│ │ │ │ +00008600: 6361 6c6c 7920 736f 6c76 6573 2073 6576  cally solves sev
│ │ │ │ +00008610: 6572 616c 2070 6f6c 796e 6f6d 6961 6c20  eral polynomial 
│ │ │ │ +00008620: 7379 7374 656d 7320 6672 6f6d 2061 2070  systems from a p
│ │ │ │ +00008630: 6172 616d 6574 6572 697a 6564 0a66 616d  arameterized.fam
│ │ │ │ +00008640: 696c 7920 6174 206f 6e63 652e 2020 5468  ily at once.  Th
│ │ │ │ +00008650: 6520 6c69 7374 2046 2069 7320 6120 7379  e list F is a sy
│ │ │ │ +00008660: 7374 656d 206f 6620 706f 6c79 6e6f 6d69  stem of polynomi
│ │ │ │ +00008670: 616c 7320 696e 2072 696e 6720 7661 7269  als in ring vari
│ │ │ │ +00008680: 6162 6c65 7320 616e 640a 7468 6520 7061  ables and.the pa
│ │ │ │ +00008690: 7261 6d65 7465 7273 206c 6973 7465 6420  rameters listed 
│ │ │ │ +000086a0: 696e 2050 2e20 2054 6865 206c 6973 7420  in P.  The list 
│ │ │ │ +000086b0: 5420 6973 2074 6865 2073 6574 206f 6620  T is the set of 
│ │ │ │ +000086c0: 7061 7261 6d65 7465 7220 7661 6c75 6573  parameter values
│ │ │ │ +000086d0: 2066 6f72 0a77 6869 6368 2073 6f6c 7574   for.which solut
│ │ │ │ +000086e0: 696f 6e73 2074 6f20 4620 6172 6520 6465  ions to F are de
│ │ │ │ +000086f0: 7369 7265 642e 2020 426f 7468 2073 7461  sired.  Both sta
│ │ │ │ +00008700: 6765 7320 6f66 2042 6572 7469 6e69 2773  ges of Bertini's
│ │ │ │ +00008710: 2070 6172 616d 6574 6572 2068 6f6d 6f74   parameter homot
│ │ │ │ +00008720: 6f70 790a 6d65 7468 6f64 2061 7265 2063  opy.method are c
│ │ │ │ +00008730: 616c 6c65 6420 7769 7468 2062 6572 7469  alled with berti
│ │ │ │ +00008740: 6e69 5061 7261 6d65 7465 7248 6f6d 6f74  niParameterHomot
│ │ │ │ +00008750: 6f70 792e 2046 6972 7374 2c20 4265 7274  opy. First, Bert
│ │ │ │ +00008760: 696e 6920 6173 7369 676e 7320 610a 7261  ini assigns a.ra
│ │ │ │ +00008770: 6e64 6f6d 2063 6f6d 706c 6578 206e 756d  ndom complex num
│ │ │ │ +00008780: 6265 7220 746f 2065 6163 6820 7061 7261  ber to each para
│ │ │ │ +00008790: 6d65 7465 7220 616e 6420 736f 6c76 6573  meter and solves
│ │ │ │ +000087a0: 2074 6865 2072 6573 756c 7469 6e67 2073   the resulting s
│ │ │ │ +000087b0: 7973 7465 6d2c 2074 6865 6e2c 0a61 6674  ystem, then,.aft
│ │ │ │ +000087c0: 6572 2074 6869 7320 696e 6974 6961 6c20  er this initial 
│ │ │ │ +000087d0: 7068 6173 652c 2042 6572 7469 6e69 2063  phase, Bertini c
│ │ │ │ +000087e0: 6f6d 7075 7465 7320 736f 6c75 7469 6f6e  omputes solution
│ │ │ │ +000087f0: 7320 666f 7220 6576 6572 7920 6769 7665  s for every give
│ │ │ │ +00008800: 6e20 6368 6f69 6365 206f 660a 7061 7261  n choice of.para
│ │ │ │ +00008810: 6d65 7465 7273 2075 7369 6e67 2061 206e  meters using a n
│ │ │ │ +00008820: 756d 6265 7220 6f66 2070 6174 6873 2065  umber of paths e
│ │ │ │ +00008830: 7175 616c 2074 6f20 7468 6520 6578 6163  qual to the exac
│ │ │ │ +00008840: 7420 726f 6f74 2063 6f75 6e74 2069 6e20  t root count in 
│ │ │ │ +00008850: 7468 6520 6669 7273 740a 7374 6167 652e  the first.stage.
│ │ │ │ +00008860: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00008870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000088a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000088b0: 0a7c 6931 203a 2052 3d43 435b 7531 2c75  .|i1 : R=CC[u1,u
│ │ │ │ -000088c0: 322c 7533 2c78 2c79 5d3b 2020 2020 2020  2,u3,x,y];      
│ │ │ │ +000088a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000088b0: 2b0a 7c69 3120 3a20 523d 4343 5b75 312c  +.|i1 : R=CC[u1,
│ │ │ │ +000088c0: 7532 2c75 332c 782c 795d 3b20 2020 2020  u2,u3,x,y];     
│ │ │ │  000088d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000088e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000088f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008900: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000088f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008900: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00008910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00008950: 0a7c 6932 203a 2066 313d 7531 2a28 792d  .|i2 : f1=u1*(y-
│ │ │ │ -00008960: 3129 2b75 322a 2879 2d32 292b 7533 2a28  1)+u2*(y-2)+u3*(
│ │ │ │ -00008970: 792d 3329 3b20 2d2d 7061 7261 6d65 7465  y-3); --paramete
│ │ │ │ -00008980: 7273 2061 7265 2075 312c 2075 322c 2061  rs are u1, u2, a
│ │ │ │ -00008990: 6e64 2075 3320 2020 2020 2020 2020 207c  nd u3          |
│ │ │ │ -000089a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008950: 2b0a 7c69 3220 3a20 6631 3d75 312a 2879  +.|i2 : f1=u1*(y
│ │ │ │ +00008960: 2d31 292b 7532 2a28 792d 3229 2b75 332a  -1)+u2*(y-2)+u3*
│ │ │ │ +00008970: 2879 2d33 293b 202d 2d70 6172 616d 6574  (y-3); --paramet
│ │ │ │ +00008980: 6572 7320 6172 6520 7531 2c20 7532 2c20  ers are u1, u2, 
│ │ │ │ +00008990: 616e 6420 7533 2020 2020 2020 2020 2020  and u3          
│ │ │ │ +000089a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000089b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000089c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000089d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000089e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000089f0: 0a7c 6933 203a 2066 323d 2878 2d31 3129  .|i3 : f2=(x-11)
│ │ │ │ -00008a00: 2a28 782d 3132 292a 2878 2d31 3329 2d75  *(x-12)*(x-13)-u
│ │ │ │ -00008a10: 313b 2020 2020 2020 2020 2020 2020 2020  1;              
│ │ │ │ +000089e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000089f0: 2b0a 7c69 3320 3a20 6632 3d28 782d 3131  +.|i3 : f2=(x-11
│ │ │ │ +00008a00: 292a 2878 2d31 3229 2a28 782d 3133 292d  )*(x-12)*(x-13)-
│ │ │ │ +00008a10: 7531 3b20 2020 2020 2020 2020 2020 2020  u1;             
│ │ │ │  00008a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008a40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008a40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00008a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00008a90: 0a7c 6934 203a 2070 6172 616d 5661 6c75  .|i4 : paramValu
│ │ │ │ -00008aa0: 6573 303d 7b31 2c30 2c30 7d3b 2020 2020  es0={1,0,0};    
│ │ │ │ +00008a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008a90: 2b0a 7c69 3420 3a20 7061 7261 6d56 616c  +.|i4 : paramVal
│ │ │ │ +00008aa0: 7565 7330 3d7b 312c 302c 307d 3b20 2020  ues0={1,0,0};   
│ │ │ │  00008ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008ae0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008ae0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00008af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00008b30: 0a7c 6935 203a 2070 6172 616d 5661 6c75  .|i5 : paramValu
│ │ │ │ -00008b40: 6573 313d 7b30 2c31 2b32 2a69 692c 307d  es1={0,1+2*ii,0}
│ │ │ │ -00008b50: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00008b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008b30: 2b0a 7c69 3520 3a20 7061 7261 6d56 616c  +.|i5 : paramVal
│ │ │ │ +00008b40: 7565 7331 3d7b 302c 312b 322a 6969 2c30  ues1={0,1+2*ii,0
│ │ │ │ +00008b50: 7d3b 2020 2020 2020 2020 2020 2020 2020  };              
│ │ │ │  00008b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008b80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008b80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00008b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00008bd0: 0a7c 6936 203a 2062 5048 3d62 6572 7469  .|i6 : bPH=berti
│ │ │ │ -00008be0: 6e69 5061 7261 6d65 7465 7248 6f6d 6f74  niParameterHomot
│ │ │ │ -00008bf0: 6f70 7928 207b 6631 2c66 327d 2c20 7b75  opy( {f1,f2}, {u
│ │ │ │ -00008c00: 312c 7532 2c75 337d 2c7b 7061 7261 6d56  1,u2,u3},{paramV
│ │ │ │ -00008c10: 616c 7565 7330 202c 7061 7261 6d56 617c  alues0 ,paramVa|
│ │ │ │ -00008c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00008bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008bd0: 2b0a 7c69 3620 3a20 6250 483d 6265 7274  +.|i6 : bPH=bert
│ │ │ │ +00008be0: 696e 6950 6172 616d 6574 6572 486f 6d6f  iniParameterHomo
│ │ │ │ +00008bf0: 746f 7079 2820 7b66 312c 6632 7d2c 207b  topy( {f1,f2}, {
│ │ │ │ +00008c00: 7531 2c75 322c 7533 7d2c 7b70 6172 616d  u1,u2,u3},{param
│ │ │ │ +00008c10: 5661 6c75 6573 3020 2c70 6172 616d 5661  Values0 ,paramVa
│ │ │ │ +00008c20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00008c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008c70: 0a7c 6f36 203d 207b 7b7b 3131 2e33 3337  .|o6 = {{{11.337
│ │ │ │ -00008c80: 362d 2e35 3632 3238 2a69 692c 2031 7d2c  6-.56228*ii, 1},
│ │ │ │ -00008c90: 207b 3131 2e33 3337 362b 2e35 3632 3238   {11.3376+.56228
│ │ │ │ -00008ca0: 2a69 692c 2031 7d2c 207b 3133 2e33 3234  *ii, 1}, {13.324
│ │ │ │ -00008cb0: 372c 2031 7d7d 2c20 7b7b 3131 2c20 207c  7, 1}}, {{11,  |
│ │ │ │ -00008cc0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00008c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008c70: 7c0a 7c6f 3620 3d20 7b7b 7b31 312e 3333  |.|o6 = {{{11.33
│ │ │ │ +00008c80: 3736 2d2e 3536 3232 382a 6969 2c20 317d  76-.56228*ii, 1}
│ │ │ │ +00008c90: 2c20 7b31 312e 3333 3736 2b2e 3536 3232  , {11.3376+.5622
│ │ │ │ +00008ca0: 382a 6969 2c20 317d 2c20 7b31 332e 3332  8*ii, 1}, {13.32
│ │ │ │ +00008cb0: 3437 2c20 317d 7d2c 207b 7b31 312c 2020  47, 1}}, {{11,  
│ │ │ │ +00008cc0: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │  00008cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00008d10: 0a7c 2020 2020 2032 7d2c 207b 3132 2c20  .|     2}, {12, 
│ │ │ │ -00008d20: 327d 2c20 7b31 332c 2032 7d7d 7d20 2020  2}, {13, 2}}}   
│ │ │ │ +00008d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008d10: 7c0a 7c20 2020 2020 327d 2c20 7b31 322c  |.|     2}, {12,
│ │ │ │ +00008d20: 2032 7d2c 207b 3133 2c20 327d 7d7d 2020   2}, {13, 2}}}  
│ │ │ │  00008d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00008d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008d60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00008d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008db0: 0a7c 6f36 203a 204c 6973 7420 2020 2020  .|o6 : List     
│ │ │ │ +00008da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008db0: 7c0a 7c6f 3620 3a20 4c69 7374 2020 2020  |.|o6 : List    
│ │ │ │  00008dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008e00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00008df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008e00: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00008e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00008e50: 0a7c 6c75 6573 3120 7d29 2020 2020 2020  .|lues1 })      
│ │ │ │ +00008e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008e50: 7c0a 7c6c 7565 7331 207d 2920 2020 2020  |.|lues1 })     
│ │ │ │  00008e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008ea0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00008e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008ea0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00008eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00008ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00008ef0: 0a7c 6937 203a 2062 5048 5f30 2d2d 7468  .|i7 : bPH_0--th
│ │ │ │ -00008f00: 6520 736f 6c75 7469 6f6e 7320 746f 2074  e solutions to t
│ │ │ │ -00008f10: 6865 2073 7973 7465 6d20 7769 7468 2070  he system with p
│ │ │ │ -00008f20: 6172 616d 6574 6572 7320 7365 7420 6571  arameters set eq
│ │ │ │ -00008f30: 7561 6c20 746f 2020 2020 2020 2020 207c  ual to         |
│ │ │ │ -00008f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00008ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00008ef0: 2b0a 7c69 3720 3a20 6250 485f 302d 2d74  +.|i7 : bPH_0--t
│ │ │ │ +00008f00: 6865 2073 6f6c 7574 696f 6e73 2074 6f20  he solutions to 
│ │ │ │ +00008f10: 7468 6520 7379 7374 656d 2077 6974 6820  the system with 
│ │ │ │ +00008f20: 7061 7261 6d65 7465 7273 2073 6574 2065  parameters set e
│ │ │ │ +00008f30: 7175 616c 2074 6f20 2020 2020 2020 2020  qual to         
│ │ │ │ +00008f40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00008f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00008f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00008f90: 0a7c 6f37 203d 207b 7b31 312e 3333 3736  .|o7 = {{11.3376
│ │ │ │ -00008fa0: 2d2e 3536 3232 382a 6969 2c20 317d 2c20  -.56228*ii, 1}, 
│ │ │ │ -00008fb0: 7b31 312e 3333 3736 2b2e 3536 3232 382a  {11.3376+.56228*
│ │ │ │ -00008fc0: 6969 2c20 317d 2c20 7b31 332e 3332 3437  ii, 1}, {13.3247
│ │ │ │ -00008fd0: 2c20 317d 7d20 2020 2020 2020 2020 207c  , 1}}          |
│ │ │ │ -00008fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00008f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00008f90: 7c0a 7c6f 3720 3d20 7b7b 3131 2e33 3337  |.|o7 = {{11.337
│ │ │ │ +00008fa0: 362d 2e35 3632 3238 2a69 692c 2031 7d2c  6-.56228*ii, 1},
│ │ │ │ +00008fb0: 207b 3131 2e33 3337 362b 2e35 3632 3238   {11.3376+.56228
│ │ │ │ +00008fc0: 2a69 692c 2031 7d2c 207b 3133 2e33 3234  *ii, 1}, {13.324
│ │ │ │ +00008fd0: 372c 2031 7d7d 2020 2020 2020 2020 2020  7, 1}}          
│ │ │ │ +00008fe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00008ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009030: 0a7c 6f37 203a 204c 6973 7420 2020 2020  .|o7 : List     
│ │ │ │ +00009020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009030: 7c0a 7c6f 3720 3a20 4c69 7374 2020 2020  |.|o7 : List    
│ │ │ │  00009040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009080: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00009070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009080: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00009090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000090a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000090b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000090c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000090d0: 0a7c 7061 7261 6d56 616c 7565 7330 2020  .|paramValues0  
│ │ │ │ +000090c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000090d0: 7c0a 7c70 6172 616d 5661 6c75 6573 3020  |.|paramValues0 
│ │ │ │  000090e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000090f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009120: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009120: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00009170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009170: 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +.+-------------
│ │ │ │  00009180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000091a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000091b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000091c0: 0a7c 6938 203a 2052 3d43 435b 782c 792c  .|i8 : R=CC[x,y,
│ │ │ │ -000091d0: 7a2c 7531 2c75 325d 2020 2020 2020 2020  z,u1,u2]        
│ │ │ │ +000091b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000091c0: 2b0a 7c69 3820 3a20 523d 4343 5b78 2c79  +.|i8 : R=CC[x,y
│ │ │ │ +000091d0: 2c7a 2c75 312c 7532 5d20 2020 2020 2020  ,z,u1,u2]       
│ │ │ │  000091e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000091f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009210: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009260: 0a7c 6f38 203d 2052 2020 2020 2020 2020  .|o8 = R        
│ │ │ │ +00009250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009260: 7c0a 7c6f 3820 3d20 5220 2020 2020 2020  |.|o8 = R       
│ │ │ │  00009270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000092a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000092b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000092a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000092b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000092c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000092d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000092e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000092f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009300: 0a7c 6f38 203a 2050 6f6c 796e 6f6d 6961  .|o8 : Polynomia
│ │ │ │ -00009310: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +000092f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009300: 7c0a 7c6f 3820 3a20 506f 6c79 6e6f 6d69  |.|o8 : Polynomi
│ │ │ │ +00009310: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │  00009320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009350: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009350: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000093a0: 0a7c 6939 203a 2066 313d 785e 322b 795e  .|i9 : f1=x^2+y^
│ │ │ │ -000093b0: 322d 7a5e 3220 2020 2020 2020 2020 2020  2-z^2           
│ │ │ │ +00009390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000093a0: 2b0a 7c69 3920 3a20 6631 3d78 5e32 2b79  +.|i9 : f1=x^2+y
│ │ │ │ +000093b0: 5e32 2d7a 5e32 2020 2020 2020 2020 2020  ^2-z^2          
│ │ │ │  000093c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000093d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000093e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000093f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000093e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000093f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009440: 0a7c 2020 2020 2020 3220 2020 2032 2020  .|      2    2  
│ │ │ │ -00009450: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00009430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009440: 7c0a 7c20 2020 2020 2032 2020 2020 3220  |.|      2    2 
│ │ │ │ +00009450: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00009460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009490: 0a7c 6f39 203d 2078 2020 2b20 7920 202d  .|o9 = x  + y  -
│ │ │ │ -000094a0: 207a 2020 2020 2020 2020 2020 2020 2020   z              
│ │ │ │ +00009480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009490: 7c0a 7c6f 3920 3d20 7820 202b 2079 2020  |.|o9 = x  + y  
│ │ │ │ +000094a0: 2d20 7a20 2020 2020 2020 2020 2020 2020  - z             
│ │ │ │  000094b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000094c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000094d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000094e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000094d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000094e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000094f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009530: 0a7c 6f39 203a 2052 2020 2020 2020 2020  .|o9 : R        
│ │ │ │ +00009520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009530: 7c0a 7c6f 3920 3a20 5220 2020 2020 2020  |.|o9 : R       
│ │ │ │  00009540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009580: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000095a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000095b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000095c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000095d0: 0a7c 6931 3020 3a20 6632 3d75 312a 782b  .|i10 : f2=u1*x+
│ │ │ │ -000095e0: 7532 2a79 2020 2020 2020 2020 2020 2020  u2*y            
│ │ │ │ +000095c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000095d0: 2b0a 7c69 3130 203a 2066 323d 7531 2a78  +.|i10 : f2=u1*x
│ │ │ │ +000095e0: 2b75 322a 7920 2020 2020 2020 2020 2020  +u2*y           
│ │ │ │  000095f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009620: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009670: 0a7c 6f31 3020 3d20 782a 7531 202b 2079  .|o10 = x*u1 + y
│ │ │ │ -00009680: 2a75 3220 2020 2020 2020 2020 2020 2020  *u2             
│ │ │ │ +00009660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009670: 7c0a 7c6f 3130 203d 2078 2a75 3120 2b20  |.|o10 = x*u1 + 
│ │ │ │ +00009680: 792a 7532 2020 2020 2020 2020 2020 2020  y*u2            
│ │ │ │  00009690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000096a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000096b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000096c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000096b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000096c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000096d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000096e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000096f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009710: 0a7c 6f31 3020 3a20 5220 2020 2020 2020  .|o10 : R       
│ │ │ │ +00009700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009710: 7c0a 7c6f 3130 203a 2052 2020 2020 2020  |.|o10 : R      
│ │ │ │  00009720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009760: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009760: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000097a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000097b0: 0a7c 6931 3120 3a20 6669 6e61 6c50 6172  .|i11 : finalPar
│ │ │ │ -000097c0: 616d 6574 6572 7330 3d7b 302c 317d 2020  ameters0={0,1}  
│ │ │ │ +000097a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000097b0: 2b0a 7c69 3131 203a 2066 696e 616c 5061  +.|i11 : finalPa
│ │ │ │ +000097c0: 7261 6d65 7465 7273 303d 7b30 2c31 7d20  rameters0={0,1} 
│ │ │ │  000097d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000097e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000097f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000097f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009800: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009850: 0a7c 6f31 3120 3d20 7b30 2c20 317d 2020  .|o11 = {0, 1}  
│ │ │ │ +00009840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009850: 7c0a 7c6f 3131 203d 207b 302c 2031 7d20  |.|o11 = {0, 1} 
│ │ │ │  00009860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000098a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000098a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000098b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000098c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000098d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000098e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000098f0: 0a7c 6f31 3120 3a20 4c69 7374 2020 2020  .|o11 : List    
│ │ │ │ +000098e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000098f0: 7c0a 7c6f 3131 203a 204c 6973 7420 2020  |.|o11 : List   
│ │ │ │  00009900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009940: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009940: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00009990: 0a7c 6931 3220 3a20 6669 6e61 6c50 6172  .|i12 : finalPar
│ │ │ │ -000099a0: 616d 6574 6572 7331 3d7b 312c 307d 2020  ameters1={1,0}  
│ │ │ │ +00009980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009990: 2b0a 7c69 3132 203a 2066 696e 616c 5061  +.|i12 : finalPa
│ │ │ │ +000099a0: 7261 6d65 7465 7273 313d 7b31 2c30 7d20  rameters1={1,0} 
│ │ │ │  000099b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000099c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000099d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000099e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000099d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000099e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000099f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009a30: 0a7c 6f31 3220 3d20 7b31 2c20 307d 2020  .|o12 = {1, 0}  
│ │ │ │ +00009a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009a30: 7c0a 7c6f 3132 203d 207b 312c 2030 7d20  |.|o12 = {1, 0} 
│ │ │ │  00009a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009a80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009a80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009ad0: 0a7c 6f31 3220 3a20 4c69 7374 2020 2020  .|o12 : List    
│ │ │ │ +00009ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009ad0: 7c0a 7c6f 3132 203a 204c 6973 7420 2020  |.|o12 : List   
│ │ │ │  00009ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009b20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009b20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00009b70: 0a7c 6931 3320 3a20 6250 483d 6265 7274  .|i13 : bPH=bert
│ │ │ │ -00009b80: 696e 6950 6172 616d 6574 6572 486f 6d6f  iniParameterHomo
│ │ │ │ -00009b90: 746f 7079 2820 7b66 312c 6632 7d2c 207b  topy( {f1,f2}, {
│ │ │ │ -00009ba0: 7531 2c75 327d 2c7b 6669 6e61 6c50 6172  u1,u2},{finalPar
│ │ │ │ -00009bb0: 616d 6574 6572 7330 202c 6669 6e61 6c7c  ameters0 ,final|
│ │ │ │ -00009bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009b70: 2b0a 7c69 3133 203a 2062 5048 3d62 6572  +.|i13 : bPH=ber
│ │ │ │ +00009b80: 7469 6e69 5061 7261 6d65 7465 7248 6f6d  tiniParameterHom
│ │ │ │ +00009b90: 6f74 6f70 7928 207b 6631 2c66 327d 2c20  otopy( {f1,f2}, 
│ │ │ │ +00009ba0: 7b75 312c 7532 7d2c 7b66 696e 616c 5061  {u1,u2},{finalPa
│ │ │ │ +00009bb0: 7261 6d65 7465 7273 3020 2c66 696e 616c  rameters0 ,final
│ │ │ │ +00009bc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009c10: 0a7c 6f31 3320 3d20 7b7b 7b31 2c20 312e  .|o13 = {{{1, 1.
│ │ │ │ -00009c20: 3034 3633 3465 2d31 372d 312e 3031 3434  04634e-17-1.0144
│ │ │ │ -00009c30: 3865 2d31 372a 6969 2c20 2d31 7d2c 207b  8e-17*ii, -1}, {
│ │ │ │ -00009c40: 312c 2031 2e33 3231 3131 652d 3137 2b36  1, 1.32111e-17+6
│ │ │ │ -00009c50: 2e34 3138 652d 3230 2a69 692c 2020 207c  .418e-20*ii,   |
│ │ │ │ -00009c60: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00009c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009c10: 7c0a 7c6f 3133 203d 207b 7b7b 312c 2031  |.|o13 = {{{1, 1
│ │ │ │ +00009c20: 2e30 3436 3334 652d 3137 2d31 2e30 3134  .04634e-17-1.014
│ │ │ │ +00009c30: 3438 652d 3137 2a69 692c 202d 317d 2c20  48e-17*ii, -1}, 
│ │ │ │ +00009c40: 7b31 2c20 312e 3332 3131 3165 2d31 372b  {1, 1.32111e-17+
│ │ │ │ +00009c50: 362e 3431 3865 2d32 302a 6969 2c20 2020  6.418e-20*ii,   
│ │ │ │ +00009c60: 7c0a 7c20 2020 2020 202d 2d2d 2d2d 2d2d  |.|      -------
│ │ │ │  00009c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00009cb0: 0a7c 2020 2020 2020 317d 7d2c 207b 7b39  .|      1}}, {{9
│ │ │ │ -00009cc0: 2e39 3738 3333 652d 3139 2b31 2e30 3931  .97833e-19+1.091
│ │ │ │ -00009cd0: 3835 652d 3138 2a69 692c 2031 2c20 317d  85e-18*ii, 1, 1}
│ │ │ │ -00009ce0: 2c20 7b2d 352e 3431 3838 3465 2d31 362b  , {-5.41884e-16+
│ │ │ │ -00009cf0: 312e 3431 3230 3165 2d31 362a 6969 2c7c  1.41201e-16*ii,|
│ │ │ │ -00009d00: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00009ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009cb0: 7c0a 7c20 2020 2020 2031 7d7d 2c20 7b7b  |.|      1}}, {{
│ │ │ │ +00009cc0: 392e 3937 3833 3365 2d31 392b 312e 3039  9.97833e-19+1.09
│ │ │ │ +00009cd0: 3138 3565 2d31 382a 6969 2c20 312c 2031  185e-18*ii, 1, 1
│ │ │ │ +00009ce0: 7d2c 207b 2d35 2e34 3138 3834 652d 3136  }, {-5.41884e-16
│ │ │ │ +00009cf0: 2b31 2e34 3132 3031 652d 3136 2a69 692c  +1.41201e-16*ii,
│ │ │ │ +00009d00: 7c0a 7c20 2020 2020 202d 2d2d 2d2d 2d2d  |.|      -------
│ │ │ │  00009d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00009d50: 0a7c 2020 2020 2020 312c 202d 317d 7d7d  .|      1, -1}}}
│ │ │ │ -00009d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009d50: 7c0a 7c20 2020 2020 2031 2c20 2d31 7d7d  |.|      1, -1}}
│ │ │ │ +00009d60: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  00009d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009d90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009da0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009da0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009df0: 0a7c 6f31 3320 3a20 4c69 7374 2020 2020  .|o13 : List    
│ │ │ │ +00009de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009df0: 7c0a 7c6f 3133 203a 204c 6973 7420 2020  |.|o13 : List   
│ │ │ │  00009e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009e40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00009e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009e40: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00009e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00009e90: 0a7c 5061 7261 6d65 7465 7273 3120 7d2c  .|Parameters1 },
│ │ │ │ -00009ea0: 486f 6d56 6172 6961 626c 6547 726f 7570  HomVariableGroup
│ │ │ │ -00009eb0: 3d3e 7b78 2c79 2c7a 7d29 2020 2020 2020  =>{x,y,z})      
│ │ │ │ +00009e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009e90: 7c0a 7c50 6172 616d 6574 6572 7331 207d  |.|Parameters1 }
│ │ │ │ +00009ea0: 2c48 6f6d 5661 7269 6162 6c65 4772 6f75  ,HomVariableGrou
│ │ │ │ +00009eb0: 703d 3e7b 782c 792c 7a7d 2920 2020 2020  p=>{x,y,z})     
│ │ │ │  00009ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009ee0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00009ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009ee0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00009f30: 0a7c 6931 3420 3a20 6250 485f 302d 2d54  .|i14 : bPH_0--T
│ │ │ │ -00009f40: 6865 2074 776f 2073 6f6c 7574 696f 6e73  he two solutions
│ │ │ │ -00009f50: 2066 6f72 2066 696e 616c 5061 7261 6d65   for finalParame
│ │ │ │ -00009f60: 7465 7273 3020 2020 2020 2020 2020 2020  ters0           
│ │ │ │ -00009f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009f80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00009f30: 2b0a 7c69 3134 203a 2062 5048 5f30 2d2d  +.|i14 : bPH_0--
│ │ │ │ +00009f40: 5468 6520 7477 6f20 736f 6c75 7469 6f6e  The two solution
│ │ │ │ +00009f50: 7320 666f 7220 6669 6e61 6c50 6172 616d  s for finalParam
│ │ │ │ +00009f60: 6574 6572 7330 2020 2020 2020 2020 2020  eters0          
│ │ │ │ +00009f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00009f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00009fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00009fd0: 0a7c 6f31 3420 3d20 7b7b 312c 2031 2e30  .|o14 = {{1, 1.0
│ │ │ │ -00009fe0: 3436 3334 652d 3137 2d31 2e30 3134 3438  4634e-17-1.01448
│ │ │ │ -00009ff0: 652d 3137 2a69 692c 202d 317d 2c20 7b31  e-17*ii, -1}, {1
│ │ │ │ -0000a000: 2c20 312e 3332 3131 3165 2d31 372b 362e  , 1.32111e-17+6.
│ │ │ │ -0000a010: 3431 3865 2d32 302a 6969 2c20 317d 7d7c  418e-20*ii, 1}}|
│ │ │ │ -0000a020: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00009fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00009fd0: 7c0a 7c6f 3134 203d 207b 7b31 2c20 312e  |.|o14 = {{1, 1.
│ │ │ │ +00009fe0: 3034 3633 3465 2d31 372d 312e 3031 3434  04634e-17-1.0144
│ │ │ │ +00009ff0: 3865 2d31 372a 6969 2c20 2d31 7d2c 207b  8e-17*ii, -1}, {
│ │ │ │ +0000a000: 312c 2031 2e33 3231 3131 652d 3137 2b36  1, 1.32111e-17+6
│ │ │ │ +0000a010: 2e34 3138 652d 3230 2a69 692c 2031 7d7d  .418e-20*ii, 1}}
│ │ │ │ +0000a020: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a070: 0a7c 6f31 3420 3a20 4c69 7374 2020 2020  .|o14 : List    
│ │ │ │ +0000a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a070: 7c0a 7c6f 3134 203a 204c 6973 7420 2020  |.|o14 : List   
│ │ │ │  0000a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a0b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a0c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a0c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000a110: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a110: 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +.+-------------
│ │ │ │  0000a120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000a160: 0a7c 6931 3520 3a20 6669 6e50 6172 616d  .|i15 : finParam
│ │ │ │ -0000a170: 5661 6c75 6573 3d7b 7b31 7d2c 7b32 7d7d  Values={{1},{2}}
│ │ │ │ -0000a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a160: 2b0a 7c69 3135 203a 2066 696e 5061 7261  +.|i15 : finPara
│ │ │ │ +0000a170: 6d56 616c 7565 733d 7b7b 317d 2c7b 327d  mValues={{1},{2}
│ │ │ │ +0000a180: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0000a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a1a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a1b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a200: 0a7c 6f31 3520 3d20 7b7b 317d 2c20 7b32  .|o15 = {{1}, {2
│ │ │ │ -0000a210: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +0000a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a200: 7c0a 7c6f 3135 203d 207b 7b31 7d2c 207b  |.|o15 = {{1}, {
│ │ │ │ +0000a210: 327d 7d20 2020 2020 2020 2020 2020 2020  2}}             
│ │ │ │  0000a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a250: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a2a0: 0a7c 6f31 3520 3a20 4c69 7374 2020 2020  .|o15 : List    
│ │ │ │ +0000a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a2a0: 7c0a 7c6f 3135 203a 204c 6973 7420 2020  |.|o15 : List   
│ │ │ │  0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a2f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000a340: 0a7c 6931 3620 3a20 6250 4831 3d62 6572  .|i16 : bPH1=ber
│ │ │ │ -0000a350: 7469 6e69 5061 7261 6d65 7465 7248 6f6d  tiniParameterHom
│ │ │ │ -0000a360: 6f74 6f70 7928 207b 2278 5e32 2d75 3122  otopy( {"x^2-u1"
│ │ │ │ -0000a370: 7d2c 2020 2020 2020 2020 2020 2020 2020  },              
│ │ │ │ -0000a380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a340: 2b0a 7c69 3136 203a 2062 5048 313d 6265  +.|i16 : bPH1=be
│ │ │ │ +0000a350: 7274 696e 6950 6172 616d 6574 6572 486f  rtiniParameterHo
│ │ │ │ +0000a360: 6d6f 746f 7079 2820 7b22 785e 322d 7531  motopy( {"x^2-u1
│ │ │ │ +0000a370: 227d 2c20 2020 2020 2020 2020 2020 2020  "},             
│ │ │ │ +0000a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a390: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a3d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a3e0: 0a7c 6f31 3620 3d20 7b7b 7b2d 317d 2c20  .|o16 = {{{-1}, 
│ │ │ │ -0000a3f0: 7b31 7d7d 2c20 7b7b 2d31 2e34 3134 3231  {1}}, {{-1.41421
│ │ │ │ -0000a400: 7d2c 207b 312e 3431 3432 317d 7d7d 2020  }, {1.41421}}}  
│ │ │ │ +0000a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a3e0: 7c0a 7c6f 3136 203d 207b 7b7b 2d31 7d2c  |.|o16 = {{{-1},
│ │ │ │ +0000a3f0: 207b 317d 7d2c 207b 7b2d 312e 3431 3432   {1}}, {{-1.4142
│ │ │ │ +0000a400: 317d 2c20 7b31 2e34 3134 3231 7d7d 7d20  1}, {1.41421}}} 
│ │ │ │  0000a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a430: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a480: 0a7c 6f31 3620 3a20 4c69 7374 2020 2020  .|o16 : List    
│ │ │ │ +0000a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a480: 7c0a 7c6f 3136 203a 204c 6973 7420 2020  |.|o16 : List   
│ │ │ │  0000a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a4c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a4d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0000a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a4d0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  0000a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0000a520: 0a7c 7b75 317d 2c66 696e 5061 7261 6d56  .|{u1},finParamV
│ │ │ │ -0000a530: 616c 7565 732c 4166 6656 6172 6961 626c  alues,AffVariabl
│ │ │ │ -0000a540: 6547 726f 7570 3d3e 7b78 7d29 2020 2020  eGroup=>{x})    
│ │ │ │ +0000a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a520: 7c0a 7c7b 7531 7d2c 6669 6e50 6172 616d  |.|{u1},finParam
│ │ │ │ +0000a530: 5661 6c75 6573 2c41 6666 5661 7269 6162  Values,AffVariab
│ │ │ │ +0000a540: 6c65 4772 6f75 703d 3e7b 787d 2920 2020  leGroup=>{x})   
│ │ │ │  0000a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a570: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a570: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000a5c0: 0a7c 6931 3720 3a20 6250 4832 3d62 6572  .|i17 : bPH2=ber
│ │ │ │ -0000a5d0: 7469 6e69 5061 7261 6d65 7465 7248 6f6d  tiniParameterHom
│ │ │ │ -0000a5e0: 6f74 6f70 7928 207b 2278 5e32 2d75 3122  otopy( {"x^2-u1"
│ │ │ │ -0000a5f0: 7d2c 2020 2020 2020 2020 2020 2020 2020  },              
│ │ │ │ -0000a600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a610: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a5c0: 2b0a 7c69 3137 203a 2062 5048 323d 6265  +.|i17 : bPH2=be
│ │ │ │ +0000a5d0: 7274 696e 6950 6172 616d 6574 6572 486f  rtiniParameterHo
│ │ │ │ +0000a5e0: 6d6f 746f 7079 2820 7b22 785e 322d 7531  motopy( {"x^2-u1
│ │ │ │ +0000a5f0: 227d 2c20 2020 2020 2020 2020 2020 2020  "},             
│ │ │ │ +0000a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a610: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a660: 0a7c 6f31 3720 3d20 7b7b 7b2d 317d 2c20  .|o17 = {{{-1}, 
│ │ │ │ -0000a670: 7b31 7d7d 2c20 7b7b 2d31 2e34 3134 3231  {1}}, {{-1.41421
│ │ │ │ -0000a680: 7d2c 207b 312e 3431 3432 317d 7d7d 2020  }, {1.41421}}}  
│ │ │ │ +0000a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a660: 7c0a 7c6f 3137 203d 207b 7b7b 2d31 7d2c  |.|o17 = {{{-1},
│ │ │ │ +0000a670: 207b 317d 7d2c 207b 7b2d 312e 3431 3432   {1}}, {{-1.4142
│ │ │ │ +0000a680: 317d 2c20 7b31 2e34 3134 3231 7d7d 7d20  1}, {1.41421}}} 
│ │ │ │  0000a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a6a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a6b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a6b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a700: 0a7c 6f31 3720 3a20 4c69 7374 2020 2020  .|o17 : List    
│ │ │ │ +0000a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a700: 7c0a 7c6f 3137 203a 204c 6973 7420 2020  |.|o17 : List   
│ │ │ │  0000a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a750: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0000a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a750: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  0000a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0000a7a0: 0a7c 7b75 317d 2c66 696e 5061 7261 6d56  .|{u1},finParamV
│ │ │ │ -0000a7b0: 616c 7565 732c 4166 6656 6172 6961 626c  alues,AffVariabl
│ │ │ │ -0000a7c0: 6547 726f 7570 3d3e 7b78 7d2c 4f75 7470  eGroup=>{x},Outp
│ │ │ │ -0000a7d0: 7574 5374 796c 653d 3e22 4f75 7453 6f6c  utStyle=>"OutSol
│ │ │ │ -0000a7e0: 7574 696f 6e73 2229 2020 2020 2020 207c  utions")       |
│ │ │ │ -0000a7f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a7a0: 7c0a 7c7b 7531 7d2c 6669 6e50 6172 616d  |.|{u1},finParam
│ │ │ │ +0000a7b0: 5661 6c75 6573 2c41 6666 5661 7269 6162  Values,AffVariab
│ │ │ │ +0000a7c0: 6c65 4772 6f75 703d 3e7b 787d 2c4f 7574  leGroup=>{x},Out
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│ │ │ │ -0000a850: 4831 5f30 5f30 2020 2020 2020 2020 2020  H1_0_0          
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│ │ │ │ -0000b3f0: 6d6f 746f 7079 3a0a 3d3d 3d3d 3d3d 3d3d  motopy:.========
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│ │ │ │  0000b400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0000b410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
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│ │ │ │ -0000b460: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -0000b470: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -0000b480: 202a 6e6f 7465 2062 6572 7469 6e69 5061   *note bertiniPa
│ │ │ │ -0000b490: 7261 6d65 7465 7248 6f6d 6f74 6f70 793a  rameterHomotopy:
│ │ │ │ -0000b4a0: 2062 6572 7469 6e69 5061 7261 6d65 7465   bertiniParamete
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│ │ │ │ -0000b4c0: 2a6e 6f74 650a 6d65 7468 6f64 2066 756e  *note.method fun
│ │ │ │ -0000b4d0: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ -0000b4e0: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
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│ │ │ │ -0000b540: 7874 3a20 6265 7274 696e 6952 6566 696e  xt: bertiniRefin
│ │ │ │ -0000b550: 6553 6f6c 732c 2050 7265 763a 2062 6572  eSols, Prev: ber
│ │ │ │ -0000b560: 7469 6e69 5061 7261 6d65 7465 7248 6f6d  tiniParameterHom
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│ │ │ │ -0000b580: 6265 7274 696e 6950 6f73 4469 6d53 6f6c  bertiniPosDimSol
│ │ │ │ -0000b590: 7665 202d 2d20 6120 6d61 696e 206d 6574  ve -- a main met
│ │ │ │ -0000b5a0: 686f 6420 7468 6174 2069 7320 7573 6564  hod that is used
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│ │ │ │ +0000b410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0000b420: 2020 2a20 2262 6572 7469 6e69 5061 7261    * "bertiniPara
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│ │ │ │ +0000b450: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +0000b460: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +0000b470: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +0000b480: 7420 2a6e 6f74 6520 6265 7274 696e 6950  t *note bertiniP
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│ │ │ │ +0000b4a0: 3a20 6265 7274 696e 6950 6172 616d 6574  : bertiniParamet
│ │ │ │ +0000b4b0: 6572 486f 6d6f 746f 7079 2c20 6973 2061  erHomotopy, is a
│ │ │ │ +0000b4c0: 202a 6e6f 7465 0a6d 6574 686f 6420 6675   *note.method fu
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│ │ │ │ +0000b4e0: 6f6e 733a 2028 4d61 6361 756c 6179 3244  ons: (Macaulay2D
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│ │ │ │ +0000b570: 6d6f 746f 7079 2c20 5570 3a20 546f 700a  motopy, Up: Top.
│ │ │ │ +0000b580: 0a62 6572 7469 6e69 506f 7344 696d 536f  .bertiniPosDimSo
│ │ │ │ +0000b590: 6c76 6520 2d2d 2061 206d 6169 6e20 6d65  lve -- a main me
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│ │ │ │ +0000b5b0: 6420 746f 2070 726f 6475 6365 2077 6974  d to produce wit
│ │ │ │ +0000b5c0: 6e65 7373 2073 6574 730a 2a2a 2a2a 2a2a  ness sets.******
│ │ │ │  0000b5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000b5e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000b5f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000b600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000b610: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -0000b620: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -0000b630: 200a 2020 2020 2020 2020 5620 3d20 6265   .        V = be
│ │ │ │ -0000b640: 7274 696e 6950 6f73 4469 6d53 6f6c 7665  rtiniPosDimSolve
│ │ │ │ -0000b650: 2049 0a20 2020 2020 2020 2056 203d 2062   I.        V = b
│ │ │ │ -0000b660: 6572 7469 6e69 506f 7344 696d 536f 6c76  ertiniPosDimSolv
│ │ │ │ -0000b670: 6520 460a 2020 2a20 496e 7075 7473 3a0a  e F.  * Inputs:.
│ │ │ │ -0000b680: 2020 2020 2020 2a20 462c 2061 202a 6e6f        * F, a *no
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│ │ │ │ -0000b6b0: 6c69 7374 206f 6620 7269 6e67 2065 6c65  list of ring ele
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│ │ │ │ -0000b6e0: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
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│ │ │ │ -0000b730: 0a20 2020 2020 202a 2042 6572 7469 6e69  .      * Bertini
│ │ │ │ -0000b740: 496e 7075 7443 6f6e 6669 6775 7261 7469  InputConfigurati
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│ │ │ │ -0000b790: 2020 202a 202a 6e6f 7465 2049 7350 726f     * *note IsPro
│ │ │ │ -0000b7a0: 6a65 6374 6976 653a 2049 7350 726f 6a65  jective: IsProje
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│ │ │ │ -0000b7d0: 206f 7074 696f 6e61 6c0a 2020 2020 2020   optional.      
│ │ │ │ -0000b7e0: 2020 6172 6775 6d65 6e74 2074 6f20 7370    argument to sp
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│ │ │ │ -0000b810: 2063 6f6f 7264 696e 6174 6573 0a20 2020   coordinates.   
│ │ │ │ -0000b820: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
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│ │ │ │ -0000b840: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70  Homotopy_lp_pd_p
│ │ │ │ -0000b850: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ -0000b860: 5f70 645f 7064 5f70 645f 7270 0a20 2020  _pd_pd_pd_rp.   
│ │ │ │ -0000b870: 2020 2020 202c 203d 3e20 2e2e 2e2c 2064       , => ..., d
│ │ │ │ -0000b880: 6566 6175 6c74 2076 616c 7565 2066 616c  efault value fal
│ │ │ │ -0000b890: 7365 2c20 4f70 7469 6f6e 2074 6f20 7369  se, Option to si
│ │ │ │ -0000b8a0: 6c65 6e63 6520 6164 6469 7469 6f6e 616c  lence additional
│ │ │ │ -0000b8b0: 206f 7574 7075 740a 2020 2a20 4f75 7470   output.  * Outp
│ │ │ │ -0000b8c0: 7574 733a 0a20 2020 2020 202a 2056 2c20  uts:.      * V, 
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│ │ │ │ -0000b8e0: 6c20 7661 7269 6574 793a 2028 4e41 4774  l variety: (NAGt
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│ │ │ │ -0000b920: 6475 6369 626c 6520 6465 636f 6d70 6f73  ducible decompos
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│ │ │ │ -0000b990: 2042 6572 7469 6e69 2074 6f20 6669 6e64   Bertini to find
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│ │ │ │ -0000b9f0: 6973 2072 6574 7572 6e65 6420 6173 2074  is returned as t
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│ │ │ │ -0000bb10: 2066 696e 6420 7769 746e 6573 7320 7375   find witness su
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│ │ │ │ -0000bb30: 6469 6d65 6e73 696f 6e2c 2028 3429 0a72  dimension, (4).r
│ │ │ │ -0000bb40: 656d 6f76 6573 2065 7874 7261 2070 6f69  emoves extra poi
│ │ │ │ -0000bb50: 6e74 7320 7573 696e 6720 6120 6d65 6d62  nts using a memb
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│ │ │ │ -0000bb80: 6573 742c 2028 3529 0a64 6566 6c61 7465  est, (5).deflate
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│ │ │ │ -0000bbd0: 6d62 696e 6174 696f 6e20 6f66 206d 6f6e  mbination of mon
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│ │ │ │ -0000bbf0: 6561 7220 7472 6163 6520 7465 7374 0a0a  ear trace test..
│ │ │ │ -0000bc00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000b610: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +0000b620: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +0000b630: 3a20 0a20 2020 2020 2020 2056 203d 2062  : .        V = b
│ │ │ │ +0000b640: 6572 7469 6e69 506f 7344 696d 536f 6c76  ertiniPosDimSolv
│ │ │ │ +0000b650: 6520 490a 2020 2020 2020 2020 5620 3d20  e I.        V = 
│ │ │ │ +0000b660: 6265 7274 696e 6950 6f73 4469 6d53 6f6c  bertiniPosDimSol
│ │ │ │ +0000b670: 7665 2046 0a20 202a 2049 6e70 7574 733a  ve F.  * Inputs:
│ │ │ │ +0000b680: 0a20 2020 2020 202a 2046 2c20 6120 2a6e  .      * F, a *n
│ │ │ │ +0000b690: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +0000b6a0: 6c61 7932 446f 6329 4c69 7374 2c2c 2061  lay2Doc)List,, a
│ │ │ │ +0000b6b0: 206c 6973 7420 6f66 2072 696e 6720 656c   list of ring el
│ │ │ │ +0000b6c0: 656d 656e 7473 2064 6566 696e 696e 670a  ements defining.
│ │ │ │ +0000b6d0: 2020 2020 2020 2020 6120 7661 7269 6574          a variet
│ │ │ │ +0000b6e0: 790a 2020 2a20 2a6e 6f74 6520 4f70 7469  y.  * *note Opti
│ │ │ │ +0000b6f0: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ +0000b700: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ +0000b710: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ +0000b720: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ +0000b730: 3a0a 2020 2020 2020 2a20 4265 7274 696e  :.      * Bertin
│ │ │ │ +0000b740: 6949 6e70 7574 436f 6e66 6967 7572 6174  iInputConfigurat
│ │ │ │ +0000b750: 696f 6e20 286d 6973 7369 6e67 2064 6f63  ion (missing doc
│ │ │ │ +0000b760: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ +0000b770: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +0000b780: 650a 2020 2020 2020 2020 7b7d 2c0a 2020  e.        {},.  
│ │ │ │ +0000b790: 2020 2020 2a20 2a6e 6f74 6520 4973 5072      * *note IsPr
│ │ │ │ +0000b7a0: 6f6a 6563 7469 7665 3a20 4973 5072 6f6a  ojective: IsProj
│ │ │ │ +0000b7b0: 6563 7469 7665 2c20 3d3e 202e 2e2e 2c20  ective, => ..., 
│ │ │ │ +0000b7c0: 6465 6661 756c 7420 7661 6c75 6520 2d31  default value -1
│ │ │ │ +0000b7d0: 2c20 6f70 7469 6f6e 616c 0a20 2020 2020  , optional.     
│ │ │ │ +0000b7e0: 2020 2061 7267 756d 656e 7420 746f 2073     argument to s
│ │ │ │ +0000b7f0: 7065 6369 6679 2077 6865 7468 6572 2074  pecify whether t
│ │ │ │ +0000b800: 6f20 7573 6520 686f 6d6f 6765 6e65 6f75  o use homogeneou
│ │ │ │ +0000b810: 7320 636f 6f72 6469 6e61 7465 730a 2020  s coordinates.  
│ │ │ │ +0000b820: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │ +0000b830: 6f73 653a 2062 6572 7469 6e69 5472 6163  ose: bertiniTrac
│ │ │ │ +0000b840: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │ +0000b850: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ +0000b860: 3e5f 7064 5f70 645f 7064 5f72 700a 2020  >_pd_pd_pd_rp.  
│ │ │ │ +0000b870: 2020 2020 2020 2c20 3d3e 202e 2e2e 2c20        , => ..., 
│ │ │ │ +0000b880: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ +0000b890: 6c73 652c 204f 7074 696f 6e20 746f 2073  lse, Option to s
│ │ │ │ +0000b8a0: 696c 656e 6365 2061 6464 6974 696f 6e61  ilence additiona
│ │ │ │ +0000b8b0: 6c20 6f75 7470 7574 0a20 202a 204f 7574  l output.  * Out
│ │ │ │ +0000b8c0: 7075 7473 3a0a 2020 2020 2020 2a20 562c  puts:.      * V,
│ │ │ │ +0000b8d0: 2061 202a 6e6f 7465 206e 756d 6572 6963   a *note numeric
│ │ │ │ +0000b8e0: 616c 2076 6172 6965 7479 3a20 284e 4147  al variety: (NAG
│ │ │ │ +0000b8f0: 7479 7065 7329 4e75 6d65 7269 6361 6c56  types)NumericalV
│ │ │ │ +0000b900: 6172 6965 7479 2c2c 2061 206e 756d 6572  ariety,, a numer
│ │ │ │ +0000b910: 6963 616c 0a20 2020 2020 2020 2069 7272  ical.        irr
│ │ │ │ +0000b920: 6564 7563 6962 6c65 2064 6563 6f6d 706f  educible decompo
│ │ │ │ +0000b930: 7369 7469 6f6e 206f 6620 7468 6520 7661  sition of the va
│ │ │ │ +0000b940: 7269 6574 7920 6465 6669 6e65 6420 6279  riety defined by
│ │ │ │ +0000b950: 2046 0a0a 4465 7363 7269 7074 696f 6e0a   F..Description.
│ │ │ │ +0000b960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0000b970: 206d 6574 686f 6420 6265 7274 696e 6950   method bertiniP
│ │ │ │ +0000b980: 6f73 4469 6d53 6f6c 7665 2063 616c 6c73  osDimSolve calls
│ │ │ │ +0000b990: 2020 4265 7274 696e 6920 746f 2066 696e    Bertini to fin
│ │ │ │ +0000b9a0: 6420 6120 6e75 6d65 7269 6361 6c20 6972  d a numerical ir
│ │ │ │ +0000b9b0: 7265 6475 6369 626c 650a 6465 636f 6d70  reducible.decomp
│ │ │ │ +0000b9c0: 6f73 6974 696f 6e20 6f66 2074 6865 207a  osition of the z
│ │ │ │ +0000b9d0: 6572 6f2d 7365 7420 6f66 2046 2e20 2054  ero-set of F.  T
│ │ │ │ +0000b9e0: 6865 2064 6563 6f6d 706f 7369 7469 6f6e  he decomposition
│ │ │ │ +0000b9f0: 2069 7320 7265 7475 726e 6564 2061 7320   is returned as 
│ │ │ │ +0000ba00: 7468 6520 2a6e 6f74 650a 4e75 6d65 7269  the *note.Numeri
│ │ │ │ +0000ba10: 6361 6c56 6172 6965 7479 3a20 284e 4147  calVariety: (NAG
│ │ │ │ +0000ba20: 7479 7065 7329 4e75 6d65 7269 6361 6c56  types)NumericalV
│ │ │ │ +0000ba30: 6172 6965 7479 2c20 4e56 2e20 2057 6974  ariety, NV.  Wit
│ │ │ │ +0000ba40: 6e65 7373 2073 6574 7320 6f66 204e 5620  ness sets of NV 
│ │ │ │ +0000ba50: 636f 6e74 6169 6e0a 6170 7072 6f78 696d  contain.approxim
│ │ │ │ +0000ba60: 6174 696f 6e73 2074 6f20 736f 6c75 7469  ations to soluti
│ │ │ │ +0000ba70: 6f6e 7320 6f66 2074 6865 2073 7973 7465  ons of the syste
│ │ │ │ +0000ba80: 6d20 463d 302e 2042 6572 7469 6e69 2028  m F=0. Bertini (
│ │ │ │ +0000ba90: 3129 2077 7269 7465 7320 7468 6520 7379  1) writes the sy
│ │ │ │ +0000baa0: 7374 656d 2074 6f0a 7465 6d70 6f72 6172  stem to.temporar
│ │ │ │ +0000bab0: 7920 6669 6c65 732c 2028 3229 2069 6e76  y files, (2) inv
│ │ │ │ +0000bac0: 6f6b 6573 2042 6572 7469 6e69 2773 2073  okes Bertini's s
│ │ │ │ +0000bad0: 6f6c 7665 7220 7769 7468 2054 7261 636b  olver with Track
│ │ │ │ +0000bae0: 5479 7065 203d 3e20 312c 2028 3329 2042  Type => 1, (3) B
│ │ │ │ +0000baf0: 6572 7469 6e69 0a75 7365 7320 6120 6361  ertini.uses a ca
│ │ │ │ +0000bb00: 7363 6164 6520 686f 6d6f 746f 7079 2074  scade homotopy t
│ │ │ │ +0000bb10: 6f20 6669 6e64 2077 6974 6e65 7373 2073  o find witness s
│ │ │ │ +0000bb20: 7570 6572 7365 7473 2069 6e20 6561 6368  upersets in each
│ │ │ │ +0000bb30: 2064 696d 656e 7369 6f6e 2c20 2834 290a   dimension, (4).
│ │ │ │ +0000bb40: 7265 6d6f 7665 7320 6578 7472 6120 706f  removes extra po
│ │ │ │ +0000bb50: 696e 7473 2075 7369 6e67 2061 206d 656d  ints using a mem
│ │ │ │ +0000bb60: 6265 7273 6869 7020 7465 7374 206f 7220  bership test or 
│ │ │ │ +0000bb70: 6c6f 6361 6c20 6469 6d65 6e73 696f 6e20  local dimension 
│ │ │ │ +0000bb80: 7465 7374 2c20 2835 290a 6465 666c 6174  test, (5).deflat
│ │ │ │ +0000bb90: 6573 2073 696e 6775 6c61 7220 7769 746e  es singular witn
│ │ │ │ +0000bba0: 6573 7320 706f 696e 7473 2c20 616e 6420  ess points, and 
│ │ │ │ +0000bbb0: 6669 6e61 6c6c 7920 2836 2920 6465 636f  finally (6) deco
│ │ │ │ +0000bbc0: 6d70 6f73 6573 2075 7369 6e67 2061 0a63  mposes using a.c
│ │ │ │ +0000bbd0: 6f6d 6269 6e61 7469 6f6e 206f 6620 6d6f  ombination of mo
│ │ │ │ +0000bbe0: 6e6f 6472 6f6d 7920 616e 6420 6120 6c69  nodromy and a li
│ │ │ │ +0000bbf0: 6e65 6172 2074 7261 6365 2074 6573 740a  near trace test.
│ │ │ │ +0000bc00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000bc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000bc50: 7c69 3120 3a20 5220 3d20 5151 5b78 2c79  |i1 : R = QQ[x,y
│ │ │ │ -0000bc60: 2c7a 5d20 2020 2020 2020 2020 2020 2020  ,z]             
│ │ │ │ +0000bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000bc50: 0a7c 6931 203a 2052 203d 2051 515b 782c  .|i1 : R = QQ[x,
│ │ │ │ +0000bc60: 792c 7a5d 2020 2020 2020 2020 2020 2020  y,z]            
│ │ │ │  0000bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bc90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000bc90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bca0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bce0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bcf0: 7c6f 3120 3d20 5220 2020 2020 2020 2020  |o1 = R         
│ │ │ │ +0000bce0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bcf0: 0a7c 6f31 203d 2052 2020 2020 2020 2020  .|o1 = R        
│ │ │ │  0000bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bd30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bd40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000bd30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bd40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bd80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bd90: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ -0000bda0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0000bd80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bd90: 0a7c 6f31 203a 2050 6f6c 796e 6f6d 6961  .|o1 : Polynomia
│ │ │ │ +0000bda0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │  0000bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bdd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bde0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000bdd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bde0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000bdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000be30: 7c69 3220 3a20 4620 3d20 7b28 795e 322b  |i2 : F = {(y^2+
│ │ │ │ -0000be40: 785e 322b 7a5e 322d 3129 2a78 2c28 795e  x^2+z^2-1)*x,(y^
│ │ │ │ -0000be50: 322b 785e 322b 7a5e 322d 3129 2a79 7d20  2+x^2+z^2-1)*y} 
│ │ │ │ +0000be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000be30: 0a7c 6932 203a 2046 203d 207b 2879 5e32  .|i2 : F = {(y^2
│ │ │ │ +0000be40: 2b78 5e32 2b7a 5e32 2d31 292a 782c 2879  +x^2+z^2-1)*x,(y
│ │ │ │ +0000be50: 5e32 2b78 5e32 2b7a 5e32 2d31 292a 797d  ^2+x^2+z^2-1)*y}
│ │ │ │  0000be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000be70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000be80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000be70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000be80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bed0: 7c20 2020 2020 2020 3320 2020 2020 2032  |       3      2
│ │ │ │ -0000bee0: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ -0000bef0: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ +0000bec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bed0: 0a7c 2020 2020 2020 2033 2020 2020 2020  .|       3      
│ │ │ │ +0000bee0: 3220 2020 2020 2032 2020 2020 2020 2032  2      2       2
│ │ │ │ +0000bef0: 2020 2020 2033 2020 2020 2020 3220 2020       3      2   
│ │ │ │  0000bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bf10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bf20: 7c6f 3220 3d20 7b78 2020 2b20 782a 7920  |o2 = {x  + x*y 
│ │ │ │ -0000bf30: 202b 2078 2a7a 2020 2d20 782c 2078 2079   + x*z  - x, x y
│ │ │ │ -0000bf40: 202b 2079 2020 2b20 792a 7a20 202d 2079   + y  + y*z  - y
│ │ │ │ -0000bf50: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -0000bf60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bf70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000bf10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bf20: 0a7c 6f32 203d 207b 7820 202b 2078 2a79  .|o2 = {x  + x*y
│ │ │ │ +0000bf30: 2020 2b20 782a 7a20 202d 2078 2c20 7820    + x*z  - x, x 
│ │ │ │ +0000bf40: 7920 2b20 7920 202b 2079 2a7a 2020 2d20  y + y  + y*z  - 
│ │ │ │ +0000bf50: 797d 2020 2020 2020 2020 2020 2020 2020  y}              
│ │ │ │ +0000bf60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bf70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bfb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000bfc0: 7c6f 3220 3a20 4c69 7374 2020 2020 2020  |o2 : List      
│ │ │ │ +0000bfb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000bfc0: 0a7c 6f32 203a 204c 6973 7420 2020 2020  .|o2 : List     
│ │ │ │  0000bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c000: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c010: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000c000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c010: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000c060: 7c69 3320 3a20 5320 3d20 6265 7274 696e  |i3 : S = bertin
│ │ │ │ -0000c070: 6950 6f73 4469 6d53 6f6c 7665 2046 2020  iPosDimSolve F  
│ │ │ │ +0000c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000c060: 0a7c 6933 203a 2053 203d 2062 6572 7469  .|i3 : S = berti
│ │ │ │ +0000c070: 6e69 506f 7344 696d 536f 6c76 6520 4620  niPosDimSolve F 
│ │ │ │  0000c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c0b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c100: 7c6f 3320 3d20 5320 2020 2020 2020 2020  |o3 = S         
│ │ │ │ +0000c0f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c100: 0a7c 6f33 203d 2053 2020 2020 2020 2020  .|o3 = S        
│ │ │ │  0000c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c150: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c150: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c1a0: 7c6f 3320 3a20 4e75 6d65 7269 6361 6c56  |o3 : NumericalV
│ │ │ │ -0000c1b0: 6172 6965 7479 2020 2020 2020 2020 2020  ariety          
│ │ │ │ +0000c190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c1a0: 0a7c 6f33 203a 204e 756d 6572 6963 616c  .|o3 : Numerical
│ │ │ │ +0000c1b0: 5661 7269 6574 7920 2020 2020 2020 2020  Variety         
│ │ │ │  0000c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c1f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000c1e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c1f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000c240: 7c69 3420 3a20 5323 315f 3023 506f 696e  |i4 : S#1_0#Poin
│ │ │ │ -0000c250: 7473 202d 2d20 315f 3020 6368 6f6f 7365  ts -- 1_0 choose
│ │ │ │ -0000c260: 7320 7468 6520 6669 7273 7420 7769 746e  s the first witn
│ │ │ │ -0000c270: 6573 7320 7365 7420 696e 2064 696d 656e  ess set in dimen
│ │ │ │ -0000c280: 7369 6f6e 2031 2020 2020 2020 2020 7c0a  sion 1        |.
│ │ │ │ -0000c290: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000c240: 0a7c 6934 203a 2053 2331 5f30 2350 6f69  .|i4 : S#1_0#Poi
│ │ │ │ +0000c250: 6e74 7320 2d2d 2031 5f30 2063 686f 6f73  nts -- 1_0 choos
│ │ │ │ +0000c260: 6573 2074 6865 2066 6972 7374 2077 6974  es the first wit
│ │ │ │ +0000c270: 6e65 7373 2073 6574 2069 6e20 6469 6d65  ness set in dime
│ │ │ │ +0000c280: 6e73 696f 6e20 3120 2020 2020 2020 207c  nsion 1        |
│ │ │ │ +0000c290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c2e0: 7c6f 3420 3d20 7b7b 322e 3634 3436 3865  |o4 = {{2.64468e
│ │ │ │ -0000c2f0: 2d35 392b 312e 3833 3934 3965 2d35 392a  -59+1.83949e-59*
│ │ │ │ -0000c300: 6969 2c20 2d31 2e30 3837 3765 2d36 302b  ii, -1.0877e-60+
│ │ │ │ -0000c310: 332e 3337 3538 3365 2d35 392a 6969 2c20  3.37583e-59*ii, 
│ │ │ │ -0000c320: 2e32 3631 3234 3620 2020 2020 2020 7c0a  .261246       |.
│ │ │ │ -0000c330: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c2d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c2e0: 0a7c 6f34 203d 207b 7b32 2e36 3434 3638  .|o4 = {{2.64468
│ │ │ │ +0000c2f0: 652d 3539 2b31 2e38 3339 3439 652d 3539  e-59+1.83949e-59
│ │ │ │ +0000c300: 2a69 692c 202d 312e 3038 3737 652d 3630  *ii, -1.0877e-60
│ │ │ │ +0000c310: 2b33 2e33 3735 3833 652d 3539 2a69 692c  +3.37583e-59*ii,
│ │ │ │ +0000c320: 202e 3236 3132 3436 2020 2020 2020 207c   .261246       |
│ │ │ │ +0000c330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c380: 7c6f 3420 3a20 5665 7274 6963 616c 4c69  |o4 : VerticalLi
│ │ │ │ -0000c390: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0000c370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c380: 0a7c 6f34 203a 2056 6572 7469 6361 6c4c  .|o4 : VerticalL
│ │ │ │ +0000c390: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0000c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c3d0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0000c3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c3d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0000c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0000c420: 7c2b 2e31 3436 3031 382a 6969 7d7d 2020  |+.146018*ii}}  
│ │ │ │ +0000c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0000c420: 0a7c 2b2e 3134 3630 3138 2a69 697d 7d20  .|+.146018*ii}} 
│ │ │ │  0000c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c470: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000c460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c470: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000c4c0: 0a45 6163 6820 2a6e 6f74 6520 5769 746e  .Each *note Witn
│ │ │ │ -0000c4d0: 6573 7353 6574 3a20 284e 4147 7479 7065  essSet: (NAGtype
│ │ │ │ -0000c4e0: 7329 5769 746e 6573 7353 6574 2c20 6973  s)WitnessSet, is
│ │ │ │ -0000c4f0: 2061 6363 6573 7365 6420 6279 2064 696d   accessed by dim
│ │ │ │ -0000c500: 656e 7369 6f6e 2061 6e64 2074 6865 6e0a  ension and then.
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│ │ │ │ -0000c520: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000c4c0: 0a0a 4561 6368 202a 6e6f 7465 2057 6974  ..Each *note Wit
│ │ │ │ +0000c4d0: 6e65 7373 5365 743a 2028 4e41 4774 7970  nessSet: (NAGtyp
│ │ │ │ +0000c4e0: 6573 2957 6974 6e65 7373 5365 742c 2069  es)WitnessSet, i
│ │ │ │ +0000c4f0: 7320 6163 6365 7373 6564 2062 7920 6469  s accessed by di
│ │ │ │ +0000c500: 6d65 6e73 696f 6e20 616e 6420 7468 656e  mension and then
│ │ │ │ +0000c510: 0a6c 6973 7420 706f 7369 7469 6f6e 2e0a  .list position..
│ │ │ │ +0000c520: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000c570: 7c69 3520 3a20 5323 3120 2d2d 6669 7273  |i5 : S#1 --firs
│ │ │ │ -0000c580: 7420 7370 6563 6966 7920 6469 6d65 6e73  t specify dimens
│ │ │ │ -0000c590: 696f 6e20 2020 2020 2020 2020 2020 2020  ion             
│ │ │ │ +0000c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000c570: 0a7c 6935 203a 2053 2331 202d 2d66 6972  .|i5 : S#1 --fir
│ │ │ │ +0000c580: 7374 2073 7065 6369 6679 2064 696d 656e  st specify dimen
│ │ │ │ +0000c590: 7369 6f6e 2020 2020 2020 2020 2020 2020  sion            
│ │ │ │  0000c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c5c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c5b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c5c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c610: 7c6f 3520 3d20 7b28 6469 6d3d 312c 6465  |o5 = {(dim=1,de
│ │ │ │ -0000c620: 673d 3129 7d20 2020 2020 2020 2020 2020  g=1)}           
│ │ │ │ +0000c600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c610: 0a7c 6f35 203d 207b 2864 696d 3d31 2c64  .|o5 = {(dim=1,d
│ │ │ │ +0000c620: 6567 3d31 297d 2020 2020 2020 2020 2020  eg=1)}          
│ │ │ │  0000c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c660: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c660: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c6a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c6b0: 7c6f 3520 3a20 4c69 7374 2020 2020 2020  |o5 : List      
│ │ │ │ +0000c6a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c6b0: 0a7c 6f35 203a 204c 6973 7420 2020 2020  .|o5 : List     
│ │ │ │  0000c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c700: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ +0000c700: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │  0000c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000c750: 7c69 3620 3a20 7065 656b 206f 6f5f 3020  |i6 : peek oo_0 
│ │ │ │ -0000c760: 2d2d 7468 656e 206c 6973 7420 706f 7369  --then list posi
│ │ │ │ -0000c770: 7469 6f6e 2020 2020 2020 2020 2020 2020  tion            
│ │ │ │ +0000c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000c750: 0a7c 6936 203a 2070 6565 6b20 6f6f 5f30  .|i6 : peek oo_0
│ │ │ │ +0000c760: 202d 2d74 6865 6e20 6c69 7374 2070 6f73   --then list pos
│ │ │ │ +0000c770: 6974 696f 6e20 2020 2020 2020 2020 2020  ition           
│ │ │ │  0000c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c7a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c7a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c7e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c7f0: 7c6f 3620 3d20 5769 746e 6573 7353 6574  |o6 = WitnessSet
│ │ │ │ -0000c800: 7b63 6163 6865 203d 3e20 4361 6368 6554  {cache => CacheT
│ │ │ │ -0000c810: 6162 6c65 7b2e 2e2e 332e 2e2e 7d20 2020  able{...3...}   
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│ │ │ │ +0000c7f0: 0a7c 6f36 203d 2057 6974 6e65 7373 5365  .|o6 = WitnessSe
│ │ │ │ +0000c800: 747b 6361 6368 6520 3d3e 2043 6163 6865  t{cache => Cache
│ │ │ │ +0000c810: 5461 626c 657b 2e2e 2e33 2e2e 2e7d 2020  Table{...3...}  
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│ │ │ │ -0000c840: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0000c850: 2045 7175 6174 696f 6e73 203d 3e20 7b2d   Equations => {-
│ │ │ │ -0000c860: 337d 207c 2078 332b 7879 322b 787a 322d  3} | x3+xy2+xz2-
│ │ │ │ -0000c870: 7820 7c20 2020 2020 2020 2020 2020 2020  x |             
│ │ │ │ -0000c880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0000c8a0: 2020 2020 2020 2020 2020 2020 2020 7b2d                {-
│ │ │ │ -0000c8b0: 337d 207c 2078 3279 2b79 332b 797a 322d  3} | x2y+y3+yz2-
│ │ │ │ -0000c8c0: 7920 7c20 2020 2020 2020 2020 2020 2020  y |             
│ │ │ │ -0000c8d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c8e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0000c8f0: 2050 6f69 6e74 7320 3d3e 207b 7b32 2e36   Points => {{2.6
│ │ │ │ -0000c900: 3434 3638 652d 3539 2b31 2e38 3339 3439  4468e-59+1.83949
│ │ │ │ -0000c910: 652d 3539 2020 2020 2020 2020 2020 2020  e-59            
│ │ │ │ -0000c920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0000c940: 2053 6c69 6365 203d 3e20 7c20 2e30 3733   Slice => | .073
│ │ │ │ -0000c950: 3838 332b 312e 3531 3332 3969 6920 312e  883+1.51329ii 1.
│ │ │ │ -0000c960: 3338 3336 2020 2020 2020 2020 2020 2020  3836            
│ │ │ │ -0000c970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000c980: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0000c830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c840: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000c850: 2020 4571 7561 7469 6f6e 7320 3d3e 207b    Equations => {
│ │ │ │ +0000c860: 2d33 7d20 7c20 7833 2b78 7932 2b78 7a32  -3} | x3+xy2+xz2
│ │ │ │ +0000c870: 2d78 207c 2020 2020 2020 2020 2020 2020  -x |            
│ │ │ │ +0000c880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000c8a0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0000c8b0: 2d33 7d20 7c20 7832 792b 7933 2b79 7a32  -3} | x2y+y3+yz2
│ │ │ │ +0000c8c0: 2d79 207c 2020 2020 2020 2020 2020 2020  -y |            
│ │ │ │ +0000c8d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c8e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000c8f0: 2020 506f 696e 7473 203d 3e20 7b7b 322e    Points => {{2.
│ │ │ │ +0000c900: 3634 3436 3865 2d35 392b 312e 3833 3934  64468e-59+1.8394
│ │ │ │ +0000c910: 3965 2d35 3920 2020 2020 2020 2020 2020  9e-59           
│ │ │ │ +0000c920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000c940: 2020 536c 6963 6520 3d3e 207c 202e 3037    Slice => | .07
│ │ │ │ +0000c950: 3338 3833 2b31 2e35 3133 3239 6969 2031  3883+1.51329ii 1
│ │ │ │ +0000c960: 2e33 3833 3620 2020 2020 2020 2020 2020  .3836           
│ │ │ │ +0000c970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c980: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0000c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0000c9d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0000c9d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ca00: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ -0000ca10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000ca20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000ca00: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +0000ca10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000ca20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ca60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000ca70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000ca60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000ca70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000cac0: 7c2a 6969 2c20 2d31 2e30 3837 3765 2d36  |*ii, -1.0877e-6
│ │ │ │ -0000cad0: 302b 332e 3337 3538 3365 2d35 392a 6969  0+3.37583e-59*ii
│ │ │ │ -0000cae0: 2c20 2e32 3631 3234 362b 2e31 3436 3031  , .261246+.14601
│ │ │ │ -0000caf0: 382a 6969 7d7d 2020 2020 2020 2020 2020  8*ii}}          
│ │ │ │ -0000cb00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000cb10: 7c2b 2e31 3836 3538 3869 6920 2d32 2e30  |+.186588ii -2.0
│ │ │ │ -0000cb20: 3231 3933 2b2e 3735 3736 3736 6969 202e  2193+.757676ii .
│ │ │ │ -0000cb30: 3633 3838 3535 2b2e 3039 3732 3939 3169  638855+.0972991i
│ │ │ │ -0000cb40: 6920 7c20 2020 2020 2020 2020 2020 2020  i |             
│ │ │ │ -0000cb50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000cb60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0000cab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000cac0: 0a7c 2a69 692c 202d 312e 3038 3737 652d  .|*ii, -1.0877e-
│ │ │ │ +0000cad0: 3630 2b33 2e33 3735 3833 652d 3539 2a69  60+3.37583e-59*i
│ │ │ │ +0000cae0: 692c 202e 3236 3132 3436 2b2e 3134 3630  i, .261246+.1460
│ │ │ │ +0000caf0: 3138 2a69 697d 7d20 2020 2020 2020 2020  18*ii}}         
│ │ │ │ +0000cb00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000cb10: 0a7c 2b2e 3138 3635 3838 6969 202d 322e  .|+.186588ii -2.
│ │ │ │ +0000cb20: 3032 3139 332b 2e37 3537 3637 3669 6920  02193+.757676ii 
│ │ │ │ +0000cb30: 2e36 3338 3835 352b 2e30 3937 3239 3931  .638855+.0972991
│ │ │ │ +0000cb40: 6969 207c 2020 2020 2020 2020 2020 2020  ii |            
│ │ │ │ +0000cb50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000cb60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0000cbb0: 0a49 6e20 7468 6520 6578 616d 706c 652c  .In the example,
│ │ │ │ -0000cbc0: 2077 6520 6669 6e64 2074 776f 2063 6f6d   we find two com
│ │ │ │ -0000cbd0: 706f 6e65 6e74 732c 206f 6e65 2063 6f6d  ponents, one com
│ │ │ │ -0000cbe0: 706f 6e65 6e74 2068 6173 2064 696d 656e  ponent has dimen
│ │ │ │ -0000cbf0: 7369 6f6e 2031 2061 6e64 0a64 6567 7265  sion 1 and.degre
│ │ │ │ -0000cc00: 6520 3120 616e 6420 7468 6520 6f74 6865  e 1 and the othe
│ │ │ │ -0000cc10: 7220 6861 7320 6469 6d65 6e73 696f 6e20  r has dimension 
│ │ │ │ -0000cc20: 3220 616e 6420 6465 6772 6565 2032 2e20  2 and degree 2. 
│ │ │ │ -0000cc30: 2057 6520 6765 7420 7468 6520 7361 6d65   We get the same
│ │ │ │ -0000cc40: 2072 6573 756c 7473 0a75 7369 6e67 2073   results.using s
│ │ │ │ -0000cc50: 796d 626f 6c69 6320 6d65 7468 6f64 732e  ymbolic methods.
│ │ │ │ -0000cc60: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0000cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0000cbb0: 0a0a 496e 2074 6865 2065 7861 6d70 6c65  ..In the example
│ │ │ │ +0000cbc0: 2c20 7765 2066 696e 6420 7477 6f20 636f  , we find two co
│ │ │ │ +0000cbd0: 6d70 6f6e 656e 7473 2c20 6f6e 6520 636f  mponents, one co
│ │ │ │ +0000cbe0: 6d70 6f6e 656e 7420 6861 7320 6469 6d65  mponent has dime
│ │ │ │ +0000cbf0: 6e73 696f 6e20 3120 616e 640a 6465 6772  nsion 1 and.degr
│ │ │ │ +0000cc00: 6565 2031 2061 6e64 2074 6865 206f 7468  ee 1 and the oth
│ │ │ │ +0000cc10: 6572 2068 6173 2064 696d 656e 7369 6f6e  er has dimension
│ │ │ │ +0000cc20: 2032 2061 6e64 2064 6567 7265 6520 322e   2 and degree 2.
│ │ │ │ +0000cc30: 2020 5765 2067 6574 2074 6865 2073 616d    We get the sam
│ │ │ │ +0000cc40: 6520 7265 7375 6c74 730a 7573 696e 6720  e results.using 
│ │ │ │ +0000cc50: 7379 6d62 6f6c 6963 206d 6574 686f 6473  symbolic methods
│ │ │ │ +0000cc60: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  0000cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000cc90: 0a7c 6937 203a 2050 443d 7072 696d 6172  .|i7 : PD=primar
│ │ │ │ -0000cca0: 7944 6563 6f6d 706f 7369 7469 6f6e 2820  yDecomposition( 
│ │ │ │ -0000ccb0: 6964 6561 6c20 4629 2020 2020 2020 7c0a  ideal F)      |.
│ │ │ │ -0000ccc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0000cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000cc90: 2b0a 7c69 3720 3a20 5044 3d70 7269 6d61  +.|i7 : PD=prima
│ │ │ │ +0000cca0: 7279 4465 636f 6d70 6f73 6974 696f 6e28  ryDecomposition(
│ │ │ │ +0000ccb0: 2069 6465 616c 2046 2920 2020 2020 207c   ideal F)      |
│ │ │ │ +0000ccc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cce0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0000ccf0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0000cd00: 2020 3220 2020 2032 2020 2020 2020 2020    2    2        
│ │ │ │ -0000cd10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0000cd20: 3720 3d20 7b69 6465 616c 2878 2020 2b20  7 = {ideal(x  + 
│ │ │ │ -0000cd30: 7920 202b 207a 2020 2d20 3129 2c20 6964  y  + z  - 1), id
│ │ │ │ -0000cd40: 6561 6c20 2879 2c20 7829 7d7c 0a7c 2020  eal (y, x)}|.|  
│ │ │ │ +0000cce0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0000ccf0: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +0000cd00: 2020 2032 2020 2020 3220 2020 2020 2020     2    2       
│ │ │ │ +0000cd10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0000cd20: 6f37 203d 207b 6964 6561 6c28 7820 202b  o7 = {ideal(x  +
│ │ │ │ +0000cd30: 2079 2020 2b20 7a20 202d 2031 292c 2069   y  + z  - 1), i
│ │ │ │ +0000cd40: 6465 616c 2028 792c 2078 297d 7c0a 7c20  deal (y, x)}|.| 
│ │ │ │  0000cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cd70: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0000cd80: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +0000cd70: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +0000cd80: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  0000cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cda0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000cda0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0000cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000cdd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -0000cde0: 6469 6d20 5044 5f30 2020 2020 2020 2020  dim PD_0        
│ │ │ │ +0000cdd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +0000cde0: 2064 696d 2050 445f 3020 2020 2020 2020   dim PD_0       
│ │ │ │  0000cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ce00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0000ce00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0000ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ce30: 2020 2020 2020 7c0a 7c6f 3820 3d20 3220        |.|o8 = 2 
│ │ │ │ +0000ce30: 2020 2020 2020 207c 0a7c 6f38 203d 2032         |.|o8 = 2
│ │ │ │  0000ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ce60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0000ce60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0000ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ce90: 2d2d 2d2d 2b0a 7c69 3920 3a20 6465 6772  ----+.|i9 : degr
│ │ │ │ -0000cea0: 6565 2050 445f 3020 2020 2020 2020 2020  ee PD_0         
│ │ │ │ +0000ce90: 2d2d 2d2d 2d2b 0a7c 6939 203a 2064 6567  -----+.|i9 : deg
│ │ │ │ +0000cea0: 7265 6520 5044 5f30 2020 2020 2020 2020  ree PD_0        
│ │ │ │  0000ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cec0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0000cec0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0000ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cef0: 2020 7c0a 7c6f 3920 3d20 3220 2020 2020    |.|o9 = 2     
│ │ │ │ +0000cef0: 2020 207c 0a7c 6f39 203d 2032 2020 2020     |.|o9 = 2    
│ │ │ │  0000cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cf20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0000cf20: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0000cf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000cf50: 2b0a 7c69 3130 203a 2064 696d 2050 445f  +.|i10 : dim PD_
│ │ │ │ -0000cf60: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -0000cf70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000cf80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000cf50: 2d2b 0a7c 6931 3020 3a20 6469 6d20 5044  -+.|i10 : dim PD
│ │ │ │ +0000cf60: 5f31 2020 2020 2020 2020 2020 2020 2020  _1              
│ │ │ │ +0000cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000cf80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cfa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000cfb0: 7c6f 3130 203d 2031 2020 2020 2020 2020  |o10 = 1        
│ │ │ │ +0000cfa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000cfb0: 0a7c 6f31 3020 3d20 3120 2020 2020 2020  .|o10 = 1       
│ │ │ │  0000cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000cfd0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0000cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000cfd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0000cfe0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0000cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0000d010: 3131 203a 2064 6567 7265 6520 5044 5f31  11 : degree PD_1
│ │ │ │ -0000d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d030: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0000d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0000d010: 6931 3120 3a20 6465 6772 6565 2050 445f  i11 : degree PD_
│ │ │ │ +0000d020: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0000d030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0000d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d060: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0000d070: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +0000d060: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0000d070: 3120 3d20 3120 2020 2020 2020 2020 2020  1 = 1           
│ │ │ │  0000d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d090: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0000d090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0000d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d0c0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320  --------+..Ways 
│ │ │ │ -0000d0d0: 746f 2075 7365 2062 6572 7469 6e69 506f  to use bertiniPo
│ │ │ │ -0000d0e0: 7344 696d 536f 6c76 653a 0a3d 3d3d 3d3d  sDimSolve:.=====
│ │ │ │ +0000d0c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973  ---------+..Ways
│ │ │ │ +0000d0d0: 2074 6f20 7573 6520 6265 7274 696e 6950   to use bertiniP
│ │ │ │ +0000d0e0: 6f73 4469 6d53 6f6c 7665 3a0a 3d3d 3d3d  osDimSolve:.====
│ │ │ │  0000d0f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0000d100: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0000d110: 2262 6572 7469 6e69 506f 7344 696d 536f  "bertiniPosDimSo
│ │ │ │ -0000d120: 6c76 6528 4964 6561 6c29 220a 2020 2a20  lve(Ideal)".  * 
│ │ │ │ -0000d130: 2262 6572 7469 6e69 506f 7344 696d 536f  "bertiniPosDimSo
│ │ │ │ -0000d140: 6c76 6528 4c69 7374 2922 0a0a 466f 7220  lve(List)"..For 
│ │ │ │ -0000d150: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +0000d100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0000d110: 2022 6265 7274 696e 6950 6f73 4469 6d53   "bertiniPosDimS
│ │ │ │ +0000d120: 6f6c 7665 2849 6465 616c 2922 0a20 202a  olve(Ideal)".  *
│ │ │ │ +0000d130: 2022 6265 7274 696e 6950 6f73 4469 6d53   "bertiniPosDimS
│ │ │ │ +0000d140: 6f6c 7665 284c 6973 7429 220a 0a46 6f72  olve(List)"..For
│ │ │ │ +0000d150: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │  0000d160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0000d170: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0000d180: 6f74 6520 6265 7274 696e 6950 6f73 4469  ote bertiniPosDi
│ │ │ │ -0000d190: 6d53 6f6c 7665 3a20 6265 7274 696e 6950  mSolve: bertiniP
│ │ │ │ -0000d1a0: 6f73 4469 6d53 6f6c 7665 2c20 6973 2061  osDimSolve, is a
│ │ │ │ -0000d1b0: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -0000d1c0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469  nction with opti
│ │ │ │ -0000d1d0: 6f6e 733a 2028 4d61 6361 756c 6179 3244  ons: (Macaulay2D
│ │ │ │ -0000d1e0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -0000d1f0: 6e57 6974 684f 7074 696f 6e73 2c2e 0a1f  nWithOptions,...
│ │ │ │ -0000d200: 0a46 696c 653a 2042 6572 7469 6e69 2e69  .File: Bertini.i
│ │ │ │ -0000d210: 6e66 6f2c 204e 6f64 653a 2062 6572 7469  nfo, Node: berti
│ │ │ │ -0000d220: 6e69 5265 6669 6e65 536f 6c73 2c20 4e65  niRefineSols, Ne
│ │ │ │ -0000d230: 7874 3a20 6265 7274 696e 6953 616d 706c  xt: bertiniSampl
│ │ │ │ -0000d240: 652c 2050 7265 763a 2062 6572 7469 6e69  e, Prev: bertini
│ │ │ │ -0000d250: 506f 7344 696d 536f 6c76 652c 2055 703a  PosDimSolve, Up:
│ │ │ │ -0000d260: 2054 6f70 0a0a 6265 7274 696e 6952 6566   Top..bertiniRef
│ │ │ │ -0000d270: 696e 6553 6f6c 7320 2d2d 2073 6861 7270  ineSols -- sharp
│ │ │ │ -0000d280: 656e 2073 6f6c 7574 696f 6e73 2074 6f20  en solutions to 
│ │ │ │ -0000d290: 6120 7072 6573 6372 6962 6564 206e 756d  a prescribed num
│ │ │ │ -0000d2a0: 6265 7220 6f66 2064 6967 6974 730a 2a2a  ber of digits.**
│ │ │ │ +0000d170: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +0000d180: 6e6f 7465 2062 6572 7469 6e69 506f 7344  note bertiniPosD
│ │ │ │ +0000d190: 696d 536f 6c76 653a 2062 6572 7469 6e69  imSolve: bertini
│ │ │ │ +0000d1a0: 506f 7344 696d 536f 6c76 652c 2069 7320  PosDimSolve, is 
│ │ │ │ +0000d1b0: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66  a *note method.f
│ │ │ │ +0000d1c0: 756e 6374 696f 6e20 7769 7468 206f 7074  unction with opt
│ │ │ │ +0000d1d0: 696f 6e73 3a20 284d 6163 6175 6c61 7932  ions: (Macaulay2
│ │ │ │ +0000d1e0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +0000d1f0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a  onWithOptions,..
│ │ │ │ +0000d200: 1f0a 4669 6c65 3a20 4265 7274 696e 692e  ..File: Bertini.
│ │ │ │ +0000d210: 696e 666f 2c20 4e6f 6465 3a20 6265 7274  info, Node: bert
│ │ │ │ +0000d220: 696e 6952 6566 696e 6553 6f6c 732c 204e  iniRefineSols, N
│ │ │ │ +0000d230: 6578 743a 2062 6572 7469 6e69 5361 6d70  ext: bertiniSamp
│ │ │ │ +0000d240: 6c65 2c20 5072 6576 3a20 6265 7274 696e  le, Prev: bertin
│ │ │ │ +0000d250: 6950 6f73 4469 6d53 6f6c 7665 2c20 5570  iPosDimSolve, Up
│ │ │ │ +0000d260: 3a20 546f 700a 0a62 6572 7469 6e69 5265  : Top..bertiniRe
│ │ │ │ +0000d270: 6669 6e65 536f 6c73 202d 2d20 7368 6172  fineSols -- shar
│ │ │ │ +0000d280: 7065 6e20 736f 6c75 7469 6f6e 7320 746f  pen solutions to
│ │ │ │ +0000d290: 2061 2070 7265 7363 7269 6265 6420 6e75   a prescribed nu
│ │ │ │ +0000d2a0: 6d62 6572 206f 6620 6469 6769 7473 0a2a  mber of digits.*
│ │ │ │  0000d2b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000d2c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000d2d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000d2e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000d2f0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ -0000d300: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ -0000d310: 6167 653a 200a 2020 2020 2020 2020 5320  age: .        S 
│ │ │ │ -0000d320: 3d20 6265 7274 696e 6952 6566 696e 6553  = bertiniRefineS
│ │ │ │ -0000d330: 6f6c 7328 4946 442c 2064 2c20 572c 204f  ols(IFD, d, W, O
│ │ │ │ -0000d340: 4644 290a 2020 2020 2020 2020 5320 3d20  FD).        S = 
│ │ │ │ -0000d350: 6265 7274 696e 6952 6566 696e 6553 6f6c  bertiniRefineSol
│ │ │ │ -0000d360: 7328 642c 2057 2c20 4f46 4429 0a20 2020  s(d, W, OFD).   
│ │ │ │ -0000d370: 2020 2020 2053 203d 2062 6572 7469 6e69       S = bertini
│ │ │ │ -0000d380: 5265 6669 6e65 536f 6c73 2864 2c20 5729  RefineSols(d, W)
│ │ │ │ -0000d390: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -0000d3a0: 2020 202a 2049 4644 2c20 6120 2a6e 6f74     * IFD, a *not
│ │ │ │ -0000d3b0: 6520 7374 7269 6e67 3a20 284d 6163 6175  e string: (Macau
│ │ │ │ -0000d3c0: 6c61 7932 446f 6329 5374 7269 6e67 2c2c  lay2Doc)String,,
│ │ │ │ -0000d3d0: 2061 2064 6972 6563 746f 7279 2077 6865   a directory whe
│ │ │ │ -0000d3e0: 7265 2074 6865 2069 6e70 7574 0a20 2020  re the input.   
│ │ │ │ -0000d3f0: 2020 2020 2066 696c 6520 6f66 2061 2062       file of a b
│ │ │ │ -0000d400: 6572 7469 6e69 5a65 726f 4469 6d20 736f  ertiniZeroDim so
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│ │ │ │ -0000d420: 6f6e 6720 7769 7468 2074 6861 7420 7275  ong with that ru
│ │ │ │ -0000d430: 6e73 206f 7574 7075 740a 2020 2020 2020  ns output.      
│ │ │ │ -0000d440: 2020 6669 6c65 730a 2020 2020 2020 2a20    files.      * 
│ │ │ │ -0000d450: 642c 2061 6e20 2a6e 6f74 6520 696e 7465  d, an *note inte
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│ │ │ │ -0000d470: 6f63 295a 5a2c 2c20 616e 2069 6e74 6567  oc)ZZ,, an integ
│ │ │ │ -0000d480: 6572 2073 7065 6369 6679 696e 6720 7468  er specifying th
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│ │ │ │ -0000d4a0: 206f 6620 6469 6769 7473 206f 6620 7072   of digits of pr
│ │ │ │ -0000d4b0: 6563 6973 696f 6e0a 2020 2020 2020 2a20  ecision.      * 
│ │ │ │ -0000d4c0: 572c 2061 202a 6e6f 7465 206c 6973 743a  W, a *note list:
│ │ │ │ -0000d4d0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
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│ │ │ │ -0000d510: 4644 2c20 6120 2a6e 6f74 6520 7374 7269  FD, a *note stri
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│ │ │ │ -0000d530: 6329 5374 7269 6e67 2c2c 2061 2064 6972  c)String,, a dir
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│ │ │ │ -0000d550: 0a20 2020 2020 2020 206f 7574 7075 7420  .        output 
│ │ │ │ -0000d560: 6669 6c65 7320 6f66 2074 6865 2072 6566  files of the ref
│ │ │ │ -0000d570: 696e 656d 656e 7420 6172 6520 7374 6f72  inement are stor
│ │ │ │ -0000d580: 6564 0a20 202a 202a 6e6f 7465 204f 7074  ed.  * *note Opt
│ │ │ │ -0000d590: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ -0000d5a0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ -0000d5b0: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ -0000d5c0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ -0000d5d0: 2c3a 0a20 2020 2020 202a 2041 6464 6974  ,:.      * Addit
│ │ │ │ -0000d5e0: 696f 6e61 6c46 696c 6573 2028 6d69 7373  ionalFiles (miss
│ │ │ │ -0000d5f0: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -0000d600: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175  n) => ..., defau
│ │ │ │ -0000d610: 6c74 2076 616c 7565 207b 7d2c 200a 2020  lt value {}, .  
│ │ │ │ -0000d620: 2020 2020 2a20 2a6e 6f74 6520 4973 5072      * *note IsPr
│ │ │ │ -0000d630: 6f6a 6563 7469 7665 3a20 4973 5072 6f6a  ojective: IsProj
│ │ │ │ -0000d640: 6563 7469 7665 2c20 3d3e 202e 2e2e 2c20  ective, => ..., 
│ │ │ │ -0000d650: 6465 6661 756c 7420 7661 6c75 6520 2d31  default value -1
│ │ │ │ -0000d660: 2c20 6f70 7469 6f6e 616c 0a20 2020 2020  , optional.     
│ │ │ │ -0000d670: 2020 2061 7267 756d 656e 7420 746f 2073     argument to s
│ │ │ │ -0000d680: 7065 6369 6679 2077 6865 7468 6572 2074  pecify whether t
│ │ │ │ -0000d690: 6f20 7573 6520 686f 6d6f 6765 6e65 6f75  o use homogeneou
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│ │ │ │ -0000d6b0: 2020 2020 2a20 4e61 6d65 4227 496e 7075      * NameB'Inpu
│ │ │ │ -0000d6c0: 7446 696c 6520 286d 6973 7369 6e67 2064  tFile (missing d
│ │ │ │ -0000d6d0: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ -0000d6e0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -0000d6f0: 6c75 6520 2269 6e70 7574 222c 200a 2020  lue "input", .  
│ │ │ │ -0000d700: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │ -0000d710: 6f73 653a 2062 6572 7469 6e69 5472 6163  ose: bertiniTrac
│ │ │ │ -0000d720: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │ -0000d730: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ -0000d740: 3e5f 7064 5f70 645f 7064 5f72 700a 2020  >_pd_pd_pd_rp.  
│ │ │ │ -0000d750: 2020 2020 2020 2c20 3d3e 202e 2e2e 2c20        , => ..., 
│ │ │ │ -0000d760: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ -0000d770: 6c73 652c 204f 7074 696f 6e20 746f 2073  lse, Option to s
│ │ │ │ -0000d780: 696c 656e 6365 2061 6464 6974 696f 6e61  ilence additiona
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│ │ │ │ -0000d7a0: 7075 7473 3a0a 2020 2020 2020 2a20 532c  puts:.      * S,
│ │ │ │ -0000d7b0: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ -0000d7c0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ -0000d7d0: 742c 2c20 6120 6c69 7374 206f 6620 736f  t,, a list of so
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│ │ │ │ -0000d7f0: 506f 696e 740a 0a44 6573 6372 6970 7469  Point..Descripti
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│ │ │ │ -0000d810: 5468 6973 206d 6574 686f 6420 7461 6b65  This method take
│ │ │ │ -0000d820: 7320 7468 6520 6c69 7374 2057 206f 6620  s the list W of 
│ │ │ │ -0000d830: 736f 6c75 7469 6f6e 7320 6672 6f6d 2061  solutions from a
│ │ │ │ -0000d840: 2062 6572 7469 6e69 5a65 726f 4469 6d53   bertiniZeroDimS
│ │ │ │ -0000d850: 6f6c 7665 2061 6e64 0a73 6861 7270 656e  olve and.sharpen
│ │ │ │ -0000d860: 7320 7468 656d 2074 6f20 6420 6469 6769  s them to d digi
│ │ │ │ -0000d870: 7473 2075 7369 6e67 2074 6865 2073 6861  ts using the sha
│ │ │ │ -0000d880: 7270 656e 696e 6720 6d6f 6475 6c65 206f  rpening module o
│ │ │ │ -0000d890: 6620 4265 7274 696e 692e 2057 6865 6e20  f Bertini. When 
│ │ │ │ -0000d8a0: 4946 4420 6973 0a6f 6d69 7474 6564 2074  IFD is.omitted t
│ │ │ │ -0000d8b0: 6865 2069 6e66 6f72 6d61 7469 6f6e 2069  he information i
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│ │ │ │ -0000d8e0: 6972 7374 2070 6f69 6e74 2069 6e20 572e  irst point in W.
│ │ │ │ -0000d8f0: 2057 6865 6e0a 4f46 4420 6973 206f 6d69   When.OFD is omi
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│ │ │ │ -0000d910: 2064 6972 6563 746f 7279 2069 7320 6372   directory is cr
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│ │ │ │ +0000d2f0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +0000d300: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +0000d310: 7361 6765 3a20 0a20 2020 2020 2020 2053  sage: .        S
│ │ │ │ +0000d320: 203d 2062 6572 7469 6e69 5265 6669 6e65   = bertiniRefine
│ │ │ │ +0000d330: 536f 6c73 2849 4644 2c20 642c 2057 2c20  Sols(IFD, d, W, 
│ │ │ │ +0000d340: 4f46 4429 0a20 2020 2020 2020 2053 203d  OFD).        S =
│ │ │ │ +0000d350: 2062 6572 7469 6e69 5265 6669 6e65 536f   bertiniRefineSo
│ │ │ │ +0000d360: 6c73 2864 2c20 572c 204f 4644 290a 2020  ls(d, W, OFD).  
│ │ │ │ +0000d370: 2020 2020 2020 5320 3d20 6265 7274 696e        S = bertin
│ │ │ │ +0000d380: 6952 6566 696e 6553 6f6c 7328 642c 2057  iRefineSols(d, W
│ │ │ │ +0000d390: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +0000d3a0: 2020 2020 2a20 4946 442c 2061 202a 6e6f      * IFD, a *no
│ │ │ │ +0000d3b0: 7465 2073 7472 696e 673a 2028 4d61 6361  te string: (Maca
│ │ │ │ +0000d3c0: 756c 6179 3244 6f63 2953 7472 696e 672c  ulay2Doc)String,
│ │ │ │ +0000d3d0: 2c20 6120 6469 7265 6374 6f72 7920 7768  , a directory wh
│ │ │ │ +0000d3e0: 6572 6520 7468 6520 696e 7075 740a 2020  ere the input.  
│ │ │ │ +0000d3f0: 2020 2020 2020 6669 6c65 206f 6620 6120        file of a 
│ │ │ │ +0000d400: 6265 7274 696e 695a 6572 6f44 696d 2073  bertiniZeroDim s
│ │ │ │ +0000d410: 6f6c 7665 2069 7320 7374 6f72 6564 2061  olve is stored a
│ │ │ │ +0000d420: 6c6f 6e67 2077 6974 6820 7468 6174 2072  long with that r
│ │ │ │ +0000d430: 756e 7320 6f75 7470 7574 0a20 2020 2020  uns output.     
│ │ │ │ +0000d440: 2020 2066 696c 6573 0a20 2020 2020 202a     files.      *
│ │ │ │ +0000d450: 2064 2c20 616e 202a 6e6f 7465 2069 6e74   d, an *note int
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│ │ │ │ +0000d490: 6865 0a20 2020 2020 2020 206e 756d 6265  he.        numbe
│ │ │ │ +0000d4a0: 7220 6f66 2064 6967 6974 7320 6f66 2070  r of digits of p
│ │ │ │ +0000d4b0: 7265 6369 7369 6f6e 0a20 2020 2020 202a  recision.      *
│ │ │ │ +0000d4c0: 2057 2c20 6120 2a6e 6f74 6520 6c69 7374   W, a *note list
│ │ │ │ +0000d4d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0000d4e0: 4c69 7374 2c2c 2061 206c 6973 7420 6f66  List,, a list of
│ │ │ │ +0000d4f0: 2070 6f69 6e74 7320 746f 2062 6520 7368   points to be sh
│ │ │ │ +0000d500: 6172 7065 6e65 640a 2020 2020 2020 2a20  arpened.      * 
│ │ │ │ +0000d510: 4f46 442c 2061 202a 6e6f 7465 2073 7472  OFD, a *note str
│ │ │ │ +0000d520: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ +0000d530: 6f63 2953 7472 696e 672c 2c20 6120 6469  oc)String,, a di
│ │ │ │ +0000d540: 7265 6374 6f72 7920 7768 6572 6520 7468  rectory where th
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│ │ │ │ +0000d560: 2066 696c 6573 206f 6620 7468 6520 7265   files of the re
│ │ │ │ +0000d570: 6669 6e65 6d65 6e74 2061 7265 2073 746f  finement are sto
│ │ │ │ +0000d580: 7265 640a 2020 2a20 2a6e 6f74 6520 4f70  red.  * *note Op
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│ │ │ │ +0000d5a0: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ +0000d5b0: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ +0000d5c0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ +0000d5d0: 732c 3a0a 2020 2020 2020 2a20 4164 6469  s,:.      * Addi
│ │ │ │ +0000d5e0: 7469 6f6e 616c 4669 6c65 7320 286d 6973  tionalFiles (mis
│ │ │ │ +0000d5f0: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
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│ │ │ │ +0000d610: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20  ult value {}, . 
│ │ │ │ +0000d620: 2020 2020 202a 202a 6e6f 7465 2049 7350       * *note IsP
│ │ │ │ +0000d630: 726f 6a65 6374 6976 653a 2049 7350 726f  rojective: IsPro
│ │ │ │ +0000d640: 6a65 6374 6976 652c 203d 3e20 2e2e 2e2c  jective, => ...,
│ │ │ │ +0000d650: 2064 6566 6175 6c74 2076 616c 7565 202d   default value -
│ │ │ │ +0000d660: 312c 206f 7074 696f 6e61 6c0a 2020 2020  1, optional.    
│ │ │ │ +0000d670: 2020 2020 6172 6775 6d65 6e74 2074 6f20      argument to 
│ │ │ │ +0000d680: 7370 6563 6966 7920 7768 6574 6865 7220  specify whether 
│ │ │ │ +0000d690: 746f 2075 7365 2068 6f6d 6f67 656e 656f  to use homogeneo
│ │ │ │ +0000d6a0: 7573 2063 6f6f 7264 696e 6174 6573 0a20  us coordinates. 
│ │ │ │ +0000d6b0: 2020 2020 202a 204e 616d 6542 2749 6e70       * NameB'Inp
│ │ │ │ +0000d6c0: 7574 4669 6c65 2028 6d69 7373 696e 6720  utFile (missing 
│ │ │ │ +0000d6d0: 646f 6375 6d65 6e74 6174 696f 6e29 203d  documentation) =
│ │ │ │ +0000d6e0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +0000d6f0: 616c 7565 2022 696e 7075 7422 2c20 0a20  alue "input", . 
│ │ │ │ +0000d700: 2020 2020 202a 202a 6e6f 7465 2056 6572       * *note Ver
│ │ │ │ +0000d710: 626f 7365 3a20 6265 7274 696e 6954 7261  bose: bertiniTra
│ │ │ │ +0000d720: 636b 486f 6d6f 746f 7079 5f6c 705f 7064  ckHomotopy_lp_pd
│ │ │ │ +0000d730: 5f70 645f 7064 5f63 6d56 6572 626f 7365  _pd_pd_cmVerbose
│ │ │ │ +0000d740: 3d3e 5f70 645f 7064 5f70 645f 7270 0a20  =>_pd_pd_pd_rp. 
│ │ │ │ +0000d750: 2020 2020 2020 202c 203d 3e20 2e2e 2e2c         , => ...,
│ │ │ │ +0000d760: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ +0000d770: 616c 7365 2c20 4f70 7469 6f6e 2074 6f20  alse, Option to 
│ │ │ │ +0000d780: 7369 6c65 6e63 6520 6164 6469 7469 6f6e  silence addition
│ │ │ │ +0000d790: 616c 206f 7574 7075 740a 2020 2a20 4f75  al output.  * Ou
│ │ │ │ +0000d7a0: 7470 7574 733a 0a20 2020 2020 202a 2053  tputs:.      * S
│ │ │ │ +0000d7b0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ +0000d7c0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ +0000d7d0: 7374 2c2c 2061 206c 6973 7420 6f66 2073  st,, a list of s
│ │ │ │ +0000d7e0: 6f6c 7574 696f 6e73 206f 6620 7479 7065  olutions of type
│ │ │ │ +0000d7f0: 2050 6f69 6e74 0a0a 4465 7363 7269 7074   Point..Descript
│ │ │ │ +0000d800: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +0000d810: 0a54 6869 7320 6d65 7468 6f64 2074 616b  .This method tak
│ │ │ │ +0000d820: 6573 2074 6865 206c 6973 7420 5720 6f66  es the list W of
│ │ │ │ +0000d830: 2073 6f6c 7574 696f 6e73 2066 726f 6d20   solutions from 
│ │ │ │ +0000d840: 6120 6265 7274 696e 695a 6572 6f44 696d  a bertiniZeroDim
│ │ │ │ +0000d850: 536f 6c76 6520 616e 640a 7368 6172 7065  Solve and.sharpe
│ │ │ │ +0000d860: 6e73 2074 6865 6d20 746f 2064 2064 6967  ns them to d dig
│ │ │ │ +0000d870: 6974 7320 7573 696e 6720 7468 6520 7368  its using the sh
│ │ │ │ +0000d880: 6172 7065 6e69 6e67 206d 6f64 756c 6520  arpening module 
│ │ │ │ +0000d890: 6f66 2042 6572 7469 6e69 2e20 5768 656e  of Bertini. When
│ │ │ │ +0000d8a0: 2049 4644 2069 730a 6f6d 6974 7465 6420   IFD is.omitted 
│ │ │ │ +0000d8b0: 7468 6520 696e 666f 726d 6174 696f 6e20  the information 
│ │ │ │ +0000d8c0: 6973 2070 756c 6c65 6420 6672 6f6d 2074  is pulled from t
│ │ │ │ +0000d8d0: 6865 2063 6163 6865 206f 6620 7468 6520  he cache of the 
│ │ │ │ +0000d8e0: 6669 7273 7420 706f 696e 7420 696e 2057  first point in W
│ │ │ │ +0000d8f0: 2e20 5768 656e 0a4f 4644 2069 7320 6f6d  . When.OFD is om
│ │ │ │ +0000d900: 6974 7465 6420 6120 7465 6d70 6f72 6172  itted a temporar
│ │ │ │ +0000d910: 7920 6469 7265 6374 6f72 7920 6973 2063  y directory is c
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│ │ │ │  0000d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000d970: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220  ------+.|i1 : R 
│ │ │ │ -0000d980: 3d20 4343 5b78 2c79 5d3b 2020 2020 2020  = CC[x,y];      
│ │ │ │ +0000d970: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2052  -------+.|i1 : R
│ │ │ │ +0000d980: 203d 2043 435b 782c 795d 3b20 2020 2020   = CC[x,y];     
│ │ │ │  0000d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000d9c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0000d9c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0000d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000da10: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 4620  ------+.|i2 : F 
│ │ │ │ -0000da20: 3d20 7b78 5e32 2d32 2c79 5e32 2d32 7d3b  = {x^2-2,y^2-2};
│ │ │ │ -0000da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000da10: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2046  -------+.|i2 : F
│ │ │ │ +0000da20: 203d 207b 785e 322d 322c 795e 322d 327d   = {x^2-2,y^2-2}
│ │ │ │ +0000da30: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  0000da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000da60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0000da60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0000da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dab0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5720  ------+.|i3 : W 
│ │ │ │ -0000dac0: 3d20 6265 7274 696e 695a 6572 6f44 696d  = bertiniZeroDim
│ │ │ │ -0000dad0: 536f 6c76 6520 2846 2920 2020 2020 2020  Solve (F)       
│ │ │ │ +0000dab0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2057  -------+.|i3 : W
│ │ │ │ +0000dac0: 203d 2062 6572 7469 6e69 5a65 726f 4469   = bertiniZeroDi
│ │ │ │ +0000dad0: 6d53 6f6c 7665 2028 4629 2020 2020 2020  mSolve (F)      
│ │ │ │  0000dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000db00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000db00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000db50: 2020 2020 2020 7c0a 7c6f 3320 3d20 7b7b        |.|o3 = {{
│ │ │ │ -0000db60: 312e 3431 3432 312c 2031 2e34 3134 3231  1.41421, 1.41421
│ │ │ │ -0000db70: 7d2c 207b 312e 3431 3432 312c 202d 312e  }, {1.41421, -1.
│ │ │ │ -0000db80: 3431 3432 317d 2c20 7b2d 312e 3431 3432  41421}, {-1.4142
│ │ │ │ -0000db90: 312c 2031 2e34 3134 3231 7d2c 2020 2020  1, 1.41421},    
│ │ │ │ -0000dba0: 2020 2020 2020 7c0a 7c20 2020 2020 2d2d        |.|     --
│ │ │ │ +0000db50: 2020 2020 2020 207c 0a7c 6f33 203d 207b         |.|o3 = {
│ │ │ │ +0000db60: 7b31 2e34 3134 3231 2c20 312e 3431 3432  {1.41421, 1.4142
│ │ │ │ +0000db70: 317d 2c20 7b31 2e34 3134 3231 2c20 2d31  1}, {1.41421, -1
│ │ │ │ +0000db80: 2e34 3134 3231 7d2c 207b 2d31 2e34 3134  .41421}, {-1.414
│ │ │ │ +0000db90: 3231 2c20 312e 3431 3432 317d 2c20 2020  21, 1.41421},   
│ │ │ │ +0000dba0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │  0000dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dbf0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7b2d  ------|.|     {-
│ │ │ │ -0000dc00: 312e 3431 3432 312c 202d 312e 3431 3432  1.41421, -1.4142
│ │ │ │ -0000dc10: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │ +0000dbf0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b  -------|.|     {
│ │ │ │ +0000dc00: 2d31 2e34 3134 3231 2c20 2d31 2e34 3134  -1.41421, -1.414
│ │ │ │ +0000dc10: 3231 7d7d 2020 2020 2020 2020 2020 2020  21}}            
│ │ │ │  0000dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dc40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000dc40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dc90: 2020 2020 2020 7c0a 7c6f 3320 3a20 4c69        |.|o3 : Li
│ │ │ │ -0000dca0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0000dc90: 2020 2020 2020 207c 0a7c 6f33 203a 204c         |.|o3 : L
│ │ │ │ +0000dca0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0000dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dce0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0000dce0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0000dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dd30: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5320  ------+.|i4 : S 
│ │ │ │ -0000dd40: 3d20 6265 7274 696e 6952 6566 696e 6553  = bertiniRefineS
│ │ │ │ -0000dd50: 6f6c 7320 2831 3030 2c57 2920 2020 2020  ols (100,W)     
│ │ │ │ +0000dd30: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2053  -------+.|i4 : S
│ │ │ │ +0000dd40: 203d 2062 6572 7469 6e69 5265 6669 6e65   = bertiniRefine
│ │ │ │ +0000dd50: 536f 6c73 2028 3130 302c 5729 2020 2020  Sols (100,W)    
│ │ │ │  0000dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dd80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000dd80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ddd0: 2020 2020 2020 7c0a 7c6f 3420 3d20 7b7b        |.|o4 = {{
│ │ │ │ -0000dde0: 312e 3431 3432 312c 2031 2e34 3134 3231  1.41421, 1.41421
│ │ │ │ -0000ddf0: 7d2c 207b 312e 3431 3432 312c 202d 312e  }, {1.41421, -1.
│ │ │ │ -0000de00: 3431 3432 317d 2c20 7b2d 312e 3431 3432  41421}, {-1.4142
│ │ │ │ -0000de10: 312c 2031 2e34 3134 3231 7d2c 2020 2020  1, 1.41421},    
│ │ │ │ -0000de20: 2020 2020 2020 7c0a 7c20 2020 2020 2d2d        |.|     --
│ │ │ │ +0000ddd0: 2020 2020 2020 207c 0a7c 6f34 203d 207b         |.|o4 = {
│ │ │ │ +0000dde0: 7b31 2e34 3134 3231 2c20 312e 3431 3432  {1.41421, 1.4142
│ │ │ │ +0000ddf0: 317d 2c20 7b31 2e34 3134 3231 2c20 2d31  1}, {1.41421, -1
│ │ │ │ +0000de00: 2e34 3134 3231 7d2c 207b 2d31 2e34 3134  .41421}, {-1.414
│ │ │ │ +0000de10: 3231 2c20 312e 3431 3432 317d 2c20 2020  21, 1.41421},   
│ │ │ │ +0000de20: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │  0000de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000de70: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7b2d  ------|.|     {-
│ │ │ │ -0000de80: 312e 3431 3432 312c 202d 312e 3431 3432  1.41421, -1.4142
│ │ │ │ -0000de90: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │ +0000de70: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b  -------|.|     {
│ │ │ │ +0000de80: 2d31 2e34 3134 3231 2c20 2d31 2e34 3134  -1.41421, -1.414
│ │ │ │ +0000de90: 3231 7d7d 2020 2020 2020 2020 2020 2020  21}}            
│ │ │ │  0000dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000dec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000dec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000df10: 2020 2020 2020 7c0a 7c6f 3420 3a20 4c69        |.|o4 : Li
│ │ │ │ -0000df20: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0000df10: 2020 2020 2020 207c 0a7c 6f34 203a 204c         |.|o4 : L
│ │ │ │ +0000df20: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0000df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000df60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0000df60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0000df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000dfb0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 636f  ------+.|i5 : co
│ │ │ │ -0000dfc0: 6f72 6473 203d 2063 6f6f 7264 696e 6174  ords = coordinat
│ │ │ │ -0000dfd0: 6573 2053 5f30 2020 2020 2020 2020 2020  es S_0          
│ │ │ │ +0000dfb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2063  -------+.|i5 : c
│ │ │ │ +0000dfc0: 6f6f 7264 7320 3d20 636f 6f72 6469 6e61  oords = coordina
│ │ │ │ +0000dfd0: 7465 7320 535f 3020 2020 2020 2020 2020  tes S_0         
│ │ │ │  0000dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e000: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000e000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e050: 2020 2020 2020 7c0a 7c6f 3520 3d20 7b31        |.|o5 = {1
│ │ │ │ -0000e060: 2e34 3134 3231 2c20 312e 3431 3432 317d  .41421, 1.41421}
│ │ │ │ -0000e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000e050: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ +0000e060: 312e 3431 3432 312c 2031 2e34 3134 3231  1.41421, 1.41421
│ │ │ │ +0000e070: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0000e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e0a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000e0a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e0f0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4c69        |.|o5 : Li
│ │ │ │ -0000e100: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0000e0f0: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
│ │ │ │ +0000e100: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0000e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e140: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0000e140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0000e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000e190: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 636f  ------+.|i6 : co
│ │ │ │ -0000e1a0: 6f72 6473 5f30 2020 2020 2020 2020 2020  ords_0          
│ │ │ │ +0000e190: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2063  -------+.|i6 : c
│ │ │ │ +0000e1a0: 6f6f 7264 735f 3020 2020 2020 2020 2020  oords_0         
│ │ │ │  0000e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e1e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000e1e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e230: 2020 2020 2020 7c0a 7c6f 3620 3d20 312e        |.|o6 = 1.
│ │ │ │ -0000e240: 3431 3432 3133 3536 3233 3733 3039 3530  4142135623730950
│ │ │ │ -0000e250: 3438 3830 3136 3838 3732 3432 3039 3639  4880168872420969
│ │ │ │ -0000e260: 3830 3738 3536 3936 3731 3837 3533 3736  8078569671875376
│ │ │ │ -0000e270: 3934 3830 3733 3137 3636 3739 3733 3739  9480731766797379
│ │ │ │ -0000e280: 3930 3733 3234 7c0a 7c20 2020 2020 3738  907324|.|     78
│ │ │ │ -0000e290: 3436 3231 3037 3033 3838 3530 3338 3735  4621070388503875
│ │ │ │ -0000e2a0: 3334 3332 3736 3431 3537 3320 2020 2020  34327641573     
│ │ │ │ +0000e230: 2020 2020 2020 207c 0a7c 6f36 203d 2031         |.|o6 = 1
│ │ │ │ +0000e240: 2e34 3134 3231 3335 3632 3337 3330 3935  .414213562373095
│ │ │ │ +0000e250: 3034 3838 3031 3638 3837 3234 3230 3936  0488016887242096
│ │ │ │ +0000e260: 3938 3037 3835 3639 3637 3138 3735 3337  9807856967187537
│ │ │ │ +0000e270: 3639 3438 3037 3331 3736 3637 3937 3337  6948073176679737
│ │ │ │ +0000e280: 3939 3037 3332 347c 0a7c 2020 2020 2037  9907324|.|     7
│ │ │ │ +0000e290: 3834 3632 3130 3730 3338 3835 3033 3837  8462107038850387
│ │ │ │ +0000e2a0: 3533 3433 3237 3634 3135 3733 2020 2020  534327641573    
│ │ │ │  0000e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000e2d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e320: 2020 2020 2020 7c0a 7c6f 3620 3a20 4343        |.|o6 : CC
│ │ │ │ -0000e330: 2028 6f66 2070 7265 6369 7369 6f6e 2033   (of precision 3
│ │ │ │ -0000e340: 3333 2920 2020 2020 2020 2020 2020 2020  33)             
│ │ │ │ +0000e320: 2020 2020 2020 207c 0a7c 6f36 203a 2043         |.|o6 : C
│ │ │ │ +0000e330: 4320 286f 6620 7072 6563 6973 696f 6e20  C (of precision 
│ │ │ │ +0000e340: 3333 3329 2020 2020 2020 2020 2020 2020  333)            
│ │ │ │  0000e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e370: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0000e370: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0000e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000e3c0: 2d2d 2d2d 2d2d 2b0a 0a2a 6e6f 7465 2062  ------+..*note b
│ │ │ │ -0000e3d0: 6572 7469 6e69 5265 6669 6e65 536f 6c73  ertiniRefineSols
│ │ │ │ -0000e3e0: 3a20 6265 7274 696e 6952 6566 696e 6553  : bertiniRefineS
│ │ │ │ -0000e3f0: 6f6c 732c 2077 696c 6c20 6f6e 6c79 2072  ols, will only r
│ │ │ │ -0000e400: 6566 696e 6520 6e6f 6e2d 7369 6e67 756c  efine non-singul
│ │ │ │ -0000e410: 6172 0a73 6f6c 7574 696f 6e73 2061 6e64  ar.solutions and
│ │ │ │ -0000e420: 2064 6f65 7320 6e6f 7420 6375 7272 656e   does not curren
│ │ │ │ -0000e430: 746c 7920 776f 726b 2066 6f72 2068 6f6d  tly work for hom
│ │ │ │ -0000e440: 6f67 656e 656f 7573 2073 7973 7465 6d73  ogeneous systems
│ │ │ │ -0000e450: 2e0a 0a57 6179 7320 746f 2075 7365 2062  ...Ways to use b
│ │ │ │ -0000e460: 6572 7469 6e69 5265 6669 6e65 536f 6c73  ertiniRefineSols
│ │ │ │ -0000e470: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0000e3c0: 2d2d 2d2d 2d2d 2d2b 0a0a 2a6e 6f74 6520  -------+..*note 
│ │ │ │ +0000e3d0: 6265 7274 696e 6952 6566 696e 6553 6f6c  bertiniRefineSol
│ │ │ │ +0000e3e0: 733a 2062 6572 7469 6e69 5265 6669 6e65  s: bertiniRefine
│ │ │ │ +0000e3f0: 536f 6c73 2c20 7769 6c6c 206f 6e6c 7920  Sols, will only 
│ │ │ │ +0000e400: 7265 6669 6e65 206e 6f6e 2d73 696e 6775  refine non-singu
│ │ │ │ +0000e410: 6c61 720a 736f 6c75 7469 6f6e 7320 616e  lar.solutions an
│ │ │ │ +0000e420: 6420 646f 6573 206e 6f74 2063 7572 7265  d does not curre
│ │ │ │ +0000e430: 6e74 6c79 2077 6f72 6b20 666f 7220 686f  ntly work for ho
│ │ │ │ +0000e440: 6d6f 6765 6e65 6f75 7320 7379 7374 656d  mogeneous system
│ │ │ │ +0000e450: 732e 0a0a 5761 7973 2074 6f20 7573 6520  s...Ways to use 
│ │ │ │ +0000e460: 6265 7274 696e 6952 6566 696e 6553 6f6c  bertiniRefineSol
│ │ │ │ +0000e470: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
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│ │ │ │ +0000e590: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +0000e5a0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ +0000e5b0: 6974 684f 7074 696f 6e73 2c2e 0a1f 0a46  ithOptions,....F
│ │ │ │ +0000e5c0: 696c 653a 2042 6572 7469 6e69 2e69 6e66  ile: Bertini.inf
│ │ │ │ +0000e5d0: 6f2c 204e 6f64 653a 2062 6572 7469 6e69  o, Node: bertini
│ │ │ │ +0000e5e0: 5361 6d70 6c65 2c20 4e65 7874 3a20 6265  Sample, Next: be
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│ │ │ │ +0000e620: 2054 6f70 0a0a 6265 7274 696e 6953 616d   Top..bertiniSam
│ │ │ │ +0000e630: 706c 6520 2d2d 2061 206d 6169 6e20 6d65  ple -- a main me
│ │ │ │ +0000e640: 7468 6f64 2074 6f20 7361 6d70 6c65 2070  thod to sample p
│ │ │ │ +0000e650: 6f69 6e74 7320 6672 6f6d 2061 6e20 6972  oints from an ir
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│ │ │ │ +0000e680: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  0000e690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000e6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000e6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000e6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000e6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
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│ │ │ │ -0000e6f0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -0000e700: 2020 2056 203d 2062 6572 7469 6e69 5361     V = bertiniSa
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│ │ │ │ -0000e720: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0000e730: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465  n, an *note inte
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│ │ │ │ -0000e940: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
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│ │ │ │ -0000e960: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70  ackHomotopy_lp_p
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│ │ │ │ -0000ea20: 7361 6d70 6c65 2070 6f69 6e74 730a 0a44  sample points..D
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│ │ │ │ -0000ea40: 3d3d 3d3d 3d3d 0a0a 5361 6d70 6c65 7320  ======..Samples 
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│ │ │ │ -0000ea60: 7272 6564 7563 6962 6c65 2063 6f6d 706f  rreducible compo
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│ │ │ │ -0000ea90: 2020 5468 650a 6972 7265 6475 6369 626c    The.irreducibl
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│ │ │ │ +0000e6f0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
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│ │ │ │ +0000e720: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
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│ │ │ │ +0000e7a0: 202a 2057 2c20 6120 2a6e 6f74 6520 7769   * W, a *note wi
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│ │ │ │ +0000e8b0: 0a20 2020 2020 202a 202a 6e6f 7465 2049  .      * *note I
│ │ │ │ +0000e8c0: 7350 726f 6a65 6374 6976 653a 2049 7350  sProjective: IsP
│ │ │ │ +0000e8d0: 726f 6a65 6374 6976 652c 203d 3e20 2e2e  rojective, => ..
│ │ │ │ +0000e8e0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +0000e8f0: 202d 312c 206f 7074 696f 6e61 6c0a 2020   -1, optional.  
│ │ │ │ +0000e900: 2020 2020 2020 6172 6775 6d65 6e74 2074        argument t
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│ │ │ │ +0000e940: 0a20 2020 2020 202a 202a 6e6f 7465 2056  .      * *note V
│ │ │ │ +0000e950: 6572 626f 7365 3a20 6265 7274 696e 6954  erbose: bertiniT
│ │ │ │ +0000e960: 7261 636b 486f 6d6f 746f 7079 5f6c 705f  rackHomotopy_lp_
│ │ │ │ +0000e970: 7064 5f70 645f 7064 5f63 6d56 6572 626f  pd_pd_pd_cmVerbo
│ │ │ │ +0000e980: 7365 3d3e 5f70 645f 7064 5f70 645f 7270  se=>_pd_pd_pd_rp
│ │ │ │ +0000e990: 0a20 2020 2020 2020 202c 203d 3e20 2e2e  .        , => ..
│ │ │ │ +0000e9a0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +0000e9b0: 2066 616c 7365 2c20 4f70 7469 6f6e 2074   false, Option t
│ │ │ │ +0000e9c0: 6f20 7369 6c65 6e63 6520 6164 6469 7469  o silence additi
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│ │ │ │ +0000e9e0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +0000e9f0: 204c 2c20 6120 2a6e 6f74 6520 6c69 7374   L, a *note list
│ │ │ │ +0000ea00: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0000ea10: 4c69 7374 2c2c 2061 206c 6973 7420 6f66  List,, a list of
│ │ │ │ +0000ea20: 2073 616d 706c 6520 706f 696e 7473 0a0a   sample points..
│ │ │ │ +0000ea30: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +0000ea40: 3d3d 3d3d 3d3d 3d0a 0a53 616d 706c 6573  =======..Samples
│ │ │ │ +0000ea50: 2070 6f69 6e74 7320 6672 6f6d 2061 6e20   points from an 
│ │ │ │ +0000ea60: 6972 7265 6475 6369 626c 6520 636f 6d70  irreducible comp
│ │ │ │ +0000ea70: 6f6e 656e 7420 6f66 2061 2076 6172 6965  onent of a varie
│ │ │ │ +0000ea80: 7479 2075 7369 6e67 2042 6572 7469 6e69  ty using Bertini
│ │ │ │ +0000ea90: 2e20 2054 6865 0a69 7272 6564 7563 6962  .  The.irreducib
│ │ │ │ +0000eaa0: 6c65 2063 6f6d 706f 6e65 6e74 206e 6565  le component nee
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│ │ │ │ +0000eac0: 6e75 6d65 7269 6361 6c20 666f 726d 2061  numerical form a
│ │ │ │ +0000ead0: 7320 6120 2a6e 6f74 6520 5769 746e 6573  s a *note Witnes
│ │ │ │ +0000eae0: 7353 6574 3a0a 284e 4147 7479 7065 7329  sSet:.(NAGtypes)
│ │ │ │ +0000eaf0: 5769 746e 6573 7353 6574 2c2e 2020 5468  WitnessSet,.  Th
│ │ │ │ +0000eb00: 6520 6d65 7468 6f64 202a 6e6f 7465 2062  e method *note b
│ │ │ │ +0000eb10: 6572 7469 6e69 506f 7344 696d 536f 6c76  ertiniPosDimSolv
│ │ │ │ +0000eb20: 653a 0a62 6572 7469 6e69 506f 7344 696d  e:.bertiniPosDim
│ │ │ │ +0000eb30: 536f 6c76 652c 2063 616e 2062 6520 7573  Solve, can be us
│ │ │ │ +0000eb40: 6564 2074 6f20 6765 6e65 7261 7465 2061  ed to generate a
│ │ │ │ +0000eb50: 2077 6974 6e65 7373 2073 6574 2066 6f72   witness set for
│ │ │ │ +0000eb60: 2074 6865 2063 6f6d 706f 6e65 6e74 2e0a   the component..
│ │ │ │ +0000eb70: 4265 7274 696e 6920 2831 2920 7772 6974  Bertini (1) writ
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│ │ │ │ +0000eb90: 6574 2074 6f20 6120 7465 6d70 6f72 6172  et to a temporar
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│ │ │ │ +0000ebb0: 6b65 7320 4265 7274 696e 6927 730a 736f  kes Bertini's.so
│ │ │ │ +0000ebc0: 6c76 6572 2077 6974 6820 6f70 7469 6f6e  lver with option
│ │ │ │ +0000ebd0: 2054 7261 636b 5479 7065 203d 3e20 322c   TrackType => 2,
│ │ │ │ +0000ebe0: 2061 6e64 2028 3320 6d6f 7665 7320 7468   and (3 moves th
│ │ │ │ +0000ebf0: 6520 6879 7065 7270 6c61 6e65 7320 6465  e hyperplanes de
│ │ │ │ +0000ec00: 6669 6e65 6420 696e 2074 6865 0a2a 6e6f  fined in the.*no
│ │ │ │ +0000ec10: 7465 2057 6974 6e65 7373 5365 743a 2028  te WitnessSet: (
│ │ │ │ +0000ec20: 4e41 4774 7970 6573 2957 6974 6e65 7373  NAGtypes)Witness
│ │ │ │ +0000ec30: 5365 742c 2057 2077 6974 6869 6e20 7468  Set, W within th
│ │ │ │ +0000ec40: 6520 7370 6163 6520 756e 7469 6c20 7468  e space until th
│ │ │ │ +0000ec50: 6520 6465 7369 7265 640a 706f 696e 7473  e desired.points
│ │ │ │ +0000ec60: 2061 7265 2073 616d 706c 6564 2c20 2834   are sampled, (4
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│ │ │ │ +0000ec80: 7075 7420 6f66 2042 6572 7469 6e69 2069  put of Bertini i
│ │ │ │ +0000ec90: 6e20 6120 7465 6d70 6f72 6172 7920 6669  n a temporary fi
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│ │ │ │ +0000ecb0: 2835 2920 7061 7273 6573 2061 6e64 206f  (5) parses and o
│ │ │ │ +0000ecc0: 7574 7075 7473 2074 6865 2073 6f6c 7574  utputs the solut
│ │ │ │ +0000ecd0: 696f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ions...+--------
│ │ │ │  0000ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ed20: 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 3d20  ----+.|i1 : R = 
│ │ │ │ -0000ed30: 4343 5b78 2c79 2c7a 5d20 2020 2020 2020  CC[x,y,z]       
│ │ │ │ +0000ed20: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +0000ed30: 2043 435b 782c 792c 7a5d 2020 2020 2020   CC[x,y,z]      
│ │ │ │  0000ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ed70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000ed70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000edc0: 2020 2020 7c0a 7c6f 3120 3d20 5220 2020      |.|o1 = R   
│ │ │ │ +0000edc0: 2020 2020 207c 0a7c 6f31 203d 2052 2020       |.|o1 = R  
│ │ │ │  0000edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ee10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000ee10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ee60: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ -0000ee70: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +0000ee60: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ +0000ee70: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │  0000ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000eeb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0000eeb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ef00: 2d2d 2d2d 2b0a 7c69 3220 3a20 4620 3d20  ----+.|i2 : F = 
│ │ │ │ -0000ef10: 7b20 2879 5e32 2b78 5e32 2b7a 5e32 2d31  { (y^2+x^2+z^2-1
│ │ │ │ -0000ef20: 292a 782c 2028 795e 322b 785e 322b 7a5e  )*x, (y^2+x^2+z^
│ │ │ │ -0000ef30: 322d 3129 2a79 207d 2020 2020 2020 2020  2-1)*y }        
│ │ │ │ +0000ef00: 2d2d 2d2d 2d2b 0a7c 6932 203a 2046 203d  -----+.|i2 : F =
│ │ │ │ +0000ef10: 207b 2028 795e 322b 785e 322b 7a5e 322d   { (y^2+x^2+z^2-
│ │ │ │ +0000ef20: 3129 2a78 2c20 2879 5e32 2b78 5e32 2b7a  1)*x, (y^2+x^2+z
│ │ │ │ +0000ef30: 5e32 2d31 292a 7920 7d20 2020 2020 2020  ^2-1)*y }       
│ │ │ │  0000ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ef50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000ef50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000efa0: 2020 2020 7c0a 7c20 2020 2020 2020 3320      |.|       3 
│ │ │ │ -0000efb0: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │ -0000efc0: 2020 2020 3220 2020 2020 3320 2020 2020      2     3     
│ │ │ │ -0000efd0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0000efa0: 2020 2020 207c 0a7c 2020 2020 2020 2033       |.|       3
│ │ │ │ +0000efb0: 2020 2020 2020 3220 2020 2020 2032 2020        2      2  
│ │ │ │ +0000efc0: 2020 2020 2032 2020 2020 2033 2020 2020       2     3    
│ │ │ │ +0000efd0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0000efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000eff0: 2020 2020 7c0a 7c6f 3220 3d20 7b78 2020      |.|o2 = {x  
│ │ │ │ -0000f000: 2b20 782a 7920 202b 2078 2a7a 2020 2d20  + x*y  + x*z  - 
│ │ │ │ -0000f010: 782c 2078 2079 202b 2079 2020 2b20 792a  x, x y + y  + y*
│ │ │ │ -0000f020: 7a20 202d 2079 7d20 2020 2020 2020 2020  z  - y}         
│ │ │ │ +0000eff0: 2020 2020 207c 0a7c 6f32 203d 207b 7820       |.|o2 = {x 
│ │ │ │ +0000f000: 202b 2078 2a79 2020 2b20 782a 7a20 202d   + x*y  + x*z  -
│ │ │ │ +0000f010: 2078 2c20 7820 7920 2b20 7920 202b 2079   x, x y + y  + y
│ │ │ │ +0000f020: 2a7a 2020 2d20 797d 2020 2020 2020 2020  *z  - y}        
│ │ │ │  0000f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f040: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f090: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │ -0000f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f090: 2020 2020 207c 0a7c 6f32 203a 204c 6973       |.|o2 : Lis
│ │ │ │ +0000f0a0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0000f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f0e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0000f0e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f130: 2d2d 2d2d 2b0a 7c69 3320 3a20 4e56 203d  ----+.|i3 : NV =
│ │ │ │ -0000f140: 2062 6572 7469 6e69 506f 7344 696d 536f   bertiniPosDimSo
│ │ │ │ -0000f150: 6c76 6528 4629 2020 2020 2020 2020 2020  lve(F)          
│ │ │ │ +0000f130: 2d2d 2d2d 2d2b 0a7c 6933 203a 204e 5620  -----+.|i3 : NV 
│ │ │ │ +0000f140: 3d20 6265 7274 696e 6950 6f73 4469 6d53  = bertiniPosDimS
│ │ │ │ +0000f150: 6f6c 7665 2846 2920 2020 2020 2020 2020  olve(F)         
│ │ │ │  0000f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f180: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f1d0: 2020 2020 7c0a 7c6f 3320 3d20 4e56 2020      |.|o3 = NV  
│ │ │ │ +0000f1d0: 2020 2020 207c 0a7c 6f33 203d 204e 5620       |.|o3 = NV 
│ │ │ │  0000f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f220: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f270: 2020 2020 7c0a 7c6f 3320 3a20 4e75 6d65      |.|o3 : Nume
│ │ │ │ -0000f280: 7269 6361 6c56 6172 6965 7479 2020 2020  ricalVariety    
│ │ │ │ +0000f270: 2020 2020 207c 0a7c 6f33 203a 204e 756d       |.|o3 : Num
│ │ │ │ +0000f280: 6572 6963 616c 5661 7269 6574 7920 2020  ericalVariety   
│ │ │ │  0000f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f2c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0000f2c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f310: 2d2d 2d2d 2b0a 7c69 3420 3a20 5720 3d20  ----+.|i4 : W = 
│ │ │ │ -0000f320: 4e56 2331 5f30 202d 2d7a 2d61 7869 7320  NV#1_0 --z-axis 
│ │ │ │ +0000f310: 2d2d 2d2d 2d2b 0a7c 6934 203a 2057 203d  -----+.|i4 : W =
│ │ │ │ +0000f320: 204e 5623 315f 3020 2d2d 7a2d 6178 6973   NV#1_0 --z-axis
│ │ │ │  0000f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f3b0: 2020 2020 7c0a 7c6f 3420 3d20 5720 2020      |.|o4 = W   
│ │ │ │ +0000f3b0: 2020 2020 207c 0a7c 6f34 203d 2057 2020       |.|o4 = W  
│ │ │ │  0000f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f400: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f400: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f450: 2020 2020 7c0a 7c6f 3420 3a20 5769 746e      |.|o4 : Witn
│ │ │ │ -0000f460: 6573 7353 6574 2020 2020 2020 2020 2020  essSet          
│ │ │ │ +0000f450: 2020 2020 207c 0a7c 6f34 203a 2057 6974       |.|o4 : Wit
│ │ │ │ +0000f460: 6e65 7373 5365 7420 2020 2020 2020 2020  nessSet         
│ │ │ │  0000f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f4a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0000f4a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f4f0: 2d2d 2d2d 2b0a 7c69 3520 3a20 6265 7274  ----+.|i5 : bert
│ │ │ │ -0000f500: 696e 6953 616d 706c 6528 342c 2057 2920  iniSample(4, W) 
│ │ │ │ +0000f4f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2062 6572  -----+.|i5 : ber
│ │ │ │ +0000f500: 7469 6e69 5361 6d70 6c65 2834 2c20 5729  tiniSample(4, W)
│ │ │ │  0000f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f540: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f590: 2020 2020 7c0a 7c6f 3520 3d20 7b7b 2d33      |.|o5 = {{-3
│ │ │ │ -0000f5a0: 2e39 3838 3935 652d 3231 2d32 2e36 3639  .98895e-21-2.669
│ │ │ │ -0000f5b0: 3033 652d 3231 2a69 692c 202d 312e 3039  03e-21*ii, -1.09
│ │ │ │ -0000f5c0: 3034 3965 2d32 302b 322e 3339 3035 3765  049e-20+2.39057e
│ │ │ │ -0000f5d0: 2d32 312a 6969 2c20 2020 2020 2020 2020  -21*ii,         
│ │ │ │ -0000f5e0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
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│ │ │ │ +0000f5e0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
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│ │ │ │  0000f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0000f630: 2d2d 2d2d 2d7c 0a7c 2020 2020 202e 3333  -----|.|     .33
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│ │ │ │  0000f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f680: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │ +0000f680: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │  0000f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f6d0: 2d2d 2d2d 7c0a 7c20 2020 2020 372e 3034  ----|.|     7.04
│ │ │ │ -0000f6e0: 3838 3565 2d32 312d 332e 3331 3433 3565  885e-21-3.31435e
│ │ │ │ -0000f6f0: 2d32 312a 6969 2c20 2e32 3335 3136 342b  -21*ii, .235164+
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│ │ │ │ +0000f6d0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2037 2e30  -----|.|     7.0
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│ │ │ │ +0000f6f0: 652d 3231 2a69 692c 202e 3233 3531 3634  e-21*ii, .235164
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│ │ │ │  0000f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f720: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │ +0000f720: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │  0000f730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f770: 2d2d 2d2d 7c0a 7c20 2020 2020 7b2d 372e  ----|.|     {-7.
│ │ │ │ -0000f780: 3136 3431 3865 2d32 312b 332e 3430 3739  16418e-21+3.4079
│ │ │ │ -0000f790: 3365 2d32 312a 6969 2c20 382e 3638 3636  3e-21*ii, 8.6866
│ │ │ │ -0000f7a0: 3565 2d32 312b 312e 3335 3435 3465 2d32  5e-21+1.35454e-2
│ │ │ │ -0000f7b0: 302a 6969 2c20 2020 2020 2020 2020 2020  0*ii,           
│ │ │ │ -0000f7c0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │ +0000f770: 2d2d 2d2d 2d7c 0a7c 2020 2020 207b 2d37  -----|.|     {-7
│ │ │ │ +0000f780: 2e31 3634 3138 652d 3231 2b33 2e34 3037  .16418e-21+3.407
│ │ │ │ +0000f790: 3933 652d 3231 2a69 692c 2038 2e36 3836  93e-21*ii, 8.686
│ │ │ │ +0000f7a0: 3635 652d 3231 2b31 2e33 3534 3534 652d  65e-21+1.35454e-
│ │ │ │ +0000f7b0: 3230 2a69 692c 2020 2020 2020 2020 2020  20*ii,          
│ │ │ │ +0000f7c0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │  0000f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f810: 2d2d 2d2d 7c0a 7c20 2020 2020 2e33 3736  ----|.|     .376
│ │ │ │ -0000f820: 3339 362b 2e31 3630 3138 382a 6969 7d2c  396+.160188*ii},
│ │ │ │ -0000f830: 207b 372e 3234 3538 3365 2d32 312b 322e   {7.24583e-21+2.
│ │ │ │ -0000f840: 3232 3631 3665 2d32 312a 6969 2c20 2020  22616e-21*ii,   
│ │ │ │ +0000f810: 2d2d 2d2d 2d7c 0a7c 2020 2020 202e 3337  -----|.|     .37
│ │ │ │ +0000f820: 3633 3936 2b2e 3136 3031 3838 2a69 697d  6396+.160188*ii}
│ │ │ │ +0000f830: 2c20 7b37 2e32 3435 3833 652d 3231 2b32  , {7.24583e-21+2
│ │ │ │ +0000f840: 2e32 3236 3136 652d 3231 2a69 692c 2020  .22616e-21*ii,  
│ │ │ │  0000f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f860: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │ +0000f860: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │  0000f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f8b0: 2d2d 2d2d 7c0a 7c20 2020 2020 352e 3234  ----|.|     5.24
│ │ │ │ -0000f8c0: 3038 3365 2d32 312b 352e 3830 3839 3965  083e-21+5.80899e
│ │ │ │ -0000f8d0: 2d32 312a 6969 2c20 2e33 3330 3038 332b  -21*ii, .330083+
│ │ │ │ -0000f8e0: 2e32 3635 3632 342a 6969 7d7d 2020 2020  .265624*ii}}    
│ │ │ │ +0000f8b0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2035 2e32  -----|.|     5.2
│ │ │ │ +0000f8c0: 3430 3833 652d 3231 2b35 2e38 3038 3939  4083e-21+5.80899
│ │ │ │ +0000f8d0: 652d 3231 2a69 692c 202e 3333 3030 3833  e-21*ii, .330083
│ │ │ │ +0000f8e0: 2b2e 3236 3536 3234 2a69 697d 7d20 2020  +.265624*ii}}   
│ │ │ │  0000f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0000f900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0000f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f950: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374      |.|o5 : List
│ │ │ │ -0000f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000f950: 2020 2020 207c 0a7c 6f35 203a 204c 6973       |.|o5 : Lis
│ │ │ │ +0000f960: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0000f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f9a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0000f9a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f9f0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ -0000fa00: 7365 2062 6572 7469 6e69 5361 6d70 6c65  se bertiniSample
│ │ │ │ -0000fa10: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -0000fa20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -0000fa30: 2a20 2262 6572 7469 6e69 5361 6d70 6c65  * "bertiniSample
│ │ │ │ -0000fa40: 285a 5a2c 5769 746e 6573 7353 6574 2922  (ZZ,WitnessSet)"
│ │ │ │ -0000fa50: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0000fa60: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0000fa70: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0000fa80: 6563 7420 2a6e 6f74 6520 6265 7274 696e  ect *note bertin
│ │ │ │ -0000fa90: 6953 616d 706c 653a 2062 6572 7469 6e69  iSample: bertini
│ │ │ │ -0000faa0: 5361 6d70 6c65 2c20 6973 2061 202a 6e6f  Sample, is a *no
│ │ │ │ -0000fab0: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -0000fac0: 6f6e 2077 6974 680a 6f70 7469 6f6e 733a  on with.options:
│ │ │ │ -0000fad0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -0000fae0: 6574 686f 6446 756e 6374 696f 6e57 6974  ethodFunctionWit
│ │ │ │ -0000faf0: 684f 7074 696f 6e73 2c2e 0a1f 0a46 696c  hOptions,....Fil
│ │ │ │ -0000fb00: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c  e: Bertini.info,
│ │ │ │ -0000fb10: 204e 6f64 653a 2062 6572 7469 6e69 5472   Node: bertiniTr
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│ │ │ │ -0000fb30: 743a 2062 6572 7469 6e69 5472 6163 6b48  t: bertiniTrackH
│ │ │ │ -0000fb40: 6f6d 6f74 6f70 795f 6c70 5f70 645f 7064  omotopy_lp_pd_pd
│ │ │ │ -0000fb50: 5f70 645f 636d 5665 7262 6f73 653d 3e5f  _pd_cmVerbose=>_
│ │ │ │ -0000fb60: 7064 5f70 645f 7064 5f72 702c 2050 7265  pd_pd_pd_rp, Pre
│ │ │ │ -0000fb70: 763a 2062 6572 7469 6e69 5361 6d70 6c65  v: bertiniSample
│ │ │ │ -0000fb80: 2c20 5570 3a20 546f 700a 0a62 6572 7469  , Up: Top..berti
│ │ │ │ -0000fb90: 6e69 5472 6163 6b48 6f6d 6f74 6f70 7920  niTrackHomotopy 
│ │ │ │ -0000fba0: 2d2d 2061 206d 6169 6e20 6d65 7468 6f64  -- a main method
│ │ │ │ -0000fbb0: 2074 6f20 7472 6163 6b20 7573 696e 6720   to track using 
│ │ │ │ -0000fbc0: 6120 7573 6572 2d64 6566 696e 6564 2068  a user-defined h
│ │ │ │ -0000fbd0: 6f6d 6f74 6f70 790a 2a2a 2a2a 2a2a 2a2a  omotopy.********
│ │ │ │ +0000f9f0: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20  -----+..Ways to 
│ │ │ │ +0000fa00: 7573 6520 6265 7274 696e 6953 616d 706c  use bertiniSampl
│ │ │ │ +0000fa10: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  e:.=============
│ │ │ │ +0000fa20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +0000fa30: 202a 2022 6265 7274 696e 6953 616d 706c   * "bertiniSampl
│ │ │ │ +0000fa40: 6528 5a5a 2c57 6974 6e65 7373 5365 7429  e(ZZ,WitnessSet)
│ │ │ │ +0000fa50: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +0000fa60: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +0000fa70: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +0000fa80: 6a65 6374 202a 6e6f 7465 2062 6572 7469  ject *note berti
│ │ │ │ +0000fa90: 6e69 5361 6d70 6c65 3a20 6265 7274 696e  niSample: bertin
│ │ │ │ +0000faa0: 6953 616d 706c 652c 2069 7320 6120 2a6e  iSample, is a *n
│ │ │ │ +0000fab0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ +0000fac0: 696f 6e20 7769 7468 0a6f 7074 696f 6e73  ion with.options
│ │ │ │ +0000fad0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0000fae0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ +0000faf0: 7468 4f70 7469 6f6e 732c 2e0a 1f0a 4669  thOptions,....Fi
│ │ │ │ +0000fb00: 6c65 3a20 4265 7274 696e 692e 696e 666f  le: Bertini.info
│ │ │ │ +0000fb10: 2c20 4e6f 6465 3a20 6265 7274 696e 6954  , Node: bertiniT
│ │ │ │ +0000fb20: 7261 636b 486f 6d6f 746f 7079 2c20 4e65  rackHomotopy, Ne
│ │ │ │ +0000fb30: 7874 3a20 6265 7274 696e 6954 7261 636b  xt: bertiniTrack
│ │ │ │ +0000fb40: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70  Homotopy_lp_pd_p
│ │ │ │ +0000fb50: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ +0000fb60: 5f70 645f 7064 5f70 645f 7270 2c20 5072  _pd_pd_pd_rp, Pr
│ │ │ │ +0000fb70: 6576 3a20 6265 7274 696e 6953 616d 706c  ev: bertiniSampl
│ │ │ │ +0000fb80: 652c 2055 703a 2054 6f70 0a0a 6265 7274  e, Up: Top..bert
│ │ │ │ +0000fb90: 696e 6954 7261 636b 486f 6d6f 746f 7079  iniTrackHomotopy
│ │ │ │ +0000fba0: 202d 2d20 6120 6d61 696e 206d 6574 686f   -- a main metho
│ │ │ │ +0000fbb0: 6420 746f 2074 7261 636b 2075 7369 6e67  d to track using
│ │ │ │ +0000fbc0: 2061 2075 7365 722d 6465 6669 6e65 6420   a user-defined 
│ │ │ │ +0000fbd0: 686f 6d6f 746f 7079 0a2a 2a2a 2a2a 2a2a  homotopy.*******
│ │ │ │  0000fbe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000fbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000fc00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0000fc10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0000fc20: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ -0000fc30: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055 7361  =======..  * Usa
│ │ │ │ -0000fc40: 6765 3a20 0a20 2020 2020 2020 2053 303d  ge: .        S0=
│ │ │ │ -0000fc50: 6265 7274 696e 6954 7261 636b 486f 6d6f  bertiniTrackHomo
│ │ │ │ -0000fc60: 746f 7079 2874 2c20 482c 2053 3129 0a20  topy(t, H, S1). 
│ │ │ │ -0000fc70: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -0000fc80: 202a 2074 2c20 6120 2a6e 6f74 6520 7269   * t, a *note ri
│ │ │ │ -0000fc90: 6e67 2065 6c65 6d65 6e74 3a20 284d 6163  ng element: (Mac
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│ │ │ │ -0000fcb0: 656d 656e 742c 2c20 6120 7061 7468 2076  ement,, a path v
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│ │ │ │ -0000fcd0: 482c 2061 202a 6e6f 7465 206c 6973 743a  H, a *note list:
│ │ │ │ -0000fce0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0000fcf0: 6973 742c 2c20 6120 6c69 7374 2070 6f6c  ist,, a list pol
│ │ │ │ -0000fd00: 796e 6f6d 6961 6c73 2074 6861 7420 6465  ynomials that de
│ │ │ │ -0000fd10: 6669 6e65 0a20 2020 2020 2020 2074 6865  fine.        the
│ │ │ │ -0000fd20: 2068 6f6d 6f74 6f70 7920 7769 7468 2072   homotopy with r
│ │ │ │ -0000fd30: 6573 7065 6374 2074 6f20 7468 6520 7061  espect to the pa
│ │ │ │ -0000fd40: 7468 2076 6172 6961 626c 650a 2020 2020  th variable.    
│ │ │ │ -0000fd50: 2020 2a20 5331 2c20 6120 2a6e 6f74 6520    * S1, a *note 
│ │ │ │ -0000fd60: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ -0000fd70: 446f 6329 4c69 7374 2c2c 2061 206c 6973  Doc)List,, a lis
│ │ │ │ -0000fd80: 7420 6f66 2073 6f6c 7574 696f 6e73 2074  t of solutions t
│ │ │ │ -0000fd90: 6f20 7468 6520 7374 6172 740a 2020 2020  o the start.    
│ │ │ │ -0000fda0: 2020 2020 7379 7374 656d 0a20 202a 202a      system.  * *
│ │ │ │ -0000fdb0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -0000fdc0: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -0000fdd0: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -0000fde0: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -0000fdf0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -0000fe00: 202a 2042 6572 7469 6e69 496e 7075 7443   * BertiniInputC
│ │ │ │ -0000fe10: 6f6e 6669 6775 7261 7469 6f6e 2028 6d69  onfiguration (mi
│ │ │ │ -0000fe20: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ -0000fe30: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566  ion) => ..., def
│ │ │ │ -0000fe40: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -0000fe50: 2020 207b 7d2c 0a20 2020 2020 202a 202a     {},.      * *
│ │ │ │ -0000fe60: 6e6f 7465 2049 7350 726f 6a65 6374 6976  note IsProjectiv
│ │ │ │ -0000fe70: 653a 2049 7350 726f 6a65 6374 6976 652c  e: IsProjective,
│ │ │ │ -0000fe80: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -0000fe90: 2076 616c 7565 202d 312c 206f 7074 696f   value -1, optio
│ │ │ │ -0000fea0: 6e61 6c0a 2020 2020 2020 2020 6172 6775  nal.        argu
│ │ │ │ -0000feb0: 6d65 6e74 2074 6f20 7370 6563 6966 7920  ment to specify 
│ │ │ │ -0000fec0: 7768 6574 6865 7220 746f 2075 7365 2068  whether to use h
│ │ │ │ -0000fed0: 6f6d 6f67 656e 656f 7573 2063 6f6f 7264  omogeneous coord
│ │ │ │ -0000fee0: 696e 6174 6573 0a20 2020 2020 202a 202a  inates.      * *
│ │ │ │ -0000fef0: 6e6f 7465 2056 6572 626f 7365 3a20 6265  note Verbose: be
│ │ │ │ -0000ff00: 7274 696e 6954 7261 636b 486f 6d6f 746f  rtiniTrackHomoto
│ │ │ │ -0000ff10: 7079 5f6c 705f 7064 5f70 645f 7064 5f63  py_lp_pd_pd_pd_c
│ │ │ │ -0000ff20: 6d56 6572 626f 7365 3d3e 5f70 645f 7064  mVerbose=>_pd_pd
│ │ │ │ -0000ff30: 5f70 645f 7270 0a20 2020 2020 2020 202c  _pd_rp.        ,
│ │ │ │ -0000ff40: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -0000ff50: 2076 616c 7565 2066 616c 7365 2c20 4f70   value false, Op
│ │ │ │ -0000ff60: 7469 6f6e 2074 6f20 7369 6c65 6e63 6520  tion to silence 
│ │ │ │ -0000ff70: 6164 6469 7469 6f6e 616c 206f 7574 7075  additional outpu
│ │ │ │ -0000ff80: 740a 2020 2a20 4f75 7470 7574 733a 0a20  t.  * Outputs:. 
│ │ │ │ -0000ff90: 2020 2020 202a 2053 302c 2061 202a 6e6f       * S0, a *no
│ │ │ │ -0000ffa0: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ -0000ffb0: 6179 3244 6f63 294c 6973 742c 2c20 6120  ay2Doc)List,, a 
│ │ │ │ -0000ffc0: 6c69 7374 206f 6620 736f 6c75 7469 6f6e  list of solution
│ │ │ │ -0000ffd0: 7320 746f 2074 6865 0a20 2020 2020 2020  s to the.       
│ │ │ │ -0000ffe0: 2074 6172 6765 7420 7379 7374 656d 0a0a   target system..
│ │ │ │ -0000fff0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00010000: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6d65  =======..This me
│ │ │ │ -00010010: 7468 6f64 2063 616c 6c73 2042 6572 7469  thod calls Berti
│ │ │ │ -00010020: 6e69 2074 6f20 7472 6163 6b20 6120 7573  ni to track a us
│ │ │ │ -00010030: 6572 2d64 6566 696e 6564 2068 6f6d 6f74  er-defined homot
│ │ │ │ -00010040: 6f70 792e 2020 5468 6520 7573 6572 206e  opy.  The user n
│ │ │ │ -00010050: 6565 6473 2074 6f0a 7370 6563 6966 7920  eeds to.specify 
│ │ │ │ -00010060: 7468 6520 686f 6d6f 746f 7079 2048 2c20  the homotopy H, 
│ │ │ │ -00010070: 7468 6520 7061 7468 2076 6172 6961 626c  the path variabl
│ │ │ │ -00010080: 6520 742c 2061 6e64 2061 206c 6973 7420  e t, and a list 
│ │ │ │ -00010090: 6f66 2073 7461 7274 2073 6f6c 7574 696f  of start solutio
│ │ │ │ -000100a0: 6e73 2053 312e 0a42 6572 7469 6e69 2028  ns S1..Bertini (
│ │ │ │ -000100b0: 3129 2077 7269 7465 7320 7468 6520 686f  1) writes the ho
│ │ │ │ -000100c0: 6d6f 746f 7079 2061 6e64 2073 7461 7274  motopy and start
│ │ │ │ -000100d0: 2073 6f6c 7574 696f 6e73 2074 6f20 7465   solutions to te
│ │ │ │ -000100e0: 6d70 6f72 6172 7920 6669 6c65 732c 2028  mporary files, (
│ │ │ │ -000100f0: 3229 0a69 6e76 6f6b 6573 2042 6572 7469  2).invokes Berti
│ │ │ │ -00010100: 6e69 2773 2073 6f6c 7665 7220 7769 7468  ni's solver with
│ │ │ │ -00010110: 2063 6f6e 6669 6775 7261 7469 6f6e 206b   configuration k
│ │ │ │ -00010120: 6579 776f 7264 2055 7365 7248 6f6d 6f74  eyword UserHomot
│ │ │ │ -00010130: 6f70 7920 3d3e 2031 2069 6e20 7468 650a  opy => 1 in the.
│ │ │ │ -00010140: 6166 6669 6e65 2063 6173 6520 616e 6420  affine case and 
│ │ │ │ -00010150: 5573 6572 486f 6d6f 746f 7079 203d 3e20  UserHomotopy => 
│ │ │ │ -00010160: 3220 696e 2074 6865 2070 726f 6a65 6374  2 in the project
│ │ │ │ -00010170: 6976 6520 7369 7475 6174 696f 6e2c 2028  ive situation, (
│ │ │ │ -00010180: 3329 2073 746f 7265 7320 7468 650a 6f75  3) stores the.ou
│ │ │ │ -00010190: 7470 7574 206f 6620 4265 7274 696e 6920  tput of Bertini 
│ │ │ │ -000101a0: 696e 2061 2074 656d 706f 7261 7279 2066  in a temporary f
│ │ │ │ -000101b0: 696c 652c 2061 6e64 2028 3429 2070 6172  ile, and (4) par
│ │ │ │ -000101c0: 7365 7320 6120 6d61 6368 696e 6520 7265  ses a machine re
│ │ │ │ -000101d0: 6164 6162 6c65 2066 696c 650a 746f 206f  adable file.to o
│ │ │ │ -000101e0: 7574 7075 7420 6120 6c69 7374 206f 6620  utput a list of 
│ │ │ │ -000101f0: 736f 6c75 7469 6f6e 732e 0a0a 2b2d 2d2d  solutions...+---
│ │ │ │ +0000fc20: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +0000fc30: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +0000fc40: 6167 653a 200a 2020 2020 2020 2020 5330  age: .        S0
│ │ │ │ +0000fc50: 3d62 6572 7469 6e69 5472 6163 6b48 6f6d  =bertiniTrackHom
│ │ │ │ +0000fc60: 6f74 6f70 7928 742c 2048 2c20 5331 290a  otopy(t, H, S1).
│ │ │ │ +0000fc70: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0000fc80: 2020 2a20 742c 2061 202a 6e6f 7465 2072    * t, a *note r
│ │ │ │ +0000fc90: 696e 6720 656c 656d 656e 743a 2028 4d61  ing element: (Ma
│ │ │ │ +0000fca0: 6361 756c 6179 3244 6f63 2952 696e 6745  caulay2Doc)RingE
│ │ │ │ +0000fcb0: 6c65 6d65 6e74 2c2c 2061 2070 6174 6820  lement,, a path 
│ │ │ │ +0000fcc0: 7661 7269 6162 6c65 0a20 2020 2020 202a  variable.      *
│ │ │ │ +0000fcd0: 2048 2c20 6120 2a6e 6f74 6520 6c69 7374   H, a *note list
│ │ │ │ +0000fce0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0000fcf0: 4c69 7374 2c2c 2061 206c 6973 7420 706f  List,, a list po
│ │ │ │ +0000fd00: 6c79 6e6f 6d69 616c 7320 7468 6174 2064  lynomials that d
│ │ │ │ +0000fd10: 6566 696e 650a 2020 2020 2020 2020 7468  efine.        th
│ │ │ │ +0000fd20: 6520 686f 6d6f 746f 7079 2077 6974 6820  e homotopy with 
│ │ │ │ +0000fd30: 7265 7370 6563 7420 746f 2074 6865 2070  respect to the p
│ │ │ │ +0000fd40: 6174 6820 7661 7269 6162 6c65 0a20 2020  ath variable.   
│ │ │ │ +0000fd50: 2020 202a 2053 312c 2061 202a 6e6f 7465     * S1, a *note
│ │ │ │ +0000fd60: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ +0000fd70: 3244 6f63 294c 6973 742c 2c20 6120 6c69  2Doc)List,, a li
│ │ │ │ +0000fd80: 7374 206f 6620 736f 6c75 7469 6f6e 7320  st of solutions 
│ │ │ │ +0000fd90: 746f 2074 6865 2073 7461 7274 0a20 2020  to the start.   
│ │ │ │ +0000fda0: 2020 2020 2073 7973 7465 6d0a 2020 2a20       system.  * 
│ │ │ │ +0000fdb0: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ +0000fdc0: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ +0000fdd0: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ +0000fde0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +0000fdf0: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ +0000fe00: 2020 2a20 4265 7274 696e 6949 6e70 7574    * BertiniInput
│ │ │ │ +0000fe10: 436f 6e66 6967 7572 6174 696f 6e20 286d  Configuration (m
│ │ │ │ +0000fe20: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ +0000fe30: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ +0000fe40: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +0000fe50: 2020 2020 7b7d 2c0a 2020 2020 2020 2a20      {},.      * 
│ │ │ │ +0000fe60: 2a6e 6f74 6520 4973 5072 6f6a 6563 7469  *note IsProjecti
│ │ │ │ +0000fe70: 7665 3a20 4973 5072 6f6a 6563 7469 7665  ve: IsProjective
│ │ │ │ +0000fe80: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +0000fe90: 7420 7661 6c75 6520 2d31 2c20 6f70 7469  t value -1, opti
│ │ │ │ +0000fea0: 6f6e 616c 0a20 2020 2020 2020 2061 7267  onal.        arg
│ │ │ │ +0000feb0: 756d 656e 7420 746f 2073 7065 6369 6679  ument to specify
│ │ │ │ +0000fec0: 2077 6865 7468 6572 2074 6f20 7573 6520   whether to use 
│ │ │ │ +0000fed0: 686f 6d6f 6765 6e65 6f75 7320 636f 6f72  homogeneous coor
│ │ │ │ +0000fee0: 6469 6e61 7465 730a 2020 2020 2020 2a20  dinates.      * 
│ │ │ │ +0000fef0: 2a6e 6f74 6520 5665 7262 6f73 653a 2062  *note Verbose: b
│ │ │ │ +0000ff00: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ +0000ff10: 6f70 795f 6c70 5f70 645f 7064 5f70 645f  opy_lp_pd_pd_pd_
│ │ │ │ +0000ff20: 636d 5665 7262 6f73 653d 3e5f 7064 5f70  cmVerbose=>_pd_p
│ │ │ │ +0000ff30: 645f 7064 5f72 700a 2020 2020 2020 2020  d_pd_rp.        
│ │ │ │ +0000ff40: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +0000ff50: 7420 7661 6c75 6520 6661 6c73 652c 204f  t value false, O
│ │ │ │ +0000ff60: 7074 696f 6e20 746f 2073 696c 656e 6365  ption to silence
│ │ │ │ +0000ff70: 2061 6464 6974 696f 6e61 6c20 6f75 7470   additional outp
│ │ │ │ +0000ff80: 7574 0a20 202a 204f 7574 7075 7473 3a0a  ut.  * Outputs:.
│ │ │ │ +0000ff90: 2020 2020 2020 2a20 5330 2c20 6120 2a6e        * S0, a *n
│ │ │ │ +0000ffa0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +0000ffb0: 6c61 7932 446f 6329 4c69 7374 2c2c 2061  lay2Doc)List,, a
│ │ │ │ +0000ffc0: 206c 6973 7420 6f66 2073 6f6c 7574 696f   list of solutio
│ │ │ │ +0000ffd0: 6e73 2074 6f20 7468 650a 2020 2020 2020  ns to the.      
│ │ │ │ +0000ffe0: 2020 7461 7267 6574 2073 7973 7465 6d0a    target system.
│ │ │ │ +0000fff0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00010000: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 206d  ========..This m
│ │ │ │ +00010010: 6574 686f 6420 6361 6c6c 7320 4265 7274  ethod calls Bert
│ │ │ │ +00010020: 696e 6920 746f 2074 7261 636b 2061 2075  ini to track a u
│ │ │ │ +00010030: 7365 722d 6465 6669 6e65 6420 686f 6d6f  ser-defined homo
│ │ │ │ +00010040: 746f 7079 2e20 2054 6865 2075 7365 7220  topy.  The user 
│ │ │ │ +00010050: 6e65 6564 7320 746f 0a73 7065 6369 6679  needs to.specify
│ │ │ │ +00010060: 2074 6865 2068 6f6d 6f74 6f70 7920 482c   the homotopy H,
│ │ │ │ +00010070: 2074 6865 2070 6174 6820 7661 7269 6162   the path variab
│ │ │ │ +00010080: 6c65 2074 2c20 616e 6420 6120 6c69 7374  le t, and a list
│ │ │ │ +00010090: 206f 6620 7374 6172 7420 736f 6c75 7469   of start soluti
│ │ │ │ +000100a0: 6f6e 7320 5331 2e0a 4265 7274 696e 6920  ons S1..Bertini 
│ │ │ │ +000100b0: 2831 2920 7772 6974 6573 2074 6865 2068  (1) writes the h
│ │ │ │ +000100c0: 6f6d 6f74 6f70 7920 616e 6420 7374 6172  omotopy and star
│ │ │ │ +000100d0: 7420 736f 6c75 7469 6f6e 7320 746f 2074  t solutions to t
│ │ │ │ +000100e0: 656d 706f 7261 7279 2066 696c 6573 2c20  emporary files, 
│ │ │ │ +000100f0: 2832 290a 696e 766f 6b65 7320 4265 7274  (2).invokes Bert
│ │ │ │ +00010100: 696e 6927 7320 736f 6c76 6572 2077 6974  ini's solver wit
│ │ │ │ +00010110: 6820 636f 6e66 6967 7572 6174 696f 6e20  h configuration 
│ │ │ │ +00010120: 6b65 7977 6f72 6420 5573 6572 486f 6d6f  keyword UserHomo
│ │ │ │ +00010130: 746f 7079 203d 3e20 3120 696e 2074 6865  topy => 1 in the
│ │ │ │ +00010140: 0a61 6666 696e 6520 6361 7365 2061 6e64  .affine case and
│ │ │ │ +00010150: 2055 7365 7248 6f6d 6f74 6f70 7920 3d3e   UserHomotopy =>
│ │ │ │ +00010160: 2032 2069 6e20 7468 6520 7072 6f6a 6563   2 in the projec
│ │ │ │ +00010170: 7469 7665 2073 6974 7561 7469 6f6e 2c20  tive situation, 
│ │ │ │ +00010180: 2833 2920 7374 6f72 6573 2074 6865 0a6f  (3) stores the.o
│ │ │ │ +00010190: 7574 7075 7420 6f66 2042 6572 7469 6e69  utput of Bertini
│ │ │ │ +000101a0: 2069 6e20 6120 7465 6d70 6f72 6172 7920   in a temporary 
│ │ │ │ +000101b0: 6669 6c65 2c20 616e 6420 2834 2920 7061  file, and (4) pa
│ │ │ │ +000101c0: 7273 6573 2061 206d 6163 6869 6e65 2072  rses a machine r
│ │ │ │ +000101d0: 6561 6461 626c 6520 6669 6c65 0a74 6f20  eadable file.to 
│ │ │ │ +000101e0: 6f75 7470 7574 2061 206c 6973 7420 6f66  output a list of
│ │ │ │ +000101f0: 2073 6f6c 7574 696f 6e73 2e0a 0a2b 2d2d   solutions...+--
│ │ │ │  00010200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010240: 2b0a 7c69 3120 3a20 5220 3d20 4343 5b78  +.|i1 : R = CC[x
│ │ │ │ -00010250: 2c74 5d3b 202d 2d20 696e 636c 7564 6520  ,t]; -- include 
│ │ │ │ -00010260: 7468 6520 7061 7468 2076 6172 6961 626c  the path variabl
│ │ │ │ -00010270: 6520 696e 2074 6865 2072 696e 6720 2020  e in the ring   
│ │ │ │ -00010280: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00010240: 2d2b 0a7c 6931 203a 2052 203d 2043 435b  -+.|i1 : R = CC[
│ │ │ │ +00010250: 782c 745d 3b20 2d2d 2069 6e63 6c75 6465  x,t]; -- include
│ │ │ │ +00010260: 2074 6865 2070 6174 6820 7661 7269 6162   the path variab
│ │ │ │ +00010270: 6c65 2069 6e20 7468 6520 7269 6e67 2020  le in the ring  
│ │ │ │ +00010280: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00010290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000102a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000102b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000102c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000102d0: 3220 3a20 4820 3d20 7b20 2878 5e32 2d31  2 : H = { (x^2-1
│ │ │ │ -000102e0: 292a 7420 2b20 2878 5e32 2d32 292a 2831  )*t + (x^2-2)*(1
│ │ │ │ -000102f0: 2d74 297d 3b20 2020 2020 2020 2020 2020  -t)};           
│ │ │ │ +000102c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000102d0: 6932 203a 2048 203d 207b 2028 785e 322d  i2 : H = { (x^2-
│ │ │ │ +000102e0: 3129 2a74 202b 2028 785e 322d 3229 2a28  1)*t + (x^2-2)*(
│ │ │ │ +000102f0: 312d 7429 7d3b 2020 2020 2020 2020 2020  1-t)};          
│ │ │ │  00010300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010310: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010310: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010350: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -00010360: 736f 6c31 203d 2070 6f69 6e74 207b 7b31  sol1 = point {{1
│ │ │ │ -00010370: 7d7d 3b20 2020 2020 2020 2020 2020 2020  }};             
│ │ │ │ +00010350: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00010360: 2073 6f6c 3120 3d20 706f 696e 7420 7b7b   sol1 = point {{
│ │ │ │ +00010370: 317d 7d3b 2020 2020 2020 2020 2020 2020  1}};            
│ │ │ │  00010380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000103a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00010390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000103a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000103b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000103c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000103d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000103e0: 2d2d 2d2d 2b0a 7c69 3420 3a20 736f 6c32  ----+.|i4 : sol2
│ │ │ │ -000103f0: 203d 2070 6f69 6e74 207b 7b2d 317d 7d3b   = point {{-1}};
│ │ │ │ -00010400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000103e0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2073 6f6c  -----+.|i4 : sol
│ │ │ │ +000103f0: 3220 3d20 706f 696e 7420 7b7b 2d31 7d7d  2 = point {{-1}}
│ │ │ │ +00010400: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  00010410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010420: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00010420: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00010430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010470: 2b0a 7c69 3520 3a20 5331 3d20 7b20 736f  +.|i5 : S1= { so
│ │ │ │ -00010480: 6c31 2c20 736f 6c32 2020 7d3b 2d2d 736f  l1, sol2  };--so
│ │ │ │ -00010490: 6c75 7469 6f6e 7320 746f 2048 2077 6865  lutions to H whe
│ │ │ │ -000104a0: 6e20 743d 3120 2020 2020 2020 2020 2020  n t=1           
│ │ │ │ -000104b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00010470: 2d2b 0a7c 6935 203a 2053 313d 207b 2073  -+.|i5 : S1= { s
│ │ │ │ +00010480: 6f6c 312c 2073 6f6c 3220 207d 3b2d 2d73  ol1, sol2  };--s
│ │ │ │ +00010490: 6f6c 7574 696f 6e73 2074 6f20 4820 7768  olutions to H wh
│ │ │ │ +000104a0: 656e 2074 3d31 2020 2020 2020 2020 2020  en t=1          
│ │ │ │ +000104b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  000104c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000104d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000104e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000104f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00010500: 3620 3a20 5330 203d 2062 6572 7469 6e69  6 : S0 = bertini
│ │ │ │ -00010510: 5472 6163 6b48 6f6d 6f74 6f70 7920 2874  TrackHomotopy (t
│ │ │ │ -00010520: 2c20 482c 2053 3129 202d 2d73 6f6c 7574  , H, S1) --solut
│ │ │ │ -00010530: 696f 6e73 2074 6f20 4820 7768 656e 2074  ions to H when t
│ │ │ │ -00010540: 3d30 7c0a 7c20 2020 2020 2020 2020 2020  =0|.|           
│ │ │ │ +000104f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00010500: 6936 203a 2053 3020 3d20 6265 7274 696e  i6 : S0 = bertin
│ │ │ │ +00010510: 6954 7261 636b 486f 6d6f 746f 7079 2028  iTrackHomotopy (
│ │ │ │ +00010520: 742c 2048 2c20 5331 2920 2d2d 736f 6c75  t, H, S1) --solu
│ │ │ │ +00010530: 7469 6f6e 7320 746f 2048 2077 6865 6e20  tions to H when 
│ │ │ │ +00010540: 743d 307c 0a7c 2020 2020 2020 2020 2020  t=0|.|          
│ │ │ │  00010550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010580: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -00010590: 7b7b 2d31 2e34 3134 3231 7d2c 207b 312e  {{-1.41421}, {1.
│ │ │ │ -000105a0: 3431 3432 317d 7d20 2020 2020 2020 2020  41421}}         
│ │ │ │ +00010580: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ +00010590: 207b 7b2d 312e 3431 3432 317d 2c20 7b31   {{-1.41421}, {1
│ │ │ │ +000105a0: 2e34 3134 3231 7d7d 2020 2020 2020 2020  .41421}}        
│ │ │ │  000105b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000105c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000105d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000105c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000105d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000105e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000105f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010610: 2020 2020 7c0a 7c6f 3620 3a20 4c69 7374      |.|o6 : List
│ │ │ │ -00010620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010610: 2020 2020 207c 0a7c 6f36 203a 204c 6973       |.|o6 : Lis
│ │ │ │ +00010620: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00010630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010650: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00010650: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00010660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000106a0: 2b0a 7c69 3720 3a20 7065 656b 2053 305f  +.|i7 : peek S0_
│ │ │ │ -000106b0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000106a0: 2d2b 0a7c 6937 203a 2070 6565 6b20 5330  -+.|i7 : peek S0
│ │ │ │ +000106b0: 5f30 2020 2020 2020 2020 2020 2020 2020  _0              
│ │ │ │  000106c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000106d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000106e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000106e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000106f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00010730: 3720 3d20 506f 696e 747b 6361 6368 6520  7 = Point{cache 
│ │ │ │ -00010740: 3d3e 2043 6163 6865 5461 626c 657b 2e2e  => CacheTable{..
│ │ │ │ -00010750: 2e39 2e2e 2e7d 7d20 2020 2020 2020 2020  .9...}}         
│ │ │ │ +00010720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00010730: 6f37 203d 2050 6f69 6e74 7b63 6163 6865  o7 = Point{cache
│ │ │ │ +00010740: 203d 3e20 4361 6368 6554 6162 6c65 7b2e   => CacheTable{.
│ │ │ │ +00010750: 2e2e 392e 2e2e 7d7d 2020 2020 2020 2020  ..9...}}        
│ │ │ │  00010760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010770: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00010780: 436f 6f72 6469 6e61 7465 7320 3d3e 207b  Coordinates => {
│ │ │ │ -00010790: 2d31 2e34 3134 3231 7d20 2020 2020 2020  -1.41421}       
│ │ │ │ +00010770: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00010780: 2043 6f6f 7264 696e 6174 6573 203d 3e20   Coordinates => 
│ │ │ │ +00010790: 7b2d 312e 3431 3432 317d 2020 2020 2020  {-1.41421}      
│ │ │ │  000107a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000107b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000107b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000107c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000107d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000107e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000107f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00010800: 0a49 6e20 7468 6520 7072 6576 696f 7573  .In the previous
│ │ │ │ -00010810: 2065 7861 6d70 6c65 2c20 7765 2073 6f6c   example, we sol
│ │ │ │ -00010820: 7665 6420 2478 5e32 2d32 2420 6279 206d  ved $x^2-2$ by m
│ │ │ │ -00010830: 6f76 696e 6720 6672 6f6d 2024 785e 322d  oving from $x^2-
│ │ │ │ -00010840: 3124 2077 6974 6820 6120 6c69 6e65 6172  1$ with a linear
│ │ │ │ -00010850: 0a68 6f6d 6f74 6f70 792e 2042 6572 7469  .homotopy. Berti
│ │ │ │ -00010860: 6e69 2074 7261 636b 7320 686f 6d6f 746f  ni tracks homoto
│ │ │ │ -00010870: 7069 6573 2073 7461 7274 696e 6720 6174  pies starting at
│ │ │ │ -00010880: 2024 743d 3124 2061 6e64 2065 6e64 696e   $t=1$ and endin
│ │ │ │ -00010890: 6720 6174 2024 743d 3024 2e0a 4669 6e61  g at $t=0$..Fina
│ │ │ │ -000108a0: 6c20 736f 6c75 7469 6f6e 7320 6172 6520  l solutions are 
│ │ │ │ -000108b0: 6f66 2074 6865 2074 7970 6520 506f 696e  of the type Poin
│ │ │ │ -000108c0: 742e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  t...+-----------
│ │ │ │ +000107f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00010800: 0a0a 496e 2074 6865 2070 7265 7669 6f75  ..In the previou
│ │ │ │ +00010810: 7320 6578 616d 706c 652c 2077 6520 736f  s example, we so
│ │ │ │ +00010820: 6c76 6564 2024 785e 322d 3224 2062 7920  lved $x^2-2$ by 
│ │ │ │ +00010830: 6d6f 7669 6e67 2066 726f 6d20 2478 5e32  moving from $x^2
│ │ │ │ +00010840: 2d31 2420 7769 7468 2061 206c 696e 6561  -1$ with a linea
│ │ │ │ +00010850: 720a 686f 6d6f 746f 7079 2e20 4265 7274  r.homotopy. Bert
│ │ │ │ +00010860: 696e 6920 7472 6163 6b73 2068 6f6d 6f74  ini tracks homot
│ │ │ │ +00010870: 6f70 6965 7320 7374 6172 7469 6e67 2061  opies starting a
│ │ │ │ +00010880: 7420 2474 3d31 2420 616e 6420 656e 6469  t $t=1$ and endi
│ │ │ │ +00010890: 6e67 2061 7420 2474 3d30 242e 0a46 696e  ng at $t=0$..Fin
│ │ │ │ +000108a0: 616c 2073 6f6c 7574 696f 6e73 2061 7265  al solutions are
│ │ │ │ +000108b0: 206f 6620 7468 6520 7479 7065 2050 6f69   of the type Poi
│ │ │ │ +000108c0: 6e74 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  nt...+----------
│ │ │ │  000108d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000108e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000108f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010910: 2d2d 2b0a 7c69 3820 3a20 523d 4343 5b78  --+.|i8 : R=CC[x
│ │ │ │ -00010920: 2c79 2c74 5d3b 202d 2d20 696e 636c 7564  ,y,t]; -- includ
│ │ │ │ -00010930: 6520 7468 6520 7061 7468 2076 6172 6961  e the path varia
│ │ │ │ -00010940: 626c 6520 696e 2074 6865 2072 696e 6720  ble in the ring 
│ │ │ │ +00010910: 2d2d 2d2b 0a7c 6938 203a 2052 3d43 435b  ---+.|i8 : R=CC[
│ │ │ │ +00010920: 782c 792c 745d 3b20 2d2d 2069 6e63 6c75  x,y,t]; -- inclu
│ │ │ │ +00010930: 6465 2074 6865 2070 6174 6820 7661 7269  de the path vari
│ │ │ │ +00010940: 6162 6c65 2069 6e20 7468 6520 7269 6e67  able in the ring
│ │ │ │  00010950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010960: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010960: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000109a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000109b0: 2d2d 2b0a 7c69 3920 3a20 6631 3d28 785e  --+.|i9 : f1=(x^
│ │ │ │ -000109c0: 322d 795e 3229 3b20 2020 2020 2020 2020  2-y^2);         
│ │ │ │ +000109b0: 2d2d 2d2b 0a7c 6939 203a 2066 313d 2878  ---+.|i9 : f1=(x
│ │ │ │ +000109c0: 5e32 2d79 5e32 293b 2020 2020 2020 2020  ^2-y^2);        
│ │ │ │  000109d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000109e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000109f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010a00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010a00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010a50: 2d2d 2b0a 7c69 3130 203a 2066 323d 2832  --+.|i10 : f2=(2
│ │ │ │ -00010a60: 2a78 5e32 2d33 2a78 2a79 2b35 2a79 5e32  *x^2-3*x*y+5*y^2
│ │ │ │ -00010a70: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +00010a50: 2d2d 2d2b 0a7c 6931 3020 3a20 6632 3d28  ---+.|i10 : f2=(
│ │ │ │ +00010a60: 322a 785e 322d 332a 782a 792b 352a 795e  2*x^2-3*x*y+5*y^
│ │ │ │ +00010a70: 3229 3b20 2020 2020 2020 2020 2020 2020  2);             
│ │ │ │  00010a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010aa0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010af0: 2d2d 2b0a 7c69 3131 203a 2048 203d 207b  --+.|i11 : H = {
│ │ │ │ -00010b00: 2066 312a 7420 2b20 6632 2a28 312d 7429   f1*t + f2*(1-t)
│ │ │ │ -00010b10: 7d3b 202d 2d48 2069 7320 6120 6c69 7374  }; --H is a list
│ │ │ │ -00010b20: 206f 6620 706f 6c79 6e6f 6d69 616c 7320   of polynomials 
│ │ │ │ -00010b30: 696e 2078 2c79 2c74 2020 2020 2020 2020  in x,y,t        
│ │ │ │ -00010b40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010af0: 2d2d 2d2b 0a7c 6931 3120 3a20 4820 3d20  ---+.|i11 : H = 
│ │ │ │ +00010b00: 7b20 6631 2a74 202b 2066 322a 2831 2d74  { f1*t + f2*(1-t
│ │ │ │ +00010b10: 297d 3b20 2d2d 4820 6973 2061 206c 6973  )}; --H is a lis
│ │ │ │ +00010b20: 7420 6f66 2070 6f6c 796e 6f6d 6961 6c73  t of polynomials
│ │ │ │ +00010b30: 2069 6e20 782c 792c 7420 2020 2020 2020   in x,y,t       
│ │ │ │ +00010b40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010b90: 2d2d 2b0a 7c69 3132 203a 2073 6f6c 313d  --+.|i12 : sol1=
│ │ │ │ -00010ba0: 2020 2020 706f 696e 747b 7b31 2c31 7d7d      point{{1,1}}
│ │ │ │ -00010bb0: 2d2d 7b7b 782c 797d 7d20 636f 6f72 6469  --{{x,y}} coordi
│ │ │ │ -00010bc0: 6e61 7465 7320 2020 2020 2020 2020 2020  nates           
│ │ │ │ +00010b90: 2d2d 2d2b 0a7c 6931 3220 3a20 736f 6c31  ---+.|i12 : sol1
│ │ │ │ +00010ba0: 3d20 2020 2070 6f69 6e74 7b7b 312c 317d  =    point{{1,1}
│ │ │ │ +00010bb0: 7d2d 2d7b 7b78 2c79 7d7d 2063 6f6f 7264  }--{{x,y}} coord
│ │ │ │ +00010bc0: 696e 6174 6573 2020 2020 2020 2020 2020  inates          
│ │ │ │  00010bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010be0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00010be0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00010bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c30: 2020 7c0a 7c6f 3132 203d 2073 6f6c 3120    |.|o12 = sol1 
│ │ │ │ +00010c30: 2020 207c 0a7c 6f31 3220 3d20 736f 6c31     |.|o12 = sol1
│ │ │ │  00010c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00010c80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00010c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010cd0: 2020 7c0a 7c6f 3132 203a 2050 6f69 6e74    |.|o12 : Point
│ │ │ │ -00010ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010cd0: 2020 207c 0a7c 6f31 3220 3a20 506f 696e     |.|o12 : Poin
│ │ │ │ +00010ce0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00010cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010d20: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010d20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010d70: 2d2d 2b0a 7c69 3133 203a 2073 6f6c 323d  --+.|i13 : sol2=
│ │ │ │ -00010d80: 2020 2020 706f 696e 747b 7b20 2d31 2c31      point{{ -1,1
│ │ │ │ -00010d90: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00010d70: 2d2d 2d2b 0a7c 6931 3320 3a20 736f 6c32  ---+.|i13 : sol2
│ │ │ │ +00010d80: 3d20 2020 2070 6f69 6e74 7b7b 202d 312c  =    point{{ -1,
│ │ │ │ +00010d90: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │  00010da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010dc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00010dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00010dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010e10: 2020 7c0a 7c6f 3133 203d 2073 6f6c 3220    |.|o13 = sol2 
│ │ │ │ +00010e10: 2020 207c 0a7c 6f31 3320 3d20 736f 6c32     |.|o13 = sol2
│ │ │ │  00010e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00010e60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00010e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010eb0: 2020 7c0a 7c6f 3133 203a 2050 6f69 6e74    |.|o13 : Point
│ │ │ │ -00010ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010eb0: 2020 207c 0a7c 6f31 3320 3a20 506f 696e     |.|o13 : Poin
│ │ │ │ +00010ec0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00010ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010f00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00010f00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00010f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010f50: 2d2d 2b0a 7c69 3134 203a 2053 313d 7b73  --+.|i14 : S1={s
│ │ │ │ -00010f60: 6f6c 312c 736f 6c32 7d2d 2d73 6f6c 7574  ol1,sol2}--solut
│ │ │ │ -00010f70: 696f 6e73 2074 6f20 4820 7768 656e 2074  ions to H when t
│ │ │ │ -00010f80: 3d31 2020 2020 2020 2020 2020 2020 2020  =1              
│ │ │ │ +00010f50: 2d2d 2d2b 0a7c 6931 3420 3a20 5331 3d7b  ---+.|i14 : S1={
│ │ │ │ +00010f60: 736f 6c31 2c73 6f6c 327d 2d2d 736f 6c75  sol1,sol2}--solu
│ │ │ │ +00010f70: 7469 6f6e 7320 746f 2048 2077 6865 6e20  tions to H when 
│ │ │ │ +00010f80: 743d 3120 2020 2020 2020 2020 2020 2020  t=1             
│ │ │ │  00010f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010fa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00010fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010ff0: 2020 7c0a 7c6f 3134 203d 207b 736f 6c31    |.|o14 = {sol1
│ │ │ │ -00011000: 2c20 736f 6c32 7d20 2020 2020 2020 2020  , sol2}         
│ │ │ │ +00010ff0: 2020 207c 0a7c 6f31 3420 3d20 7b73 6f6c     |.|o14 = {sol
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│ │ │ │  00011010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011040: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00011040: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00011050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011090: 2020 7c0a 7c6f 3134 203a 204c 6973 7420    |.|o14 : List 
│ │ │ │ +00011090: 2020 207c 0a7c 6f31 3420 3a20 4c69 7374     |.|o14 : List
│ │ │ │  000110a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000110b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000110c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000110d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000110e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000110f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011130: 2d2d 2b0a 7c69 3135 203a 2053 303d 6265  --+.|i15 : S0=be
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│ │ │ │ +00011130: 2d2d 2d2b 0a7c 6931 3520 3a20 5330 3d62  ---+.|i15 : S0=b
│ │ │ │ +00011140: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
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│ │ │ │ +00011160: 5072 6f6a 6563 7469 7665 3d3e 3129 202d  Projective=>1) -
│ │ │ │ +00011170: 2d73 6f6c 7574 696f 6e73 2074 6f20 4820  -solutions to H 
│ │ │ │ +00011180: 7768 657c 0a7c 2020 2020 2020 2020 2020  whe|.|          
│ │ │ │  00011190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000111a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000111b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000111c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000111d0: 2020 7c0a 7c6f 3135 203d 207b 7b2d 2e34    |.|o15 = {{-.4
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│ │ │ │ -000111f0: 2c20 2e32 3135 3034 382d 2e34 3632 3233  , .215048-.46223
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│ │ │ │ -00011220: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d    |.|      -----
│ │ │ │ +000111d0: 2020 207c 0a7c 6f31 3520 3d20 7b7b 2d2e     |.|o15 = {{-.
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│ │ │ │ +00011220: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d     |.|      ----
│ │ │ │  00011230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011270: 2d2d 7c0a 7c20 2020 2020 202d 312e 3338  --|.|      -1.38
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│ │ │ │ +00011270: 2d2d 2d7c 0a7c 2020 2020 2020 2d31 2e33  ---|.|      -1.3
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│ │ │ │  00011290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000112a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000112b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000112c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000112c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000112d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000112e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000112f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011310: 2020 7c0a 7c6f 3135 203a 204c 6973 7420    |.|o15 : List 
│ │ │ │ +00011310: 2020 207c 0a7c 6f31 3520 3a20 4c69 7374     |.|o15 : List
│ │ │ │  00011320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011360: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +00011360: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │  00011370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000113a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000113b0: 2d2d 7c0a 7c6e 2074 3d30 2020 2020 2020  --|.|n t=0      
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│ │ │ │  000113d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00011400: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00011410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00011510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
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│ │ │ │ -00011560: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -00011570: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
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│ │ │ │ -000115a0: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ -000115b0: 6f70 792c 2069 7320 6120 2a6e 6f74 6520  opy, is a *note 
│ │ │ │ -000115c0: 6d65 7468 6f64 0a66 756e 6374 696f 6e20  method.function 
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│ │ │ │ -00011670: 6f6d 6f74 6f70 792c 2050 7265 763a 2062  omotopy, Prev: b
│ │ │ │ -00011680: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
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│ │ │ │ -000116a0: 7274 696e 6954 7261 636b 486f 6d6f 746f  rtiniTrackHomoto
│ │ │ │ -000116b0: 7079 282e 2e2e 2c56 6572 626f 7365 3d3e  py(...,Verbose=>
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│ │ │ │ +00011510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00011520: 202a 2022 6265 7274 696e 6954 7261 636b   * "bertiniTrack
│ │ │ │ +00011530: 486f 6d6f 746f 7079 2852 696e 6745 6c65  Homotopy(RingEle
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│ │ │ │ +00011550: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
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│ │ │ │ +00011570: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
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│ │ │ │ +000115a0: 6265 7274 696e 6954 7261 636b 486f 6d6f  bertiniTrackHomo
│ │ │ │ +000115b0: 746f 7079 2c20 6973 2061 202a 6e6f 7465  topy, is a *note
│ │ │ │ +000115c0: 206d 6574 686f 640a 6675 6e63 7469 6f6e   method.function
│ │ │ │ +000115d0: 2077 6974 6820 6f70 7469 6f6e 733a 2028   with options: (
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│ │ │ │ +00011630: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
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│ │ │ │ +00011680: 6265 7274 696e 6954 7261 636b 486f 6d6f  bertiniTrackHomo
│ │ │ │ +00011690: 746f 7079 2c20 5570 3a20 546f 700a 0a62  topy, Up: Top..b
│ │ │ │ +000116a0: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ +000116b0: 6f70 7928 2e2e 2e2c 5665 7262 6f73 653d  opy(...,Verbose=
│ │ │ │ +000116c0: 3e2e 2e2e 2920 2d2d 204f 7074 696f 6e20  >...) -- Option 
│ │ │ │ +000116d0: 746f 2073 696c 656e 6365 2061 6464 6974  to silence addit
│ │ │ │ +000116e0: 696f 6e61 6c20 6f75 7470 7574 0a2a 2a2a  ional output.***
│ │ │ │  000116f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00011700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00011710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00011720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011730: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ -00011740: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ -00011750: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -00011760: 2020 6265 7274 696e 6954 7261 636b 486f    bertiniTrackHo
│ │ │ │ -00011770: 6d6f 746f 7079 5665 7262 6f73 6528 2e2e  motopyVerbose(..
│ │ │ │ -00011780: 2e2c 5665 7262 6f73 653d 3e42 6f6f 6c65  .,Verbose=>Boole
│ │ │ │ -00011790: 616e 290a 2020 2020 2020 2020 6265 7274  an).        bert
│ │ │ │ -000117a0: 696e 6955 7365 7248 6f6d 6f74 6f70 7956  iniUserHomotopyV
│ │ │ │ -000117b0: 6572 626f 7365 282e 2e2e 2c56 6572 626f  erbose(...,Verbo
│ │ │ │ -000117c0: 7365 3d3e 426f 6f6c 6561 6e29 0a20 2020  se=>Boolean).   
│ │ │ │ -000117d0: 2020 2020 2062 6572 7469 6e69 506f 7344       bertiniPosD
│ │ │ │ -000117e0: 696d 536f 6c76 6528 2e2e 2e2c 5665 7262  imSolve(...,Verb
│ │ │ │ -000117f0: 6f73 653d 3e42 6f6f 6c65 616e 290a 2020  ose=>Boolean).  
│ │ │ │ -00011800: 2020 2020 2020 6265 7274 696e 6952 6566        bertiniRef
│ │ │ │ -00011810: 696e 6553 6f6c 7328 2e2e 2e2c 5665 7262  ineSols(...,Verb
│ │ │ │ -00011820: 6f73 653d 3e42 6f6f 6c65 616e 290a 2020  ose=>Boolean).  
│ │ │ │ -00011830: 2020 2020 2020 6265 7274 696e 6953 616d        bertiniSam
│ │ │ │ -00011840: 706c 6528 2e2e 2e2c 5665 7262 6f73 653d  ple(...,Verbose=
│ │ │ │ -00011850: 3e42 6f6f 6c65 616e 290a 2020 2020 2020  >Boolean).      
│ │ │ │ -00011860: 2020 6265 7274 696e 695a 6572 6f44 696d    bertiniZeroDim
│ │ │ │ -00011870: 536f 6c76 6528 2e2e 2e2c 5665 7262 6f73  Solve(...,Verbos
│ │ │ │ -00011880: 653d 3e42 6f6f 6c65 616e 290a 2020 2020  e=>Boolean).    
│ │ │ │ -00011890: 2020 2020 6265 7274 696e 6950 6172 616d      bertiniParam
│ │ │ │ -000118a0: 6574 6572 486f 6d6f 746f 7079 282e 2e2e  eterHomotopy(...
│ │ │ │ -000118b0: 2c56 6572 626f 7365 3d3e 426f 6f6c 6561  ,Verbose=>Boolea
│ │ │ │ -000118c0: 6e29 0a20 2020 2020 2020 206d 616b 6542  n).        makeB
│ │ │ │ -000118d0: 2749 6e70 7574 4669 6c65 282e 2e2e 2c56  'InputFile(...,V
│ │ │ │ -000118e0: 6572 626f 7365 3d3e 426f 6f6c 6561 6e29  erbose=>Boolean)
│ │ │ │ -000118f0: 0a20 2020 2020 2020 206d 616b 654d 656d  .        makeMem
│ │ │ │ -00011900: 6265 7273 6869 7046 696c 6528 2e2e 2e2c  bershipFile(...,
│ │ │ │ -00011910: 5665 7262 6f73 653d 3e42 6f6f 6c65 616e  Verbose=>Boolean
│ │ │ │ -00011920: 290a 2020 2020 2020 2020 6227 5048 4761  ).        b'PHGa
│ │ │ │ -00011930: 6c6f 6973 4772 6f75 7028 2e2e 2e2c 5665  loisGroup(...,Ve
│ │ │ │ -00011940: 7262 6f73 653d 3e42 6f6f 6c65 616e 290a  rbose=>Boolean).
│ │ │ │ -00011950: 2020 2020 2020 2020 6227 5048 4d6f 6e6f          b'PHMono
│ │ │ │ -00011960: 6472 6f6d 7943 6f6c 6c65 6374 282e 2e2e  dromyCollect(...
│ │ │ │ -00011970: 2c56 6572 626f 7365 3d3e 426f 6f6c 6561  ,Verbose=>Boolea
│ │ │ │ -00011980: 6e29 0a20 2020 2020 2020 2069 6d70 6f72  n).        impor
│ │ │ │ -00011990: 7449 6e63 6964 656e 6365 4d61 7472 6978  tIncidenceMatrix
│ │ │ │ -000119a0: 282e 2e2e 2c56 6572 626f 7365 3d3e 426f  (...,Verbose=>Bo
│ │ │ │ -000119b0: 6f6c 6561 6e29 0a20 2020 2020 2020 2069  olean).        i
│ │ │ │ -000119c0: 6d70 6f72 744d 6169 6e44 6174 6146 696c  mportMainDataFil
│ │ │ │ -000119d0: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e42  e(...,Verbose=>B
│ │ │ │ -000119e0: 6f6f 6c65 616e 290a 2020 2020 2020 2020  oolean).        
│ │ │ │ -000119f0: 696d 706f 7274 536c 6963 6546 696c 6528  importSliceFile(
│ │ │ │ -00011a00: 2e2e 2e2c 5665 7262 6f73 653d 3e42 6f6f  ...,Verbose=>Boo
│ │ │ │ -00011a10: 6c65 616e 290a 2020 2020 2020 2020 696d  lean).        im
│ │ │ │ -00011a20: 706f 7274 536f 6c75 7469 6f6e 7346 696c  portSolutionsFil
│ │ │ │ -00011a30: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e42  e(...,Verbose=>B
│ │ │ │ -00011a40: 6f6f 6c65 616e 290a 2020 2020 2020 2020  oolean).        
│ │ │ │ -00011a50: 7275 6e42 6572 7469 6e69 282e 2e2e 2c56  runBertini(...,V
│ │ │ │ -00011a60: 6572 626f 7365 3d3e 426f 6f6c 6561 6e29  erbose=>Boolean)
│ │ │ │ -00011a70: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00011a80: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 2056  =========..Use V
│ │ │ │ -00011a90: 6572 626f 7365 3d3e 6661 6c73 6520 746f  erbose=>false to
│ │ │ │ -00011aa0: 2073 696c 656e 6365 2061 6464 6974 696f   silence additio
│ │ │ │ -00011ab0: 6e61 6c20 6f75 7470 7574 2e0a 0a46 7572  nal output...Fur
│ │ │ │ -00011ac0: 7468 6572 2069 6e66 6f72 6d61 7469 6f6e  ther information
│ │ │ │ -00011ad0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00011ae0: 3d3d 3d3d 0a0a 2020 2a20 4465 6661 756c  ====..  * Defaul
│ │ │ │ -00011af0: 7420 7661 6c75 653a 202a 6e6f 7465 2066  t value: *note f
│ │ │ │ -00011b00: 616c 7365 3a20 284d 6163 6175 6c61 7932  alse: (Macaulay2
│ │ │ │ -00011b10: 446f 6329 6661 6c73 652c 0a20 202a 2046  Doc)false,.  * F
│ │ │ │ -00011b20: 756e 6374 696f 6e3a 202a 6e6f 7465 2062  unction: *note b
│ │ │ │ -00011b30: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ -00011b40: 6f70 793a 2062 6572 7469 6e69 5472 6163  opy: bertiniTrac
│ │ │ │ -00011b50: 6b48 6f6d 6f74 6f70 792c 202d 2d20 6120  kHomotopy, -- a 
│ │ │ │ -00011b60: 6d61 696e 0a20 2020 206d 6574 686f 6420  main.    method 
│ │ │ │ -00011b70: 746f 2074 7261 636b 2075 7369 6e67 2061  to track using a
│ │ │ │ -00011b80: 2075 7365 722d 6465 6669 6e65 6420 686f   user-defined ho
│ │ │ │ -00011b90: 6d6f 746f 7079 0a20 202a 204f 7074 696f  motopy.  * Optio
│ │ │ │ -00011ba0: 6e20 6b65 793a 202a 6e6f 7465 2056 6572  n key: *note Ver
│ │ │ │ -00011bb0: 626f 7365 3a20 284d 6163 6175 6c61 7932  bose: (Macaulay2
│ │ │ │ -00011bc0: 446f 6329 5665 7262 6f73 652c 202d 2d20  Doc)Verbose, -- 
│ │ │ │ -00011bd0: 7265 7175 6573 7420 7665 7262 6f73 650a  request verbose.
│ │ │ │ -00011be0: 2020 2020 6665 6564 6261 636b 0a0a 4675      feedback..Fu
│ │ │ │ -00011bf0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -00011c00: 696f 6e61 6c20 6172 6775 6d65 6e74 206e  ional argument n
│ │ │ │ -00011c10: 616d 6564 2056 6572 626f 7365 3a0a 3d3d  amed Verbose:.==
│ │ │ │ +00011730: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +00011740: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +00011750: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00011760: 2020 2062 6572 7469 6e69 5472 6163 6b48     bertiniTrackH
│ │ │ │ +00011770: 6f6d 6f74 6f70 7956 6572 626f 7365 282e  omotopyVerbose(.
│ │ │ │ +00011780: 2e2e 2c56 6572 626f 7365 3d3e 426f 6f6c  ..,Verbose=>Bool
│ │ │ │ +00011790: 6561 6e29 0a20 2020 2020 2020 2062 6572  ean).        ber
│ │ │ │ +000117a0: 7469 6e69 5573 6572 486f 6d6f 746f 7079  tiniUserHomotopy
│ │ │ │ +000117b0: 5665 7262 6f73 6528 2e2e 2e2c 5665 7262  Verbose(...,Verb
│ │ │ │ +000117c0: 6f73 653d 3e42 6f6f 6c65 616e 290a 2020  ose=>Boolean).  
│ │ │ │ +000117d0: 2020 2020 2020 6265 7274 696e 6950 6f73        bertiniPos
│ │ │ │ +000117e0: 4469 6d53 6f6c 7665 282e 2e2e 2c56 6572  DimSolve(...,Ver
│ │ │ │ +000117f0: 626f 7365 3d3e 426f 6f6c 6561 6e29 0a20  bose=>Boolean). 
│ │ │ │ +00011800: 2020 2020 2020 2062 6572 7469 6e69 5265         bertiniRe
│ │ │ │ +00011810: 6669 6e65 536f 6c73 282e 2e2e 2c56 6572  fineSols(...,Ver
│ │ │ │ +00011820: 626f 7365 3d3e 426f 6f6c 6561 6e29 0a20  bose=>Boolean). 
│ │ │ │ +00011830: 2020 2020 2020 2062 6572 7469 6e69 5361         bertiniSa
│ │ │ │ +00011840: 6d70 6c65 282e 2e2e 2c56 6572 626f 7365  mple(...,Verbose
│ │ │ │ +00011850: 3d3e 426f 6f6c 6561 6e29 0a20 2020 2020  =>Boolean).     
│ │ │ │ +00011860: 2020 2062 6572 7469 6e69 5a65 726f 4469     bertiniZeroDi
│ │ │ │ +00011870: 6d53 6f6c 7665 282e 2e2e 2c56 6572 626f  mSolve(...,Verbo
│ │ │ │ +00011880: 7365 3d3e 426f 6f6c 6561 6e29 0a20 2020  se=>Boolean).   
│ │ │ │ +00011890: 2020 2020 2062 6572 7469 6e69 5061 7261       bertiniPara
│ │ │ │ +000118a0: 6d65 7465 7248 6f6d 6f74 6f70 7928 2e2e  meterHomotopy(..
│ │ │ │ +000118b0: 2e2c 5665 7262 6f73 653d 3e42 6f6f 6c65  .,Verbose=>Boole
│ │ │ │ +000118c0: 616e 290a 2020 2020 2020 2020 6d61 6b65  an).        make
│ │ │ │ +000118d0: 4227 496e 7075 7446 696c 6528 2e2e 2e2c  B'InputFile(...,
│ │ │ │ +000118e0: 5665 7262 6f73 653d 3e42 6f6f 6c65 616e  Verbose=>Boolean
│ │ │ │ +000118f0: 290a 2020 2020 2020 2020 6d61 6b65 4d65  ).        makeMe
│ │ │ │ +00011900: 6d62 6572 7368 6970 4669 6c65 282e 2e2e  mbershipFile(...
│ │ │ │ +00011910: 2c56 6572 626f 7365 3d3e 426f 6f6c 6561  ,Verbose=>Boolea
│ │ │ │ +00011920: 6e29 0a20 2020 2020 2020 2062 2750 4847  n).        b'PHG
│ │ │ │ +00011930: 616c 6f69 7347 726f 7570 282e 2e2e 2c56  aloisGroup(...,V
│ │ │ │ +00011940: 6572 626f 7365 3d3e 426f 6f6c 6561 6e29  erbose=>Boolean)
│ │ │ │ +00011950: 0a20 2020 2020 2020 2062 2750 484d 6f6e  .        b'PHMon
│ │ │ │ +00011960: 6f64 726f 6d79 436f 6c6c 6563 7428 2e2e  odromyCollect(..
│ │ │ │ +00011970: 2e2c 5665 7262 6f73 653d 3e42 6f6f 6c65  .,Verbose=>Boole
│ │ │ │ +00011980: 616e 290a 2020 2020 2020 2020 696d 706f  an).        impo
│ │ │ │ +00011990: 7274 496e 6369 6465 6e63 654d 6174 7269  rtIncidenceMatri
│ │ │ │ +000119a0: 7828 2e2e 2e2c 5665 7262 6f73 653d 3e42  x(...,Verbose=>B
│ │ │ │ +000119b0: 6f6f 6c65 616e 290a 2020 2020 2020 2020  oolean).        
│ │ │ │ +000119c0: 696d 706f 7274 4d61 696e 4461 7461 4669  importMainDataFi
│ │ │ │ +000119d0: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e  le(...,Verbose=>
│ │ │ │ +000119e0: 426f 6f6c 6561 6e29 0a20 2020 2020 2020  Boolean).       
│ │ │ │ +000119f0: 2069 6d70 6f72 7453 6c69 6365 4669 6c65   importSliceFile
│ │ │ │ +00011a00: 282e 2e2e 2c56 6572 626f 7365 3d3e 426f  (...,Verbose=>Bo
│ │ │ │ +00011a10: 6f6c 6561 6e29 0a20 2020 2020 2020 2069  olean).        i
│ │ │ │ +00011a20: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669  mportSolutionsFi
│ │ │ │ +00011a30: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e  le(...,Verbose=>
│ │ │ │ +00011a40: 426f 6f6c 6561 6e29 0a20 2020 2020 2020  Boolean).       
│ │ │ │ +00011a50: 2072 756e 4265 7274 696e 6928 2e2e 2e2c   runBertini(...,
│ │ │ │ +00011a60: 5665 7262 6f73 653d 3e42 6f6f 6c65 616e  Verbose=>Boolean
│ │ │ │ +00011a70: 290a 0a44 6573 6372 6970 7469 6f6e 0a3d  )..Description.=
│ │ │ │ +00011a80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5573 6520  ==========..Use 
│ │ │ │ +00011a90: 5665 7262 6f73 653d 3e66 616c 7365 2074  Verbose=>false t
│ │ │ │ +00011aa0: 6f20 7369 6c65 6e63 6520 6164 6469 7469  o silence additi
│ │ │ │ +00011ab0: 6f6e 616c 206f 7574 7075 742e 0a0a 4675  onal output...Fu
│ │ │ │ +00011ac0: 7274 6865 7220 696e 666f 726d 6174 696f  rther informatio
│ │ │ │ +00011ad0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n.==============
│ │ │ │ +00011ae0: 3d3d 3d3d 3d0a 0a20 202a 2044 6566 6175  =====..  * Defau
│ │ │ │ +00011af0: 6c74 2076 616c 7565 3a20 2a6e 6f74 6520  lt value: *note 
│ │ │ │ +00011b00: 6661 6c73 653a 2028 4d61 6361 756c 6179  false: (Macaulay
│ │ │ │ +00011b10: 3244 6f63 2966 616c 7365 2c0a 2020 2a20  2Doc)false,.  * 
│ │ │ │ +00011b20: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +00011b30: 6265 7274 696e 6954 7261 636b 486f 6d6f  bertiniTrackHomo
│ │ │ │ +00011b40: 746f 7079 3a20 6265 7274 696e 6954 7261  topy: bertiniTra
│ │ │ │ +00011b50: 636b 486f 6d6f 746f 7079 2c20 2d2d 2061  ckHomotopy, -- a
│ │ │ │ +00011b60: 206d 6169 6e0a 2020 2020 6d65 7468 6f64   main.    method
│ │ │ │ +00011b70: 2074 6f20 7472 6163 6b20 7573 696e 6720   to track using 
│ │ │ │ +00011b80: 6120 7573 6572 2d64 6566 696e 6564 2068  a user-defined h
│ │ │ │ +00011b90: 6f6d 6f74 6f70 790a 2020 2a20 4f70 7469  omotopy.  * Opti
│ │ │ │ +00011ba0: 6f6e 206b 6579 3a20 2a6e 6f74 6520 5665  on key: *note Ve
│ │ │ │ +00011bb0: 7262 6f73 653a 2028 4d61 6361 756c 6179  rbose: (Macaulay
│ │ │ │ +00011bc0: 3244 6f63 2956 6572 626f 7365 2c20 2d2d  2Doc)Verbose, --
│ │ │ │ +00011bd0: 2072 6571 7565 7374 2076 6572 626f 7365   request verbose
│ │ │ │ +00011be0: 0a20 2020 2066 6565 6462 6163 6b0a 0a46  .    feedback..F
│ │ │ │ +00011bf0: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +00011c00: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ +00011c10: 6e61 6d65 6420 5665 7262 6f73 653a 0a3d  named Verbose:.=
│ │ │ │  00011c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00011c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00011c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00011c50: 202a 2022 6265 7274 696e 6943 6f6d 706f   * "bertiniCompo
│ │ │ │ -00011c60: 6e65 6e74 4d65 6d62 6572 5465 7374 282e  nentMemberTest(.
│ │ │ │ -00011c70: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29  ..,Verbose=>...)
│ │ │ │ -00011c80: 220a 2020 2a20 2262 6572 7469 6e69 5061  ".  * "bertiniPa
│ │ │ │ -00011c90: 7261 6d65 7465 7248 6f6d 6f74 6f70 7928  rameterHomotopy(
│ │ │ │ -00011ca0: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ -00011cb0: 2922 0a20 202a 2022 6265 7274 696e 6950  )".  * "bertiniP
│ │ │ │ -00011cc0: 6f73 4469 6d53 6f6c 7665 282e 2e2e 2c56  osDimSolve(...,V
│ │ │ │ -00011cd0: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020  erbose=>...)".  
│ │ │ │ -00011ce0: 2a20 2262 6572 7469 6e69 5265 6669 6e65  * "bertiniRefine
│ │ │ │ -00011cf0: 536f 6c73 282e 2e2e 2c56 6572 626f 7365  Sols(...,Verbose
│ │ │ │ -00011d00: 3d3e 2e2e 2e29 220a 2020 2a20 2262 6572  =>...)".  * "ber
│ │ │ │ -00011d10: 7469 6e69 5361 6d70 6c65 282e 2e2e 2c56  tiniSample(...,V
│ │ │ │ -00011d20: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020  erbose=>...)".  
│ │ │ │ -00011d30: 2a20 2a6e 6f74 6520 6265 7274 696e 6954  * *note bertiniT
│ │ │ │ -00011d40: 7261 636b 486f 6d6f 746f 7079 282e 2e2e  rackHomotopy(...
│ │ │ │ -00011d50: 2c56 6572 626f 7365 3d3e 2e2e 2e29 3a0a  ,Verbose=>...):.
│ │ │ │ -00011d60: 2020 2020 6265 7274 696e 6954 7261 636b      bertiniTrack
│ │ │ │ -00011d70: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70  Homotopy_lp_pd_p
│ │ │ │ -00011d80: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ -00011d90: 5f70 645f 7064 5f70 645f 7270 2c20 2d2d  _pd_pd_pd_rp, --
│ │ │ │ -00011da0: 204f 7074 696f 6e20 746f 0a20 2020 2073   Option to.    s
│ │ │ │ -00011db0: 696c 656e 6365 2061 6464 6974 696f 6e61  ilence additiona
│ │ │ │ -00011dc0: 6c20 6f75 7470 7574 0a20 202a 2022 6265  l output.  * "be
│ │ │ │ -00011dd0: 7274 696e 6955 7365 7248 6f6d 6f74 6f70  rtiniUserHomotop
│ │ │ │ -00011de0: 7928 2e2e 2e2c 5665 7262 6f73 653d 3e2e  y(...,Verbose=>.
│ │ │ │ -00011df0: 2e2e 2922 0a20 202a 2022 6265 7274 696e  ..)".  * "bertin
│ │ │ │ -00011e00: 695a 6572 6f44 696d 536f 6c76 6528 2e2e  iZeroDimSolve(..
│ │ │ │ -00011e10: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922  .,Verbose=>...)"
│ │ │ │ -00011e20: 0a20 202a 2022 696d 706f 7274 496e 6369  .  * "importInci
│ │ │ │ -00011e30: 6465 6e63 654d 6174 7269 7828 2e2e 2e2c  denceMatrix(...,
│ │ │ │ -00011e40: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ -00011e50: 202a 2022 696d 706f 7274 4d61 696e 4461   * "importMainDa
│ │ │ │ -00011e60: 7461 4669 6c65 282e 2e2e 2c56 6572 626f  taFile(...,Verbo
│ │ │ │ -00011e70: 7365 3d3e 2e2e 2e29 220a 2020 2a20 2269  se=>...)".  * "i
│ │ │ │ -00011e80: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669  mportSolutionsFi
│ │ │ │ -00011e90: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e  le(...,Verbose=>
│ │ │ │ -00011ea0: 2e2e 2e29 220a 2020 2a20 226d 616b 6542  ...)".  * "makeB
│ │ │ │ -00011eb0: 2749 6e70 7574 4669 6c65 282e 2e2e 2c56  'InputFile(...,V
│ │ │ │ -00011ec0: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020  erbose=>...)".  
│ │ │ │ -00011ed0: 2a20 226d 616b 654d 656d 6265 7273 6869  * "makeMembershi
│ │ │ │ -00011ee0: 7046 696c 6528 2e2e 2e2c 5665 7262 6f73  pFile(...,Verbos
│ │ │ │ -00011ef0: 653d 3e2e 2e2e 2922 0a20 202a 2022 6d61  e=>...)".  * "ma
│ │ │ │ -00011f00: 6b65 5361 6d70 6c65 536f 6c75 7469 6f6e  keSampleSolution
│ │ │ │ -00011f10: 7346 696c 6528 2e2e 2e2c 5665 7262 6f73  sFile(...,Verbos
│ │ │ │ -00011f20: 653d 3e2e 2e2e 2922 0a20 202a 2022 7275  e=>...)".  * "ru
│ │ │ │ -00011f30: 6e42 6572 7469 6e69 282e 2e2e 2c56 6572  nBertini(...,Ver
│ │ │ │ -00011f40: 626f 7365 3d3e 2e2e 2e29 220a 2020 2a20  bose=>...)".  * 
│ │ │ │ -00011f50: 2263 6865 636b 282e 2e2e 2c56 6572 626f  "check(...,Verbo
│ │ │ │ -00011f60: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565  se=>...)" -- see
│ │ │ │ -00011f70: 202a 6e6f 7465 2063 6865 636b 3a20 284d   *note check: (M
│ │ │ │ -00011f80: 6163 6175 6c61 7932 446f 6329 6368 6563  acaulay2Doc)chec
│ │ │ │ -00011f90: 6b2c 202d 2d0a 2020 2020 7065 7266 6f72  k, --.    perfor
│ │ │ │ -00011fa0: 6d20 7465 7374 7320 6f66 2061 2070 6163  m tests of a pac
│ │ │ │ -00011fb0: 6b61 6765 0a20 202a 2022 6368 6563 6b44  kage.  * "checkD
│ │ │ │ -00011fc0: 6567 7265 6573 282e 2e2e 2c56 6572 626f  egrees(...,Verbo
│ │ │ │ -00011fd0: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565  se=>...)" -- see
│ │ │ │ -00011fe0: 202a 6e6f 7465 2063 6865 636b 4465 6772   *note checkDegr
│ │ │ │ -00011ff0: 6565 733a 0a20 2020 2028 4973 6f6d 6f72  ees:.    (Isomor
│ │ │ │ -00012000: 7068 6973 6d29 6368 6563 6b44 6567 7265  phism)checkDegre
│ │ │ │ -00012010: 6573 2c20 2d2d 2063 6f6d 7061 7265 7320  es, -- compares 
│ │ │ │ -00012020: 7468 6520 6465 6772 6565 7320 6f66 2067  the degrees of g
│ │ │ │ -00012030: 656e 6572 6174 6f72 7320 6f66 2074 776f  enerators of two
│ │ │ │ -00012040: 0a20 2020 206d 6f64 756c 6573 0a20 202a  .    modules.  *
│ │ │ │ -00012050: 2022 636f 7079 4469 7265 6374 6f72 7928   "copyDirectory(
│ │ │ │ -00012060: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ -00012070: 2922 202d 2d20 7365 6520 2a6e 6f74 650a  )" -- see *note.
│ │ │ │ -00012080: 2020 2020 636f 7079 4469 7265 6374 6f72      copyDirector
│ │ │ │ -00012090: 7928 5374 7269 6e67 2c53 7472 696e 6729  y(String,String)
│ │ │ │ -000120a0: 3a0a 2020 2020 284d 6163 6175 6c61 7932  :.    (Macaulay2
│ │ │ │ -000120b0: 446f 6329 636f 7079 4469 7265 6374 6f72  Doc)copyDirector
│ │ │ │ -000120c0: 795f 6c70 5374 7269 6e67 5f63 6d53 7472  y_lpString_cmStr
│ │ │ │ -000120d0: 696e 675f 7270 2c0a 2020 2a20 2263 6f70  ing_rp,.  * "cop
│ │ │ │ -000120e0: 7946 696c 6528 2e2e 2e2c 5665 7262 6f73  yFile(...,Verbos
│ │ │ │ -000120f0: 653d 3e2e 2e2e 2922 202d 2d20 7365 6520  e=>...)" -- see 
│ │ │ │ -00012100: 2a6e 6f74 6520 636f 7079 4669 6c65 2853  *note copyFile(S
│ │ │ │ -00012110: 7472 696e 672c 5374 7269 6e67 293a 0a20  tring,String):. 
│ │ │ │ -00012120: 2020 2028 4d61 6361 756c 6179 3244 6f63     (Macaulay2Doc
│ │ │ │ -00012130: 2963 6f70 7946 696c 655f 6c70 5374 7269  )copyFile_lpStri
│ │ │ │ -00012140: 6e67 5f63 6d53 7472 696e 675f 7270 2c0a  ng_cmString_rp,.
│ │ │ │ -00012150: 2020 2a20 2266 696e 6450 726f 6772 616d    * "findProgram
│ │ │ │ -00012160: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e  (...,Verbose=>..
│ │ │ │ -00012170: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ -00012180: 2066 696e 6450 726f 6772 616d 3a0a 2020   findProgram:.  
│ │ │ │ -00012190: 2020 284d 6163 6175 6c61 7932 446f 6329    (Macaulay2Doc)
│ │ │ │ -000121a0: 6669 6e64 5072 6f67 7261 6d2c 202d 2d20  findProgram, -- 
│ │ │ │ -000121b0: 6c6f 6164 2065 7874 6572 6e61 6c20 7072  load external pr
│ │ │ │ -000121c0: 6f67 7261 6d0a 2020 2a20 2269 6e73 7461  ogram.  * "insta
│ │ │ │ -000121d0: 6c6c 5061 636b 6167 6528 2e2e 2e2c 5665  llPackage(...,Ve
│ │ │ │ -000121e0: 7262 6f73 653d 3e2e 2e2e 2922 202d 2d20  rbose=>...)" -- 
│ │ │ │ -000121f0: 7365 6520 2a6e 6f74 6520 696e 7374 616c  see *note instal
│ │ │ │ -00012200: 6c50 6163 6b61 6765 3a0a 2020 2020 284d  lPackage:.    (M
│ │ │ │ -00012210: 6163 6175 6c61 7932 446f 6329 696e 7374  acaulay2Doc)inst
│ │ │ │ -00012220: 616c 6c50 6163 6b61 6765 2c20 2d2d 206c  allPackage, -- l
│ │ │ │ -00012230: 6f61 6420 616e 6420 696e 7374 616c 6c20  oad and install 
│ │ │ │ -00012240: 6120 7061 636b 6167 6520 616e 6420 6974  a package and it
│ │ │ │ -00012250: 730a 2020 2020 646f 6375 6d65 6e74 6174  s.    documentat
│ │ │ │ -00012260: 696f 6e0a 2020 2a20 2269 7349 736f 6d6f  ion.  * "isIsomo
│ │ │ │ -00012270: 7270 6869 6328 2e2e 2e2c 5665 7262 6f73  rphic(...,Verbos
│ │ │ │ -00012280: 653d 3e2e 2e2e 2922 202d 2d20 7365 6520  e=>...)" -- see 
│ │ │ │ -00012290: 2a6e 6f74 6520 6973 4973 6f6d 6f72 7068  *note isIsomorph
│ │ │ │ -000122a0: 6963 3a0a 2020 2020 2849 736f 6d6f 7270  ic:.    (Isomorp
│ │ │ │ -000122b0: 6869 736d 2969 7349 736f 6d6f 7270 6869  hism)isIsomorphi
│ │ │ │ -000122c0: 632c 202d 2d20 5072 6f62 6162 696c 6973  c, -- Probabilis
│ │ │ │ -000122d0: 7469 6320 7465 7374 2066 6f72 2069 736f  tic test for iso
│ │ │ │ -000122e0: 6d6f 7270 6869 736d 206f 6620 6d6f 6475  morphism of modu
│ │ │ │ -000122f0: 6c65 730a 2020 2a20 226d 6f76 6546 696c  les.  * "moveFil
│ │ │ │ -00012300: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e2e  e(...,Verbose=>.
│ │ │ │ -00012310: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ -00012320: 6520 6d6f 7665 4669 6c65 2853 7472 696e  e moveFile(Strin
│ │ │ │ -00012330: 672c 5374 7269 6e67 293a 0a20 2020 2028  g,String):.    (
│ │ │ │ -00012340: 4d61 6361 756c 6179 3244 6f63 296d 6f76  Macaulay2Doc)mov
│ │ │ │ -00012350: 6546 696c 655f 6c70 5374 7269 6e67 5f63  eFile_lpString_c
│ │ │ │ -00012360: 6d53 7472 696e 675f 7270 2c0a 2020 2a20  mString_rp,.  * 
│ │ │ │ -00012370: 2272 756e 5072 6f67 7261 6d28 2e2e 2e2c  "runProgram(...,
│ │ │ │ -00012380: 5665 7262 6f73 653d 3e2e 2e2e 2922 202d  Verbose=>...)" -
│ │ │ │ -00012390: 2d20 7365 6520 2a6e 6f74 6520 7275 6e50  - see *note runP
│ │ │ │ -000123a0: 726f 6772 616d 3a0a 2020 2020 284d 6163  rogram:.    (Mac
│ │ │ │ -000123b0: 6175 6c61 7932 446f 6329 7275 6e50 726f  aulay2Doc)runPro
│ │ │ │ -000123c0: 6772 616d 2c20 2d2d 2072 756e 2061 6e20  gram, -- run an 
│ │ │ │ -000123d0: 6578 7465 726e 616c 2070 726f 6772 616d  external program
│ │ │ │ -000123e0: 0a20 202a 2022 7379 6d6c 696e 6b44 6972  .  * "symlinkDir
│ │ │ │ -000123f0: 6563 746f 7279 282e 2e2e 2c56 6572 626f  ectory(...,Verbo
│ │ │ │ -00012400: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565  se=>...)" -- see
│ │ │ │ -00012410: 202a 6e6f 7465 0a20 2020 2073 796d 6c69   *note.    symli
│ │ │ │ -00012420: 6e6b 4469 7265 6374 6f72 7928 5374 7269  nkDirectory(Stri
│ │ │ │ -00012430: 6e67 2c53 7472 696e 6729 3a0a 2020 2020  ng,String):.    
│ │ │ │ -00012440: 284d 6163 6175 6c61 7932 446f 6329 7379  (Macaulay2Doc)sy
│ │ │ │ -00012450: 6d6c 696e 6b44 6972 6563 746f 7279 5f6c  mlinkDirectory_l
│ │ │ │ -00012460: 7053 7472 696e 675f 636d 5374 7269 6e67  pString_cmString
│ │ │ │ -00012470: 5f72 702c 202d 2d20 6d61 6b65 2073 796d  _rp, -- make sym
│ │ │ │ -00012480: 626f 6c69 6320 6c69 6e6b 730a 2020 2020  bolic links.    
│ │ │ │ -00012490: 666f 7220 616c 6c20 6669 6c65 7320 696e  for all files in
│ │ │ │ -000124a0: 2061 2064 6972 6563 746f 7279 2074 7265   a directory tre
│ │ │ │ -000124b0: 650a 1f0a 4669 6c65 3a20 4265 7274 696e  e...File: Bertin
│ │ │ │ -000124c0: 692e 696e 666f 2c20 4e6f 6465 3a20 6265  i.info, Node: be
│ │ │ │ -000124d0: 7274 696e 6955 7365 7248 6f6d 6f74 6f70  rtiniUserHomotop
│ │ │ │ -000124e0: 792c 204e 6578 743a 2062 6572 7469 6e69  y, Next: bertini
│ │ │ │ -000124f0: 5a65 726f 4469 6d53 6f6c 7665 2c20 5072  ZeroDimSolve, Pr
│ │ │ │ -00012500: 6576 3a20 6265 7274 696e 6954 7261 636b  ev: bertiniTrack
│ │ │ │ -00012510: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70  Homotopy_lp_pd_p
│ │ │ │ -00012520: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ -00012530: 5f70 645f 7064 5f70 645f 7270 2c20 5570  _pd_pd_pd_rp, Up
│ │ │ │ -00012540: 3a20 546f 700a 0a62 6572 7469 6e69 5573  : Top..bertiniUs
│ │ │ │ -00012550: 6572 486f 6d6f 746f 7079 202d 2d20 6120  erHomotopy -- a 
│ │ │ │ -00012560: 6d61 696e 206d 6574 686f 6420 746f 2074  main method to t
│ │ │ │ -00012570: 7261 636b 2061 2075 7365 722d 6465 6669  rack a user-defi
│ │ │ │ -00012580: 6e65 6420 686f 6d6f 746f 7079 0a2a 2a2a  ned homotopy.***
│ │ │ │ +00011c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00011c50: 2020 2a20 2262 6572 7469 6e69 436f 6d70    * "bertiniComp
│ │ │ │ +00011c60: 6f6e 656e 744d 656d 6265 7254 6573 7428  onentMemberTest(
│ │ │ │ +00011c70: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ +00011c80: 2922 0a20 202a 2022 6265 7274 696e 6950  )".  * "bertiniP
│ │ │ │ +00011c90: 6172 616d 6574 6572 486f 6d6f 746f 7079  arameterHomotopy
│ │ │ │ +00011ca0: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e  (...,Verbose=>..
│ │ │ │ +00011cb0: 2e29 220a 2020 2a20 2262 6572 7469 6e69  .)".  * "bertini
│ │ │ │ +00011cc0: 506f 7344 696d 536f 6c76 6528 2e2e 2e2c  PosDimSolve(...,
│ │ │ │ +00011cd0: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ +00011ce0: 202a 2022 6265 7274 696e 6952 6566 696e   * "bertiniRefin
│ │ │ │ +00011cf0: 6553 6f6c 7328 2e2e 2e2c 5665 7262 6f73  eSols(...,Verbos
│ │ │ │ +00011d00: 653d 3e2e 2e2e 2922 0a20 202a 2022 6265  e=>...)".  * "be
│ │ │ │ +00011d10: 7274 696e 6953 616d 706c 6528 2e2e 2e2c  rtiniSample(...,
│ │ │ │ +00011d20: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ +00011d30: 202a 202a 6e6f 7465 2062 6572 7469 6e69   * *note bertini
│ │ │ │ +00011d40: 5472 6163 6b48 6f6d 6f74 6f70 7928 2e2e  TrackHomotopy(..
│ │ │ │ +00011d50: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 293a  .,Verbose=>...):
│ │ │ │ +00011d60: 0a20 2020 2062 6572 7469 6e69 5472 6163  .    bertiniTrac
│ │ │ │ +00011d70: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │ +00011d80: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ +00011d90: 3e5f 7064 5f70 645f 7064 5f72 702c 202d  >_pd_pd_pd_rp, -
│ │ │ │ +00011da0: 2d20 4f70 7469 6f6e 2074 6f0a 2020 2020  - Option to.    
│ │ │ │ +00011db0: 7369 6c65 6e63 6520 6164 6469 7469 6f6e  silence addition
│ │ │ │ +00011dc0: 616c 206f 7574 7075 740a 2020 2a20 2262  al output.  * "b
│ │ │ │ +00011dd0: 6572 7469 6e69 5573 6572 486f 6d6f 746f  ertiniUserHomoto
│ │ │ │ +00011de0: 7079 282e 2e2e 2c56 6572 626f 7365 3d3e  py(...,Verbose=>
│ │ │ │ +00011df0: 2e2e 2e29 220a 2020 2a20 2262 6572 7469  ...)".  * "berti
│ │ │ │ +00011e00: 6e69 5a65 726f 4469 6d53 6f6c 7665 282e  niZeroDimSolve(.
│ │ │ │ +00011e10: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29  ..,Verbose=>...)
│ │ │ │ +00011e20: 220a 2020 2a20 2269 6d70 6f72 7449 6e63  ".  * "importInc
│ │ │ │ +00011e30: 6964 656e 6365 4d61 7472 6978 282e 2e2e  idenceMatrix(...
│ │ │ │ +00011e40: 2c56 6572 626f 7365 3d3e 2e2e 2e29 220a  ,Verbose=>...)".
│ │ │ │ +00011e50: 2020 2a20 2269 6d70 6f72 744d 6169 6e44    * "importMainD
│ │ │ │ +00011e60: 6174 6146 696c 6528 2e2e 2e2c 5665 7262  ataFile(...,Verb
│ │ │ │ +00011e70: 6f73 653d 3e2e 2e2e 2922 0a20 202a 2022  ose=>...)".  * "
│ │ │ │ +00011e80: 696d 706f 7274 536f 6c75 7469 6f6e 7346  importSolutionsF
│ │ │ │ +00011e90: 696c 6528 2e2e 2e2c 5665 7262 6f73 653d  ile(...,Verbose=
│ │ │ │ +00011ea0: 3e2e 2e2e 2922 0a20 202a 2022 6d61 6b65  >...)".  * "make
│ │ │ │ +00011eb0: 4227 496e 7075 7446 696c 6528 2e2e 2e2c  B'InputFile(...,
│ │ │ │ +00011ec0: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ +00011ed0: 202a 2022 6d61 6b65 4d65 6d62 6572 7368   * "makeMembersh
│ │ │ │ +00011ee0: 6970 4669 6c65 282e 2e2e 2c56 6572 626f  ipFile(...,Verbo
│ │ │ │ +00011ef0: 7365 3d3e 2e2e 2e29 220a 2020 2a20 226d  se=>...)".  * "m
│ │ │ │ +00011f00: 616b 6553 616d 706c 6553 6f6c 7574 696f  akeSampleSolutio
│ │ │ │ +00011f10: 6e73 4669 6c65 282e 2e2e 2c56 6572 626f  nsFile(...,Verbo
│ │ │ │ +00011f20: 7365 3d3e 2e2e 2e29 220a 2020 2a20 2272  se=>...)".  * "r
│ │ │ │ +00011f30: 756e 4265 7274 696e 6928 2e2e 2e2c 5665  unBertini(...,Ve
│ │ │ │ +00011f40: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a  rbose=>...)".  *
│ │ │ │ +00011f50: 2022 6368 6563 6b28 2e2e 2e2c 5665 7262   "check(...,Verb
│ │ │ │ +00011f60: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365  ose=>...)" -- se
│ │ │ │ +00011f70: 6520 2a6e 6f74 6520 6368 6563 6b3a 2028  e *note check: (
│ │ │ │ +00011f80: 4d61 6361 756c 6179 3244 6f63 2963 6865  Macaulay2Doc)che
│ │ │ │ +00011f90: 636b 2c20 2d2d 0a20 2020 2070 6572 666f  ck, --.    perfo
│ │ │ │ +00011fa0: 726d 2074 6573 7473 206f 6620 6120 7061  rm tests of a pa
│ │ │ │ +00011fb0: 636b 6167 650a 2020 2a20 2263 6865 636b  ckage.  * "check
│ │ │ │ +00011fc0: 4465 6772 6565 7328 2e2e 2e2c 5665 7262  Degrees(...,Verb
│ │ │ │ +00011fd0: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365  ose=>...)" -- se
│ │ │ │ +00011fe0: 6520 2a6e 6f74 6520 6368 6563 6b44 6567  e *note checkDeg
│ │ │ │ +00011ff0: 7265 6573 3a0a 2020 2020 2849 736f 6d6f  rees:.    (Isomo
│ │ │ │ +00012000: 7270 6869 736d 2963 6865 636b 4465 6772  rphism)checkDegr
│ │ │ │ +00012010: 6565 732c 202d 2d20 636f 6d70 6172 6573  ees, -- compares
│ │ │ │ +00012020: 2074 6865 2064 6567 7265 6573 206f 6620   the degrees of 
│ │ │ │ +00012030: 6765 6e65 7261 746f 7273 206f 6620 7477  generators of tw
│ │ │ │ +00012040: 6f0a 2020 2020 6d6f 6475 6c65 730a 2020  o.    modules.  
│ │ │ │ +00012050: 2a20 2263 6f70 7944 6972 6563 746f 7279  * "copyDirectory
│ │ │ │ +00012060: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e  (...,Verbose=>..
│ │ │ │ +00012070: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ +00012080: 0a20 2020 2063 6f70 7944 6972 6563 746f  .    copyDirecto
│ │ │ │ +00012090: 7279 2853 7472 696e 672c 5374 7269 6e67  ry(String,String
│ │ │ │ +000120a0: 293a 0a20 2020 2028 4d61 6361 756c 6179  ):.    (Macaulay
│ │ │ │ +000120b0: 3244 6f63 2963 6f70 7944 6972 6563 746f  2Doc)copyDirecto
│ │ │ │ +000120c0: 7279 5f6c 7053 7472 696e 675f 636d 5374  ry_lpString_cmSt
│ │ │ │ +000120d0: 7269 6e67 5f72 702c 0a20 202a 2022 636f  ring_rp,.  * "co
│ │ │ │ +000120e0: 7079 4669 6c65 282e 2e2e 2c56 6572 626f  pyFile(...,Verbo
│ │ │ │ +000120f0: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565  se=>...)" -- see
│ │ │ │ +00012100: 202a 6e6f 7465 2063 6f70 7946 696c 6528   *note copyFile(
│ │ │ │ +00012110: 5374 7269 6e67 2c53 7472 696e 6729 3a0a  String,String):.
│ │ │ │ +00012120: 2020 2020 284d 6163 6175 6c61 7932 446f      (Macaulay2Do
│ │ │ │ +00012130: 6329 636f 7079 4669 6c65 5f6c 7053 7472  c)copyFile_lpStr
│ │ │ │ +00012140: 696e 675f 636d 5374 7269 6e67 5f72 702c  ing_cmString_rp,
│ │ │ │ +00012150: 0a20 202a 2022 6669 6e64 5072 6f67 7261  .  * "findProgra
│ │ │ │ +00012160: 6d28 2e2e 2e2c 5665 7262 6f73 653d 3e2e  m(...,Verbose=>.
│ │ │ │ +00012170: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ +00012180: 6520 6669 6e64 5072 6f67 7261 6d3a 0a20  e findProgram:. 
│ │ │ │ +00012190: 2020 2028 4d61 6361 756c 6179 3244 6f63     (Macaulay2Doc
│ │ │ │ +000121a0: 2966 696e 6450 726f 6772 616d 2c20 2d2d  )findProgram, --
│ │ │ │ +000121b0: 206c 6f61 6420 6578 7465 726e 616c 2070   load external p
│ │ │ │ +000121c0: 726f 6772 616d 0a20 202a 2022 696e 7374  rogram.  * "inst
│ │ │ │ +000121d0: 616c 6c50 6163 6b61 6765 282e 2e2e 2c56  allPackage(...,V
│ │ │ │ +000121e0: 6572 626f 7365 3d3e 2e2e 2e29 2220 2d2d  erbose=>...)" --
│ │ │ │ +000121f0: 2073 6565 202a 6e6f 7465 2069 6e73 7461   see *note insta
│ │ │ │ +00012200: 6c6c 5061 636b 6167 653a 0a20 2020 2028  llPackage:.    (
│ │ │ │ +00012210: 4d61 6361 756c 6179 3244 6f63 2969 6e73  Macaulay2Doc)ins
│ │ │ │ +00012220: 7461 6c6c 5061 636b 6167 652c 202d 2d20  tallPackage, -- 
│ │ │ │ +00012230: 6c6f 6164 2061 6e64 2069 6e73 7461 6c6c  load and install
│ │ │ │ +00012240: 2061 2070 6163 6b61 6765 2061 6e64 2069   a package and i
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│ │ │ │ +00012280: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565  se=>...)" -- see
│ │ │ │ +00012290: 202a 6e6f 7465 2069 7349 736f 6d6f 7270   *note isIsomorp
│ │ │ │ +000122a0: 6869 633a 0a20 2020 2028 4973 6f6d 6f72  hic:.    (Isomor
│ │ │ │ +000122b0: 7068 6973 6d29 6973 4973 6f6d 6f72 7068  phism)isIsomorph
│ │ │ │ +000122c0: 6963 2c20 2d2d 2050 726f 6261 6269 6c69  ic, -- Probabili
│ │ │ │ +000122d0: 7374 6963 2074 6573 7420 666f 7220 6973  stic test for is
│ │ │ │ +000122e0: 6f6d 6f72 7068 6973 6d20 6f66 206d 6f64  omorphism of mod
│ │ │ │ +000122f0: 756c 6573 0a20 202a 2022 6d6f 7665 4669  ules.  * "moveFi
│ │ │ │ +00012300: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e  le(...,Verbose=>
│ │ │ │ +00012310: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ +00012320: 7465 206d 6f76 6546 696c 6528 5374 7269  te moveFile(Stri
│ │ │ │ +00012330: 6e67 2c53 7472 696e 6729 3a0a 2020 2020  ng,String):.    
│ │ │ │ +00012340: 284d 6163 6175 6c61 7932 446f 6329 6d6f  (Macaulay2Doc)mo
│ │ │ │ +00012350: 7665 4669 6c65 5f6c 7053 7472 696e 675f  veFile_lpString_
│ │ │ │ +00012360: 636d 5374 7269 6e67 5f72 702c 0a20 202a  cmString_rp,.  *
│ │ │ │ +00012370: 2022 7275 6e50 726f 6772 616d 282e 2e2e   "runProgram(...
│ │ │ │ +00012380: 2c56 6572 626f 7365 3d3e 2e2e 2e29 2220  ,Verbose=>...)" 
│ │ │ │ +00012390: 2d2d 2073 6565 202a 6e6f 7465 2072 756e  -- see *note run
│ │ │ │ +000123a0: 5072 6f67 7261 6d3a 0a20 2020 2028 4d61  Program:.    (Ma
│ │ │ │ +000123b0: 6361 756c 6179 3244 6f63 2972 756e 5072  caulay2Doc)runPr
│ │ │ │ +000123c0: 6f67 7261 6d2c 202d 2d20 7275 6e20 616e  ogram, -- run an
│ │ │ │ +000123d0: 2065 7874 6572 6e61 6c20 7072 6f67 7261   external progra
│ │ │ │ +000123e0: 6d0a 2020 2a20 2273 796d 6c69 6e6b 4469  m.  * "symlinkDi
│ │ │ │ +000123f0: 7265 6374 6f72 7928 2e2e 2e2c 5665 7262  rectory(...,Verb
│ │ │ │ +00012400: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365  ose=>...)" -- se
│ │ │ │ +00012410: 6520 2a6e 6f74 650a 2020 2020 7379 6d6c  e *note.    syml
│ │ │ │ +00012420: 696e 6b44 6972 6563 746f 7279 2853 7472  inkDirectory(Str
│ │ │ │ +00012430: 696e 672c 5374 7269 6e67 293a 0a20 2020  ing,String):.   
│ │ │ │ +00012440: 2028 4d61 6361 756c 6179 3244 6f63 2973   (Macaulay2Doc)s
│ │ │ │ +00012450: 796d 6c69 6e6b 4469 7265 6374 6f72 795f  ymlinkDirectory_
│ │ │ │ +00012460: 6c70 5374 7269 6e67 5f63 6d53 7472 696e  lpString_cmStrin
│ │ │ │ +00012470: 675f 7270 2c20 2d2d 206d 616b 6520 7379  g_rp, -- make sy
│ │ │ │ +00012480: 6d62 6f6c 6963 206c 696e 6b73 0a20 2020  mbolic links.   
│ │ │ │ +00012490: 2066 6f72 2061 6c6c 2066 696c 6573 2069   for all files i
│ │ │ │ +000124a0: 6e20 6120 6469 7265 6374 6f72 7920 7472  n a directory tr
│ │ │ │ +000124b0: 6565 0a1f 0a46 696c 653a 2042 6572 7469  ee...File: Berti
│ │ │ │ +000124c0: 6e69 2e69 6e66 6f2c 204e 6f64 653a 2062  ni.info, Node: b
│ │ │ │ +000124d0: 6572 7469 6e69 5573 6572 486f 6d6f 746f  ertiniUserHomoto
│ │ │ │ +000124e0: 7079 2c20 4e65 7874 3a20 6265 7274 696e  py, Next: bertin
│ │ │ │ +000124f0: 695a 6572 6f44 696d 536f 6c76 652c 2050  iZeroDimSolve, P
│ │ │ │ +00012500: 7265 763a 2062 6572 7469 6e69 5472 6163  rev: bertiniTrac
│ │ │ │ +00012510: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │ +00012520: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ +00012530: 3e5f 7064 5f70 645f 7064 5f72 702c 2055  >_pd_pd_pd_rp, U
│ │ │ │ +00012540: 703a 2054 6f70 0a0a 6265 7274 696e 6955  p: Top..bertiniU
│ │ │ │ +00012550: 7365 7248 6f6d 6f74 6f70 7920 2d2d 2061  serHomotopy -- a
│ │ │ │ +00012560: 206d 6169 6e20 6d65 7468 6f64 2074 6f20   main method to 
│ │ │ │ +00012570: 7472 6163 6b20 6120 7573 6572 2d64 6566  track a user-def
│ │ │ │ +00012580: 696e 6564 2068 6f6d 6f74 6f70 790a 2a2a  ined homotopy.**
│ │ │ │  00012590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000125a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000125b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000125c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000125d0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ -000125e0: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ -000125f0: 3a20 0a20 2020 2020 2020 2053 303d 6265  : .        S0=be
│ │ │ │ -00012600: 7274 696e 6955 7365 7248 6f6d 6f74 6f70  rtiniUserHomotop
│ │ │ │ -00012610: 7928 742c 2050 2c20 482c 2053 3129 0a20  y(t, P, H, S1). 
│ │ │ │ -00012620: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00012630: 202a 2074 2c20 6120 2a6e 6f74 6520 7269   * t, a *note ri
│ │ │ │ -00012640: 6e67 2065 6c65 6d65 6e74 3a20 284d 6163  ng element: (Mac
│ │ │ │ -00012650: 6175 6c61 7932 446f 6329 5269 6e67 456c  aulay2Doc)RingEl
│ │ │ │ -00012660: 656d 656e 742c 2c20 6120 7061 7468 2076  ement,, a path v
│ │ │ │ -00012670: 6172 6961 626c 650a 2020 2020 2020 2a20  ariable.      * 
│ │ │ │ -00012680: 502c 2061 202a 6e6f 7465 206c 6973 743a  P, a *note list:
│ │ │ │ -00012690: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -000126a0: 6973 742c 2c20 6120 6c69 7374 206f 6620  ist,, a list of 
│ │ │ │ -000126b0: 6f70 7469 6f6e 7320 7468 6174 2073 6574  options that set
│ │ │ │ -000126c0: 2074 6865 0a20 2020 2020 2020 2070 6172   the.        par
│ │ │ │ -000126d0: 616d 6574 6572 730a 2020 2020 2020 2a20  ameters.      * 
│ │ │ │ -000126e0: 482c 2061 202a 6e6f 7465 206c 6973 743a  H, a *note list:
│ │ │ │ -000126f0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -00012700: 6973 742c 2c20 6120 6c69 7374 206f 6620  ist,, a list of 
│ │ │ │ -00012710: 706f 6c79 6e6f 6d69 616c 7320 7468 6174  polynomials that
│ │ │ │ -00012720: 2064 6566 696e 650a 2020 2020 2020 2020   define.        
│ │ │ │ -00012730: 7468 6520 686f 6d6f 746f 7079 0a20 2020  the homotopy.   
│ │ │ │ -00012740: 2020 202a 2053 312c 2061 202a 6e6f 7465     * S1, a *note
│ │ │ │ -00012750: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00012760: 3244 6f63 294c 6973 742c 2c20 6120 6c69  2Doc)List,, a li
│ │ │ │ -00012770: 7374 206f 6620 736f 6c75 7469 6f6e 7320  st of solutions 
│ │ │ │ -00012780: 746f 2074 6865 2073 7461 7274 0a20 2020  to the start.   
│ │ │ │ -00012790: 2020 2020 2073 7973 7465 6d0a 2020 2a20       system.  * 
│ │ │ │ -000127a0: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ -000127b0: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ -000127c0: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ -000127d0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ -000127e0: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ -000127f0: 2020 2a20 2a6e 6f74 6520 4166 6656 6172    * *note AffVar
│ │ │ │ -00012800: 6961 626c 6547 726f 7570 3a20 5661 7269  iableGroup: Vari
│ │ │ │ -00012810: 6162 6c65 2067 726f 7570 732c 203d 3e20  able groups, => 
│ │ │ │ -00012820: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -00012830: 7565 207b 7d2c 2061 6e0a 2020 2020 2020  ue {}, an.      
│ │ │ │ -00012840: 2020 6f70 7469 6f6e 2074 6f20 6772 6f75    option to grou
│ │ │ │ -00012850: 7020 7661 7269 6162 6c65 7320 616e 6420  p variables and 
│ │ │ │ -00012860: 7573 6520 6d75 6c74 6968 6f6d 6f67 656e  use multihomogen
│ │ │ │ -00012870: 656f 7573 2068 6f6d 6f74 6f70 6965 730a  eous homotopies.
│ │ │ │ -00012880: 2020 2020 2020 2a20 4227 436f 6e73 7461        * B'Consta
│ │ │ │ -00012890: 6e74 7320 286d 6973 7369 6e67 2064 6f63  nts (missing doc
│ │ │ │ -000128a0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ -000128b0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -000128c0: 6520 7b7d 2c20 0a20 2020 2020 202a 2042  e {}, .      * B
│ │ │ │ -000128d0: 2746 756e 6374 696f 6e73 2028 6d69 7373  'Functions (miss
│ │ │ │ -000128e0: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -000128f0: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175  n) => ..., defau
│ │ │ │ -00012900: 6c74 2076 616c 7565 207b 7d2c 200a 2020  lt value {}, .  
│ │ │ │ -00012910: 2020 2020 2a20 4265 7274 696e 6949 6e70      * BertiniInp
│ │ │ │ -00012920: 7574 436f 6e66 6967 7572 6174 696f 6e20  utConfiguration 
│ │ │ │ -00012930: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ -00012940: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20  tation) => ..., 
│ │ │ │ -00012950: 6465 6661 756c 7420 7661 6c75 650a 2020  default value.  
│ │ │ │ -00012960: 2020 2020 2020 7b7d 2c0a 2020 2020 2020        {},.      
│ │ │ │ -00012970: 2a20 486f 6d56 6172 6961 626c 6547 726f  * HomVariableGro
│ │ │ │ -00012980: 7570 2028 6d69 7373 696e 6720 646f 6375  up (missing docu
│ │ │ │ -00012990: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e  mentation) => ..
│ │ │ │ -000129a0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -000129b0: 207b 7d2c 200a 2020 2020 2020 2a20 4d32   {}, .      * M2
│ │ │ │ -000129c0: 5072 6563 6973 696f 6e20 286d 6973 7369  Precision (missi
│ │ │ │ -000129d0: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ -000129e0: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ -000129f0: 7420 7661 6c75 6520 3533 2c20 0a20 2020  t value 53, .   
│ │ │ │ -00012a00: 2020 202a 204f 7574 7075 7453 7479 6c65     * OutputStyle
│ │ │ │ -00012a10: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ -00012a20: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ -00012a30: 2064 6566 6175 6c74 2076 616c 7565 2022   default value "
│ │ │ │ -00012a40: 4f75 7450 6f69 6e74 7322 2c20 0a20 2020  OutPoints", .   
│ │ │ │ -00012a50: 2020 202a 2052 616e 646f 6d43 6f6d 706c     * RandomCompl
│ │ │ │ -00012a60: 6578 2028 6d69 7373 696e 6720 646f 6375  ex (missing docu
│ │ │ │ -00012a70: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e  mentation) => ..
│ │ │ │ -00012a80: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00012a90: 207b 7d2c 200a 2020 2020 2020 2a20 5261   {}, .      * Ra
│ │ │ │ -00012aa0: 6e64 6f6d 5265 616c 2028 6d69 7373 696e  ndomReal (missin
│ │ │ │ -00012ab0: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ -00012ac0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00012ad0: 2076 616c 7565 207b 7d2c 200a 2020 2020   value {}, .    
│ │ │ │ -00012ae0: 2020 2a20 2a6e 6f74 6520 546f 7044 6972    * *note TopDir
│ │ │ │ -00012af0: 6563 746f 7279 3a20 546f 7044 6972 6563  ectory: TopDirec
│ │ │ │ -00012b00: 746f 7279 2c20 3d3e 202e 2e2e 2c20 6465  tory, => ..., de
│ │ │ │ -00012b10: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -00012b20: 2020 2020 222f 746d 702f 4d32 2d37 3134      "/tmp/M2-714
│ │ │ │ -00012b30: 3831 2d30 2f30 222c 204f 7074 696f 6e20  81-0/0", Option 
│ │ │ │ -00012b40: 746f 2063 6861 6e67 6520 6469 7265 6374  to change direct
│ │ │ │ -00012b50: 6f72 7920 666f 7220 6669 6c65 2073 746f  ory for file sto
│ │ │ │ -00012b60: 7261 6765 2e0a 2020 2020 2020 2a20 2a6e  rage..      * *n
│ │ │ │ -00012b70: 6f74 6520 5665 7262 6f73 653a 2062 6572  ote Verbose: ber
│ │ │ │ -00012b80: 7469 6e69 5472 6163 6b48 6f6d 6f74 6f70  tiniTrackHomotop
│ │ │ │ -00012b90: 795f 6c70 5f70 645f 7064 5f70 645f 636d  y_lp_pd_pd_pd_cm
│ │ │ │ -00012ba0: 5665 7262 6f73 653d 3e5f 7064 5f70 645f  Verbose=>_pd_pd_
│ │ │ │ -00012bb0: 7064 5f72 700a 2020 2020 2020 2020 2c20  pd_rp.        , 
│ │ │ │ -00012bc0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00012bd0: 7661 6c75 6520 6661 6c73 652c 204f 7074  value false, Opt
│ │ │ │ -00012be0: 696f 6e20 746f 2073 696c 656e 6365 2061  ion to silence a
│ │ │ │ -00012bf0: 6464 6974 696f 6e61 6c20 6f75 7470 7574  dditional output
│ │ │ │ -00012c00: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00012c10: 2020 2020 2a20 5330 2c20 6120 2a6e 6f74      * S0, a *not
│ │ │ │ -00012c20: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ -00012c30: 7932 446f 6329 4c69 7374 2c2c 2061 206c  y2Doc)List,, a l
│ │ │ │ -00012c40: 6973 7420 6f66 2073 6f6c 7574 696f 6e73  ist of solutions
│ │ │ │ -00012c50: 2074 6f20 7468 650a 2020 2020 2020 2020   to the.        
│ │ │ │ -00012c60: 7461 7267 6574 2073 7973 7465 6d0a 0a44  target system..D
│ │ │ │ -00012c70: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00012c80: 3d3d 3d3d 3d3d 0a0a 5468 6973 206d 6574  ======..This met
│ │ │ │ -00012c90: 686f 6420 6361 6c6c 7320 4265 7274 696e  hod calls Bertin
│ │ │ │ -00012ca0: 6920 746f 2074 7261 636b 2061 2075 7365  i to track a use
│ │ │ │ -00012cb0: 722d 6465 6669 6e65 6420 686f 6d6f 746f  r-defined homoto
│ │ │ │ -00012cc0: 7079 2e20 2054 6865 2075 7365 7220 6e65  py.  The user ne
│ │ │ │ -00012cd0: 6564 7320 746f 0a73 7065 6369 6679 2074  eds to.specify t
│ │ │ │ -00012ce0: 6865 2068 6f6d 6f74 6f70 7920 482c 2074  he homotopy H, t
│ │ │ │ -00012cf0: 6865 2070 6174 6820 7661 7269 6162 6c65  he path variable
│ │ │ │ -00012d00: 2074 2c20 616e 6420 6120 6c69 7374 206f   t, and a list o
│ │ │ │ -00012d10: 6620 7374 6172 7420 736f 6c75 7469 6f6e  f start solution
│ │ │ │ -00012d20: 7320 5331 2e0a 4265 7274 696e 6920 2831  s S1..Bertini (1
│ │ │ │ -00012d30: 2920 7772 6974 6573 2074 6865 2068 6f6d  ) writes the hom
│ │ │ │ -00012d40: 6f74 6f70 7920 616e 6420 7374 6172 7420  otopy and start 
│ │ │ │ -00012d50: 736f 6c75 7469 6f6e 7320 746f 2074 656d  solutions to tem
│ │ │ │ -00012d60: 706f 7261 7279 2066 696c 6573 2c20 2832  porary files, (2
│ │ │ │ -00012d70: 290a 696e 766f 6b65 7320 4265 7274 696e  ).invokes Bertin
│ │ │ │ -00012d80: 6927 7320 736f 6c76 6572 2077 6974 6820  i's solver with 
│ │ │ │ -00012d90: 636f 6e66 6967 7572 6174 696f 6e20 6b65  configuration ke
│ │ │ │ -00012da0: 7977 6f72 6420 5573 6572 486f 6d6f 746f  yword UserHomoto
│ │ │ │ -00012db0: 7079 203d 3e20 322c 2028 3329 0a73 746f  py => 2, (3).sto
│ │ │ │ -00012dc0: 7265 7320 7468 6520 6f75 7470 7574 206f  res the output o
│ │ │ │ -00012dd0: 6620 4265 7274 696e 6920 696e 2061 2074  f Bertini in a t
│ │ │ │ -00012de0: 656d 706f 7261 7279 2066 696c 652c 2061  emporary file, a
│ │ │ │ -00012df0: 6e64 2028 3429 2070 6172 7365 7320 6120  nd (4) parses a 
│ │ │ │ -00012e00: 6d61 6368 696e 650a 7265 6164 6162 6c65  machine.readable
│ │ │ │ -00012e10: 2066 696c 6520 746f 206f 7574 7075 7420   file to output 
│ │ │ │ -00012e20: 6120 6c69 7374 206f 6620 736f 6c75 7469  a list of soluti
│ │ │ │ -00012e30: 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ons...+---------
│ │ │ │ +000125d0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +000125e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ +000125f0: 653a 200a 2020 2020 2020 2020 5330 3d62  e: .        S0=b
│ │ │ │ +00012600: 6572 7469 6e69 5573 6572 486f 6d6f 746f  ertiniUserHomoto
│ │ │ │ +00012610: 7079 2874 2c20 502c 2048 2c20 5331 290a  py(t, P, H, S1).
│ │ │ │ +00012620: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00012630: 2020 2a20 742c 2061 202a 6e6f 7465 2072    * t, a *note r
│ │ │ │ +00012640: 696e 6720 656c 656d 656e 743a 2028 4d61  ing element: (Ma
│ │ │ │ +00012650: 6361 756c 6179 3244 6f63 2952 696e 6745  caulay2Doc)RingE
│ │ │ │ +00012660: 6c65 6d65 6e74 2c2c 2061 2070 6174 6820  lement,, a path 
│ │ │ │ +00012670: 7661 7269 6162 6c65 0a20 2020 2020 202a  variable.      *
│ │ │ │ +00012680: 2050 2c20 6120 2a6e 6f74 6520 6c69 7374   P, a *note list
│ │ │ │ +00012690: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000126a0: 4c69 7374 2c2c 2061 206c 6973 7420 6f66  List,, a list of
│ │ │ │ +000126b0: 206f 7074 696f 6e73 2074 6861 7420 7365   options that se
│ │ │ │ +000126c0: 7420 7468 650a 2020 2020 2020 2020 7061  t the.        pa
│ │ │ │ +000126d0: 7261 6d65 7465 7273 0a20 2020 2020 202a  rameters.      *
│ │ │ │ +000126e0: 2048 2c20 6120 2a6e 6f74 6520 6c69 7374   H, a *note list
│ │ │ │ +000126f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00012700: 4c69 7374 2c2c 2061 206c 6973 7420 6f66  List,, a list of
│ │ │ │ +00012710: 2070 6f6c 796e 6f6d 6961 6c73 2074 6861   polynomials tha
│ │ │ │ +00012720: 7420 6465 6669 6e65 0a20 2020 2020 2020  t define.       
│ │ │ │ +00012730: 2074 6865 2068 6f6d 6f74 6f70 790a 2020   the homotopy.  
│ │ │ │ +00012740: 2020 2020 2a20 5331 2c20 6120 2a6e 6f74      * S1, a *not
│ │ │ │ +00012750: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +00012760: 7932 446f 6329 4c69 7374 2c2c 2061 206c  y2Doc)List,, a l
│ │ │ │ +00012770: 6973 7420 6f66 2073 6f6c 7574 696f 6e73  ist of solutions
│ │ │ │ +00012780: 2074 6f20 7468 6520 7374 6172 740a 2020   to the start.  
│ │ │ │ +00012790: 2020 2020 2020 7379 7374 656d 0a20 202a        system.  *
│ │ │ │ +000127a0: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +000127b0: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +000127c0: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +000127d0: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +000127e0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +000127f0: 2020 202a 202a 6e6f 7465 2041 6666 5661     * *note AffVa
│ │ │ │ +00012800: 7269 6162 6c65 4772 6f75 703a 2056 6172  riableGroup: Var
│ │ │ │ +00012810: 6961 626c 6520 6772 6f75 7073 2c20 3d3e  iable groups, =>
│ │ │ │ +00012820: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +00012830: 6c75 6520 7b7d 2c20 616e 0a20 2020 2020  lue {}, an.     
│ │ │ │ +00012840: 2020 206f 7074 696f 6e20 746f 2067 726f     option to gro
│ │ │ │ +00012850: 7570 2076 6172 6961 626c 6573 2061 6e64  up variables and
│ │ │ │ +00012860: 2075 7365 206d 756c 7469 686f 6d6f 6765   use multihomoge
│ │ │ │ +00012870: 6e65 6f75 7320 686f 6d6f 746f 7069 6573  neous homotopies
│ │ │ │ +00012880: 0a20 2020 2020 202a 2042 2743 6f6e 7374  .      * B'Const
│ │ │ │ +00012890: 616e 7473 2028 6d69 7373 696e 6720 646f  ants (missing do
│ │ │ │ +000128a0: 6375 6d65 6e74 6174 696f 6e29 203d 3e20  cumentation) => 
│ │ │ │ +000128b0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +000128c0: 7565 207b 7d2c 200a 2020 2020 2020 2a20  ue {}, .      * 
│ │ │ │ +000128d0: 4227 4675 6e63 7469 6f6e 7320 286d 6973  B'Functions (mis
│ │ │ │ +000128e0: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
│ │ │ │ +000128f0: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661  on) => ..., defa
│ │ │ │ +00012900: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20  ult value {}, . 
│ │ │ │ +00012910: 2020 2020 202a 2042 6572 7469 6e69 496e       * BertiniIn
│ │ │ │ +00012920: 7075 7443 6f6e 6669 6775 7261 7469 6f6e  putConfiguration
│ │ │ │ +00012930: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00012940: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ +00012950: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ +00012960: 2020 2020 2020 207b 7d2c 0a20 2020 2020         {},.     
│ │ │ │ +00012970: 202a 2048 6f6d 5661 7269 6162 6c65 4772   * HomVariableGr
│ │ │ │ +00012980: 6f75 7020 286d 6973 7369 6e67 2064 6f63  oup (missing doc
│ │ │ │ +00012990: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ +000129a0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +000129b0: 6520 7b7d 2c20 0a20 2020 2020 202a 204d  e {}, .      * M
│ │ │ │ +000129c0: 3250 7265 6369 7369 6f6e 2028 6d69 7373  2Precision (miss
│ │ │ │ +000129d0: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ +000129e0: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175  n) => ..., defau
│ │ │ │ +000129f0: 6c74 2076 616c 7565 2035 332c 200a 2020  lt value 53, .  
│ │ │ │ +00012a00: 2020 2020 2a20 4f75 7470 7574 5374 796c      * OutputStyl
│ │ │ │ +00012a10: 6520 286d 6973 7369 6e67 2064 6f63 756d  e (missing docum
│ │ │ │ +00012a20: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e  entation) => ...
│ │ │ │ +00012a30: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00012a40: 224f 7574 506f 696e 7473 222c 200a 2020  "OutPoints", .  
│ │ │ │ +00012a50: 2020 2020 2a20 5261 6e64 6f6d 436f 6d70      * RandomComp
│ │ │ │ +00012a60: 6c65 7820 286d 6973 7369 6e67 2064 6f63  lex (missing doc
│ │ │ │ +00012a70: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ +00012a80: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +00012a90: 6520 7b7d 2c20 0a20 2020 2020 202a 2052  e {}, .      * R
│ │ │ │ +00012aa0: 616e 646f 6d52 6561 6c20 286d 6973 7369  andomReal (missi
│ │ │ │ +00012ab0: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ +00012ac0: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ +00012ad0: 7420 7661 6c75 6520 7b7d 2c20 0a20 2020  t value {}, .   
│ │ │ │ +00012ae0: 2020 202a 202a 6e6f 7465 2054 6f70 4469     * *note TopDi
│ │ │ │ +00012af0: 7265 6374 6f72 793a 2054 6f70 4469 7265  rectory: TopDire
│ │ │ │ +00012b00: 6374 6f72 792c 203d 3e20 2e2e 2e2c 2064  ctory, => ..., d
│ │ │ │ +00012b10: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ +00012b20: 2020 2020 2022 2f74 6d70 2f4d 322d 3132       "/tmp/M2-12
│ │ │ │ +00012b30: 3430 3334 2d30 2f30 222c 204f 7074 696f  4034-0/0", Optio
│ │ │ │ +00012b40: 6e20 746f 2063 6861 6e67 6520 6469 7265  n to change dire
│ │ │ │ +00012b50: 6374 6f72 7920 666f 7220 6669 6c65 2073  ctory for file s
│ │ │ │ +00012b60: 746f 7261 6765 2e0a 2020 2020 2020 2a20  torage..      * 
│ │ │ │ +00012b70: 2a6e 6f74 6520 5665 7262 6f73 653a 2062  *note Verbose: b
│ │ │ │ +00012b80: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ +00012b90: 6f70 795f 6c70 5f70 645f 7064 5f70 645f  opy_lp_pd_pd_pd_
│ │ │ │ +00012ba0: 636d 5665 7262 6f73 653d 3e5f 7064 5f70  cmVerbose=>_pd_p
│ │ │ │ +00012bb0: 645f 7064 5f72 700a 2020 2020 2020 2020  d_pd_rp.        
│ │ │ │ +00012bc0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +00012bd0: 7420 7661 6c75 6520 6661 6c73 652c 204f  t value false, O
│ │ │ │ +00012be0: 7074 696f 6e20 746f 2073 696c 656e 6365  ption to silence
│ │ │ │ +00012bf0: 2061 6464 6974 696f 6e61 6c20 6f75 7470   additional outp
│ │ │ │ +00012c00: 7574 0a20 202a 204f 7574 7075 7473 3a0a  ut.  * Outputs:.
│ │ │ │ +00012c10: 2020 2020 2020 2a20 5330 2c20 6120 2a6e        * S0, a *n
│ │ │ │ +00012c20: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00012c30: 6c61 7932 446f 6329 4c69 7374 2c2c 2061  lay2Doc)List,, a
│ │ │ │ +00012c40: 206c 6973 7420 6f66 2073 6f6c 7574 696f   list of solutio
│ │ │ │ +00012c50: 6e73 2074 6f20 7468 650a 2020 2020 2020  ns to the.      
│ │ │ │ +00012c60: 2020 7461 7267 6574 2073 7973 7465 6d0a    target system.
│ │ │ │ +00012c70: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00012c80: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 206d  ========..This m
│ │ │ │ +00012c90: 6574 686f 6420 6361 6c6c 7320 4265 7274  ethod calls Bert
│ │ │ │ +00012ca0: 696e 6920 746f 2074 7261 636b 2061 2075  ini to track a u
│ │ │ │ +00012cb0: 7365 722d 6465 6669 6e65 6420 686f 6d6f  ser-defined homo
│ │ │ │ +00012cc0: 746f 7079 2e20 2054 6865 2075 7365 7220  topy.  The user 
│ │ │ │ +00012cd0: 6e65 6564 7320 746f 0a73 7065 6369 6679  needs to.specify
│ │ │ │ +00012ce0: 2074 6865 2068 6f6d 6f74 6f70 7920 482c   the homotopy H,
│ │ │ │ +00012cf0: 2074 6865 2070 6174 6820 7661 7269 6162   the path variab
│ │ │ │ +00012d00: 6c65 2074 2c20 616e 6420 6120 6c69 7374  le t, and a list
│ │ │ │ +00012d10: 206f 6620 7374 6172 7420 736f 6c75 7469   of start soluti
│ │ │ │ +00012d20: 6f6e 7320 5331 2e0a 4265 7274 696e 6920  ons S1..Bertini 
│ │ │ │ +00012d30: 2831 2920 7772 6974 6573 2074 6865 2068  (1) writes the h
│ │ │ │ +00012d40: 6f6d 6f74 6f70 7920 616e 6420 7374 6172  omotopy and star
│ │ │ │ +00012d50: 7420 736f 6c75 7469 6f6e 7320 746f 2074  t solutions to t
│ │ │ │ +00012d60: 656d 706f 7261 7279 2066 696c 6573 2c20  emporary files, 
│ │ │ │ +00012d70: 2832 290a 696e 766f 6b65 7320 4265 7274  (2).invokes Bert
│ │ │ │ +00012d80: 696e 6927 7320 736f 6c76 6572 2077 6974  ini's solver wit
│ │ │ │ +00012d90: 6820 636f 6e66 6967 7572 6174 696f 6e20  h configuration 
│ │ │ │ +00012da0: 6b65 7977 6f72 6420 5573 6572 486f 6d6f  keyword UserHomo
│ │ │ │ +00012db0: 746f 7079 203d 3e20 322c 2028 3329 0a73  topy => 2, (3).s
│ │ │ │ +00012dc0: 746f 7265 7320 7468 6520 6f75 7470 7574  tores the output
│ │ │ │ +00012dd0: 206f 6620 4265 7274 696e 6920 696e 2061   of Bertini in a
│ │ │ │ +00012de0: 2074 656d 706f 7261 7279 2066 696c 652c   temporary file,
│ │ │ │ +00012df0: 2061 6e64 2028 3429 2070 6172 7365 7320   and (4) parses 
│ │ │ │ +00012e00: 6120 6d61 6368 696e 650a 7265 6164 6162  a machine.readab
│ │ │ │ +00012e10: 6c65 2066 696c 6520 746f 206f 7574 7075  le file to outpu
│ │ │ │ +00012e20: 7420 6120 6c69 7374 206f 6620 736f 6c75  t a list of solu
│ │ │ │ +00012e30: 7469 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d  tions...+-------
│ │ │ │  00012e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012e80: 2b0a 7c69 3120 3a20 5220 3d20 4343 5b78  +.|i1 : R = CC[x
│ │ │ │ -00012e90: 2c61 2c74 5d3b 202d 2d20 696e 636c 7564  ,a,t]; -- includ
│ │ │ │ -00012ea0: 6520 7468 6520 7061 7468 2076 6172 6961  e the path varia
│ │ │ │ -00012eb0: 626c 6520 696e 2074 6865 2072 696e 6720  ble in the ring 
│ │ │ │ -00012ec0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00012ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012e80: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 4343  --+.|i1 : R = CC
│ │ │ │ +00012e90: 5b78 2c61 2c74 5d3b 202d 2d20 696e 636c  [x,a,t]; -- incl
│ │ │ │ +00012ea0: 7564 6520 7468 6520 7061 7468 2076 6172  ude the path var
│ │ │ │ +00012eb0: 6961 626c 6520 696e 2074 6865 2072 696e  iable in the rin
│ │ │ │ +00012ec0: 6720 2020 2020 2020 2020 2020 2020 7c0a  g             |.
│ │ │ │ +00012ed0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00012ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00012f20: 4820 3d20 7b20 2878 5e32 2d31 292a 6120  H = { (x^2-1)*a 
│ │ │ │ -00012f30: 2b20 2878 5e32 2d32 292a 2831 2d61 297d  + (x^2-2)*(1-a)}
│ │ │ │ -00012f40: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00012f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00012f20: 3a20 4820 3d20 7b20 2878 5e32 2d31 292a  : H = { (x^2-1)*
│ │ │ │ +00012f30: 6120 2b20 2878 5e32 2d32 292a 2831 2d61  a + (x^2-2)*(1-a
│ │ │ │ +00012f40: 297d 3b20 2020 2020 2020 2020 2020 2020  )};             
│ │ │ │  00012f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012f60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00012f60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00012f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012fb0: 2b0a 7c69 3320 3a20 736f 6c31 203d 2070  +.|i3 : sol1 = p
│ │ │ │ -00012fc0: 6f69 6e74 207b 7b31 7d7d 3b20 2020 2020  oint {{1}};     
│ │ │ │ +00012fb0: 2d2d 2b0a 7c69 3320 3a20 736f 6c31 203d  --+.|i3 : sol1 =
│ │ │ │ +00012fc0: 2070 6f69 6e74 207b 7b31 7d7d 3b20 2020   point {{1}};   
│ │ │ │  00012fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012ff0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013000: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013040: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00013050: 736f 6c32 203d 2070 6f69 6e74 207b 7b2d  sol2 = point {{-
│ │ │ │ -00013060: 317d 7d3b 2020 2020 2020 2020 2020 2020  1}};            
│ │ │ │ +00013040: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +00013050: 3a20 736f 6c32 203d 2070 6f69 6e74 207b  : sol2 = point {
│ │ │ │ +00013060: 7b2d 317d 7d3b 2020 2020 2020 2020 2020  {-1}};          
│ │ │ │  00013070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013090: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00013090: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000130a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000130b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000130c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000130d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130e0: 2b0a 7c69 3520 3a20 5331 3d20 7b20 736f  +.|i5 : S1= { so
│ │ │ │ -000130f0: 6c31 2c20 736f 6c32 2020 7d3b 2d2d 736f  l1, sol2  };--so
│ │ │ │ -00013100: 6c75 7469 6f6e 7320 746f 2048 2077 6865  lutions to H whe
│ │ │ │ -00013110: 6e20 743d 3120 2020 2020 2020 2020 2020  n t=1           
│ │ │ │ -00013120: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000130e0: 2d2d 2b0a 7c69 3520 3a20 5331 3d20 7b20  --+.|i5 : S1= { 
│ │ │ │ +000130f0: 736f 6c31 2c20 736f 6c32 2020 7d3b 2d2d  sol1, sol2  };--
│ │ │ │ +00013100: 736f 6c75 7469 6f6e 7320 746f 2048 2077  solutions to H w
│ │ │ │ +00013110: 6865 6e20 743d 3120 2020 2020 2020 2020  hen t=1         
│ │ │ │ +00013120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013130: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013170: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00013180: 5330 203d 2062 6572 7469 6e69 5573 6572  S0 = bertiniUser
│ │ │ │ -00013190: 486f 6d6f 746f 7079 2028 742c 7b61 3d3e  Homotopy (t,{a=>
│ │ │ │ -000131a0: 747d 2c20 482c 2053 3129 202d 2d73 6f6c  t}, H, S1) --sol
│ │ │ │ -000131b0: 7574 696f 6e73 2074 6f20 4820 7768 656e  utions to H when
│ │ │ │ -000131c0: 2074 3d30 7c0a 7c20 2020 2020 2020 2020   t=0|.|         
│ │ │ │ +00013170: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +00013180: 3a20 5330 203d 2062 6572 7469 6e69 5573  : S0 = bertiniUs
│ │ │ │ +00013190: 6572 486f 6d6f 746f 7079 2028 742c 7b61  erHomotopy (t,{a
│ │ │ │ +000131a0: 3d3e 747d 2c20 482c 2053 3129 202d 2d73  =>t}, H, S1) --s
│ │ │ │ +000131b0: 6f6c 7574 696f 6e73 2074 6f20 4820 7768  olutions to H wh
│ │ │ │ +000131c0: 656e 2074 3d30 7c0a 7c20 2020 2020 2020  en t=0|.|       
│ │ │ │  000131d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000131e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013210: 7c0a 7c6f 3620 3d20 7b7b 312e 3431 3432  |.|o6 = {{1.4142
│ │ │ │ -00013220: 317d 2c20 7b2d 312e 3431 3432 317d 7d20  1}, {-1.41421}} 
│ │ │ │ -00013230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013210: 2020 7c0a 7c6f 3620 3d20 7b7b 312e 3431    |.|o6 = {{1.41
│ │ │ │ +00013220: 3432 317d 2c20 7b2d 312e 3431 3432 317d  421}, {-1.41421}
│ │ │ │ +00013230: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  00013240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00013260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013260: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -000132b0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +000132a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +000132b0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │  000132c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000132d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000132e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000132f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00013300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013340: 2b0a 7c69 3720 3a20 7065 656b 2053 305f  +.|i7 : peek S0_
│ │ │ │ -00013350: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00013340: 2d2d 2b0a 7c69 3720 3a20 7065 656b 2053  --+.|i7 : peek S
│ │ │ │ +00013350: 305f 3020 2020 2020 2020 2020 2020 2020  0_0             
│ │ │ │  00013360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00013390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013390: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000133a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000133b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000133c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000133d0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -000133e0: 506f 696e 747b 6361 6368 6520 3d3e 2043  Point{cache => C
│ │ │ │ -000133f0: 6163 6865 5461 626c 657b 2e2e 2e31 342e  acheTable{...14.
│ │ │ │ -00013400: 2e2e 7d7d 2020 2020 2020 2020 2020 2020  ..}}            
│ │ │ │ +000133d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +000133e0: 3d20 506f 696e 747b 6361 6368 6520 3d3e  = Point{cache =>
│ │ │ │ +000133f0: 2043 6163 6865 5461 626c 657b 2e2e 2e31   CacheTable{...1
│ │ │ │ +00013400: 342e 2e2e 7d7d 2020 2020 2020 2020 2020  4...}}          
│ │ │ │  00013410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013420: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00013430: 2020 436f 6f72 6469 6e61 7465 7320 3d3e    Coordinates =>
│ │ │ │ -00013440: 207b 312e 3431 3432 317d 2020 2020 2020   {1.41421}      
│ │ │ │ +00013420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00013430: 2020 2020 436f 6f72 6469 6e61 7465 7320      Coordinates 
│ │ │ │ +00013440: 3d3e 207b 312e 3431 3432 317d 2020 2020  => {1.41421}    
│ │ │ │  00013450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013470: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00013470: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00013480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000134a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000134b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d  ------------+.+-
│ │ │ │ -000134c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000134b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000134c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000134d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000134e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000134f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00013510: 3820 3a20 523d 4343 5b78 2c79 2c74 2c61  8 : R=CC[x,y,t,a
│ │ │ │ -00013520: 5d3b 202d 2d20 696e 636c 7564 6520 7468  ]; -- include th
│ │ │ │ -00013530: 6520 7061 7468 2076 6172 6961 626c 6520  e path variable 
│ │ │ │ -00013540: 696e 2074 6865 2072 696e 6720 2020 2020  in the ring     
│ │ │ │ -00013550: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013510: 7c69 3820 3a20 523d 4343 5b78 2c79 2c74  |i8 : R=CC[x,y,t
│ │ │ │ +00013520: 2c61 5d3b 202d 2d20 696e 636c 7564 6520  ,a]; -- include 
│ │ │ │ +00013530: 7468 6520 7061 7468 2076 6172 6961 626c  the path variabl
│ │ │ │ +00013540: 6520 696e 2074 6865 2072 696e 6720 2020  e in the ring   
│ │ │ │ +00013550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013560: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000135a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000135b0: 3920 3a20 6631 3d28 785e 322d 795e 3229  9 : f1=(x^2-y^2)
│ │ │ │ -000135c0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +000135a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000135b0: 7c69 3920 3a20 6631 3d28 785e 322d 795e  |i9 : f1=(x^2-y^
│ │ │ │ +000135c0: 3229 3b20 2020 2020 2020 2020 2020 2020  2);             
│ │ │ │  000135d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000135e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000135f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000135f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013600: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00013650: 3130 203a 2066 323d 2832 2a78 5e32 2d33  10 : f2=(2*x^2-3
│ │ │ │ -00013660: 2a78 2a79 2b35 2a79 5e32 293b 2020 2020  *x*y+5*y^2);    
│ │ │ │ +00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013650: 7c69 3130 203a 2066 323d 2832 2a78 5e32  |i10 : f2=(2*x^2
│ │ │ │ +00013660: 2d33 2a78 2a79 2b35 2a79 5e32 293b 2020  -3*x*y+5*y^2);  
│ │ │ │  00013670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013690: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000136a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000136a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000136b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000136c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000136d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000136e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000136f0: 3131 203a 2048 203d 207b 2066 312a 6120  11 : H = { f1*a 
│ │ │ │ -00013700: 2b20 6632 2a28 312d 6129 7d3b 202d 2d48  + f2*(1-a)}; --H
│ │ │ │ -00013710: 2069 7320 6120 6c69 7374 206f 6620 706f   is a list of po
│ │ │ │ -00013720: 6c79 6e6f 6d69 616c 7320 696e 2078 2c79  lynomials in x,y
│ │ │ │ -00013730: 2c74 2020 2020 2020 2020 2020 7c0a 2b2d  ,t          |.+-
│ │ │ │ -00013740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000136e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000136f0: 7c69 3131 203a 2048 203d 207b 2066 312a  |i11 : H = { f1*
│ │ │ │ +00013700: 6120 2b20 6632 2a28 312d 6129 7d3b 202d  a + f2*(1-a)}; -
│ │ │ │ +00013710: 2d48 2069 7320 6120 6c69 7374 206f 6620  -H is a list of 
│ │ │ │ +00013720: 706f 6c79 6e6f 6d69 616c 7320 696e 2078  polynomials in x
│ │ │ │ +00013730: 2c79 2c74 2020 2020 2020 2020 2020 7c0a  ,y,t          |.
│ │ │ │ +00013740: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00013790: 3132 203a 2073 6f6c 313d 2020 2020 706f  12 : sol1=    po
│ │ │ │ -000137a0: 696e 747b 7b31 2c31 7d7d 2d2d 7b7b 782c  int{{1,1}}--{{x,
│ │ │ │ -000137b0: 797d 7d20 636f 6f72 6469 6e61 7465 7320  y}} coordinates 
│ │ │ │ -000137c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000137d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000137e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013790: 7c69 3132 203a 2073 6f6c 313d 2020 2020  |i12 : sol1=    
│ │ │ │ +000137a0: 706f 696e 747b 7b31 2c31 7d7d 2d2d 7b7b  point{{1,1}}--{{
│ │ │ │ +000137b0: 782c 797d 7d20 636f 6f72 6469 6e61 7465  x,y}} coordinate
│ │ │ │ +000137c0: 7320 2020 2020 2020 2020 2020 2020 2020  s               
│ │ │ │ +000137d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000137e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000137f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00013830: 3132 203d 2073 6f6c 3120 2020 2020 2020  12 = sol1       
│ │ │ │ +00013820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013830: 7c6f 3132 203d 2073 6f6c 3120 2020 2020  |o12 = sol1     
│ │ │ │  00013840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00013880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013880: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000138a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000138b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000138c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000138d0: 3132 203a 2050 6f69 6e74 2020 2020 2020  12 : Point      
│ │ │ │ +000138c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000138d0: 7c6f 3132 203a 2050 6f69 6e74 2020 2020  |o12 : Point    
│ │ │ │  000138e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000138f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013910: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013920: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00013970: 3133 203a 2073 6f6c 323d 2020 2020 706f  13 : sol2=    po
│ │ │ │ -00013980: 696e 747b 7b20 2d31 2c31 7d7d 2020 2020  int{{ -1,1}}    
│ │ │ │ +00013960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013970: 7c69 3133 203a 2073 6f6c 323d 2020 2020  |i13 : sol2=    
│ │ │ │ +00013980: 706f 696e 747b 7b20 2d31 2c31 7d7d 2020  point{{ -1,1}}  
│ │ │ │  00013990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000139b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000139c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000139b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000139c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000139d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00013a10: 3133 203d 2073 6f6c 3220 2020 2020 2020  13 = sol2       
│ │ │ │ +00013a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013a10: 7c6f 3133 203d 2073 6f6c 3220 2020 2020  |o13 = sol2     
│ │ │ │  00013a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00013a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013a60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00013ab0: 3133 203a 2050 6f69 6e74 2020 2020 2020  13 : Point      
│ │ │ │ +00013aa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013ab0: 7c6f 3133 203a 2050 6f69 6e74 2020 2020  |o13 : Point    
│ │ │ │  00013ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013af0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013af0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013b00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00013b50: 3134 203a 2053 313d 7b73 6f6c 312c 736f  14 : S1={sol1,so
│ │ │ │ -00013b60: 6c32 7d2d 2d73 6f6c 7574 696f 6e73 2074  l2}--solutions t
│ │ │ │ -00013b70: 6f20 4820 7768 656e 2074 3d31 2020 2020  o H when t=1    
│ │ │ │ +00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013b50: 7c69 3134 203a 2053 313d 7b73 6f6c 312c  |i14 : S1={sol1,
│ │ │ │ +00013b60: 736f 6c32 7d2d 2d73 6f6c 7574 696f 6e73  sol2}--solutions
│ │ │ │ +00013b70: 2074 6f20 4820 7768 656e 2074 3d31 2020   to H when t=1  
│ │ │ │  00013b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013b90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013ba0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00013bf0: 3134 203d 207b 736f 6c31 2c20 736f 6c32  14 = {sol1, sol2
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│ │ │ │ +00013be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013bf0: 7c6f 3134 203d 207b 736f 6c31 2c20 736f  |o14 = {sol1, so
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│ │ │ │  00013c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00013e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
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│ │ │ │ -00014590: 6566 6175 6c74 2076 616c 7565 207b 7d2c  efault value {},
│ │ │ │ -000145a0: 2061 6e0a 2020 2020 2020 2020 6f70 7469   an.        opti
│ │ │ │ -000145b0: 6f6e 2074 6f20 6772 6f75 7020 7661 7269  on to group vari
│ │ │ │ -000145c0: 6162 6c65 7320 616e 6420 7573 6520 6d75  ables and use mu
│ │ │ │ -000145d0: 6c74 6968 6f6d 6f67 656e 656f 7573 2068  ltihomogeneous h
│ │ │ │ -000145e0: 6f6d 6f74 6f70 6965 730a 2020 2020 2020  omotopies.      
│ │ │ │ -000145f0: 2a20 4973 5072 6f6a 6563 7469 7665 2028  * IsProjective (
│ │ │ │ -00014600: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ -00014610: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064  ation) => ..., d
│ │ │ │ -00014620: 6566 6175 6c74 2076 616c 7565 202d 312c  efault value -1,
│ │ │ │ -00014630: 200a 2020 2020 2020 2a20 4d32 5072 6563   .      * M2Prec
│ │ │ │ -00014640: 6973 696f 6e20 286d 6973 7369 6e67 2064  ision (missing d
│ │ │ │ -00014650: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ -00014660: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -00014670: 6c75 6520 3533 2c20 0a20 2020 2020 202a  lue 53, .      *
│ │ │ │ -00014680: 204e 616d 654d 6169 6e44 6174 6146 696c   NameMainDataFil
│ │ │ │ -00014690: 6520 286d 6973 7369 6e67 2064 6f63 756d  e (missing docum
│ │ │ │ -000146a0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e  entation) => ...
│ │ │ │ -000146b0: 2c20 6465 6661 756c 7420 7661 6c75 650a  , default value.
│ │ │ │ -000146c0: 2020 2020 2020 2020 226d 6169 6e5f 6461          "main_da
│ │ │ │ -000146d0: 7461 222c 0a20 2020 2020 202a 204e 616d  ta",.      * Nam
│ │ │ │ -000146e0: 6553 6f6c 7574 696f 6e73 4669 6c65 2028  eSolutionsFile (
│ │ │ │ -000146f0: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ -00014700: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064  ation) => ..., d
│ │ │ │ -00014710: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ -00014720: 2020 2020 2022 7261 775f 736f 6c75 7469       "raw_soluti
│ │ │ │ -00014730: 6f6e 7322 2c0a 2020 2020 2020 2a20 4f75  ons",.      * Ou
│ │ │ │ -00014740: 7470 7574 5374 796c 6520 286d 6973 7369  tputStyle (missi
│ │ │ │ -00014750: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ -00014760: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ -00014770: 7420 7661 6c75 6520 224f 7574 506f 696e  t value "OutPoin
│ │ │ │ -00014780: 7473 222c 200a 2020 2020 2020 2a20 2a6e  ts", .      * *n
│ │ │ │ -00014790: 6f74 6520 5261 6e64 6f6d 436f 6d70 6c65  ote RandomComple
│ │ │ │ -000147a0: 783a 2042 6572 7469 6e69 2069 6e70 7574  x: Bertini input
│ │ │ │ -000147b0: 2066 696c 6520 6465 636c 6172 6174 696f   file declaratio
│ │ │ │ -000147c0: 6e73 5f63 6f20 7261 6e64 6f6d 206e 756d  ns_co random num
│ │ │ │ -000147d0: 6265 7273 2c0a 2020 2020 2020 2020 3d3e  bers,.        =>
│ │ │ │ -000147e0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -000147f0: 6c75 6520 7b7d 2c20 616e 206f 7074 696f  lue {}, an optio
│ │ │ │ -00014800: 6e20 7768 6963 6820 6465 7369 676e 6174  n which designat
│ │ │ │ -00014810: 6573 0a20 2020 2020 2020 2073 796d 626f  es.        symbo
│ │ │ │ -00014820: 6c73 2f73 7472 696e 6773 2f76 6172 6961  ls/strings/varia
│ │ │ │ -00014830: 626c 6573 2074 6861 7420 7769 6c6c 2062  bles that will b
│ │ │ │ -00014840: 6520 7365 7420 746f 2062 6520 6120 7261  e set to be a ra
│ │ │ │ -00014850: 6e64 6f6d 2072 6561 6c20 6e75 6d62 6572  ndom real number
│ │ │ │ -00014860: 0a20 2020 2020 2020 206f 7220 7261 6e64  .        or rand
│ │ │ │ -00014870: 6f6d 2063 6f6d 706c 6578 206e 756d 6265  om complex numbe
│ │ │ │ -00014880: 720a 2020 2020 2020 2a20 2a6e 6f74 6520  r.      * *note 
│ │ │ │ -00014890: 5261 6e64 6f6d 5265 616c 3a20 4265 7274  RandomReal: Bert
│ │ │ │ -000148a0: 696e 6920 696e 7075 7420 6669 6c65 2064  ini input file d
│ │ │ │ -000148b0: 6563 6c61 7261 7469 6f6e 735f 636f 2072  eclarations_co r
│ │ │ │ -000148c0: 616e 646f 6d20 6e75 6d62 6572 732c 203d  andom numbers, =
│ │ │ │ -000148d0: 3e0a 2020 2020 2020 2020 2e2e 2e2c 2064  >.        ..., d
│ │ │ │ -000148e0: 6566 6175 6c74 2076 616c 7565 207b 7d2c  efault value {},
│ │ │ │ -000148f0: 2061 6e20 6f70 7469 6f6e 2077 6869 6368   an option which
│ │ │ │ -00014900: 2064 6573 6967 6e61 7465 730a 2020 2020   designates.    
│ │ │ │ -00014910: 2020 2020 7379 6d62 6f6c 732f 7374 7269      symbols/stri
│ │ │ │ -00014920: 6e67 732f 7661 7269 6162 6c65 7320 7468  ngs/variables th
│ │ │ │ -00014930: 6174 2077 696c 6c20 6265 2073 6574 2074  at will be set t
│ │ │ │ -00014940: 6f20 6265 2061 2072 616e 646f 6d20 7265  o be a random re
│ │ │ │ -00014950: 616c 206e 756d 6265 720a 2020 2020 2020  al number.      
│ │ │ │ -00014960: 2020 6f72 2072 616e 646f 6d20 636f 6d70    or random comp
│ │ │ │ -00014970: 6c65 7820 6e75 6d62 6572 0a20 2020 2020  lex number.     
│ │ │ │ -00014980: 202a 202a 6e6f 7465 2054 6f70 4469 7265   * *note TopDire
│ │ │ │ -00014990: 6374 6f72 793a 2054 6f70 4469 7265 6374  ctory: TopDirect
│ │ │ │ -000149a0: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566  ory, => ..., def
│ │ │ │ -000149b0: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -000149c0: 2020 2022 2f74 6d70 2f4d 322d 3731 3438     "/tmp/M2-7148
│ │ │ │ -000149d0: 312d 302f 3022 2c20 4f70 7469 6f6e 2074  1-0/0", Option t
│ │ │ │ -000149e0: 6f20 6368 616e 6765 2064 6972 6563 746f  o change directo
│ │ │ │ -000149f0: 7279 2066 6f72 2066 696c 6520 7374 6f72  ry for file stor
│ │ │ │ -00014a00: 6167 652e 0a20 2020 2020 202a 2055 7365  age..      * Use
│ │ │ │ -00014a10: 5265 6765 6e65 7261 7469 6f6e 2028 6d69  Regeneration (mi
│ │ │ │ -00014a20: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ -00014a30: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566  ion) => ..., def
│ │ │ │ -00014a40: 6175 6c74 2076 616c 7565 202d 312c 200a  ault value -1, .
│ │ │ │ -00014a50: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
│ │ │ │ -00014a60: 7262 6f73 653a 2062 6572 7469 6e69 5472  rbose: bertiniTr
│ │ │ │ -00014a70: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70  ackHomotopy_lp_p
│ │ │ │ -00014a80: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ -00014a90: 653d 3e5f 7064 5f70 645f 7064 5f72 700a  e=>_pd_pd_pd_rp.
│ │ │ │ -00014aa0: 2020 2020 2020 2020 2c20 3d3e 202e 2e2e          , => ...
│ │ │ │ -00014ab0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00014ac0: 6661 6c73 652c 204f 7074 696f 6e20 746f  false, Option to
│ │ │ │ -00014ad0: 2073 696c 656e 6365 2061 6464 6974 696f   silence additio
│ │ │ │ -00014ae0: 6e61 6c20 6f75 7470 7574 0a20 202a 204f  nal output.  * O
│ │ │ │ -00014af0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -00014b00: 532c 2061 202a 6e6f 7465 206c 6973 743a  S, a *note list:
│ │ │ │ -00014b10: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -00014b20: 6973 742c 2c20 6120 6c69 7374 206f 6620  ist,, a list of 
│ │ │ │ -00014b30: 706f 696e 7473 2074 6861 7420 6172 650a  points that are.
│ │ │ │ -00014b40: 2020 2020 2020 2020 636f 6e74 6169 6e65          containe
│ │ │ │ -00014b50: 6420 696e 2074 6865 2076 6172 6965 7479  d in the variety
│ │ │ │ -00014b60: 206f 6620 460a 0a44 6573 6372 6970 7469   of F..Descripti
│ │ │ │ -00014b70: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00014b80: 5468 6973 206d 6574 686f 6420 6669 6e64  This method find
│ │ │ │ -00014b90: 7320 6973 6f6c 6174 6564 2073 6f6c 7574  s isolated solut
│ │ │ │ -00014ba0: 696f 6e73 2074 6f20 7468 6520 7379 7374  ions to the syst
│ │ │ │ -00014bb0: 656d 2046 2076 6961 206e 756d 6572 6963  em F via numeric
│ │ │ │ -00014bc0: 616c 2070 6f6c 796e 6f6d 6961 6c0a 686f  al polynomial.ho
│ │ │ │ -00014bd0: 6d6f 746f 7079 2063 6f6e 7469 6e75 6174  motopy continuat
│ │ │ │ -00014be0: 696f 6e20 6279 2028 3129 2062 7569 6c64  ion by (1) build
│ │ │ │ -00014bf0: 696e 6720 6120 4265 7274 696e 6920 696e  ing a Bertini in
│ │ │ │ -00014c00: 7075 7420 6669 6c65 2066 726f 6d20 7468  put file from th
│ │ │ │ -00014c10: 6520 7379 7374 656d 2046 2c0a 2832 2920  e system F,.(2) 
│ │ │ │ -00014c20: 6361 6c6c 696e 6720 4265 7274 696e 6920  calling Bertini 
│ │ │ │ -00014c30: 6f6e 2074 6869 7320 696e 7075 7420 6669  on this input fi
│ │ │ │ -00014c40: 6c65 2c20 2833 2920 7265 7475 726e 696e  le, (3) returnin
│ │ │ │ -00014c50: 6720 736f 6c75 7469 6f6e 7320 6672 6f6d  g solutions from
│ │ │ │ -00014c60: 2061 206d 6163 6869 6e65 0a72 6561 6461   a machine.reada
│ │ │ │ -00014c70: 626c 6520 6669 6c65 2074 6861 7420 6973  ble file that is
│ │ │ │ -00014c80: 2061 6e20 6f75 7470 7574 2066 726f 6d20   an output from 
│ │ │ │ -00014c90: 4265 7274 696e 692e 0a0a 2b2d 2d2d 2d2d  Bertini...+-----
│ │ │ │ +000141e0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +000141f0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +00014200: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00014210: 2020 2053 203d 2062 6572 7469 6e69 5a65     S = bertiniZe
│ │ │ │ +00014220: 726f 4469 6d53 6f6c 7665 2046 0a20 2020  roDimSolve F.   
│ │ │ │ +00014230: 2020 2020 2053 203d 2062 6572 7469 6e69       S = bertini
│ │ │ │ +00014240: 5a65 726f 4469 6d53 6f6c 7665 2049 0a20  ZeroDimSolve I. 
│ │ │ │ +00014250: 2020 2020 2020 2053 203d 2062 6572 7469         S = berti
│ │ │ │ +00014260: 6e69 5a65 726f 4469 6d53 6f6c 7665 2849  niZeroDimSolve(I
│ │ │ │ +00014270: 2c20 5573 6552 6567 656e 6572 6174 696f  , UseRegeneratio
│ │ │ │ +00014280: 6e3d 3e31 290a 2020 2a20 496e 7075 7473  n=>1).  * Inputs
│ │ │ │ +00014290: 3a0a 2020 2020 2020 2a20 462c 2061 202a  :.      * F, a *
│ │ │ │ +000142a0: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ +000142b0: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ +000142c0: 6120 6c69 7374 206f 6620 7269 6e67 2065  a list of ring e
│ │ │ │ +000142d0: 6c65 6d65 6e74 7320 2873 7973 7465 6d0a  lements (system.
│ │ │ │ +000142e0: 2020 2020 2020 2020 6e65 6564 206e 6f74          need not
│ │ │ │ +000142f0: 2062 6520 7371 7561 7265 290a 2020 2020   be square).    
│ │ │ │ +00014300: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ +00014310: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00014320: 3244 6f63 2949 6465 616c 2c2c 2061 6e20  2Doc)Ideal,, an 
│ │ │ │ +00014330: 6964 6561 6c20 6465 6669 6e69 6e67 2061  ideal defining a
│ │ │ │ +00014340: 2076 6172 6965 7479 0a20 202a 202a 6e6f   variety.  * *no
│ │ │ │ +00014350: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ +00014360: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ +00014370: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ +00014380: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +00014390: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ +000143a0: 202a 6e6f 7465 2041 6666 5661 7269 6162   *note AffVariab
│ │ │ │ +000143b0: 6c65 4772 6f75 703a 2056 6172 6961 626c  leGroup: Variabl
│ │ │ │ +000143c0: 6520 6772 6f75 7073 2c20 3d3e 202e 2e2e  e groups, => ...
│ │ │ │ +000143d0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +000143e0: 7b7d 2c20 616e 0a20 2020 2020 2020 206f  {}, an.        o
│ │ │ │ +000143f0: 7074 696f 6e20 746f 2067 726f 7570 2076  ption to group v
│ │ │ │ +00014400: 6172 6961 626c 6573 2061 6e64 2075 7365  ariables and use
│ │ │ │ +00014410: 206d 756c 7469 686f 6d6f 6765 6e65 6f75   multihomogeneou
│ │ │ │ +00014420: 7320 686f 6d6f 746f 7069 6573 0a20 2020  s homotopies.   
│ │ │ │ +00014430: 2020 202a 202a 6e6f 7465 2042 2743 6f6e     * *note B'Con
│ │ │ │ +00014440: 7374 616e 7473 3a20 4227 436f 6e73 7461  stants: B'Consta
│ │ │ │ +00014450: 6e74 732c 203d 3e20 2e2e 2e2c 2064 6566  nts, => ..., def
│ │ │ │ +00014460: 6175 6c74 2076 616c 7565 207b 7d2c 2061  ault value {}, a
│ │ │ │ +00014470: 6e20 6f70 7469 6f6e 2074 6f0a 2020 2020  n option to.    
│ │ │ │ +00014480: 2020 2020 6465 7369 676e 6174 6520 7468      designate th
│ │ │ │ +00014490: 6520 636f 6e73 7461 6e74 7320 666f 7220  e constants for 
│ │ │ │ +000144a0: 6120 4265 7274 696e 6920 496e 7075 7420  a Bertini Input 
│ │ │ │ +000144b0: 6669 6c65 0a20 2020 2020 202a 2042 2746  file.      * B'F
│ │ │ │ +000144c0: 756e 6374 696f 6e73 2028 6d69 7373 696e  unctions (missin
│ │ │ │ +000144d0: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ +000144e0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +000144f0: 2076 616c 7565 207b 7d2c 200a 2020 2020   value {}, .    
│ │ │ │ +00014500: 2020 2a20 4265 7274 696e 6949 6e70 7574    * BertiniInput
│ │ │ │ +00014510: 436f 6e66 6967 7572 6174 696f 6e20 286d  Configuration (m
│ │ │ │ +00014520: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ +00014530: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ +00014540: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +00014550: 2020 2020 7b7d 2c0a 2020 2020 2020 2a20      {},.      * 
│ │ │ │ +00014560: 2a6e 6f74 6520 486f 6d56 6172 6961 626c  *note HomVariabl
│ │ │ │ +00014570: 6547 726f 7570 3a20 5661 7269 6162 6c65  eGroup: Variable
│ │ │ │ +00014580: 2067 726f 7570 732c 203d 3e20 2e2e 2e2c   groups, => ...,
│ │ │ │ +00014590: 2064 6566 6175 6c74 2076 616c 7565 207b   default value {
│ │ │ │ +000145a0: 7d2c 2061 6e0a 2020 2020 2020 2020 6f70  }, an.        op
│ │ │ │ +000145b0: 7469 6f6e 2074 6f20 6772 6f75 7020 7661  tion to group va
│ │ │ │ +000145c0: 7269 6162 6c65 7320 616e 6420 7573 6520  riables and use 
│ │ │ │ +000145d0: 6d75 6c74 6968 6f6d 6f67 656e 656f 7573  multihomogeneous
│ │ │ │ +000145e0: 2068 6f6d 6f74 6f70 6965 730a 2020 2020   homotopies.    
│ │ │ │ +000145f0: 2020 2a20 4973 5072 6f6a 6563 7469 7665    * IsProjective
│ │ │ │ +00014600: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00014610: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ +00014620: 2064 6566 6175 6c74 2076 616c 7565 202d   default value -
│ │ │ │ +00014630: 312c 200a 2020 2020 2020 2a20 4d32 5072  1, .      * M2Pr
│ │ │ │ +00014640: 6563 6973 696f 6e20 286d 6973 7369 6e67  ecision (missing
│ │ │ │ +00014650: 2064 6f63 756d 656e 7461 7469 6f6e 2920   documentation) 
│ │ │ │ +00014660: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00014670: 7661 6c75 6520 3533 2c20 0a20 2020 2020  value 53, .     
│ │ │ │ +00014680: 202a 204e 616d 654d 6169 6e44 6174 6146   * NameMainDataF
│ │ │ │ +00014690: 696c 6520 286d 6973 7369 6e67 2064 6f63  ile (missing doc
│ │ │ │ +000146a0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ +000146b0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +000146c0: 650a 2020 2020 2020 2020 226d 6169 6e5f  e.        "main_
│ │ │ │ +000146d0: 6461 7461 222c 0a20 2020 2020 202a 204e  data",.      * N
│ │ │ │ +000146e0: 616d 6553 6f6c 7574 696f 6e73 4669 6c65  ameSolutionsFile
│ │ │ │ +000146f0: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00014700: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ +00014710: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ +00014720: 2020 2020 2020 2022 7261 775f 736f 6c75         "raw_solu
│ │ │ │ +00014730: 7469 6f6e 7322 2c0a 2020 2020 2020 2a20  tions",.      * 
│ │ │ │ +00014740: 4f75 7470 7574 5374 796c 6520 286d 6973  OutputStyle (mis
│ │ │ │ +00014750: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
│ │ │ │ +00014760: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661  on) => ..., defa
│ │ │ │ +00014770: 756c 7420 7661 6c75 6520 224f 7574 506f  ult value "OutPo
│ │ │ │ +00014780: 696e 7473 222c 200a 2020 2020 2020 2a20  ints", .      * 
│ │ │ │ +00014790: 2a6e 6f74 6520 5261 6e64 6f6d 436f 6d70  *note RandomComp
│ │ │ │ +000147a0: 6c65 783a 2042 6572 7469 6e69 2069 6e70  lex: Bertini inp
│ │ │ │ +000147b0: 7574 2066 696c 6520 6465 636c 6172 6174  ut file declarat
│ │ │ │ +000147c0: 696f 6e73 5f63 6f20 7261 6e64 6f6d 206e  ions_co random n
│ │ │ │ +000147d0: 756d 6265 7273 2c0a 2020 2020 2020 2020  umbers,.        
│ │ │ │ +000147e0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +000147f0: 7661 6c75 6520 7b7d 2c20 616e 206f 7074  value {}, an opt
│ │ │ │ +00014800: 696f 6e20 7768 6963 6820 6465 7369 676e  ion which design
│ │ │ │ +00014810: 6174 6573 0a20 2020 2020 2020 2073 796d  ates.        sym
│ │ │ │ +00014820: 626f 6c73 2f73 7472 696e 6773 2f76 6172  bols/strings/var
│ │ │ │ +00014830: 6961 626c 6573 2074 6861 7420 7769 6c6c  iables that will
│ │ │ │ +00014840: 2062 6520 7365 7420 746f 2062 6520 6120   be set to be a 
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│ │ │ │ +00014890: 6520 5261 6e64 6f6d 5265 616c 3a20 4265  e RandomReal: Be
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│ │ │ │ +00014960: 2020 2020 6f72 2072 616e 646f 6d20 636f      or random co
│ │ │ │ +00014970: 6d70 6c65 7820 6e75 6d62 6572 0a20 2020  mplex number.   
│ │ │ │ +00014980: 2020 202a 202a 6e6f 7465 2054 6f70 4469     * *note TopDi
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│ │ │ │ +000149d0: 3430 3334 2d30 2f30 222c 204f 7074 696f  4034-0/0", Optio
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│ │ │ │ +00014a50: 2c20 0a20 2020 2020 202a 202a 6e6f 7465  , .      * *note
│ │ │ │ +00014a60: 2056 6572 626f 7365 3a20 6265 7274 696e   Verbose: bertin
│ │ │ │ +00014a70: 6954 7261 636b 486f 6d6f 746f 7079 5f6c  iTrackHomotopy_l
│ │ │ │ +00014a80: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ +00014a90: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ +00014aa0: 7270 0a20 2020 2020 2020 202c 203d 3e20  rp.        , => 
│ │ │ │ +00014ab0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00014ac0: 7565 2066 616c 7365 2c20 4f70 7469 6f6e  ue false, Option
│ │ │ │ +00014ad0: 2074 6f20 7369 6c65 6e63 6520 6164 6469   to silence addi
│ │ │ │ +00014ae0: 7469 6f6e 616c 206f 7574 7075 740a 2020  tional output.  
│ │ │ │ +00014af0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
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│ │ │ │ +00014b20: 6329 4c69 7374 2c2c 2061 206c 6973 7420  c)List,, a list 
│ │ │ │ +00014b30: 6f66 2070 6f69 6e74 7320 7468 6174 2061  of points that a
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│ │ │ │ +00014b80: 3d0a 0a54 6869 7320 6d65 7468 6f64 2066  =..This method f
│ │ │ │ +00014b90: 696e 6473 2069 736f 6c61 7465 6420 736f  inds isolated so
│ │ │ │ +00014ba0: 6c75 7469 6f6e 7320 746f 2074 6865 2073  lutions to the s
│ │ │ │ +00014bb0: 7973 7465 6d20 4620 7669 6120 6e75 6d65  ystem F via nume
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│ │ │ │ +00014bd0: 0a68 6f6d 6f74 6f70 7920 636f 6e74 696e  .homotopy contin
│ │ │ │ +00014be0: 7561 7469 6f6e 2062 7920 2831 2920 6275  uation by (1) bu
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│ │ │ │ +00014c00: 2069 6e70 7574 2066 696c 6520 6672 6f6d   input file from
│ │ │ │ +00014c10: 2074 6865 2073 7973 7465 6d20 462c 0a28   the system F,.(
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│ │ │ │  00014ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00014ce0: 3120 3a20 5220 3d20 4343 5b78 2c79 5d3b  1 : R = CC[x,y];
│ │ │ │ -00014cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014ce0: 0a7c 6931 203a 2052 203d 2043 435b 782c  .|i1 : R = CC[x,
│ │ │ │ +00014cf0: 795d 3b20 2020 2020 2020 2020 2020 2020  y];             
│ │ │ │  00014d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00014d20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00014d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d60: 2d2d 2d2d 2b0a 7c69 3220 3a20 4620 3d20  ----+.|i2 : F = 
│ │ │ │ -00014d70: 7b78 5e32 2d31 2c79 5e32 2d32 7d3b 2020  {x^2-1,y^2-2};  
│ │ │ │ -00014d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00014d70: 203d 207b 785e 322d 312c 795e 322d 327d   = {x^2-1,y^2-2}
│ │ │ │ +00014d80: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  00014d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014da0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00014da0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00014db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00014df0: 3320 3a20 5320 3d20 6265 7274 696e 695a  3 : S = bertiniZ
│ │ │ │ -00014e00: 6572 6f44 696d 536f 6c76 6520 4620 2020  eroDimSolve F   
│ │ │ │ +00014de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014df0: 0a7c 6933 203a 2053 203d 2062 6572 7469  .|i3 : S = berti
│ │ │ │ +00014e00: 6e69 5a65 726f 4469 6d53 6f6c 7665 2046  niZeroDimSolve F
│ │ │ │  00014e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00014e30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00014e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e70: 2020 2020 7c0a 7c6f 3320 3d20 7b7b 312c      |.|o3 = {{1,
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│ │ │ │ -00014ea0: 2e34 3134 3231 7d2c 207b 2d31 2c20 2d31  .41421}, {-1, -1
│ │ │ │ -00014eb0: 2e34 3134 3231 7d7d 7c0a 7c20 2020 2020  .41421}}|.|     
│ │ │ │ +00014e70: 2020 2020 2020 207c 0a7c 6f33 203d 207b         |.|o3 = {
│ │ │ │ +00014e80: 7b31 2c20 312e 3431 3432 317d 2c20 7b31  {1, 1.41421}, {1
│ │ │ │ +00014e90: 2c20 2d31 2e34 3134 3231 7d2c 207b 2d31  , -1.41421}, {-1
│ │ │ │ +00014ea0: 2c20 312e 3431 3432 317d 2c20 7b2d 312c  , 1.41421}, {-1,
│ │ │ │ +00014eb0: 202d 312e 3431 3432 317d 7d7c 0a7c 2020   -1.41421}}|.|  
│ │ │ │  00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00014f00: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │ +00014ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014f00: 0a7c 6f33 203a 204c 6973 7420 2020 2020  .|o3 : List     
│ │ │ │  00014f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00014f40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00014f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f80: 2d2d 2d2d 2b0a 0a45 6163 6820 736f 6c75  ----+..Each solu
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│ │ │ │ -00014fa0: 2a6e 6f74 6520 506f 696e 743a 2028 4e41  *note Point: (NA
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│ │ │ │ -00014fe0: 626f 7574 2074 6865 2073 6f6c 7574 696f  bout the solutio
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│ │ │ │ -00015000: 6420 6279 2075 7369 6e67 202a 6e6f 7465  d by using *note
│ │ │ │ -00015010: 2070 6565 6b3a 0a28 4d61 6361 756c 6179   peek:.(Macaulay
│ │ │ │ -00015020: 3244 6f63 2970 6565 6b2c 2e0a 0a2b 2d2d  2Doc)peek,...+--
│ │ │ │ -00015030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00014fa0: 7065 202a 6e6f 7465 2050 6f69 6e74 3a20  pe *note Point: 
│ │ │ │ +00014fb0: 284e 4147 7479 7065 7329 4162 7374 7261  (NAGtypes)Abstra
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│ │ │ │ +00015000: 7373 6564 2062 7920 7573 696e 6720 2a6e  ssed by using *n
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│ │ │ │ +00015030: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00015040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015050: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2070  -------+.|i4 : p
│ │ │ │ -00015060: 6565 6b20 535f 3020 2020 2020 2020 2020  eek S_0         
│ │ │ │ +00015050: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +00015060: 3a20 7065 656b 2053 5f30 2020 2020 2020  : peek S_0      
│ │ │ │  00015070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015080: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015080: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000150b0: 0a7c 6f34 203d 2050 6f69 6e74 7b63 6163  .|o4 = Point{cac
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│ │ │ │ -000150e0: 2020 2020 2020 2020 2043 6f6f 7264 696e           Coordin
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│ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000150e0: 7c20 2020 2020 2020 2020 2020 436f 6f72  |           Coor
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│ │ │ │ +00015100: 2e34 3134 3231 7d20 2020 7c0a 2b2d 2d2d  .41421}   |.+---
│ │ │ │  00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015130: 2d2d 2d2b 0a0a 4265 7274 696e 6920 7573  ---+..Bertini us
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│ │ │ │ -00015160: 7320 6120 6465 6661 756c 742c 2062 7574  s a default, but
│ │ │ │ -00015170: 2072 6567 656e 6572 6174 696f 6e20 6361   regeneration ca
│ │ │ │ -00015180: 6e20 6265 0a64 6570 6c6f 7965 6420 7769  n be.deployed wi
│ │ │ │ -00015190: 7468 2074 6865 206f 7074 696f 6e20 5573  th the option Us
│ │ │ │ -000151a0: 6552 6567 656e 6572 6174 696f 6e3d 3e31  eRegeneration=>1
│ │ │ │ -000151b0: 202e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d   ...+-----------
│ │ │ │ +00015130: 2d2d 2d2d 2d2d 2b0a 0a42 6572 7469 6e69  ------+..Bertini
│ │ │ │ +00015140: 2075 7365 7320 6120 6d75 6c74 6968 6f6d   uses a multihom
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│ │ │ │ +00015190: 2077 6974 6820 7468 6520 6f70 7469 6f6e   with the option
│ │ │ │ +000151a0: 2055 7365 5265 6765 6e65 7261 7469 6f6e   UseRegeneration
│ │ │ │ +000151b0: 3d3e 3120 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  =>1 ...+--------
│ │ │ │  000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000151e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2052  -------+.|i5 : R
│ │ │ │ -000151f0: 203d 2043 435b 785d 3b20 2020 2020 2020   = CC[x];       
│ │ │ │ +000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +000151f0: 3a20 5220 3d20 4343 5b78 5d3b 2020 2020  : R = CC[x];    
│ │ │ │  00015200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00015220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015220: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │  00015240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015250: 2d2b 0a7c 6936 203a 2046 203d 207b 785e  -+.|i6 : F = {x^
│ │ │ │ -00015260: 322a 2878 2d31 297d 3b20 2020 2020 2020  2*(x-1)};       
│ │ │ │ +00015250: 2d2d 2d2d 2b0a 7c69 3620 3a20 4620 3d20  ----+.|i6 : F = 
│ │ │ │ +00015260: 7b78 5e32 2a28 782d 3129 7d3b 2020 2020  {x^2*(x-1)};    
│ │ │ │  00015270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015280: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00015280: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  00015290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000152a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000152b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -000152c0: 203a 2053 203d 2062 6572 7469 6e69 5a65   : S = bertiniZe
│ │ │ │ -000152d0: 726f 4469 6d53 6f6c 7665 2046 2020 2020  roDimSolve F    
│ │ │ │ +000152b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000152c0: 7c69 3720 3a20 5320 3d20 6265 7274 696e  |i7 : S = bertin
│ │ │ │ +000152d0: 695a 6572 6f44 696d 536f 6c76 6520 4620  iZeroDimSolve F 
│ │ │ │  000152e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000152f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015320: 2020 2020 207c 0a7c 6f37 203d 207b 7b31       |.|o7 = {{1
│ │ │ │ -00015330: 7d2c 207b 2d31 2e35 3535 3839 652d 3135  }, {-1.55589e-15
│ │ │ │ -00015340: 2d32 2e34 3630 3531 652d 3135 2a69 697d  -2.46051e-15*ii}
│ │ │ │ -00015350: 7d20 2020 2020 2020 2020 7c0a 7c20 2020  }         |.|   
│ │ │ │ +00015320: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +00015330: 7b7b 317d 2c20 7b2d 312e 3535 3538 3965  {{1}, {-1.55589e
│ │ │ │ +00015340: 2d31 352d 322e 3436 3035 3165 2d31 352a  -15-2.46051e-15*
│ │ │ │ +00015350: 6969 7d7d 2020 2020 2020 2020 207c 0a7c  ii}}         |.|
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│ │ │ │  00015370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015390: 0a7c 6f37 203a 204c 6973 7420 2020 2020  .|o7 : List     
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│ │ │ │ +00015390: 2020 7c0a 7c6f 3720 3a20 4c69 7374 2020    |.|o7 : List  
│ │ │ │  000153a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000153c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │  000153e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000153f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -00015400: 2042 203d 2062 6572 7469 6e69 5a65 726f   B = bertiniZero
│ │ │ │ -00015410: 4469 6d53 6f6c 7665 2846 2c55 7365 5265  DimSolve(F,UseRe
│ │ │ │ -00015420: 6765 6e65 7261 7469 6f6e 3d3e 3129 7c0a  generation=>1)|.
│ │ │ │ -00015430: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00015400: 3820 3a20 4220 3d20 6265 7274 696e 695a  8 : B = bertiniZ
│ │ │ │ +00015410: 6572 6f44 696d 536f 6c76 6528 462c 5573  eroDimSolve(F,Us
│ │ │ │ +00015420: 6552 6567 656e 6572 6174 696f 6e3d 3e31  eRegeneration=>1
│ │ │ │ +00015430: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │  00015440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015460: 2020 207c 0a7c 6f38 203d 207b 7b31 7d7d     |.|o8 = {{1}}
│ │ │ │ -00015470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015460: 2020 2020 2020 7c0a 7c6f 3820 3d20 7b7b        |.|o8 = {{
│ │ │ │ +00015470: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │  00015480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015490: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00015490: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000154a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000154b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000154d0: 6f38 203a 204c 6973 7420 2020 2020 2020  o8 : List       
│ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000154d0: 7c0a 7c6f 3820 3a20 4c69 7374 2020 2020  |.|o8 : List    
│ │ │ │  000154e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000154f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015500: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00015500: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
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│ │ │ │  00015520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015530: 2d2d 2d2d 2d2d 2d2b 0a0a 5661 7269 6162  -------+..Variab
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│ │ │ │ +000155e0: 6c76 6520 6f6e 6c79 2066 696e 6473 206e  lve only finds n
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│ │ │ │ +00015610: 7320 746f 2075 7365 2062 6572 7469 6e69  s to use bertini
│ │ │ │ +00015620: 5a65 726f 4469 6d53 6f6c 7665 3a0a 3d3d  ZeroDimSolve:.==
│ │ │ │  00015630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00015640: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00015650: 2022 6265 7274 696e 695a 6572 6f44 696d   "bertiniZeroDim
│ │ │ │ -00015660: 536f 6c76 6528 4964 6561 6c29 220a 2020  Solve(Ideal)".  
│ │ │ │ -00015670: 2a20 2262 6572 7469 6e69 5a65 726f 4469  * "bertiniZeroDi
│ │ │ │ -00015680: 6d53 6f6c 7665 284c 6973 7429 220a 0a46  mSolve(List)"..F
│ │ │ │ -00015690: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
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│ │ │ │ -000156b0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -000156c0: 202a 6e6f 7465 2062 6572 7469 6e69 5a65   *note bertiniZe
│ │ │ │ -000156d0: 726f 4469 6d53 6f6c 7665 3a20 6265 7274  roDimSolve: bert
│ │ │ │ -000156e0: 696e 695a 6572 6f44 696d 536f 6c76 652c  iniZeroDimSolve,
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│ │ │ │ -00015d90: 4265 7274 696e 6920 7072 6f64 7563 6573  Bertini produces
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│ │ │ │ -00015fa0: 746f 2069 6d70 6f72 7420 6669 6c65 7320  to import files 
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│ │ │ │ +00015b00: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
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│ │ │ │ +00015b20: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
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│ │ │ │ +00015b60: 2061 202a 6e6f 7465 2073 7472 696e 673a   a *note string:
│ │ │ │ +00015b70: 2028 4d61 6361 756c 6179 3244 6f63 2953   (Macaulay2Doc)S
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│ │ │ │ +00015c10: 4e61 6d65 496e 6369 6465 6e63 654d 6174  NameIncidenceMat
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│ │ │ │ +00015c50: 7661 6c75 650a 2020 2020 2020 2020 2269  value.        "i
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│ │ │ │ +00015c70: 2c0a 2020 2020 2020 2a20 5374 6f72 6167  ,.      * Storag
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│ │ │ │ +00015cb0: 7661 6c75 6520 6e75 6c6c 2c20 0a20 2020  value null, .   
│ │ │ │ +00015cc0: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
│ │ │ │ +00015cd0: 7365 3a20 6265 7274 696e 6954 7261 636b  se: bertiniTrack
│ │ │ │ +00015ce0: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70  Homotopy_lp_pd_p
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│ │ │ │ -00016ac0: 7420 286d 6973 7369 6e67 2064 6f63 756d  t (missing docum
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│ │ │ │ -00016b40: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
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│ │ │ │ -00016b60: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70  ackHomotopy_lp_p
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│ │ │ │ +00016900: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
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│ │ │ │ +00016b40: 2c20 0a20 2020 2020 202a 202a 6e6f 7465  , .      * *note
│ │ │ │ +00016b50: 2056 6572 626f 7365 3a20 6265 7274 696e   Verbose: bertin
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│ │ │ │ -00016e40: 0a7c 2020 2020 2020 2042 6572 7469 6e69  .|       Bertini
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│ │ │ │ -00016f30: 0a7c 6932 203a 2072 756e 4265 7274 696e  .|i2 : runBertin
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│ │ │ │ -00018b20: 2c20 6973 2073 6574 2074 6f20 3520 7468  , is set to 5 th
│ │ │ │ -00018b30: 656e 0a22 7261 775f 736f 6c75 7469 6f6e  en."raw_solution
│ │ │ │ -00018b40: 7322 2069 7320 696d 706f 7274 6564 2e0a  s" is imported..
│ │ │ │ -00018b50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000185a0: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +000185b0: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +000185c0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000185d0: 696d 706f 7274 536f 6c75 7469 6f6e 7346  importSolutionsF
│ │ │ │ +000185e0: 696c 6528 7329 0a20 202a 2049 6e70 7574  ile(s).  * Input
│ │ │ │ +000185f0: 733a 0a20 2020 2020 202a 2073 2c20 6120  s:.      * s, a 
│ │ │ │ +00018600: 2a6e 6f74 6520 7374 7269 6e67 3a20 284d  *note string: (M
│ │ │ │ +00018610: 6163 6175 6c61 7932 446f 6329 5374 7269  acaulay2Doc)Stri
│ │ │ │ +00018620: 6e67 2c2c 2054 6865 2064 6972 6563 746f  ng,, The directo
│ │ │ │ +00018630: 7279 2077 6865 7265 2074 6865 2066 696c  ry where the fil
│ │ │ │ +00018640: 650a 2020 2020 2020 2020 6973 2073 746f  e.        is sto
│ │ │ │ +00018650: 7265 642e 0a20 202a 202a 6e6f 7465 204f  red..  * *note O
│ │ │ │ +00018660: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ +00018670: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ +00018680: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ +00018690: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ +000186a0: 7473 2c3a 0a20 2020 2020 202a 204d 3250  ts,:.      * M2P
│ │ │ │ +000186b0: 7265 6369 7369 6f6e 2028 6d69 7373 696e  recision (missin
│ │ │ │ +000186c0: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ +000186d0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +000186e0: 2076 616c 7565 2035 332c 200a 2020 2020   value 53, .    
│ │ │ │ +000186f0: 2020 2a20 4e61 6d65 536f 6c75 7469 6f6e    * NameSolution
│ │ │ │ +00018700: 7346 696c 6520 286d 6973 7369 6e67 2064  sFile (missing d
│ │ │ │ +00018710: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ +00018720: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +00018730: 6c75 650a 2020 2020 2020 2020 2272 6177  lue.        "raw
│ │ │ │ +00018740: 5f73 6f6c 7574 696f 6e73 222c 0a20 2020  _solutions",.   
│ │ │ │ +00018750: 2020 202a 204f 7264 6572 5061 7468 7320     * OrderPaths 
│ │ │ │ +00018760: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ +00018770: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20  tation) => ..., 
│ │ │ │ +00018780: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ +00018790: 6c73 652c 200a 2020 2020 2020 2a20 5374  lse, .      * St
│ │ │ │ +000187a0: 6f72 6167 6546 6f6c 6465 7220 286d 6973  orageFolder (mis
│ │ │ │ +000187b0: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
│ │ │ │ +000187c0: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661  on) => ..., defa
│ │ │ │ +000187d0: 756c 7420 7661 6c75 6520 6e75 6c6c 2c20  ult value null, 
│ │ │ │ +000187e0: 0a20 2020 2020 202a 202a 6e6f 7465 2056  .      * *note V
│ │ │ │ +000187f0: 6572 626f 7365 3a20 6265 7274 696e 6954  erbose: bertiniT
│ │ │ │ +00018800: 7261 636b 486f 6d6f 746f 7079 5f6c 705f  rackHomotopy_lp_
│ │ │ │ +00018810: 7064 5f70 645f 7064 5f63 6d56 6572 626f  pd_pd_pd_cmVerbo
│ │ │ │ +00018820: 7365 3d3e 5f70 645f 7064 5f70 645f 7270  se=>_pd_pd_pd_rp
│ │ │ │ +00018830: 0a20 2020 2020 2020 202c 203d 3e20 2e2e  .        , => ..
│ │ │ │ +00018840: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +00018850: 2066 616c 7365 2c20 4f70 7469 6f6e 2074   false, Option t
│ │ │ │ +00018860: 6f20 7369 6c65 6e63 6520 6164 6469 7469  o silence additi
│ │ │ │ +00018870: 6f6e 616c 206f 7574 7075 740a 0a44 6573  onal output..Des
│ │ │ │ +00018880: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00018890: 3d3d 3d3d 0a0a 4166 7465 7220 4265 7274  ====..After Bert
│ │ │ │ +000188a0: 696e 6920 646f 6573 2061 2072 756e 206d  ini does a run m
│ │ │ │ +000188b0: 616e 7920 6669 6c65 7320 6172 6520 6372  any files are cr
│ │ │ │ +000188c0: 6561 7465 642e 2054 6869 7320 6675 6e63  eated. This func
│ │ │ │ +000188d0: 7469 6f6e 2069 6d70 6f72 7473 2074 6865  tion imports the
│ │ │ │ +000188e0: 0a63 6f6f 7264 696e 6174 6573 206f 6620  .coordinates of 
│ │ │ │ +000188f0: 736f 6c75 7469 6f6e 7320 6672 6f6d 2074  solutions from t
│ │ │ │ +00018900: 6865 2073 696d 706c 6520 2272 6177 5f73  he simple "raw_s
│ │ │ │ +00018910: 6f6c 7574 696f 6e73 2220 6669 6c65 2e20  olutions" file. 
│ │ │ │ +00018920: 4279 2075 7369 6e67 2074 6865 0a6f 7074  By using the.opt
│ │ │ │ +00018930: 696f 6e20 4e61 6d65 536f 6c75 7469 6f6e  ion NameSolution
│ │ │ │ +00018940: 7346 696c 653d 3e22 7265 616c 5f66 696e  sFile=>"real_fin
│ │ │ │ +00018950: 6974 655f 736f 6c75 7469 6f6e 7322 2077  ite_solutions" w
│ │ │ │ +00018960: 6520 776f 756c 6420 696d 706f 7274 2073  e would import s
│ │ │ │ +00018970: 6f6c 7574 696f 6e73 0a66 726f 6d20 7265  olutions.from re
│ │ │ │ +00018980: 616c 2066 696e 6974 6520 736f 6c75 7469  al finite soluti
│ │ │ │ +00018990: 6f6e 732e 204f 7468 6572 2063 6f6d 6d6f  ons. Other commo
│ │ │ │ +000189a0: 6e20 6669 6c65 206e 616d 6573 2061 7265  n file names are
│ │ │ │ +000189b0: 0a22 6e6f 6e73 696e 6775 6c61 725f 736f  ."nonsingular_so
│ │ │ │ +000189c0: 6c75 7469 6f6e 7322 2c20 2266 696e 6974  lutions", "finit
│ │ │ │ +000189d0: 655f 736f 6c75 7469 6f6e 7322 2c20 2269  e_solutions", "i
│ │ │ │ +000189e0: 6e66 696e 6974 655f 736f 6c75 7469 6f6e  nfinite_solution
│ │ │ │ +000189f0: 7322 2c20 616e 640a 2273 696e 6775 6c61  s", and."singula
│ │ │ │ +00018a00: 725f 736f 6c75 7469 6f6e 7322 2e0a 0a49  r_solutions"...I
│ │ │ │ +00018a10: 6620 7468 6520 4e61 6d65 536f 6c75 7469  f the NameSoluti
│ │ │ │ +00018a20: 6f6e 7346 696c 6520 6f70 7469 6f6e 2069  onsFile option i
│ │ │ │ +00018a30: 7320 7365 7420 746f 2030 2074 6865 6e20  s set to 0 then 
│ │ │ │ +00018a40: 226e 6f6e 7369 6e67 756c 6172 5f73 6f6c  "nonsingular_sol
│ │ │ │ +00018a50: 7574 696f 6e73 2220 6973 0a69 6d70 6f72  utions" is.impor
│ │ │ │ +00018a60: 7465 642c 2069 7320 7365 7420 746f 2031  ted, is set to 1
│ │ │ │ +00018a70: 2074 6865 6e20 2272 6561 6c5f 6669 6e69   then "real_fini
│ │ │ │ +00018a80: 7465 5f73 6f6c 7574 696f 6e73 2220 6973  te_solutions" is
│ │ │ │ +00018a90: 2069 6d70 6f72 7465 642c 2069 7320 7365   imported, is se
│ │ │ │ +00018aa0: 7420 746f 2032 0a74 6865 6e20 2269 6e66  t to 2.then "inf
│ │ │ │ +00018ab0: 696e 6974 655f 736f 6c75 7469 6f6e 7322  inite_solutions"
│ │ │ │ +00018ac0: 2069 7320 696d 706f 7274 6564 2c20 6973   is imported, is
│ │ │ │ +00018ad0: 2073 6574 2074 6f20 3320 7468 656e 2022   set to 3 then "
│ │ │ │ +00018ae0: 6669 6e69 7465 5f73 6f6c 7574 696f 6e73  finite_solutions
│ │ │ │ +00018af0: 2220 6973 0a69 6d70 6f72 7465 642c 2069  " is.imported, i
│ │ │ │ +00018b00: 7320 7365 7420 746f 2034 2074 6865 6e20  s set to 4 then 
│ │ │ │ +00018b10: 2273 7461 7274 2220 6973 2069 6d70 6f72  "start" is impor
│ │ │ │ +00018b20: 7465 642c 2069 7320 7365 7420 746f 2035  ted, is set to 5
│ │ │ │ +00018b30: 2074 6865 6e0a 2272 6177 5f73 6f6c 7574   then."raw_solut
│ │ │ │ +00018b40: 696f 6e73 2220 6973 2069 6d70 6f72 7465  ions" is importe
│ │ │ │ +00018b50: 642e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  d...+-----------
│ │ │ │  00018b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00018ba0: 0a7c 6931 203a 2052 3d51 515b 782c 795d  .|i1 : R=QQ[x,y]
│ │ │ │ -00018bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ba0: 2d2d 2b0a 7c69 3120 3a20 523d 5151 5b78  --+.|i1 : R=QQ[x
│ │ │ │ +00018bb0: 2c79 5d20 2020 2020 2020 2020 2020 2020  ,y]             
│ │ │ │  00018bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018bf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00018c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018c40: 0a7c 6f31 203d 2052 2020 2020 2020 2020  .|o1 = R        
│ │ │ │ +00018c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018c40: 2020 7c0a 7c6f 3120 3d20 5220 2020 2020    |.|o1 = R     
│ │ │ │  00018c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018c90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018c90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00018ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018ce0: 0a7c 6f31 203a 2050 6f6c 796e 6f6d 6961  .|o1 : Polynomia
│ │ │ │ -00018cf0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +00018cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ce0: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f    |.|o1 : Polyno
│ │ │ │ +00018cf0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  00018d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018d30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00018d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00018d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00018d80: 0a7c 6932 203a 206d 616b 6542 2749 6e70  .|i2 : makeB'Inp
│ │ │ │ -00018d90: 7574 4669 6c65 2873 746f 7265 424d 3246  utFile(storeBM2F
│ │ │ │ -00018da0: 696c 6573 2c20 2020 2020 2020 2020 2020  iles,           
│ │ │ │ +00018d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018d80: 2d2d 2b0a 7c69 3220 3a20 6d61 6b65 4227  --+.|i2 : makeB'
│ │ │ │ +00018d90: 496e 7075 7446 696c 6528 7374 6f72 6542  InputFile(storeB
│ │ │ │ +00018da0: 4d32 4669 6c65 732c 2020 2020 2020 2020  M2Files,        
│ │ │ │  00018db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018dd0: 0a7c 2020 2020 2020 2020 2041 6666 5661  .|         AffVa
│ │ │ │ -00018de0: 7269 6162 6c65 4772 6f75 703d 3e7b 7b78  riableGroup=>{{x
│ │ │ │ -00018df0: 2c79 7d7d 2c20 2020 2020 2020 2020 2020  ,y}},           
│ │ │ │ +00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018dd0: 2020 7c0a 7c20 2020 2020 2020 2020 4166    |.|         Af
│ │ │ │ +00018de0: 6656 6172 6961 626c 6547 726f 7570 3d3e  fVariableGroup=>
│ │ │ │ +00018df0: 7b7b 782c 797d 7d2c 2020 2020 2020 2020  {{x,y}},        
│ │ │ │  00018e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018e20: 0a7c 2020 2020 2020 2020 2042 2750 6f6c  .|         B'Pol
│ │ │ │ -00018e30: 796e 6f6d 6961 6c73 3d3e 7b78 5e32 2d31  ynomials=>{x^2-1
│ │ │ │ -00018e40: 2c79 5e33 2d31 7d29 3b20 2020 2020 2020  ,y^3-1});       
│ │ │ │ +00018e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018e20: 2020 7c0a 7c20 2020 2020 2020 2020 4227    |.|         B'
│ │ │ │ +00018e30: 506f 6c79 6e6f 6d69 616c 733d 3e7b 785e  Polynomials=>{x^
│ │ │ │ +00018e40: 322d 312c 795e 332d 317d 293b 2020 2020  2-1,y^3-1});    
│ │ │ │  00018e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018e70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00018e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018e70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00018ec0: 0a7c 6933 203a 2072 756e 4265 7274 696e  .|i3 : runBertin
│ │ │ │ -00018ed0: 6928 7374 6f72 6542 4d32 4669 6c65 7329  i(storeBM2Files)
│ │ │ │ -00018ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ec0: 2d2d 2b0a 7c69 3320 3a20 7275 6e42 6572  --+.|i3 : runBer
│ │ │ │ +00018ed0: 7469 6e69 2873 746f 7265 424d 3246 696c  tini(storeBM2Fil
│ │ │ │ +00018ee0: 6573 2920 2020 2020 2020 2020 2020 2020  es)             
│ │ │ │  00018ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018f10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00018f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018f10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00018f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00018f60: 0a7c 6934 203a 2069 6d70 6f72 7453 6f6c  .|i4 : importSol
│ │ │ │ -00018f70: 7574 696f 6e73 4669 6c65 2873 746f 7265  utionsFile(store
│ │ │ │ -00018f80: 424d 3246 696c 6573 2920 2020 2020 2020  BM2Files)       
│ │ │ │ +00018f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f60: 2d2d 2b0a 7c69 3420 3a20 696d 706f 7274  --+.|i4 : import
│ │ │ │ +00018f70: 536f 6c75 7469 6f6e 7346 696c 6528 7374  SolutionsFile(st
│ │ │ │ +00018f80: 6f72 6542 4d32 4669 6c65 7329 2020 2020  oreBM2Files)    
│ │ │ │  00018f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00018fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00019000: 0a7c 6f34 203d 207b 7b31 2c20 317d 2c20  .|o4 = {{1, 1}, 
│ │ │ │ -00019010: 7b31 2c20 2d2e 352b 2e38 3636 3032 352a  {1, -.5+.866025*
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│ │ │ │ -00019040: 7d2c 207b 2d31 2c20 2020 2020 2020 207c  }, {-1,        |
│ │ │ │ -00019050: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00018ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019000: 2020 7c0a 7c6f 3420 3d20 7b7b 312c 2031    |.|o4 = {{1, 1
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│ │ │ │ +00019020: 3235 2a69 697d 2c20 7b31 2c20 2d2e 352d  25*ii}, {1, -.5-
│ │ │ │ +00019030: 2e38 3636 3032 352a 6969 7d2c 207b 2d31  .866025*ii}, {-1
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│ │ │ │ -00019920: 3d3d 3d3d 0a0a 5768 656e 2073 6574 2074  ====..When set t
│ │ │ │ -00019930: 6f20 312c 2074 6869 7320 6f70 7469 6f6e  o 1, this option
│ │ │ │ -00019940: 2069 6e64 6963 6174 6573 2074 6861 7420   indicates that 
│ │ │ │ -00019950: 7468 6520 696e 7075 7420 7379 7374 656d  the input system
│ │ │ │ -00019960: 2069 7320 686f 6d6f 6765 6e69 7a65 6420   is homogenized 
│ │ │ │ -00019970: 616e 640a 7468 6520 6f75 7470 7574 2073  and.the output s
│ │ │ │ -00019980: 686f 756c 6420 6265 2067 6976 656e 2069  hould be given i
│ │ │ │ -00019990: 6e20 7072 6f6a 6563 7469 7665 2073 7061  n projective spa
│ │ │ │ -000199a0: 6365 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ce...+----------
│ │ │ │ +00019900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00019910: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ +00019960: 7465 6d20 6973 2068 6f6d 6f67 656e 697a  tem is homogeniz
│ │ │ │ +00019970: 6564 2061 6e64 0a74 6865 206f 7574 7075  ed and.the outpu
│ │ │ │ +00019980: 7420 7368 6f75 6c64 2062 6520 6769 7665  t should be give
│ │ │ │ +00019990: 6e20 696e 2070 726f 6a65 6374 6976 6520  n in projective 
│ │ │ │ +000199a0: 7370 6163 652e 0a0a 2b2d 2d2d 2d2d 2d2d  space...+-------
│ │ │ │  000199b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000199c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000199d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -000199e0: 2052 203d 2043 435b 782c 792c 7a5d 3b20   R = CC[x,y,z]; 
│ │ │ │ -000199f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00019a10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000199d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000199e0: 3120 3a20 5220 3d20 4343 5b78 2c79 2c7a  1 : R = CC[x,y,z
│ │ │ │ +000199f0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ +00019a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019a10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00019a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a40: 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 203d  -----+.|i2 : f =
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│ │ │ │ -00019a60: 287a 2d78 292c 2878 5e32 2b79 5e32 2d7a  (z-x),(x^2+y^2-z
│ │ │ │ -00019a70: 5e32 292a 287a 2b79 297d 3b7c 0a2b 2d2d  ^2)*(z+y)};|.+--
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│ │ │ │ +00019a40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00019a50: 6620 3d20 7b28 785e 322b 795e 322d 7a5e  f = {(x^2+y^2-z^
│ │ │ │ +00019a60: 3229 2a28 7a2d 7829 2c28 785e 322b 795e  2)*(z-x),(x^2+y^
│ │ │ │ +00019a70: 322d 7a5e 3229 2a28 7a2b 7929 7d3b 7c0a  2-z^2)*(z+y)};|.
│ │ │ │ +00019a80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00019a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ab0: 2d2b 0a7c 6933 203a 2062 6572 7469 6e69  -+.|i3 : bertini
│ │ │ │ -00019ac0: 506f 7344 696d 536f 6c76 6528 662c 4973  PosDimSolve(f,Is
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│ │ │ │ -00019ae0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00019ab0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6265 7274  ----+.|i3 : bert
│ │ │ │ +00019ac0: 696e 6950 6f73 4469 6d53 6f6c 7665 2866  iniPosDimSolve(f
│ │ │ │ +00019ad0: 2c49 7350 726f 6a65 6374 6976 653d 3e31  ,IsProjective=>1
│ │ │ │ +00019ae0: 293b 2020 2020 2020 2020 7c0a 2b2d 2d2d  );        |.+---
│ │ │ │  00019af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00019b20: 4675 6e63 7469 6f6e 7320 7769 7468 206f  Functions with o
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│ │ │ │ +00019b40: 656e 7420 6e61 6d65 6420 4973 5072 6f6a  ent named IsProj
│ │ │ │ +00019b50: 6563 7469 7665 3a0a 3d3d 3d3d 3d3d 3d3d  ective:.========
│ │ │ │  00019b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00019b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00019b80: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2062  =========..  * b
│ │ │ │ -00019b90: 6572 7469 6e69 5a65 726f 4469 6d53 6f6c  ertiniZeroDimSol
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│ │ │ │ -00019bb0: 6976 653d 3e2e 2e2e 2920 286d 6973 7369  ive=>...) (missi
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│ │ │ │ -00019c40: 6572 7469 6e69 5265 6669 6e65 536f 6c73  ertiniRefineSols
│ │ │ │ -00019c50: 282e 2e2e 2c49 7350 726f 6a65 6374 6976  (...,IsProjectiv
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│ │ │ │ -00019c70: 7274 696e 6953 616d 706c 6528 2e2e 2e2c  rtiniSample(...,
│ │ │ │ -00019c80: 4973 5072 6f6a 6563 7469 7665 3d3e 2e2e  IsProjective=>..
│ │ │ │ -00019c90: 2e29 220a 2020 2a20 2262 6572 7469 6e69  .)".  * "bertini
│ │ │ │ -00019ca0: 5472 6163 6b48 6f6d 6f74 6f70 7928 2e2e  TrackHomotopy(..
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│ │ │ │ -00019cc0: 2e2e 2e29 220a 0a46 6f72 2074 6865 2070  ...)"..For the p
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│ │ │ │ -00019de0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00019e30: 6f74 6520 7379 6d62 6f6c 3a0a 284d 6163  ote symbol:.(Mac
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│ │ │ │ -00019e70: 616b 6542 2749 6e70 7574 4669 6c65 2c20  akeB'InputFile, 
│ │ │ │ -00019e80: 4e65 7874 3a20 6d61 6b65 4227 5365 6374  Next: makeB'Sect
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│ │ │ │ -00019ec0: 7574 4669 6c65 202d 2d20 7772 6974 6520  utFile -- write 
│ │ │ │ -00019ed0: 6120 4265 7274 696e 6920 696e 7075 7420  a Bertini input 
│ │ │ │ -00019ee0: 6669 6c65 2069 6e20 6120 6469 7265 6374  file in a direct
│ │ │ │ -00019ef0: 6f72 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ory.************
│ │ │ │ +00019b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00019b90: 2a20 6265 7274 696e 695a 6572 6f44 696d  * bertiniZeroDim
│ │ │ │ +00019ba0: 536f 6c76 6528 2e2e 2e2c 4973 5072 6f6a  Solve(...,IsProj
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│ │ │ │ +00019bf0: 5465 7374 282e 2e2e 2c49 7350 726f 6a65  Test(...,IsProje
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│ │ │ │ +00019c10: 2022 6265 7274 696e 6950 6f73 4469 6d53   "bertiniPosDimS
│ │ │ │ +00019c20: 6f6c 7665 282e 2e2e 2c49 7350 726f 6a65  olve(...,IsProje
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│ │ │ │ +00019c40: 2022 6265 7274 696e 6952 6566 696e 6553   "bertiniRefineS
│ │ │ │ +00019c50: 6f6c 7328 2e2e 2e2c 4973 5072 6f6a 6563  ols(...,IsProjec
│ │ │ │ +00019c60: 7469 7665 3d3e 2e2e 2e29 220a 2020 2a20  tive=>...)".  * 
│ │ │ │ +00019c70: 2262 6572 7469 6e69 5361 6d70 6c65 282e  "bertiniSample(.
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│ │ │ │ +00019dd0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
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│ │ │ │ +00019e40: 4d61 6361 756c 6179 3244 6f63 2953 796d  Macaulay2Doc)Sym
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│ │ │ │ +00019ec0: 496e 7075 7446 696c 6520 2d2d 2077 7269  InputFile -- wri
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│ │ │ │ +00019ef0: 6563 746f 7279 0a2a 2a2a 2a2a 2a2a 2a2a  ectory.*********
│ │ │ │  00019f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00019f10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00019f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f30: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ -00019f40: 3d3d 3d0a 0a20 202a 2055 7361 6765 3a20  ===..  * Usage: 
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│ │ │ │ -00019f60: 6e70 7574 4669 6c65 2873 290a 2020 2a20  nputFile(s).  * 
│ │ │ │ -00019f70: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00019f80: 732c 2061 202a 6e6f 7465 2073 7472 696e  s, a *note strin
│ │ │ │ -00019f90: 673a 2028 4d61 6361 756c 6179 3244 6f63  g: (Macaulay2Doc
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│ │ │ │ -00019fe0: 656e 0a20 202a 202a 6e6f 7465 204f 7074  en.  * *note Opt
│ │ │ │ -00019ff0: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ -0001a000: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ -0001a010: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ -0001a020: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ -0001a030: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465  ,:.      * *note
│ │ │ │ -0001a040: 2041 6666 5661 7269 6162 6c65 4772 6f75   AffVariableGrou
│ │ │ │ -0001a050: 703a 2056 6172 6961 626c 6520 6772 6f75  p: Variable grou
│ │ │ │ -0001a060: 7073 2c20 3d3e 202e 2e2e 2c20 6465 6661  ps, => ..., defa
│ │ │ │ -0001a070: 756c 7420 7661 6c75 6520 7b7d 2c20 616e  ult value {}, an
│ │ │ │ -0001a080: 0a20 2020 2020 2020 206f 7074 696f 6e20  .        option 
│ │ │ │ -0001a090: 746f 2067 726f 7570 2076 6172 6961 626c  to group variabl
│ │ │ │ -0001a0a0: 6573 2061 6e64 2075 7365 206d 756c 7469  es and use multi
│ │ │ │ -0001a0b0: 686f 6d6f 6765 6e65 6f75 7320 686f 6d6f  homogeneous homo
│ │ │ │ -0001a0c0: 746f 7069 6573 0a20 2020 2020 202a 202a  topies.      * *
│ │ │ │ -0001a0d0: 6e6f 7465 2042 2743 6f6e 7374 616e 7473  note B'Constants
│ │ │ │ -0001a0e0: 3a20 4227 436f 6e73 7461 6e74 732c 203d  : B'Constants, =
│ │ │ │ -0001a0f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -0001a100: 616c 7565 207b 7d2c 2061 6e20 6f70 7469  alue {}, an opti
│ │ │ │ -0001a110: 6f6e 2074 6f0a 2020 2020 2020 2020 6465  on to.        de
│ │ │ │ -0001a120: 7369 676e 6174 6520 7468 6520 636f 6e73  signate the cons
│ │ │ │ -0001a130: 7461 6e74 7320 666f 7220 6120 4265 7274  tants for a Bert
│ │ │ │ -0001a140: 696e 6920 496e 7075 7420 6669 6c65 0a20  ini Input file. 
│ │ │ │ -0001a150: 2020 2020 202a 2042 2746 756e 6374 696f       * B'Functio
│ │ │ │ -0001a160: 6e73 2028 6d69 7373 696e 6720 646f 6375  ns (missing docu
│ │ │ │ -0001a170: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e  mentation) => ..
│ │ │ │ -0001a180: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0001a190: 207b 7d2c 200a 2020 2020 2020 2a20 4227   {}, .      * B'
│ │ │ │ -0001a1a0: 506f 6c79 6e6f 6d69 616c 7320 286d 6973  Polynomials (mis
│ │ │ │ -0001a1b0: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
│ │ │ │ -0001a1c0: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661  on) => ..., defa
│ │ │ │ -0001a1d0: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20  ult value {}, . 
│ │ │ │ -0001a1e0: 2020 2020 202a 2042 6572 7469 6e69 496e       * BertiniIn
│ │ │ │ -0001a1f0: 7075 7443 6f6e 6669 6775 7261 7469 6f6e  putConfiguration
│ │ │ │ -0001a200: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ -0001a210: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ -0001a220: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ -0001a230: 2020 2020 2020 207b 7d2c 0a20 2020 2020         {},.     
│ │ │ │ -0001a240: 202a 202a 6e6f 7465 2048 6f6d 5661 7269   * *note HomVari
│ │ │ │ -0001a250: 6162 6c65 4772 6f75 703a 2056 6172 6961  ableGroup: Varia
│ │ │ │ -0001a260: 626c 6520 6772 6f75 7073 2c20 3d3e 202e  ble groups, => .
│ │ │ │ -0001a270: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -0001a280: 6520 7b7d 2c20 616e 0a20 2020 2020 2020  e {}, an.       
│ │ │ │ -0001a290: 206f 7074 696f 6e20 746f 2067 726f 7570   option to group
│ │ │ │ -0001a2a0: 2076 6172 6961 626c 6573 2061 6e64 2075   variables and u
│ │ │ │ -0001a2b0: 7365 206d 756c 7469 686f 6d6f 6765 6e65  se multihomogene
│ │ │ │ -0001a2c0: 6f75 7320 686f 6d6f 746f 7069 6573 0a20  ous homotopies. 
│ │ │ │ -0001a2d0: 2020 2020 202a 204e 616d 6542 2749 6e70       * NameB'Inp
│ │ │ │ -0001a2e0: 7574 4669 6c65 2028 6d69 7373 696e 6720  utFile (missing 
│ │ │ │ -0001a2f0: 646f 6375 6d65 6e74 6174 696f 6e29 203d  documentation) =
│ │ │ │ -0001a300: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -0001a310: 616c 7565 2022 696e 7075 7422 2c20 0a20  alue "input", . 
│ │ │ │ -0001a320: 2020 2020 202a 204e 616d 6550 6f6c 796e       * NamePolyn
│ │ │ │ -0001a330: 6f6d 6961 6c73 2028 6d69 7373 696e 6720  omials (missing 
│ │ │ │ -0001a340: 646f 6375 6d65 6e74 6174 696f 6e29 203d  documentation) =
│ │ │ │ -0001a350: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -0001a360: 616c 7565 207b 7d2c 200a 2020 2020 2020  alue {}, .      
│ │ │ │ -0001a370: 2a20 5061 7261 6d65 7465 7247 726f 7570  * ParameterGroup
│ │ │ │ -0001a380: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ -0001a390: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ -0001a3a0: 2064 6566 6175 6c74 2076 616c 7565 207b   default value {
│ │ │ │ -0001a3b0: 7d2c 200a 2020 2020 2020 2a20 5061 7468  }, .      * Path
│ │ │ │ -0001a3c0: 5661 7269 6162 6c65 2028 6d69 7373 696e  Variable (missin
│ │ │ │ -0001a3d0: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ -0001a3e0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -0001a3f0: 2076 616c 7565 207b 7d2c 200a 2020 2020   value {}, .    
│ │ │ │ -0001a400: 2020 2a20 2a6e 6f74 6520 5261 6e64 6f6d    * *note Random
│ │ │ │ -0001a410: 436f 6d70 6c65 783a 2042 6572 7469 6e69  Complex: Bertini
│ │ │ │ -0001a420: 2069 6e70 7574 2066 696c 6520 6465 636c   input file decl
│ │ │ │ -0001a430: 6172 6174 696f 6e73 5f63 6f20 7261 6e64  arations_co rand
│ │ │ │ -0001a440: 6f6d 206e 756d 6265 7273 2c0a 2020 2020  om numbers,.    
│ │ │ │ -0001a450: 2020 2020 3d3e 202e 2e2e 2c20 6465 6661      => ..., defa
│ │ │ │ -0001a460: 756c 7420 7661 6c75 6520 7b7d 2c20 616e  ult value {}, an
│ │ │ │ -0001a470: 206f 7074 696f 6e20 7768 6963 6820 6465   option which de
│ │ │ │ -0001a480: 7369 676e 6174 6573 0a20 2020 2020 2020  signates.       
│ │ │ │ -0001a490: 2073 796d 626f 6c73 2f73 7472 696e 6773   symbols/strings
│ │ │ │ -0001a4a0: 2f76 6172 6961 626c 6573 2074 6861 7420  /variables that 
│ │ │ │ -0001a4b0: 7769 6c6c 2062 6520 7365 7420 746f 2062  will be set to b
│ │ │ │ -0001a4c0: 6520 6120 7261 6e64 6f6d 2072 6561 6c20  e a random real 
│ │ │ │ -0001a4d0: 6e75 6d62 6572 0a20 2020 2020 2020 206f  number.        o
│ │ │ │ -0001a4e0: 7220 7261 6e64 6f6d 2063 6f6d 706c 6578  r random complex
│ │ │ │ -0001a4f0: 206e 756d 6265 720a 2020 2020 2020 2a20   number.      * 
│ │ │ │ -0001a500: 2a6e 6f74 6520 5261 6e64 6f6d 5265 616c  *note RandomReal
│ │ │ │ -0001a510: 3a20 4265 7274 696e 6920 696e 7075 7420  : Bertini input 
│ │ │ │ -0001a520: 6669 6c65 2064 6563 6c61 7261 7469 6f6e  file declaration
│ │ │ │ -0001a530: 735f 636f 2072 616e 646f 6d20 6e75 6d62  s_co random numb
│ │ │ │ -0001a540: 6572 732c 203d 3e0a 2020 2020 2020 2020  ers, =>.        
│ │ │ │ -0001a550: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -0001a560: 7565 207b 7d2c 2061 6e20 6f70 7469 6f6e  ue {}, an option
│ │ │ │ -0001a570: 2077 6869 6368 2064 6573 6967 6e61 7465   which designate
│ │ │ │ -0001a580: 730a 2020 2020 2020 2020 7379 6d62 6f6c  s.        symbol
│ │ │ │ -0001a590: 732f 7374 7269 6e67 732f 7661 7269 6162  s/strings/variab
│ │ │ │ -0001a5a0: 6c65 7320 7468 6174 2077 696c 6c20 6265  les that will be
│ │ │ │ -0001a5b0: 2073 6574 2074 6f20 6265 2061 2072 616e   set to be a ran
│ │ │ │ -0001a5c0: 646f 6d20 7265 616c 206e 756d 6265 720a  dom real number.
│ │ │ │ -0001a5d0: 2020 2020 2020 2020 6f72 2072 616e 646f          or rando
│ │ │ │ -0001a5e0: 6d20 636f 6d70 6c65 7820 6e75 6d62 6572  m complex number
│ │ │ │ -0001a5f0: 0a20 2020 2020 202a 2053 6574 5061 7261  .      * SetPara
│ │ │ │ -0001a600: 6d65 7465 7247 726f 7570 2028 6d69 7373  meterGroup (miss
│ │ │ │ -0001a610: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -0001a620: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175  n) => ..., defau
│ │ │ │ -0001a630: 6c74 2076 616c 7565 207b 7d2c 200a 2020  lt value {}, .  
│ │ │ │ -0001a640: 2020 2020 2a20 5374 6f72 6167 6546 6f6c      * StorageFol
│ │ │ │ -0001a650: 6465 7220 286d 6973 7369 6e67 2064 6f63  der (missing doc
│ │ │ │ -0001a660: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ -0001a670: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -0001a680: 6520 6e75 6c6c 2c20 0a20 2020 2020 202a  e null, .      *
│ │ │ │ -0001a690: 2056 6172 6961 626c 654c 6973 7420 286d   VariableList (m
│ │ │ │ -0001a6a0: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ -0001a6b0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ -0001a6c0: 6661 756c 7420 7661 6c75 6520 7b7d 2c20  fault value {}, 
│ │ │ │ -0001a6d0: 0a20 2020 2020 202a 202a 6e6f 7465 2056  .      * *note V
│ │ │ │ -0001a6e0: 6572 626f 7365 3a20 6265 7274 696e 6954  erbose: bertiniT
│ │ │ │ -0001a6f0: 7261 636b 486f 6d6f 746f 7079 5f6c 705f  rackHomotopy_lp_
│ │ │ │ -0001a700: 7064 5f70 645f 7064 5f63 6d56 6572 626f  pd_pd_pd_cmVerbo
│ │ │ │ -0001a710: 7365 3d3e 5f70 645f 7064 5f70 645f 7270  se=>_pd_pd_pd_rp
│ │ │ │ -0001a720: 0a20 2020 2020 2020 202c 203d 3e20 2e2e  .        , => ..
│ │ │ │ -0001a730: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0001a740: 2066 616c 7365 2c20 4f70 7469 6f6e 2074   false, Option t
│ │ │ │ -0001a750: 6f20 7369 6c65 6e63 6520 6164 6469 7469  o silence additi
│ │ │ │ -0001a760: 6f6e 616c 206f 7574 7075 740a 0a44 6573  onal output..Des
│ │ │ │ -0001a770: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0001a780: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374  ====..This funct
│ │ │ │ -0001a790: 696f 6e20 7772 6974 6573 2061 2042 6572  ion writes a Ber
│ │ │ │ -0001a7a0: 7469 6e69 2069 6e70 7574 2066 696c 652e  tini input file.
│ │ │ │ -0001a7b0: 2054 6865 2075 7365 7220 6361 6e20 7370   The user can sp
│ │ │ │ -0001a7c0: 6563 6966 7920 434f 4e46 4947 5320 666f  ecify CONFIGS fo
│ │ │ │ -0001a7d0: 7220 7468 650a 6669 6c65 2075 7369 6e67  r the.file using
│ │ │ │ -0001a7e0: 2074 6865 2042 6572 7469 6e69 496e 7075   the BertiniInpu
│ │ │ │ -0001a7f0: 7443 6f6e 6669 6775 7261 7469 6f6e 206f  tConfiguration o
│ │ │ │ -0001a800: 7074 696f 6e2e 2054 6865 2075 7365 7220  ption. The user 
│ │ │ │ -0001a810: 7368 6f75 6c64 2073 7065 6369 6679 0a76  should specify.v
│ │ │ │ -0001a820: 6172 6961 626c 6520 6772 6f75 7073 2077  ariable groups w
│ │ │ │ -0001a830: 6974 6820 7468 6520 4166 6656 6172 6961  ith the AffVaria
│ │ │ │ -0001a840: 626c 6547 726f 7570 2028 6166 6669 6e65  bleGroup (affine
│ │ │ │ -0001a850: 2076 6172 6961 626c 6520 6772 6f75 7029   variable group)
│ │ │ │ -0001a860: 206f 7074 696f 6e20 6f72 0a48 6f6d 5661   option or.HomVa
│ │ │ │ -0001a870: 7269 6162 6c65 4772 6f75 7020 2868 6f6d  riableGroup (hom
│ │ │ │ -0001a880: 6f67 656e 656f 7573 2076 6172 6961 626c  ogeneous variabl
│ │ │ │ -0001a890: 6520 6772 6f75 7029 206f 7074 696f 6e2e  e group) option.
│ │ │ │ -0001a8a0: 2054 6865 2075 7365 7220 7368 6f75 6c64   The user should
│ │ │ │ -0001a8b0: 2073 7065 6369 6679 0a74 6865 2070 6f6c   specify.the pol
│ │ │ │ -0001a8c0: 796e 6f6d 6961 6c20 7379 7374 656d 2074  ynomial system t
│ │ │ │ -0001a8d0: 6865 7920 7761 6e74 2074 6f20 736f 6c76  hey want to solv
│ │ │ │ -0001a8e0: 6520 7769 7468 2074 6865 2020 4227 506f  e with the  B'Po
│ │ │ │ -0001a8f0: 6c79 6e6f 6d69 616c 7320 6f70 7469 6f6e  lynomials option
│ │ │ │ -0001a900: 206f 720a 4227 4675 6e63 7469 6f6e 7320   or.B'Functions 
│ │ │ │ -0001a910: 6f70 7469 6f6e 2e20 4966 2042 2750 6f6c  option. If B'Pol
│ │ │ │ -0001a920: 796e 6f6d 6961 6c73 2069 7320 6e6f 7420  ynomials is not 
│ │ │ │ -0001a930: 7573 6564 2074 6865 6e20 7468 6520 7573  used then the us
│ │ │ │ -0001a940: 6572 2073 686f 756c 6420 7573 6520 7468  er should use th
│ │ │ │ -0001a950: 650a 4e61 6d65 506f 6c79 6e6f 6d69 616c  e.NamePolynomial
│ │ │ │ -0001a960: 7320 6f70 7469 6f6e 2e0a 0a2b 2d2d 2d2d  s option...+----
│ │ │ │ +00019f30: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +00019f40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ +00019f50: 653a 200a 2020 2020 2020 2020 6d61 6b65  e: .        make
│ │ │ │ +00019f60: 4227 496e 7075 7446 696c 6528 7329 0a20  B'InputFile(s). 
│ │ │ │ +00019f70: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00019f80: 202a 2073 2c20 6120 2a6e 6f74 6520 7374   * s, a *note st
│ │ │ │ +00019f90: 7269 6e67 3a20 284d 6163 6175 6c61 7932  ring: (Macaulay2
│ │ │ │ +00019fa0: 446f 6329 5374 7269 6e67 2c2c 2061 2064  Doc)String,, a d
│ │ │ │ +00019fb0: 6972 6563 746f 7279 2077 6865 7265 2074  irectory where t
│ │ │ │ +00019fc0: 6865 2069 6e70 7574 0a20 2020 2020 2020  he input.       
│ │ │ │ +00019fd0: 2066 696c 6520 7769 6c6c 2062 6520 7772   file will be wr
│ │ │ │ +00019fe0: 6974 7465 6e0a 2020 2a20 2a6e 6f74 6520  itten.  * *note 
│ │ │ │ +00019ff0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ +0001a000: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ +0001a010: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ +0001a020: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ +0001a030: 7574 732c 3a0a 2020 2020 2020 2a20 2a6e  uts,:.      * *n
│ │ │ │ +0001a040: 6f74 6520 4166 6656 6172 6961 626c 6547  ote AffVariableG
│ │ │ │ +0001a050: 726f 7570 3a20 5661 7269 6162 6c65 2067  roup: Variable g
│ │ │ │ +0001a060: 726f 7570 732c 203d 3e20 2e2e 2e2c 2064  roups, => ..., d
│ │ │ │ +0001a070: 6566 6175 6c74 2076 616c 7565 207b 7d2c  efault value {},
│ │ │ │ +0001a080: 2061 6e0a 2020 2020 2020 2020 6f70 7469   an.        opti
│ │ │ │ +0001a090: 6f6e 2074 6f20 6772 6f75 7020 7661 7269  on to group vari
│ │ │ │ +0001a0a0: 6162 6c65 7320 616e 6420 7573 6520 6d75  ables and use mu
│ │ │ │ +0001a0b0: 6c74 6968 6f6d 6f67 656e 656f 7573 2068  ltihomogeneous h
│ │ │ │ +0001a0c0: 6f6d 6f74 6f70 6965 730a 2020 2020 2020  omotopies.      
│ │ │ │ +0001a0d0: 2a20 2a6e 6f74 6520 4227 436f 6e73 7461  * *note B'Consta
│ │ │ │ +0001a0e0: 6e74 733a 2042 2743 6f6e 7374 616e 7473  nts: B'Constants
│ │ │ │ +0001a0f0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +0001a100: 7420 7661 6c75 6520 7b7d 2c20 616e 206f  t value {}, an o
│ │ │ │ +0001a110: 7074 696f 6e20 746f 0a20 2020 2020 2020  ption to.       
│ │ │ │ +0001a120: 2064 6573 6967 6e61 7465 2074 6865 2063   designate the c
│ │ │ │ +0001a130: 6f6e 7374 616e 7473 2066 6f72 2061 2042  onstants for a B
│ │ │ │ +0001a140: 6572 7469 6e69 2049 6e70 7574 2066 696c  ertini Input fil
│ │ │ │ +0001a150: 650a 2020 2020 2020 2a20 4227 4675 6e63  e.      * B'Func
│ │ │ │ +0001a160: 7469 6f6e 7320 286d 6973 7369 6e67 2064  tions (missing d
│ │ │ │ +0001a170: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ +0001a180: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0001a190: 6c75 6520 7b7d 2c20 0a20 2020 2020 202a  lue {}, .      *
│ │ │ │ +0001a1a0: 2042 2750 6f6c 796e 6f6d 6961 6c73 2028   B'Polynomials (
│ │ │ │ +0001a1b0: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ +0001a1c0: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064  ation) => ..., d
│ │ │ │ +0001a1d0: 6566 6175 6c74 2076 616c 7565 207b 7d2c  efault value {},
│ │ │ │ +0001a1e0: 200a 2020 2020 2020 2a20 4265 7274 696e   .      * Bertin
│ │ │ │ +0001a1f0: 6949 6e70 7574 436f 6e66 6967 7572 6174  iInputConfigurat
│ │ │ │ +0001a200: 696f 6e20 286d 6973 7369 6e67 2064 6f63  ion (missing doc
│ │ │ │ +0001a210: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ +0001a220: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +0001a230: 650a 2020 2020 2020 2020 7b7d 2c0a 2020  e.        {},.  
│ │ │ │ +0001a240: 2020 2020 2a20 2a6e 6f74 6520 486f 6d56      * *note HomV
│ │ │ │ +0001a250: 6172 6961 626c 6547 726f 7570 3a20 5661  ariableGroup: Va
│ │ │ │ +0001a260: 7269 6162 6c65 2067 726f 7570 732c 203d  riable groups, =
│ │ │ │ +0001a270: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +0001a280: 616c 7565 207b 7d2c 2061 6e0a 2020 2020  alue {}, an.    
│ │ │ │ +0001a290: 2020 2020 6f70 7469 6f6e 2074 6f20 6772      option to gr
│ │ │ │ +0001a2a0: 6f75 7020 7661 7269 6162 6c65 7320 616e  oup variables an
│ │ │ │ +0001a2b0: 6420 7573 6520 6d75 6c74 6968 6f6d 6f67  d use multihomog
│ │ │ │ +0001a2c0: 656e 656f 7573 2068 6f6d 6f74 6f70 6965  eneous homotopie
│ │ │ │ +0001a2d0: 730a 2020 2020 2020 2a20 4e61 6d65 4227  s.      * NameB'
│ │ │ │ +0001a2e0: 496e 7075 7446 696c 6520 286d 6973 7369  InputFile (missi
│ │ │ │ +0001a2f0: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ +0001a300: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
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│ │ │ │ +0001ae50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ae60: 0a7c 6f35 203a 2050 6f6c 796e 6f6d 6961  .|o5 : Polynomia
│ │ │ │ +0001ae70: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
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│ │ │ │ +0001ae90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
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│ │ │ │ -0001aed0: 3a20 6d61 6b65 4227 496e 7075 7446 696c  : makeB'InputFil
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│ │ │ │ +0001aed0: 6936 203a 206d 616b 6542 2749 6e70 7574  i6 : makeB'Input
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│ │ │ │  0001aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001af10: 2042 6572 7469 6e69 496e 7075 7443 6f6e   BertiniInputCon
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│ │ │ │ -0001af70: 0a7c 2020 2020 2020 2020 2020 2020 204e  .|             N
│ │ │ │ -0001af80: 616d 6550 6f6c 796e 6f6d 6961 6c73 3d3e  amePolynomials=>
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│ │ │ │ -0001afa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001afb0: 2020 2020 2020 4227 4675 6e63 7469 6f6e        B'Function
│ │ │ │ -0001afc0: 733d 3e7b 2020 2020 2020 2020 2020 2020  s=>{            
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│ │ │ │ -0001afe0: 2020 2020 2020 2020 2020 2020 7b58 2c78              {X,x
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│ │ │ │ +0001af00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001af10: 2020 2020 4265 7274 696e 6949 6e70 7574      BertiniInput
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│ │ │ │ +0001af70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001af80: 2020 4e61 6d65 506f 6c79 6e6f 6d69 616c    NamePolynomial
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│ │ │ │ +0001afb0: 2020 2020 2020 2020 2042 2746 756e 6374           B'Funct
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│ │ │ │ +0001afe0: 7c0a 7c20 2020 2020 2020 2020 2020 207b  |.|            {
│ │ │ │ +0001aff0: 582c 7831 2b78 322b 317d 2c20 2020 2020  X,x1+x2+1},     
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│ │ │ │ -0001b020: 2020 207b 6631 2c79 2a58 5e32 2b31 7d2c     {f1,y*X^2+1},
│ │ │ │ -0001b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b040: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001b050: 2020 2020 2020 2020 2020 7b66 322c 7831            {f2,x1
│ │ │ │ -0001b060: 2d78 322b 317d 2c20 2020 2020 2020 2020  -x2+1},         
│ │ │ │ +0001b010: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b020: 2020 2020 2020 7b66 312c 792a 585e 322b        {f1,y*X^2+
│ │ │ │ +0001b030: 317d 2c20 2020 2020 2020 2020 2020 2020  1},             
│ │ │ │ +0001b040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001b050: 7c20 2020 2020 2020 2020 2020 207b 6632  |            {f2
│ │ │ │ +0001b060: 2c78 312d 7832 2b31 7d2c 2020 2020 2020  ,x1-x2+1},      
│ │ │ │  0001b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b080: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001b090: 207b 6633 2c79 2d32 7d7d 293b 2020 2020   {f3,y-2}});    
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│ │ │ │  0001b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b160: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b120: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0001b130: 3a20 523d 5151 5b78 312c 7832 2c79 2c58  : R=QQ[x1,x2,y,X
│ │ │ │ +0001b140: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0001b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b160: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b190: 2020 2020 207c 0a7c 6f37 203d 2052 2020       |.|o7 = R  
│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b190: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +0001b1a0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b1c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b1d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b200: 2020 207c 0a7c 6f37 203a 2050 6f6c 796e     |.|o7 : Polyn
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│ │ │ │ +0001b230: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001b2b0: 2020 2020 2020 2020 2042 6572 7469 6e69           Bertini
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│ │ │ │ -0001b310: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b320: 2020 2020 2020 2020 4227 506f 6c79 6e6f          B'Polyno
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│ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b380: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b390: 2020 2020 2020 2020 2020 7b58 2c78 312b            {X,x1+
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│ │ │ │ -0001b3b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ +0001b2b0: 2020 2020 2020 2020 2020 2020 4265 7274              Bert
│ │ │ │ +0001b2c0: 696e 6949 6e70 7574 436f 6e66 6967 7572  iniInputConfigur
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│ │ │ │ +0001b310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b320: 2020 2020 2020 2020 2020 2042 2750 6f6c             B'Pol
│ │ │ │ +0001b330: 796e 6f6d 6961 6c73 3d3e 7b79 2a58 5e32  ynomials=>{y*X^2
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│ │ │ │ +0001b350: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001b360: 2020 2042 2746 756e 6374 696f 6e73 3d3e     B'Functions=>
│ │ │ │ +0001b370: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +0001b380: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b390: 2020 2020 2020 2020 2020 2020 207b 582c               {X,
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│ │ │ │ +0001b3b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001b3c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd90: 2020 2020 2020 7c0a 7c6f 3120 3a20 4227        |.|o1 : B'
│ │ │ │ -0001bda0: 5365 6374 696f 6e20 2020 2020 2020 2020  Section         
│ │ │ │ +0001bd90: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ +0001bda0: 2042 2753 6563 7469 6f6e 2020 2020 2020   B'Section      
│ │ │ │  0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bde0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001bde0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001bdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001be30: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 636c  ------+.|i2 : cl
│ │ │ │ -0001be40: 6173 7320 7320 2020 2020 2020 2020 2020  ass s           
│ │ │ │ +0001be30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +0001be40: 2063 6c61 7373 2073 2020 2020 2020 2020   class s        
│ │ │ │  0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001be80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bed0: 2020 2020 2020 7c0a 7c6f 3220 3d20 4227        |.|o2 = B'
│ │ │ │ -0001bee0: 5365 6374 696f 6e20 2020 2020 2020 2020  Section         
│ │ │ │ +0001bed0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ +0001bee0: 2042 2753 6563 7469 6f6e 2020 2020 2020   B'Section      
│ │ │ │  0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bf20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf70: 2020 2020 2020 7c0a 7c6f 3220 3a20 5479        |.|o2 : Ty
│ │ │ │ -0001bf80: 7065 2020 2020 2020 2020 2020 2020 2020  pe              
│ │ │ │ +0001bf70: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ +0001bf80: 2054 7970 6520 2020 2020 2020 2020 2020   Type           
│ │ │ │  0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001bfc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c010: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7261  ------+.|i3 : ra
│ │ │ │ -0001c020: 6e64 6f6d 5265 616c 436f 6566 6669 6369  ndomRealCoeffici
│ │ │ │ -0001c030: 656e 7447 656e 6572 6174 6f72 3d28 292d  entGenerator=()-
│ │ │ │ -0001c040: 3e72 616e 646f 6d28 5252 2920 2020 2020  >random(RR)     
│ │ │ │ +0001c010: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0001c020: 2072 616e 646f 6d52 6561 6c43 6f65 6666   randomRealCoeff
│ │ │ │ +0001c030: 6963 6965 6e74 4765 6e65 7261 746f 723d  icientGenerator=
│ │ │ │ +0001c040: 2829 2d3e 7261 6e64 6f6d 2852 5229 2020  ()->random(RR)  
│ │ │ │  0001c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c060: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0b0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7261        |.|o3 = ra
│ │ │ │ -0001c0c0: 6e64 6f6d 5265 616c 436f 6566 6669 6369  ndomRealCoeffici
│ │ │ │ -0001c0d0: 656e 7447 656e 6572 6174 6f72 2020 2020  entGenerator    
│ │ │ │ +0001c0b0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +0001c0c0: 2072 616e 646f 6d52 6561 6c43 6f65 6666   randomRealCoeff
│ │ │ │ +0001c0d0: 6963 6965 6e74 4765 6e65 7261 746f 7220  icientGenerator 
│ │ │ │  0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c100: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c150: 2020 2020 2020 7c0a 7c6f 3320 3a20 4675        |.|o3 : Fu
│ │ │ │ -0001c160: 6e63 7469 6f6e 436c 6f73 7572 6520 2020  nctionClosure   
│ │ │ │ +0001c150: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +0001c160: 2046 756e 6374 696f 6e43 6c6f 7375 7265   FunctionClosure
│ │ │ │  0001c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c1a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c1f0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7352  ------+.|i4 : sR
│ │ │ │ -0001c200: 6561 6c3d 6d61 6b65 4227 5365 6374 696f  eal=makeB'Sectio
│ │ │ │ -0001c210: 6e28 7b78 2c79 2c7a 7d2c 5261 6e64 6f6d  n({x,y,z},Random
│ │ │ │ -0001c220: 436f 6566 6669 6369 656e 7447 656e 6572  CoefficientGener
│ │ │ │ -0001c230: 6174 6f72 3d3e 2020 2020 2020 2020 2020  ator=>          
│ │ │ │ -0001c240: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c1f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +0001c200: 2073 5265 616c 3d6d 616b 6542 2753 6563   sReal=makeB'Sec
│ │ │ │ +0001c210: 7469 6f6e 287b 782c 792c 7a7d 2c52 616e  tion({x,y,z},Ran
│ │ │ │ +0001c220: 646f 6d43 6f65 6666 6963 6965 6e74 4765  domCoefficientGe
│ │ │ │ +0001c230: 6e65 7261 746f 723d 3e20 2020 2020 2020  nerator=>       
│ │ │ │ +0001c240: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c290: 2020 2020 2020 7c0a 7c6f 3420 3d20 4227        |.|o4 = B'
│ │ │ │ -0001c2a0: 5365 6374 696f 6e7b 2e2e 2e32 2e2e 2e7d  Section{...2...}
│ │ │ │ -0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c290: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +0001c2a0: 2042 2753 6563 7469 6f6e 7b2e 2e2e 322e   B'Section{...2.
│ │ │ │ +0001c2b0: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │  0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c2e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c330: 2020 2020 2020 7c0a 7c6f 3420 3a20 4227        |.|o4 : B'
│ │ │ │ -0001c340: 5365 6374 696f 6e20 2020 2020 2020 2020  Section         
│ │ │ │ +0001c330: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0001c340: 2042 2753 6563 7469 6f6e 2020 2020 2020   B'Section      
│ │ │ │  0001c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c380: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001c380: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0001c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c3d0: 2d2d 2d2d 2d2d 7c0a 7c72 616e 646f 6d52  ------|.|randomR
│ │ │ │ -0001c3e0: 6561 6c43 6f65 6666 6963 6965 6e74 4765  ealCoefficientGe
│ │ │ │ -0001c3f0: 6e65 7261 746f 7229 2020 2020 2020 2020  nerator)        
│ │ │ │ +0001c3d0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7261 6e64  ---------|.|rand
│ │ │ │ +0001c3e0: 6f6d 5265 616c 436f 6566 6669 6369 656e  omRealCoefficien
│ │ │ │ +0001c3f0: 7447 656e 6572 6174 6f72 2920 2020 2020  tGenerator)     
│ │ │ │  0001c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c420: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c420: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c470: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7352  ------+.|i5 : sR
│ │ │ │ -0001c480: 6561 6c23 4227 4e75 6d62 6572 436f 6566  eal#B'NumberCoef
│ │ │ │ -0001c490: 6669 6369 656e 7473 2020 2020 2020 2020  ficients        
│ │ │ │ +0001c470: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0001c480: 2073 5265 616c 2342 274e 756d 6265 7243   sReal#B'NumberC
│ │ │ │ +0001c490: 6f65 6666 6963 6965 6e74 7320 2020 2020  oefficients     
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c4c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c510: 2020 2020 2020 7c0a 7c6f 3520 3d20 7b2e        |.|o5 = {.
│ │ │ │ -0001c520: 3037 3431 3833 352c 202e 3830 3836 3934  0741835, .808694
│ │ │ │ -0001c530: 2c20 2e33 3632 3833 357d 2020 2020 2020  , .362835}      
│ │ │ │ +0001c510: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +0001c520: 207b 2e30 3734 3138 3335 2c20 2e38 3038   {.0741835, .808
│ │ │ │ +0001c530: 3639 342c 202e 3336 3238 3335 7d20 2020  694, .362835}   
│ │ │ │  0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c560: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4c69        |.|o5 : Li
│ │ │ │ -0001c5c0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0001c5b0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +0001c5c0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c600: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c600: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c650: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 7261  ------+.|i6 : ra
│ │ │ │ -0001c660: 6e64 6f6d 5261 7469 6f6e 616c 436f 6566  ndomRationalCoef
│ │ │ │ -0001c670: 6669 6369 656e 7447 656e 6572 6174 6f72  ficientGenerator
│ │ │ │ -0001c680: 3d28 292d 3e72 616e 646f 6d28 5151 2920  =()->random(QQ) 
│ │ │ │ -0001c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c650: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0001c660: 2072 616e 646f 6d52 6174 696f 6e61 6c43   randomRationalC
│ │ │ │ +0001c670: 6f65 6666 6963 6965 6e74 4765 6e65 7261  oefficientGenera
│ │ │ │ +0001c680: 746f 723d 2829 2d3e 7261 6e64 6f6d 2851  tor=()->random(Q
│ │ │ │ +0001c690: 5129 2020 2020 2020 2020 2020 2020 2020  Q)              
│ │ │ │ +0001c6a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6f0: 2020 2020 2020 7c0a 7c6f 3620 3d20 7261        |.|o6 = ra
│ │ │ │ -0001c700: 6e64 6f6d 5261 7469 6f6e 616c 436f 6566  ndomRationalCoef
│ │ │ │ -0001c710: 6669 6369 656e 7447 656e 6572 6174 6f72  ficientGenerator
│ │ │ │ -0001c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6f0: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ +0001c700: 2072 616e 646f 6d52 6174 696f 6e61 6c43   randomRationalC
│ │ │ │ +0001c710: 6f65 6666 6963 6965 6e74 4765 6e65 7261  oefficientGenera
│ │ │ │ +0001c720: 746f 7220 2020 2020 2020 2020 2020 2020  tor             
│ │ │ │  0001c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c740: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c790: 2020 2020 2020 7c0a 7c6f 3620 3a20 4675        |.|o6 : Fu
│ │ │ │ -0001c7a0: 6e63 7469 6f6e 436c 6f73 7572 6520 2020  nctionClosure   
│ │ │ │ +0001c790: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ +0001c7a0: 2046 756e 6374 696f 6e43 6c6f 7375 7265   FunctionClosure
│ │ │ │  0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c7e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c830: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 7352  ------+.|i7 : sR
│ │ │ │ -0001c840: 6174 696f 6e61 6c3d 6d61 6b65 4227 5365  ational=makeB'Se
│ │ │ │ -0001c850: 6374 696f 6e28 7b78 2c79 2c7a 7d2c 5261  ction({x,y,z},Ra
│ │ │ │ -0001c860: 6e64 6f6d 436f 6566 6669 6369 656e 7447  ndomCoefficientG
│ │ │ │ -0001c870: 656e 6572 6174 6f72 3d3e 2020 2020 2020  enerator=>      
│ │ │ │ -0001c880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c830: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ +0001c840: 2073 5261 7469 6f6e 616c 3d6d 616b 6542   sRational=makeB
│ │ │ │ +0001c850: 2753 6563 7469 6f6e 287b 782c 792c 7a7d  'Section({x,y,z}
│ │ │ │ +0001c860: 2c52 616e 646f 6d43 6f65 6666 6963 6965  ,RandomCoefficie
│ │ │ │ +0001c870: 6e74 4765 6e65 7261 746f 723d 3e20 2020  ntGenerator=>   
│ │ │ │ +0001c880: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8d0: 2020 2020 2020 7c0a 7c6f 3720 3d20 4227        |.|o7 = B'
│ │ │ │ -0001c8e0: 5365 6374 696f 6e7b 2e2e 2e32 2e2e 2e7d  Section{...2...}
│ │ │ │ -0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8d0: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +0001c8e0: 2042 2753 6563 7469 6f6e 7b2e 2e2e 322e   B'Section{...2.
│ │ │ │ +0001c8f0: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │  0001c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c920: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c920: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c970: 2020 2020 2020 7c0a 7c6f 3720 3a20 4227        |.|o7 : B'
│ │ │ │ -0001c980: 5365 6374 696f 6e20 2020 2020 2020 2020  Section         
│ │ │ │ +0001c970: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0001c980: 2042 2753 6563 7469 6f6e 2020 2020 2020   B'Section      
│ │ │ │  0001c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9c0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001c9c0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0001c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ca10: 2d2d 2d2d 2d2d 7c0a 7c72 616e 646f 6d52  ------|.|randomR
│ │ │ │ -0001ca20: 6174 696f 6e61 6c43 6f65 6666 6963 6965  ationalCoefficie
│ │ │ │ -0001ca30: 6e74 4765 6e65 7261 746f 7229 2020 2020  ntGenerator)    
│ │ │ │ +0001ca10: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7261 6e64  ---------|.|rand
│ │ │ │ +0001ca20: 6f6d 5261 7469 6f6e 616c 436f 6566 6669  omRationalCoeffi
│ │ │ │ +0001ca30: 6369 656e 7447 656e 6572 6174 6f72 2920  cientGenerator) 
│ │ │ │  0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001ca60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cab0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 7352  ------+.|i8 : sR
│ │ │ │ -0001cac0: 6174 696f 6e61 6c23 4227 4e75 6d62 6572  ational#B'Number
│ │ │ │ -0001cad0: 436f 6566 6669 6369 656e 7473 2020 2020  Coefficients    
│ │ │ │ +0001cab0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +0001cac0: 2073 5261 7469 6f6e 616c 2342 274e 756d   sRational#B'Num
│ │ │ │ +0001cad0: 6265 7243 6f65 6666 6963 6965 6e74 7320  berCoefficients 
│ │ │ │  0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cb00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001cb60: 3720 2031 2020 2037 2020 2020 2020 2020  7  1   7        
│ │ │ │ +0001cb50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001cb60: 2020 2037 2020 3120 2020 3720 2020 2020     7  1   7     
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cba0: 2020 2020 2020 7c0a 7c6f 3820 3d20 7b2d        |.|o8 = {-
│ │ │ │ -0001cbb0: 2d2c 202d 2c20 2d2d 7d20 2020 2020 2020  -, -, --}       
│ │ │ │ +0001cba0: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +0001cbb0: 207b 2d2d 2c20 2d2c 202d 2d7d 2020 2020   {--, -, --}    
│ │ │ │  0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbf0: 2020 2020 2020 7c0a 7c20 2020 2020 2031        |.|      1
│ │ │ │ -0001cc00: 3020 2032 2020 3130 2020 2020 2020 2020  0  2  10        
│ │ │ │ +0001cbf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001cc00: 2020 3130 2020 3220 2031 3020 2020 2020    10  2  10     
│ │ │ │  0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cc40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc90: 2020 2020 2020 7c0a 7c6f 3820 3a20 4c69        |.|o8 : Li
│ │ │ │ -0001cca0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0001cc90: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ +0001cca0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cce0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001cce0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d  ------+.+-------
│ │ │ │ +0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d 2d2d  ---------+.+----
│ │ │ │  0001cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd60: 2d2d 2d2d 2b0a 7c69 3920 3a20 6166 6669  ----+.|i9 : affi
│ │ │ │ -0001cd70: 6e65 5365 6374 696f 6e3d 6d61 6b65 4227  neSection=makeB'
│ │ │ │ -0001cd80: 5365 6374 696f 6e28 7b78 2c79 2c7a 2c31  Section({x,y,z,1
│ │ │ │ -0001cd90: 7d29 7c0a 7c20 2020 2020 2020 2020 2020  })|.|           
│ │ │ │ +0001cd60: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2061  -------+.|i9 : a
│ │ │ │ +0001cd70: 6666 696e 6553 6563 7469 6f6e 3d6d 616b  ffineSection=mak
│ │ │ │ +0001cd80: 6542 2753 6563 7469 6f6e 287b 782c 792c  eB'Section({x,y,
│ │ │ │ +0001cd90: 7a2c 317d 297c 0a7c 2020 2020 2020 2020  z,1})|.|        
│ │ │ │  0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdc0: 7c0a 7c6f 3920 3d20 4227 5365 6374 696f  |.|o9 = B'Sectio
│ │ │ │ -0001cdd0: 6e7b 2e2e 2e32 2e2e 2e7d 2020 2020 2020  n{...2...}      
│ │ │ │ -0001cde0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001cdf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001cdc0: 2020 207c 0a7c 6f39 203d 2042 2753 6563     |.|o9 = B'Sec
│ │ │ │ +0001cdd0: 7469 6f6e 7b2e 2e2e 322e 2e2e 7d20 2020  tion{...2...}   
│ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001ce20: 3920 3a20 4227 5365 6374 696f 6e20 2020  9 : B'Section   
│ │ │ │ +0001ce10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ce20: 0a7c 6f39 203a 2042 2753 6563 7469 6f6e  .|o9 : B'Section
│ │ │ │  0001ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001ce40: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce70: 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d  --------+.+-----
│ │ │ │ +0001ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d  -----------+.+--
│ │ │ │  0001ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cec0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ -0001ced0: 2058 3d7b 782c 792c 7a7d 2020 2020 2020   X={x,y,z}      
│ │ │ │ +0001cec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001ced0: 3020 3a20 583d 7b78 2c79 2c7a 7d20 2020  0 : X={x,y,z}   
│ │ │ │  0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf60: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -0001cf70: 207b 782c 2079 2c20 7a7d 2020 2020 2020   {x, y, z}      
│ │ │ │ +0001cf60: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001cf70: 3020 3d20 7b78 2c20 792c 207a 7d20 2020  0 = {x, y, z}   
│ │ │ │  0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cfb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001cfb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d000: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ -0001d010: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +0001d000: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d010: 3020 3a20 4c69 7374 2020 2020 2020 2020  0 : List        
│ │ │ │  0001d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d050: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001d050: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001d060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d0a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ -0001d0b0: 2050 3d7b 312c 322c 337d 2020 2020 2020   P={1,2,3}      
│ │ │ │ +0001d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001d0b0: 3120 3a20 503d 7b31 2c32 2c33 7d20 2020  1 : P={1,2,3}   
│ │ │ │  0001d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d0f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d140: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ -0001d150: 207b 312c 2032 2c20 337d 2020 2020 2020   {1, 2, 3}      
│ │ │ │ +0001d140: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d150: 3120 3d20 7b31 2c20 322c 2033 7d20 2020  1 = {1, 2, 3}   
│ │ │ │  0001d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d190: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d1e0: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ -0001d1f0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +0001d1e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d1f0: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │  0001d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d230: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001d230: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d280: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │ -0001d290: 2061 6666 696e 6543 6f6e 7461 696e 696e   affineContainin
│ │ │ │ -0001d2a0: 6750 6f69 6e74 3d6d 616b 6542 2753 6563  gPoint=makeB'Sec
│ │ │ │ -0001d2b0: 7469 6f6e 287b 782c 792c 7a7d 2c43 6f6e  tion({x,y,z},Con
│ │ │ │ -0001d2c0: 7461 696e 7350 6f69 6e74 3d3e 5029 2020  tainsPoint=>P)  
│ │ │ │ -0001d2d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001d290: 3220 3a20 6166 6669 6e65 436f 6e74 6169  2 : affineContai
│ │ │ │ +0001d2a0: 6e69 6e67 506f 696e 743d 6d61 6b65 4227  ningPoint=makeB'
│ │ │ │ +0001d2b0: 5365 6374 696f 6e28 7b78 2c79 2c7a 7d2c  Section({x,y,z},
│ │ │ │ +0001d2c0: 436f 6e74 6169 6e73 506f 696e 743d 3e50  ContainsPoint=>P
│ │ │ │ +0001d2d0: 2920 2020 2020 2020 2020 207c 0a7c 2020  )          |.|  
│ │ │ │  0001d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d320: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ -0001d330: 2042 2753 6563 7469 6f6e 7b2e 2e2e 332e   B'Section{...3.
│ │ │ │ -0001d340: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │ +0001d320: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d330: 3220 3d20 4227 5365 6374 696f 6e7b 2e2e  2 = B'Section{..
│ │ │ │ +0001d340: 2e33 2e2e 2e7d 2020 2020 2020 2020 2020  .3...}          
│ │ │ │  0001d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d370: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d3c0: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ -0001d3d0: 2042 2753 6563 7469 6f6e 2020 2020 2020   B'Section      
│ │ │ │ +0001d3c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d3d0: 3220 3a20 4227 5365 6374 696f 6e20 2020  2 : B'Section   
│ │ │ │  0001d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001d470: 3320 3a20 723d 2061 6666 696e 6543 6f6e  3 : r= affineCon
│ │ │ │ +0001d480: 7461 696e 696e 6750 6f69 6e74 2342 2753  tainingPoint#B'S
│ │ │ │ +0001d490: 6563 7469 6f6e 5374 7269 6e67 2020 2020  ectionString    
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│ │ │ │  0001d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d500: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
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│ │ │ │ -0001d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d500: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d510: 3320 3d20 2831 2e31 3839 3231 2b2e 3834  3 = (1.18921+.84
│ │ │ │ +0001d520: 3935 3339 2a69 6929 2a28 782d 2831 292a  9539*ii)*(x-(1)*
│ │ │ │ +0001d530: 2831 2929 2b28 2d2e 3534 3233 3731 2b2e  (1))+(-.542371+.
│ │ │ │ +0001d540: 3330 3731 3337 2a69 6929 2a28 792d 2831  307137*ii)*(y-(1
│ │ │ │ +0001d550: 292a 2832 2929 2b28 2020 207c 0a7c 2020  )*(2))+(   |.|  
│ │ │ │ +0001d560: 2020 2020 312e 3336 3934 352b 2e30 3135      1.36945+.015
│ │ │ │ +0001d570: 3633 332a 6969 292a 287a 2d28 3129 2a28  633*ii)*(z-(1)*(
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│ │ │ │  0001d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001d600: 2070 7269 6e74 2072 2020 2020 2020 2020   print r        
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│ │ │ │ +0001d600: 3420 3a20 7072 696e 7420 7220 2020 2020  4 : print r     
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│ │ │ │  0001d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d640: 2020 2020 2020 2020 7c0a 7c28 312e 3138          |.|(1.18
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│ │ │ │ -0001d660: 2878 2d28 3129 2a28 3129 292b 282d 2e35  (x-(1)*(1))+(-.5
│ │ │ │ -0001d670: 3432 3337 312b 2e33 3037 3133 372a 6969  42371+.307137*ii
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│ │ │ │ +0001d660: 6929 2a28 782d 2831 292a 2831 2929 2b28  i)*(x-(1)*(1))+(
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│ │ │ │  0001d6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d6e0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c2b 2e30 3135  --------|.|+.015
│ │ │ │ -0001d6f0: 3633 332a 6969 292a 287a 2d28 3129 2a28  633*ii)*(z-(1)*(
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│ │ │ │ +0001d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2b2e  -----------|.|+.
│ │ │ │ +0001d6f0: 3031 3536 3333 2a69 6929 2a28 7a2d 2831  015633*ii)*(z-(1
│ │ │ │ +0001d700: 292a 2833 2929 2020 2020 2020 2020 2020  )*(3))          
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│ │ │ │ -0001d780: 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d  --------+.+-----
│ │ │ │ +0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d  -----------+.+--
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│ │ │ │  0001d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d7d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a  --------+.|i15 :
│ │ │ │ -0001d7e0: 2072 486f 6d6f 6765 5365 6374 696f 6e3d   rHomogeSection=
│ │ │ │ -0001d7f0: 206d 616b 6542 2753 6563 7469 6f6e 287b   makeB'Section({
│ │ │ │ -0001d800: 782c 792c 7a7d 2c43 6f6e 7461 696e 7350  x,y,z},ContainsP
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│ │ │ │ +0001d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001d7e0: 3520 3a20 7248 6f6d 6f67 6553 6563 7469  5 : rHomogeSecti
│ │ │ │ +0001d7f0: 6f6e 3d20 6d61 6b65 4227 5365 6374 696f  on= makeB'Sectio
│ │ │ │ +0001d800: 6e28 7b78 2c79 2c7a 7d2c 436f 6e74 6169  n({x,y,z},Contai
│ │ │ │ +0001d810: 6e73 506f 696e 743d 3e50 2c42 2748 6f6d  nsPoint=>P,B'Hom
│ │ │ │ +0001d820: 6f67 656e 697a 6174 696f 6e7c 0a7c 2020  ogenization|.|  
│ │ │ │  0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d870: 2020 2020 2020 2020 7c0a 7c6f 3135 203d          |.|o15 =
│ │ │ │ -0001d880: 2042 2753 6563 7469 6f6e 7b2e 2e2e 332e   B'Section{...3.
│ │ │ │ -0001d890: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
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│ │ │ │ +0001d880: 3520 3d20 4227 5365 6374 696f 6e7b 2e2e  5 = B'Section{..
│ │ │ │ +0001d890: 2e33 2e2e 2e7d 2020 2020 2020 2020 2020  .3...}          
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│ │ │ │  0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001d910: 2020 2020 2020 2020 7c0a 7c6f 3135 203a          |.|o15 :
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│ │ │ │ -0001d9b0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c3d 3e22 782b  --------|.|=>"x+
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│ │ │ │ +0001d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3d3e  -----------|.|=>
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│ │ │ │  0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001da00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │  0001da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001da60: 2070 6565 6b20 7248 6f6d 6f67 6553 6563   peek rHomogeSec
│ │ │ │ -0001da70: 7469 6f6e 2020 2020 2020 2020 2020 2020  tion            
│ │ │ │ +0001da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001da60: 3620 3a20 7065 656b 2072 486f 6d6f 6765  6 : peek rHomoge
│ │ │ │ +0001da70: 5365 6374 696f 6e20 2020 2020 2020 2020  Section         
│ │ │ │  0001da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001daa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001daa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001db00: 2042 2753 6563 7469 6f6e 7b42 2748 6f6d   B'Section{B'Hom
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│ │ │ │ +0001daf0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001db00: 3620 3d20 4227 5365 6374 696f 6e7b 4227  6 = B'Section{B'
│ │ │ │ +0001db10: 486f 6d6f 6765 6e69 7a61 7469 6f6e 203d  Homogenization =
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│ │ │ │  0001db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001db50: 2020 2020 2020 2020 2020 2042 274e 756d             B'Num
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│ │ │ │ -0001db80: 3934 352a 6969 2c20 2e34 3236 3730 3420  945*ii, .426704 
│ │ │ │ -0001db90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001dba0: 2020 2020 2020 2020 2020 2042 2753 6563             B'Sec
│ │ │ │ -0001dbb0: 7469 6f6e 5374 7269 6e67 203d 3e20 282e  tionString => (.
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│ │ │ │ +0001db50: 2020 2020 2020 2020 2020 2020 2020 4227                B'
│ │ │ │ +0001db60: 4e75 6d62 6572 436f 6566 6669 6369 656e  NumberCoefficien
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│ │ │ │ +0001db90: 3034 2020 2020 2020 2020 207c 0a7c 2020  04         |.|  
│ │ │ │ +0001dba0: 2020 2020 2020 2020 2020 2020 2020 4227                B'
│ │ │ │ +0001dbb0: 5365 6374 696f 6e53 7472 696e 6720 3d3e  SectionString =>
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│ │ │ │ +0001dbd0: 352a 6969 292a 2878 2d28 782b 792b 7a29  5*ii)*(x-(x+y+z)
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│ │ │ │  0001dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001dce0: 2031 2929 2b28 2e34 3236 3730 342d 2e39   1))+(.426704-.9
│ │ │ │ -0001dcf0: 3735 3339 2a69 6929 2a28 792d 2878 2b79  7539*ii)*(y-(x+y
│ │ │ │ -0001dd00: 2b7a 292a 2832 2929 2b28 2d2e 3437 3838  +z)*(2))+(-.4788
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│ │ │ │ +0001dce0: 2020 2020 3129 292b 282e 3432 3637 3034      1))+(.426704
│ │ │ │ +0001dcf0: 2d2e 3937 3533 392a 6969 292a 2879 2d28  -.97539*ii)*(y-(
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│ │ │ │ +0001e530: 696e 692e 696e 666f 2c20 4e6f 6465 3a20  ini.info, Node: 
│ │ │ │ +0001e540: 6d61 6b65 4227 536c 6963 652c 204e 6578  makeB'Slice, Nex
│ │ │ │ +0001e550: 743a 206d 6f76 6542 2746 696c 652c 2050  t: moveB'File, P
│ │ │ │ +0001e560: 7265 763a 206d 616b 6542 2753 6563 7469  rev: makeB'Secti
│ │ │ │ +0001e570: 6f6e 2c20 5570 3a20 546f 700a 0a6d 616b  on, Up: Top..mak
│ │ │ │ +0001e580: 6542 2753 6c69 6365 202d 2d20 6d61 6b65  eB'Slice -- make
│ │ │ │ +0001e590: 4227 536c 6963 6520 6372 6561 7465 7320  B'Slice creates 
│ │ │ │ +0001e5a0: 6120 6861 7368 2074 6162 6c65 2074 6861  a hash table tha
│ │ │ │ +0001e5b0: 7420 7265 7072 6573 656e 7473 2061 206c  t represents a l
│ │ │ │ +0001e5c0: 696e 6561 7220 736c 6963 652e 0a2a 2a2a  inear slice..***
│ │ │ │  0001e5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001e5e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001e5f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001e600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e610: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ -0001e620: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ -0001e630: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0001e640: 2020 6d61 6b65 4227 536c 6963 6528 736c    makeB'Slice(sl
│ │ │ │ -0001e650: 6963 6554 7970 652c 7661 7269 6162 6c65  iceType,variable
│ │ │ │ -0001e660: 4772 6f75 7073 290a 2020 2a20 496e 7075  Groups).  * Inpu
│ │ │ │ -0001e670: 7473 3a0a 2020 2020 2020 2a20 736c 6963  ts:.      * slic
│ │ │ │ -0001e680: 6554 7970 652c 2061 202a 6e6f 7465 206c  eType, a *note l
│ │ │ │ -0001e690: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ -0001e6a0: 6f63 294c 6973 742c 2c20 4120 6c69 7374  oc)List,, A list
│ │ │ │ -0001e6b0: 206f 6620 696e 7465 6765 7273 206f 720a   of integers or.
│ │ │ │ -0001e6c0: 2020 2020 2020 2020 696e 7465 6765 722e          integer.
│ │ │ │ -0001e6d0: 0a20 2020 2020 202a 2076 6172 6961 626c  .      * variabl
│ │ │ │ -0001e6e0: 6547 726f 7570 732c 2061 202a 6e6f 7465  eGroups, a *note
│ │ │ │ -0001e6f0: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -0001e700: 3244 6f63 294c 6973 742c 2c20 4120 6c69  2Doc)List,, A li
│ │ │ │ -0001e710: 7374 206f 6620 6c69 7374 206f 660a 2020  st of list of.  
│ │ │ │ -0001e720: 2020 2020 2020 7661 7269 6162 6c65 7320        variables 
│ │ │ │ -0001e730: 6f72 206c 6973 7420 6f66 2076 6172 6961  or list of varia
│ │ │ │ -0001e740: 626c 6573 2e0a 2020 2a20 2a6e 6f74 6520  bles..  * *note 
│ │ │ │ -0001e750: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ -0001e760: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ -0001e770: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
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│ │ │ │ -0001e7a0: 6f74 6520 4227 486f 6d6f 6765 6e69 7a61  ote B'Homogeniza
│ │ │ │ -0001e7b0: 7469 6f6e 3a20 6d61 6b65 4227 5365 6374  tion: makeB'Sect
│ │ │ │ -0001e7c0: 696f 6e2c 203d 3e20 2e2e 2e2c 2064 6566  ion, => ..., def
│ │ │ │ -0001e7d0: 6175 6c74 2076 616c 7565 207b 7d2c 0a20  ault value {},. 
│ │ │ │ -0001e7e0: 2020 2020 2020 206d 616b 6542 2753 6563         makeB'Sec
│ │ │ │ -0001e7f0: 7469 6f6e 2063 7265 6174 6573 2061 2068  tion creates a h
│ │ │ │ -0001e800: 6173 6820 7461 626c 6520 7468 6174 2072  ash table that r
│ │ │ │ -0001e810: 6570 7265 7365 6e74 7320 6120 6879 7065  epresents a hype
│ │ │ │ -0001e820: 7270 6c61 6e65 2e0a 2020 2020 2020 2a20  rplane..      * 
│ │ │ │ -0001e830: 4227 4e75 6d62 6572 436f 6566 6669 6369  B'NumberCoeffici
│ │ │ │ -0001e840: 656e 7473 203d 3e20 2e2e 2e2c 2064 6566  ents => ..., def
│ │ │ │ -0001e850: 6175 6c74 2076 616c 7565 207b 7d0a 2020  ault value {}.  
│ │ │ │ -0001e860: 2020 2020 2a20 436f 6e74 6169 6e73 4d75      * ContainsMu
│ │ │ │ -0001e870: 6c74 6950 726f 6a65 6374 6976 6550 6f69  ltiProjectivePoi
│ │ │ │ -0001e880: 6e74 203d 3e20 2e2e 2e2c 2064 6566 6175  nt => ..., defau
│ │ │ │ -0001e890: 6c74 2076 616c 7565 207b 7d0a 2020 2020  lt value {}.    
│ │ │ │ -0001e8a0: 2020 2a20 436f 6e74 6169 6e73 506f 696e    * ContainsPoin
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│ │ │ │ -0001e8c0: 7420 7661 6c75 6520 7b7d 0a20 2020 2020  t value {}.     
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│ │ │ │ -0001e900: 2a20 5261 6e64 6f6d 436f 6566 6669 6369  * RandomCoeffici
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│ │ │ │ -0001e970: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
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│ │ │ │ -0001e990: 616c 6c6f 7773 2066 6f72 2065 6173 7920  allows for easy 
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│ │ │ │ +0001e610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ +0001e620: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ +0001e630: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
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│ │ │ │ +0001e6a0: 7932 446f 6329 4c69 7374 2c2c 2041 206c  y2Doc)List,, A l
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│ │ │ │ +0001e6c0: 6f72 0a20 2020 2020 2020 2069 6e74 6567  or.        integ
│ │ │ │ +0001e6d0: 6572 2e0a 2020 2020 2020 2a20 7661 7269  er..      * vari
│ │ │ │ +0001e6e0: 6162 6c65 4772 6f75 7073 2c20 6120 2a6e  ableGroups, a *n
│ │ │ │ +0001e6f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
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│ │ │ │ +0001e7a0: 202a 6e6f 7465 2042 2748 6f6d 6f67 656e   *note B'Homogen
│ │ │ │ +0001e7b0: 697a 6174 696f 6e3a 206d 616b 6542 2753  ization: makeB'S
│ │ │ │ +0001e7c0: 6563 7469 6f6e 2c20 3d3e 202e 2e2e 2c20  ection, => ..., 
│ │ │ │ +0001e7d0: 6465 6661 756c 7420 7661 6c75 6520 7b7d  default value {}
│ │ │ │ +0001e7e0: 2c0a 2020 2020 2020 2020 6d61 6b65 4227  ,.        makeB'
│ │ │ │ +0001e7f0: 5365 6374 696f 6e20 6372 6561 7465 7320  Section creates 
│ │ │ │ +0001e800: 6120 6861 7368 2074 6162 6c65 2074 6861  a hash table tha
│ │ │ │ +0001e810: 7420 7265 7072 6573 656e 7473 2061 2068  t represents a h
│ │ │ │ +0001e820: 7970 6572 706c 616e 652e 0a20 2020 2020  yperplane..     
│ │ │ │ +0001e830: 202a 2042 274e 756d 6265 7243 6f65 6666   * B'NumberCoeff
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│ │ │ │ +0001e850: 6465 6661 756c 7420 7661 6c75 6520 7b7d  default value {}
│ │ │ │ +0001e860: 0a20 2020 2020 202a 2043 6f6e 7461 696e  .      * Contain
│ │ │ │ +0001e870: 734d 756c 7469 5072 6f6a 6563 7469 7665  sMultiProjective
│ │ │ │ +0001e880: 506f 696e 7420 3d3e 202e 2e2e 2c20 6465  Point => ..., de
│ │ │ │ +0001e890: 6661 756c 7420 7661 6c75 6520 7b7d 0a20  fault value {}. 
│ │ │ │ +0001e8a0: 2020 2020 202a 2043 6f6e 7461 696e 7350       * ContainsP
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│ │ │ │ +0001e8c0: 6175 6c74 2076 616c 7565 207b 7d0a 2020  ault value {}.  
│ │ │ │ +0001e8d0: 2020 2020 2a20 4e61 6d65 4227 536c 6963      * NameB'Slic
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│ │ │ │ +0001e900: 2020 202a 2052 616e 646f 6d43 6f65 6666     * RandomCoeff
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│ │ │ │ +0001ed30: 2c79 312c 7932 7d7d 2020 2020 2020 2020  ,y1,y2}}        
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│ │ │ │ -0001edb0: 3d20 7b7b 7830 2c20 7831 7d2c 207b 7930  = {{x0, x1}, {y0
│ │ │ │ -0001edc0: 2c20 7931 2c20 7932 7d7d 2020 2020 2020  , y1, y2}}      
│ │ │ │ +0001eda0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001edb0: 6f32 203d 207b 7b78 302c 2078 317d 2c20  o2 = {{x0, x1}, 
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│ │ │ │  0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee40: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -0001ee50: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
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│ │ │ │ +0001ee50: 6f32 203a 204c 6973 7420 2020 2020 2020  o2 : List       
│ │ │ │  0001ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001eef0: 3a20 7879 536c 6963 653d 6d61 6b65 4227  : xySlice=makeB'
│ │ │ │ -0001ef00: 536c 6963 6528 736c 6963 6554 7970 652c  Slice(sliceType,
│ │ │ │ -0001ef10: 7661 7269 6162 6c65 4772 6f75 7073 2920  variableGroups) 
│ │ │ │ -0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001eef0: 6933 203a 2078 7953 6c69 6365 3d6d 616b  i3 : xySlice=mak
│ │ │ │ +0001ef00: 6542 2753 6c69 6365 2873 6c69 6365 5479  eB'Slice(sliceTy
│ │ │ │ +0001ef10: 7065 2c76 6172 6961 626c 6547 726f 7570  pe,variableGroup
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│ │ │ │ -0001ef90: 3d20 4227 536c 6963 657b 2e2e 2e34 2e2e  = B'Slice{...4..
│ │ │ │ -0001efa0: 2e7d 2020 2020 2020 2020 2020 2020 2020  .}              
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│ │ │ │ +0001ef90: 6f33 203d 2042 2753 6c69 6365 7b2e 2e2e  o3 = B'Slice{...
│ │ │ │ +0001efa0: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
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│ │ │ │ -0001f030: 3a20 4227 536c 6963 6520 2020 2020 2020  : B'Slice       
│ │ │ │ +0001f020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f030: 6f33 203a 2042 2753 6c69 6365 2020 2020  o3 : B'Slice    
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│ │ │ │  0001f0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001f0d0: 3a20 7065 656b 2078 7953 6c69 6365 2020  : peek xySlice  
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│ │ │ │ +0001f0d0: 6934 203a 2070 6565 6b20 7879 536c 6963  i4 : peek xySlic
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│ │ │ │ -0001f1b0: 2b31 2e31 3938 362a 2020 7c0a 7c20 2020  +1.1986*  |.|   
│ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 4227 5365 6374            B'Sect
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│ │ │ │ -0001f1f0: 6929 2a28 7830 292b 282d 2e38 3430 3131  i)*(x0)+(-.84011
│ │ │ │ -0001f200: 332b 312e 3139 3836 2020 7c0a 7c20 2020  3+1.1986  |.|   
│ │ │ │ -0001f210: 2020 2020 2020 2020 2020 4c69 7374 4227            ListB'
│ │ │ │ -0001f220: 5365 6374 696f 6e73 203d 3e20 7b42 2753  Sections => {B'S
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│ │ │ │ -0001f240: 2042 2753 6563 7469 6f6e 7b2e 2e2e 322e   B'Section{...2.
│ │ │ │ -0001f250: 2e2e 7d7d 2020 2020 2020 7c0a 7c20 2020  ..}}      |.|   
│ │ │ │ -0001f260: 2020 2020 2020 2020 2020 4e61 6d65 4227            NameB'
│ │ │ │ -0001f270: 536c 6963 6520 3d3e 206e 756c 6c20 2020  Slice => null   
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│ │ │ │ +0001f170: 6f34 203d 2042 2753 6c69 6365 7b42 274e  o4 = B'Slice{B'N
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│ │ │ │ +0001f1a0: 3731 3338 3436 2a69 692c 202d 2e38 3430  713846*ii, -.840
│ │ │ │ +0001f1b0: 3131 332b 312e 3139 3836 2a20 207c 0a7c  113+1.1986*  |.|
│ │ │ │ +0001f1c0: 2020 2020 2020 2020 2020 2020 2042 2753               B'S
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│ │ │ │ +0001f1f0: 362a 6969 292a 2878 3029 2b28 2d2e 3834  6*ii)*(x0)+(-.84
│ │ │ │ +0001f200: 3031 3133 2b31 2e31 3938 3620 207c 0a7c  0113+1.1986  |.|
│ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 204c 6973               Lis
│ │ │ │ +0001f220: 7442 2753 6563 7469 6f6e 7320 3d3e 207b  tB'Sections => {
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│ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 204e 616d               Nam
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│ │ │ │ +0001f350: 2a69 6929 2a28 7831 292c 2028 2e30 3134  *ii)*(x1), (.014
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│ │ │ │ +0001f380: 3931 3132 3933 2a69 6929 2a28 7931 292b  911293*ii)*(y1)+
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│ │ │ │ -0001f500: 696e 6720 646f 2070 7269 6e74 2069 2020  ing do print i  
│ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001f530: 3439 3134 342b 2e37 3133 3834 362a 6969  49144+.713846*ii
│ │ │ │ -0001f540: 292a 2878 3029 2b28 2d2e 3834 3031 3133  )*(x0)+(-.840113
│ │ │ │ -0001f550: 2b31 2e31 3938 362a 6969 292a 2878 3129  +1.1986*ii)*(x1)
│ │ │ │ -0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001f4e0: 6935 203a 2066 6f72 2069 2069 6e20 2078  i5 : for i in  x
│ │ │ │ +0001f4f0: 7953 6c69 6365 2342 2753 6563 7469 6f6e  ySlice#B'Section
│ │ │ │ +0001f500: 5374 7269 6e67 2064 6f20 7072 696e 7420  String do print 
│ │ │ │ +0001f510: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │ +0001f520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f530: 2831 2e34 3931 3434 2b2e 3731 3338 3436  (1.49144+.713846
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│ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f580: 282e 3031 3438 3432 2b31 2e32 3335 3438  (.014842+1.23548
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│ │ │ │ -0001f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d  ----------+.+---
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│ │ │ │  0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0001f710: 3a20 6153 6c69 6365 3d6d 616b 6542 2753  : aSlice=makeB'S
│ │ │ │ -0001f720: 6c69 6365 2833 2c7b 782c 792c 7a2c 317d  lice(3,{x,y,z,1}
│ │ │ │ -0001f730: 2c4e 616d 6542 2753 6c69 6365 3d3e 2266  ,NameB'Slice=>"f
│ │ │ │ -0001f740: 2229 3b20 2020 2020 2020 2020 2020 2020  ");             
│ │ │ │ -0001f750: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001f710: 6936 203a 2061 536c 6963 653d 6d61 6b65  i6 : aSlice=make
│ │ │ │ +0001f720: 4227 536c 6963 6528 332c 7b78 2c79 2c7a  B'Slice(3,{x,y,z
│ │ │ │ +0001f730: 2c31 7d2c 4e61 6d65 4227 536c 6963 653d  ,1},NameB'Slice=
│ │ │ │ +0001f740: 3e22 6622 293b 2020 2020 2020 2020 2020  >"f");          
│ │ │ │ +0001f750: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001f760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -0001f7b0: 3a20 6153 6c69 6365 234e 616d 6542 2753  : aSlice#NameB'S
│ │ │ │ -0001f7c0: 6c69 6365 2020 2020 2020 2020 2020 2020  lice            
│ │ │ │ +0001f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001f7b0: 6937 203a 2061 536c 6963 6523 4e61 6d65  i7 : aSlice#Name
│ │ │ │ +0001f7c0: 4227 536c 6963 6520 2020 2020 2020 2020  B'Slice         
│ │ │ │  0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f840: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0001f850: 3d20 7b66 302c 2066 312c 2066 327d 2020  = {f0, f1, f2}  
│ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f850: 6f37 203d 207b 6630 2c20 6631 2c20 6632  o7 = {f0, f1, f2
│ │ │ │ +0001f860: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0001f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f890: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0001f8f0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +0001f8e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f8f0: 6f37 203a 204c 6973 7420 2020 2020 2020  o7 : List       
│ │ │ │  0001f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f930: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001f930: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -0001f990: 3a20 6d61 6b65 4227 496e 7075 7446 696c  : makeB'InputFil
│ │ │ │ -0001f9a0: 6528 7374 6f72 6542 4d32 4669 6c65 732c  e(storeBM2Files,
│ │ │ │ -0001f9b0: 4166 6656 6172 6961 626c 6547 726f 7570  AffVariableGroup
│ │ │ │ -0001f9c0: 3d3e 7b78 2c79 2c7a 7d2c 4227 4675 6e63  =>{x,y,z},B'Func
│ │ │ │ -0001f9d0: 7469 6f6e 733d 3e7b 2020 7c0a 7c2d 2d2d  tions=>{  |.|---
│ │ │ │ +0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001f990: 6938 203a 206d 616b 6542 2749 6e70 7574  i8 : makeB'Input
│ │ │ │ +0001f9a0: 4669 6c65 2873 746f 7265 424d 3246 696c  File(storeBM2Fil
│ │ │ │ +0001f9b0: 6573 2c41 6666 5661 7269 6162 6c65 4772  es,AffVariableGr
│ │ │ │ +0001f9c0: 6f75 703d 3e7b 782c 792c 7a7d 2c42 2746  oup=>{x,y,z},B'F
│ │ │ │ +0001f9d0: 756e 6374 696f 6e73 3d3e 7b20 207c 0a7c  unctions=>{  |.|
│ │ │ │  0001f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c61 536c  ----------|.|aSl
│ │ │ │ -0001fa30: 6963 657d 2c4e 616d 6550 6f6c 796e 6f6d  ice},NamePolynom
│ │ │ │ -0001fa40: 6961 6c73 3d3e 7b22 6630 222c 2266 3122  ials=>{"f0","f1"
│ │ │ │ -0001fa50: 2c22 6632 227d 293b 2020 2020 2020 2020  ,"f2"});        
│ │ │ │ +0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0001fa30: 6153 6c69 6365 7d2c 4e61 6d65 506f 6c79  aSlice},NamePoly
│ │ │ │ +0001fa40: 6e6f 6d69 616c 733d 3e7b 2266 3022 2c22  nomials=>{"f0","
│ │ │ │ +0001fa50: 6631 222c 2266 3222 7d29 3b20 2020 2020  f1","f2"});     
│ │ │ │  0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001fa70: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001faa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d  ----------+.+---
│ │ │ │ +0001fac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b  -------------+.+
│ │ │ │  0001fad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fb00: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 6631  ------+.|i9 : f1
│ │ │ │ -0001fb10: 3d22 7830 2a79 302b 7831 2a79 302b 7832  ="x0*y0+x1*y0+x2
│ │ │ │ -0001fb20: 2a79 3222 2020 2020 2020 2020 2020 2020  *y2"            
│ │ │ │ +0001fb00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ +0001fb10: 2066 313d 2278 302a 7930 2b78 312a 7930   f1="x0*y0+x1*y0
│ │ │ │ +0001fb20: 2b78 322a 7932 2220 2020 2020 2020 2020  +x2*y2"         
│ │ │ │  0001fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fb40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fb80: 7c6f 3920 3d20 7830 2a79 302b 7831 2a79  |o9 = x0*y0+x1*y
│ │ │ │ -0001fb90: 302b 7832 2a79 3220 2020 2020 2020 2020  0+x2*y2         
│ │ │ │ +0001fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb80: 207c 0a7c 6f39 203d 2078 302a 7930 2b78   |.|o9 = x0*y0+x
│ │ │ │ +0001fb90: 312a 7930 2b78 322a 7932 2020 2020 2020  1*y0+x2*y2      
│ │ │ │  0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001fbb0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fbf0: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2066  ------+.|i10 : f
│ │ │ │ -0001fc00: 323d 2278 302a 7930 5e32 2b78 312a 7931  2="x0*y0^2+x1*y1
│ │ │ │ -0001fc10: 2a79 322b 7832 2a79 302a 7932 2220 2020  *y2+x2*y0*y2"   
│ │ │ │ +0001fbf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ +0001fc00: 3a20 6632 3d22 7830 2a79 305e 322b 7831  : f2="x0*y0^2+x1
│ │ │ │ +0001fc10: 2a79 312a 7932 2b78 322a 7930 2a79 3222  *y1*y2+x2*y0*y2"
│ │ │ │  0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fc30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fc70: 7c6f 3130 203d 2078 302a 7930 5e32 2b78  |o10 = x0*y0^2+x
│ │ │ │ -0001fc80: 312a 7931 2a79 322b 7832 2a79 302a 7932  1*y1*y2+x2*y0*y2
│ │ │ │ -0001fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fca0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc70: 207c 0a7c 6f31 3020 3d20 7830 2a79 305e   |.|o10 = x0*y0^
│ │ │ │ +0001fc80: 322b 7831 2a79 312a 7932 2b78 322a 7930  2+x1*y1*y2+x2*y0
│ │ │ │ +0001fc90: 2a79 3220 2020 2020 2020 2020 2020 2020  *y2             
│ │ │ │ +0001fca0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fce0: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2076  ------+.|i11 : v
│ │ │ │ -0001fcf0: 6172 6961 626c 6547 726f 7570 733d 7b7b  ariableGroups={{
│ │ │ │ -0001fd00: 7830 2c78 312c 7832 7d2c 7b79 302c 7931  x0,x1,x2},{y0,y1
│ │ │ │ -0001fd10: 2c79 327d 7d20 2020 2020 2020 2020 2020  ,y2}}           
│ │ │ │ -0001fd20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fce0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +0001fcf0: 3a20 7661 7269 6162 6c65 4772 6f75 7073  : variableGroups
│ │ │ │ +0001fd00: 3d7b 7b78 302c 7831 2c78 327d 2c7b 7930  ={{x0,x1,x2},{y0
│ │ │ │ +0001fd10: 2c79 312c 7932 7d7d 2020 2020 2020 2020  ,y1,y2}}        
│ │ │ │ +0001fd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fd60: 7c6f 3131 203d 207b 7b78 302c 2078 312c  |o11 = {{x0, x1,
│ │ │ │ -0001fd70: 2078 327d 2c20 7b79 302c 2079 312c 2079   x2}, {y0, y1, y
│ │ │ │ -0001fd80: 327d 7d20 2020 2020 2020 2020 2020 2020  2}}             
│ │ │ │ -0001fd90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd60: 207c 0a7c 6f31 3120 3d20 7b7b 7830 2c20   |.|o11 = {{x0, 
│ │ │ │ +0001fd70: 7831 2c20 7832 7d2c 207b 7930 2c20 7931  x1, x2}, {y0, y1
│ │ │ │ +0001fd80: 2c20 7932 7d7d 2020 2020 2020 2020 2020  , y2}}          
│ │ │ │ +0001fd90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fdd0: 2020 2020 2020 7c0a 7c6f 3131 203a 204c        |.|o11 : L
│ │ │ │ -0001fde0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0001fdd0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ +0001fde0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │  0001fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001fe10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0001fe20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001fe50: 7c69 3132 203a 2078 7853 6c69 6365 3d6d  |i12 : xxSlice=m
│ │ │ │ -0001fe60: 616b 6542 2753 6c69 6365 287b 322c 307d  akeB'Slice({2,0}
│ │ │ │ -0001fe70: 2c76 6172 6961 626c 6547 726f 7570 7329  ,variableGroups)
│ │ │ │ -0001fe80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001fe50: 2d2b 0a7c 6931 3220 3a20 7878 536c 6963  -+.|i12 : xxSlic
│ │ │ │ +0001fe60: 653d 6d61 6b65 4227 536c 6963 6528 7b32  e=makeB'Slice({2
│ │ │ │ +0001fe70: 2c30 7d2c 7661 7269 6162 6c65 4772 6f75  ,0},variableGrou
│ │ │ │ +0001fe80: 7073 2920 2020 2020 2020 2020 207c 0a7c  ps)          |.|
│ │ │ │  0001fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fec0: 2020 2020 2020 7c0a 7c6f 3132 203d 2042        |.|o12 = B
│ │ │ │ -0001fed0: 2753 6c69 6365 7b2e 2e2e 342e 2e2e 7d20  'Slice{...4...} 
│ │ │ │ -0001fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fec0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ +0001fed0: 3d20 4227 536c 6963 657b 2e2e 2e34 2e2e  = B'Slice{...4..
│ │ │ │ +0001fee0: 2e7d 2020 2020 2020 2020 2020 2020 2020  .}              
│ │ │ │  0001fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001ff00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ff40: 7c6f 3132 203a 2042 2753 6c69 6365 2020  |o12 : B'Slice  
│ │ │ │ -0001ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff40: 207c 0a7c 6f31 3220 3a20 4227 536c 6963   |.|o12 : B'Slic
│ │ │ │ +0001ff50: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  0001ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001ff70: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001ff80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ff90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ffb0: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2078  ------+.|i13 : x
│ │ │ │ -0001ffc0: 7953 6c69 6365 3d6d 616b 6542 2753 6c69  ySlice=makeB'Sli
│ │ │ │ -0001ffd0: 6365 287b 312c 317d 2c76 6172 6961 626c  ce({1,1},variabl
│ │ │ │ -0001ffe0: 6547 726f 7570 7329 2020 2020 2020 2020  eGroups)        
│ │ │ │ -0001fff0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001ffb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ +0001ffc0: 3a20 7879 536c 6963 653d 6d61 6b65 4227  : xySlice=makeB'
│ │ │ │ +0001ffd0: 536c 6963 6528 7b31 2c31 7d2c 7661 7269  Slice({1,1},vari
│ │ │ │ +0001ffe0: 6162 6c65 4772 6f75 7073 2920 2020 2020  ableGroups)     
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│ │ │ │ -00020040: 2e2e 342e 2e2e 7d20 2020 2020 2020 2020  ..4...}         
│ │ │ │ +00020020: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020040: 657b 2e2e 2e34 2e2e 2e7d 2020 2020 2020  e{...4...}      
│ │ │ │  00020050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00020090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200a0: 2020 2020 2020 7c0a 7c6f 3133 203a 2042        |.|o13 : B
│ │ │ │ -000200b0: 2753 6c69 6365 2020 2020 2020 2020 2020  'Slice          
│ │ │ │ +000200a0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +000200b0: 3a20 4227 536c 6963 6520 2020 2020 2020  : B'Slice       
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│ │ │ │  000200f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ -00020130: 616b 6542 2753 6c69 6365 287b 302c 327d  akeB'Slice({0,2}
│ │ │ │ -00020140: 2c76 6172 6961 626c 6547 726f 7570 7329  ,variableGroups)
│ │ │ │ -00020150: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020120: 2d2b 0a7c 6931 3420 3a20 7979 536c 6963  -+.|i14 : yySlic
│ │ │ │ +00020130: 653d 6d61 6b65 4227 536c 6963 6528 7b30  e=makeB'Slice({0
│ │ │ │ +00020140: 2c32 7d2c 7661 7269 6162 6c65 4772 6f75  ,2},variableGrou
│ │ │ │ +00020150: 7073 2920 2020 2020 2020 2020 207c 0a7c  ps)          |.|
│ │ │ │  00020160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020190: 2020 2020 2020 7c0a 7c6f 3134 203d 2042        |.|o14 = B
│ │ │ │ -000201a0: 2753 6c69 6365 7b2e 2e2e 342e 2e2e 7d20  'Slice{...4...} 
│ │ │ │ -000201b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020190: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ +000201a0: 3d20 4227 536c 6963 657b 2e2e 2e34 2e2e  = B'Slice{...4..
│ │ │ │ +000201b0: 2e7d 2020 2020 2020 2020 2020 2020 2020  .}              
│ │ │ │  000201c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000201d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000201e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020210: 7c6f 3134 203a 2042 2753 6c69 6365 2020  |o14 : B'Slice  
│ │ │ │ -00020220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020210: 207c 0a7c 6f31 3420 3a20 4227 536c 6963   |.|o14 : B'Slic
│ │ │ │ +00020220: 6520 2020 2020 2020 2020 2020 2020 2020  e               
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│ │ │ │  00020260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020280: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 206d  ------+.|i15 : m
│ │ │ │ -00020290: 616b 6542 2749 6e70 7574 4669 6c65 2873  akeB'InputFile(s
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│ │ │ │ +00020280: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520  ---------+.|i15 
│ │ │ │ +00020290: 3a20 6d61 6b65 4227 496e 7075 7446 696c  : makeB'InputFil
│ │ │ │ +000202a0: 6528 7374 6f72 6542 4d32 4669 6c65 732c  e(storeBM2Files,
│ │ │ │  000202b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202c0: 2020 7c0a 7c20 2020 2020 2020 2020 2048    |.|          H
│ │ │ │ -000202d0: 6f6d 5661 7269 6162 6c65 4772 6f75 703d  omVariableGroup=
│ │ │ │ -000202e0: 3e76 6172 6961 626c 6547 726f 7570 732c  >variableGroups,
│ │ │ │ -000202f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020300: 7c20 2020 2020 2020 2020 2042 2750 6f6c  |          B'Pol
│ │ │ │ -00020310: 796e 6f6d 6961 6c73 3d3e 7b66 312c 6632  ynomials=>{f1,f2
│ │ │ │ -00020320: 7d7c 7878 536c 6963 6523 4c69 7374 4227  }|xxSlice#ListB'
│ │ │ │ -00020330: 5365 6374 696f 6e73 293b 7c0a 2b2d 2d2d  Sections);|.+---
│ │ │ │ +000202c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000202d0: 2020 486f 6d56 6172 6961 626c 6547 726f    HomVariableGro
│ │ │ │ +000202e0: 7570 3d3e 7661 7269 6162 6c65 4772 6f75  up=>variableGrou
│ │ │ │ +000202f0: 7073 2c20 2020 2020 2020 2020 2020 2020  ps,             
│ │ │ │ +00020300: 207c 0a7c 2020 2020 2020 2020 2020 4227   |.|          B'
│ │ │ │ +00020310: 506f 6c79 6e6f 6d69 616c 733d 3e7b 6631  Polynomials=>{f1
│ │ │ │ +00020320: 2c66 327d 7c78 7853 6c69 6365 234c 6973  ,f2}|xxSlice#Lis
│ │ │ │ +00020330: 7442 2753 6563 7469 6f6e 7329 3b7c 0a2b  tB'Sections);|.+
│ │ │ │  00020340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00020360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020380: 756e 4265 7274 696e 6928 7374 6f72 6542  unBertini(storeB
│ │ │ │ -00020390: 4d32 4669 6c65 7329 2020 2020 2020 2020  M2Files)        
│ │ │ │ +00020370: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ +00020380: 3a20 7275 6e42 6572 7469 6e69 2873 746f  : runBertini(sto
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│ │ │ │ +000203b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000203c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000203d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000203e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000203f0: 7c69 3137 203a 2078 7844 6567 7265 653d  |i17 : xxDegree=
│ │ │ │ -00020400: 2369 6d70 6f72 7453 6f6c 7574 696f 6e73  #importSolutions
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│ │ │ │ +000203e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000203f0: 2d2b 0a7c 6931 3720 3a20 7878 4465 6772  -+.|i17 : xxDegr
│ │ │ │ +00020400: 6565 3d23 696d 706f 7274 536f 6c75 7469  ee=#importSoluti
│ │ │ │ +00020410: 6f6e 7346 696c 6528 7374 6f72 6542 4d32  onsFile(storeBM2
│ │ │ │ +00020420: 4669 6c65 7329 2020 2020 2020 207c 0a7c  Files)       |.|
│ │ │ │  00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020460: 2020 2020 2020 7c0a 7c6f 3137 203d 2032        |.|o17 = 2
│ │ │ │ -00020470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020460: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ +00020470: 3d20 3220 2020 2020 2020 2020 2020 2020  = 2             
│ │ │ │  00020480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000204a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000204a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000204c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000204e0: 7c69 3138 203a 206d 616b 6542 2749 6e70  |i18 : makeB'Inp
│ │ │ │ -000204f0: 7574 4669 6c65 2873 746f 7265 424d 3246  utFile(storeBM2F
│ │ │ │ -00020500: 696c 6573 2c20 2020 2020 2020 2020 2020  iles,           
│ │ │ │ -00020510: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020520: 2020 2020 2020 2048 6f6d 5661 7269 6162         HomVariab
│ │ │ │ -00020530: 6c65 4772 6f75 703d 3e76 6172 6961 626c  leGroup=>variabl
│ │ │ │ -00020540: 6547 726f 7570 732c 2020 2020 2020 2020  eGroups,        
│ │ │ │ -00020550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00020560: 2020 2042 2750 6f6c 796e 6f6d 6961 6c73     B'Polynomials
│ │ │ │ -00020570: 3d3e 7b66 312c 6632 7d7c 7879 536c 6963  =>{f1,f2}|xySlic
│ │ │ │ -00020580: 6523 4c69 7374 4227 5365 6374 696f 6e73  e#ListB'Sections
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│ │ │ │ +000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000204e0: 2d2b 0a7c 6931 3820 3a20 6d61 6b65 4227  -+.|i18 : makeB'
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│ │ │ │ +00020500: 4d32 4669 6c65 732c 2020 2020 2020 2020  M2Files,        
│ │ │ │ +00020510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020520: 2020 2020 2020 2020 2020 486f 6d56 6172            HomVar
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│ │ │ │ +00020540: 6162 6c65 4772 6f75 7073 2c20 2020 2020  ableGroups,     
│ │ │ │ +00020550: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020560: 2020 2020 2020 4227 506f 6c79 6e6f 6d69        B'Polynomi
│ │ │ │ +00020570: 616c 733d 3e7b 6631 2c66 327d 7c78 7953  als=>{f1,f2}|xyS
│ │ │ │ +00020580: 6c69 6365 234c 6973 7442 2753 6563 7469  lice#ListB'Secti
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│ │ │ │  000205a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000205b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000205c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000205d0: 7c69 3139 203a 2072 756e 4265 7274 696e  |i19 : runBertin
│ │ │ │ -000205e0: 6928 7374 6f72 6542 4d32 4669 6c65 7329  i(storeBM2Files)
│ │ │ │ -000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000205c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205d0: 2d2b 0a7c 6931 3920 3a20 7275 6e42 6572  -+.|i19 : runBer
│ │ │ │ +000205e0: 7469 6e69 2873 746f 7265 424d 3246 696c  tini(storeBM2Fil
│ │ │ │ +000205f0: 6573 2920 2020 2020 2020 2020 2020 2020  es)             
│ │ │ │ +00020600: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  00020610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020640: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 2078  ------+.|i20 : x
│ │ │ │ -00020650: 7944 6567 7265 653d 2369 6d70 6f72 7453  yDegree=#importS
│ │ │ │ -00020660: 6f6c 7574 696f 6e73 4669 6c65 2873 746f  olutionsFile(sto
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│ │ │ │ -00020680: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00020640: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ +00020650: 3a20 7879 4465 6772 6565 3d23 696d 706f  : xyDegree=#impo
│ │ │ │ +00020660: 7274 536f 6c75 7469 6f6e 7346 696c 6528  rtSolutionsFile(
│ │ │ │ +00020670: 7374 6f72 6542 4d32 4669 6c65 7329 2020  storeBM2Files)  
│ │ │ │ +00020680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000206a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000206b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000206c0: 7c6f 3230 203d 2033 2020 2020 2020 2020  |o20 = 3        
│ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206c0: 207c 0a7c 6f32 3020 3d20 3320 2020 2020   |.|o20 = 3     
│ │ │ │  000206d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000206e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000206f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │  00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020730: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 206d  ------+.|i21 : m
│ │ │ │ -00020740: 616b 6542 2749 6e70 7574 4669 6c65 2873  akeB'InputFile(s
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│ │ │ │ +00020730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120  ---------+.|i21 
│ │ │ │ +00020740: 3a20 6d61 6b65 4227 496e 7075 7446 696c  : makeB'InputFil
│ │ │ │ +00020750: 6528 7374 6f72 6542 4d32 4669 6c65 732c  e(storeBM2Files,
│ │ │ │  00020760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020770: 2020 7c0a 7c20 2020 2020 2020 2020 2048    |.|          H
│ │ │ │ -00020780: 6f6d 5661 7269 6162 6c65 4772 6f75 703d  omVariableGroup=
│ │ │ │ -00020790: 3e76 6172 6961 626c 6547 726f 7570 732c  >variableGroups,
│ │ │ │ -000207a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000207b0: 7c20 2020 2020 2020 2020 2042 2750 6f6c  |          B'Pol
│ │ │ │ -000207c0: 796e 6f6d 6961 6c73 3d3e 7b66 312c 6632  ynomials=>{f1,f2
│ │ │ │ -000207d0: 7d7c 7979 536c 6963 6523 4c69 7374 4227  }|yySlice#ListB'
│ │ │ │ -000207e0: 5365 6374 696f 6e73 293b 7c0a 2b2d 2d2d  Sections);|.+---
│ │ │ │ +00020770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020780: 2020 486f 6d56 6172 6961 626c 6547 726f    HomVariableGro
│ │ │ │ +00020790: 7570 3d3e 7661 7269 6162 6c65 4772 6f75  up=>variableGrou
│ │ │ │ +000207a0: 7073 2c20 2020 2020 2020 2020 2020 2020  ps,             
│ │ │ │ +000207b0: 207c 0a7c 2020 2020 2020 2020 2020 4227   |.|          B'
│ │ │ │ +000207c0: 506f 6c79 6e6f 6d69 616c 733d 3e7b 6631  Polynomials=>{f1
│ │ │ │ +000207d0: 2c66 327d 7c79 7953 6c69 6365 234c 6973  ,f2}|yySlice#Lis
│ │ │ │ +000207e0: 7442 2753 6563 7469 6f6e 7329 3b7c 0a2b  tB'Sections);|.+
│ │ │ │  000207f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020820: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2072  ------+.|i22 : r
│ │ │ │ -00020830: 756e 4265 7274 696e 6928 7374 6f72 6542  unBertini(storeB
│ │ │ │ -00020840: 4d32 4669 6c65 7329 2020 2020 2020 2020  M2Files)        
│ │ │ │ +00020820: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220  ---------+.|i22 
│ │ │ │ +00020830: 3a20 7275 6e42 6572 7469 6e69 2873 746f  : runBertini(sto
│ │ │ │ +00020840: 7265 424d 3246 696c 6573 2920 2020 2020  reBM2Files)     
│ │ │ │  00020850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020860: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │ -00020ab0: 4e65 7874 3a20 4e75 6d62 6572 546f 4227  Next: NumberToB'
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│ │ │ │ -00020ad0: 6b65 4227 536c 6963 652c 2055 703a 2054  keB'Slice, Up: T
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│ │ │ │ +000209c0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65  =====..  * "make
│ │ │ │ +000209d0: 4227 536c 6963 6528 5468 696e 672c 4c69  B'Slice(Thing,Li
│ │ │ │ +000209e0: 7374 2922 0a0a 466f 7220 7468 6520 7072  st)"..For the pr
│ │ │ │ +000209f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00020a00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +00020a10: 206f 626a 6563 7420 2a6e 6f74 6520 6d61   object *note ma
│ │ │ │ +00020a20: 6b65 4227 536c 6963 653a 206d 616b 6542  keB'Slice: makeB
│ │ │ │ +00020a30: 2753 6c69 6365 2c20 6973 2061 202a 6e6f  'Slice, is a *no
│ │ │ │ +00020a40: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
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│ │ │ │ +00020a60: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00020a70: 6574 686f 6446 756e 6374 696f 6e57 6974  ethodFunctionWit
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│ │ │ │ +00020a90: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c  e: Bertini.info,
│ │ │ │ +00020aa0: 204e 6f64 653a 206d 6f76 6542 2746 696c   Node: moveB'Fil
│ │ │ │ +00020ab0: 652c 204e 6578 743a 204e 756d 6265 7254  e, Next: NumberT
│ │ │ │ +00020ac0: 6f42 2753 7472 696e 672c 2050 7265 763a  oB'String, Prev:
│ │ │ │ +00020ad0: 206d 616b 6542 2753 6c69 6365 2c20 5570   makeB'Slice, Up
│ │ │ │ +00020ae0: 3a20 546f 700a 0a6d 6f76 6542 2746 696c  : Top..moveB'Fil
│ │ │ │ +00020af0: 6520 2d2d 204d 6f76 6520 6f72 2063 6f70  e -- Move or cop
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│ │ │ │  00020b10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00020b20: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ -00020b30: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ -00020b40: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00020b50: 6d6f 7665 4227 4669 6c65 2873 2c66 2c6e  moveB'File(s,f,n
│ │ │ │ -00020b60: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -00020b70: 2020 2020 2a20 732c 2061 202a 6e6f 7465      * s, a *note
│ │ │ │ -00020b80: 2073 7472 696e 673a 2028 4d61 6361 756c   string: (Macaul
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│ │ │ │ -00020ba0: 4120 7374 7269 6e67 2067 6976 696e 6720  A string giving 
│ │ │ │ -00020bb0: 6120 6469 7265 6374 6f72 792e 0a20 2020  a directory..   
│ │ │ │ -00020bc0: 2020 202a 2066 2c20 6120 2a6e 6f74 6520     * f, a *note 
│ │ │ │ -00020bd0: 7374 7269 6e67 3a20 284d 6163 6175 6c61  string: (Macaula
│ │ │ │ -00020be0: 7932 446f 6329 5374 7269 6e67 2c2c 2041  y2Doc)String,, A
│ │ │ │ -00020bf0: 206e 616d 6520 6f66 2061 2066 696c 652e   name of a file.
│ │ │ │ -00020c00: 0a20 2020 2020 202a 2073 2c20 6120 2a6e  .      * s, a *n
│ │ │ │ -00020c10: 6f74 6520 7374 7269 6e67 3a20 284d 6163  ote string: (Mac
│ │ │ │ -00020c20: 6175 6c61 7932 446f 6329 5374 7269 6e67  aulay2Doc)String
│ │ │ │ -00020c30: 2c2c 2041 206e 6577 206e 616d 6520 666f  ,, A new name fo
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│ │ │ │ -00020c50: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
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│ │ │ │ -00020ca0: 2020 2a20 2a6e 6f74 6520 436f 7079 4227    * *note CopyB'
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│ │ │ │ -00020d30: 4d6f 7665 546f 4469 7265 6374 6f72 7920  MoveToDirectory 
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│ │ │ │ -00020d90: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
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│ │ │ │ -00020de0: 6974 2074 6f20 6e2e 0a0a 2b2d 2d2d 2d2d  it to n...+-----
│ │ │ │ +00020b20: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +00020b30: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +00020b40: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00020b50: 2020 206d 6f76 6542 2746 696c 6528 732c     moveB'File(s,
│ │ │ │ +00020b60: 662c 6e29 0a20 202a 2049 6e70 7574 733a  f,n).  * Inputs:
│ │ │ │ +00020b70: 0a20 2020 2020 202a 2073 2c20 6120 2a6e  .      * s, a *n
│ │ │ │ +00020b80: 6f74 6520 7374 7269 6e67 3a20 284d 6163  ote string: (Mac
│ │ │ │ +00020b90: 6175 6c61 7932 446f 6329 5374 7269 6e67  aulay2Doc)String
│ │ │ │ +00020ba0: 2c2c 2041 2073 7472 696e 6720 6769 7669  ,, A string givi
│ │ │ │ +00020bb0: 6e67 2061 2064 6972 6563 746f 7279 2e0a  ng a directory..
│ │ │ │ +00020bc0: 2020 2020 2020 2a20 662c 2061 202a 6e6f        * f, a *no
│ │ │ │ +00020bd0: 7465 2073 7472 696e 673a 2028 4d61 6361  te string: (Maca
│ │ │ │ +00020be0: 756c 6179 3244 6f63 2953 7472 696e 672c  ulay2Doc)String,
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│ │ │ │ +00020c20: 4d61 6361 756c 6179 3244 6f63 2953 7472  Macaulay2Doc)Str
│ │ │ │ +00020c30: 696e 672c 2c20 4120 6e65 7720 6e61 6d65  ing,, A new name
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│ │ │ │ +00020c50: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
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│ │ │ │ +00020ca0: 2020 2020 202a 202a 6e6f 7465 2043 6f70       * *note Cop
│ │ │ │ +00020cb0: 7942 2746 696c 653a 2043 6f70 7942 2746  yB'File: CopyB'F
│ │ │ │ +00020cc0: 696c 652c 203d 3e20 2e2e 2e2c 2064 6566  ile, => ..., def
│ │ │ │ +00020cd0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00020ce0: 2c20 616e 206f 7074 696f 6e61 6c0a 2020  , an optional.  
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│ │ │ │ +00020d30: 202a 204d 6f76 6554 6f44 6972 6563 746f   * MoveToDirecto
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│ │ │ │ +00020d70: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
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│ │ │ │ +00020d90: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
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│ │ │ │ -00020f60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +00020f70: 207c 0a7c 6f32 203d 2074 7275 6520 2020   |.|o2 = true   
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│ │ │ │  00020f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020fb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │ +00021010: 6933 203a 206d 6f76 6542 2746 696c 6528  i3 : moveB'File(
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│ │ │ │ -000210b0: 6c65 4578 6973 7473 2873 746f 7265 424d  leExists(storeBM
│ │ │ │ -000210c0: 3246 696c 6573 7c22 2f66 696e 616c 5f70  2Files|"/final_p
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│ │ │ │  000210e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021150: 7365 2020 2020 2020 2020 2020 2020 2020  se              
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│ │ │ │ -00021230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000211f0: 7374 7328 7374 6f72 6542 4d32 4669 6c65  sts(storeBM2File
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│ │ │ │  00021240: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00021260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021270: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00021280: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │ +00021270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021280: 6f35 203d 2074 7275 6520 2020 2020 2020  o5 = true       
│ │ │ │  00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000212b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000212f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00021320: 7665 4227 4669 6c65 2873 746f 7265 424d  veB'File(storeBM
│ │ │ │ -00021330: 3246 696c 6573 2c22 7374 6172 745f 7061  2Files,"start_pa
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│ │ │ │ +00021310: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +00021320: 206d 6f76 6542 2746 696c 6528 7374 6f72   moveB'File(stor
│ │ │ │ +00021330: 6542 4d32 4669 6c65 732c 2273 7461 7274  eBM2Files,"start
│ │ │ │ +00021340: 5f70 6172 616d 6574 6572 7322 2c22 6261  _parameters","ba
│ │ │ │ +00021350: 636b 7570 222c 436f 7079 4227 4669 6c65  ckup",CopyB'File
│ │ │ │ +00021360: 3d3e 7472 7565 297c 0a2b 2d2d 2d2d 2d2d  =>true)|.+------
│ │ │ │  00021370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000213b0: 2d2d 2b0a 7c69 3720 3a20 6669 6c65 4578  --+.|i7 : fileEx
│ │ │ │ -000213c0: 6973 7473 2873 746f 7265 424d 3246 696c  ists(storeBM2Fil
│ │ │ │ -000213d0: 6573 7c22 2f73 7461 7274 5f70 6172 616d  es|"/start_param
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│ │ │ │ +000213b0: 2d2d 2d2d 2d2b 0a7c 6937 203a 2066 696c  -----+.|i7 : fil
│ │ │ │ +000213c0: 6545 7869 7374 7328 7374 6f72 6542 4d32  eExists(storeBM2
│ │ │ │ +000213d0: 4669 6c65 737c 222f 7374 6172 745f 7061  Files|"/start_pa
│ │ │ │ +000213e0: 7261 6d65 7465 7273 2229 2020 2020 2020  rameters")      
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│ │ │ │ -00021440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021450: 7c6f 3720 3d20 7472 7565 2020 2020 2020  |o7 = true      
│ │ │ │ +00021440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021450: 207c 0a7c 6f37 203d 2074 7275 6520 2020   |.|o7 = true   
│ │ │ │  00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00021580: 2020 2020 2020 7c0a 7c6f 3820 3d20 7472        |.|o8 = tr
│ │ │ │ -00021590: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00021580: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +00021590: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000215f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000216b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000216e0: 2074 656d 706f 7261 7279 4669 6c65 4e61   temporaryFileNa
│ │ │ │ +000216f0: 6d65 2829 3b20 2020 2020 2020 2020 2020  me();           
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│ │ │ │ -00021770: 0a7c 6931 3020 3a20 6d61 6b65 4469 7265  .|i10 : makeDire
│ │ │ │ -00021780: 6374 6f72 7920 4469 7231 2020 2020 2020  ctory Dir1      
│ │ │ │ +00021760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021770: 2d2d 2b0a 7c69 3130 203a 206d 616b 6544  --+.|i10 : makeD
│ │ │ │ +00021780: 6972 6563 746f 7279 2044 6972 3120 2020  irectory Dir1   
│ │ │ │  00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00021830: 424d 3246 696c 6573 2c7b 322c 332c 352c  BM2Files,{2,3,5,
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│ │ │ │ +00021810: 2d2d 2b0a 7c69 3131 203a 2077 7269 7465  --+.|i11 : write
│ │ │ │ +00021820: 5061 7261 6d65 7465 7246 696c 6528 7374  ParameterFile(st
│ │ │ │ +00021830: 6f72 6542 4d32 4669 6c65 732c 7b32 2c33  oreBM2Files,{2,3
│ │ │ │ +00021840: 2c35 2c37 7d29 3b20 2020 2020 2020 2020  ,5,7});         
│ │ │ │ +00021850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021860: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │  00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000218a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000218b0: 0a7c 6931 3220 3a20 6d6f 7665 4227 4669  .|i12 : moveB'Fi
│ │ │ │ -000218c0: 6c65 2873 746f 7265 424d 3246 696c 6573  le(storeBM2Files
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│ │ │ │ -00021900: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │ +000218c0: 2746 696c 6528 7374 6f72 6542 4d32 4669  'File(storeBM2Fi
│ │ │ │ +000218d0: 6c65 732c 2266 696e 616c 5f70 6172 616d  les,"final_param
│ │ │ │ +000218e0: 6574 6572 7322 2c22 7374 6172 745f 7061  eters","start_pa
│ │ │ │ +000218f0: 7261 6d65 7465 7273 222c 2020 2020 2020  rameters",      
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│ │ │ │ -00021950: 0a7c 4d6f 7665 546f 4469 7265 6374 6f72  .|MoveToDirector
│ │ │ │ -00021960: 793d 3e44 6972 3129 2020 2020 2020 2020  y=>Dir1)        
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│ │ │ │ +00021950: 2d2d 7c0a 7c4d 6f76 6554 6f44 6972 6563  --|.|MoveToDirec
│ │ │ │ +00021960: 746f 7279 3d3e 4469 7231 2920 2020 2020  tory=>Dir1)     
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│ │ │ │ -000219f0: 0a7c 6931 3320 3a20 6669 6c65 4578 6973  .|i13 : fileExis
│ │ │ │ -00021a00: 7473 2844 6972 317c 222f 7374 6172 745f  ts(Dir1|"/start_
│ │ │ │ -00021a10: 7061 7261 6d65 7465 7273 2229 2020 2020  parameters")    
│ │ │ │ +000219e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00021a00: 7869 7374 7328 4469 7231 7c22 2f73 7461  xists(Dir1|"/sta
│ │ │ │ +00021a10: 7274 5f70 6172 616d 6574 6572 7322 2920  rt_parameters") 
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│ │ │ │ -00021a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00021a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00021a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021a90: 0a7c 6f31 3320 3d20 7472 7565 2020 2020  .|o13 = true    
│ │ │ │ +00021a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a90: 2020 7c0a 7c6f 3133 203d 2074 7275 6520    |.|o13 = true 
│ │ │ │  00021aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00021b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00021b80: 0a7c 6931 3420 3a20 6d61 6b65 4469 7265  .|i14 : makeDire
│ │ │ │ -00021b90: 6374 6f72 7920 2873 746f 7265 424d 3246  ctory (storeBM2F
│ │ │ │ -00021ba0: 696c 6573 7c22 2f44 6972 3222 2920 2020  iles|"/Dir2")   
│ │ │ │ +00021b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021b80: 2d2d 2b0a 7c69 3134 203a 206d 616b 6544  --+.|i14 : makeD
│ │ │ │ +00021b90: 6972 6563 746f 7279 2028 7374 6f72 6542  irectory (storeB
│ │ │ │ +00021ba0: 4d32 4669 6c65 737c 222f 4469 7232 2229  M2Files|"/Dir2")
│ │ │ │  00021bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021bd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021bd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00021be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00021c20: 0a7c 6931 3520 3a20 7772 6974 6550 6172  .|i15 : writePar
│ │ │ │ -00021c30: 616d 6574 6572 4669 6c65 2873 746f 7265  ameterFile(store
│ │ │ │ -00021c40: 424d 3246 696c 6573 2c7b 322c 332c 352c  BM2Files,{2,3,5,
│ │ │ │ -00021c50: 377d 293b 2020 2020 2020 2020 2020 2020  7});            
│ │ │ │ -00021c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021c70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021c20: 2d2d 2b0a 7c69 3135 203a 2077 7269 7465  --+.|i15 : write
│ │ │ │ +00021c30: 5061 7261 6d65 7465 7246 696c 6528 7374  ParameterFile(st
│ │ │ │ +00021c40: 6f72 6542 4d32 4669 6c65 732c 7b32 2c33  oreBM2Files,{2,3
│ │ │ │ +00021c50: 2c35 2c37 7d29 3b20 2020 2020 2020 2020  ,5,7});         
│ │ │ │ +00021c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00021c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00021cc0: 0a7c 6931 3620 3a20 6d6f 7665 4227 4669  .|i16 : moveB'Fi
│ │ │ │ -00021cd0: 6c65 2873 746f 7265 424d 3246 696c 6573  le(storeBM2Files
│ │ │ │ -00021ce0: 2c22 6669 6e61 6c5f 7061 7261 6d65 7465  ,"final_paramete
│ │ │ │ -00021cf0: 7273 222c 2273 7461 7274 5f70 6172 616d  rs","start_param
│ │ │ │ -00021d00: 6574 6572 7322 2c20 2020 2020 2020 207c  eters",        |
│ │ │ │ -00021d10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00021cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021cc0: 2d2d 2b0a 7c69 3136 203a 206d 6f76 6542  --+.|i16 : moveB
│ │ │ │ +00021cd0: 2746 696c 6528 7374 6f72 6542 4d32 4669  'File(storeBM2Fi
│ │ │ │ +00021ce0: 6c65 732c 2266 696e 616c 5f70 6172 616d  les,"final_param
│ │ │ │ +00021cf0: 6574 6572 7322 2c22 7374 6172 745f 7061  eters","start_pa
│ │ │ │ +00021d00: 7261 6d65 7465 7273 222c 2020 2020 2020  rameters",      
│ │ │ │ +00021d10: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00021d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00021d60: 0a7c 5375 6246 6f6c 6465 723d 3e22 4469  .|SubFolder=>"Di
│ │ │ │ -00021d70: 7232 2229 2020 2020 2020 2020 2020 2020  r2")            
│ │ │ │ +00021d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021d60: 2d2d 7c0a 7c53 7562 466f 6c64 6572 3d3e  --|.|SubFolder=>
│ │ │ │ +00021d70: 2244 6972 3222 2920 2020 2020 2020 2020  "Dir2")         
│ │ │ │  00021d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021db0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │ -00021df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00021e00: 0a7c 6931 3720 3a20 6669 6c65 4578 6973  .|i17 : fileExis
│ │ │ │ -00021e10: 7473 2873 746f 7265 424d 3246 696c 6573  ts(storeBM2Files
│ │ │ │ -00021e20: 7c22 2f44 6972 322f 7374 6172 745f 7061  |"/Dir2/start_pa
│ │ │ │ -00021e30: 7261 6d65 7465 7273 2229 2020 2020 2020  rameters")      
│ │ │ │ -00021e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021e50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +00021e20: 6c65 737c 222f 4469 7232 2f73 7461 7274  les|"/Dir2/start
│ │ │ │ +00021e30: 5f70 6172 616d 6574 6572 7322 2920 2020  _parameters")   
│ │ │ │ +00021e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00021e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00021e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021ea0: 0a7c 6f31 3720 3d20 7472 7565 2020 2020  .|o17 = true    
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│ │ │ │ +00022000: 6675 6e63 7469 6f6e 2077 6974 680a 6f70  function with.op
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│ │ │ │  00022110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00022120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -00022130: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -00022140: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00022150: 2020 2020 204e 756d 6265 7254 6f42 2753       NumberToB'S
│ │ │ │ -00022160: 7472 696e 6728 6e29 0a20 202a 2049 6e70  tring(n).  * Inp
│ │ │ │ -00022170: 7574 733a 0a20 2020 2020 202a 206e 2c20  uts:.      * n, 
│ │ │ │ -00022180: 6120 2a6e 6f74 6520 7468 696e 673a 2028  a *note thing: (
│ │ │ │ -00022190: 4d61 6361 756c 6179 3244 6f63 2954 6869  Macaulay2Doc)Thi
│ │ │ │ -000221a0: 6e67 2c2c 206e 2069 7320 6120 6e75 6d62  ng,, n is a numb
│ │ │ │ -000221b0: 6572 2e0a 2020 2a20 2a6e 6f74 6520 4f70  er..  * *note Op
│ │ │ │ -000221c0: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -000221d0: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -000221e0: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -000221f0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -00022200: 732c 3a0a 2020 2020 2020 2a20 4d32 5072  s,:.      * M2Pr
│ │ │ │ -00022210: 6563 6973 696f 6e20 286d 6973 7369 6e67  ecision (missing
│ │ │ │ -00022220: 2064 6f63 756d 656e 7461 7469 6f6e 2920   documentation) 
│ │ │ │ -00022230: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00022240: 7661 6c75 6520 3533 2c20 0a0a 4465 7363  value 53, ..Desc
│ │ │ │ -00022250: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -00022260: 3d3d 3d0a 0a54 6869 7320 6675 6e63 7469  ===..This functi
│ │ │ │ -00022270: 6f6e 2074 616b 6573 2061 206e 756d 6265  on takes a numbe
│ │ │ │ -00022280: 7220 6173 2061 6e20 696e 7075 7420 7468  r as an input th
│ │ │ │ -00022290: 656e 206f 7574 7075 7473 2061 2073 7472  en outputs a str
│ │ │ │ -000222a0: 696e 6720 746f 2072 6570 7265 7365 6e74  ing to represent
│ │ │ │ -000222b0: 0a74 6869 7320 6e75 6d62 6572 2074 6f20  .this number to 
│ │ │ │ -000222c0: 4265 7274 696e 692e 2054 6865 206e 756d  Bertini. The num
│ │ │ │ -000222d0: 6265 7273 2061 7265 2063 6f6e 7665 7274  bers are convert
│ │ │ │ -000222e0: 6564 2074 6f20 666c 6f61 7469 6e67 2070  ed to floating p
│ │ │ │ -000222f0: 6f69 6e74 2074 6f0a 7072 6563 6973 696f  oint to.precisio
│ │ │ │ -00022300: 6e20 6465 7465 726d 696e 6564 2062 7920  n determined by 
│ │ │ │ -00022310: 7468 6520 6f70 7469 6f6e 204d 3250 7265  the option M2Pre
│ │ │ │ -00022320: 6369 7369 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d  cision...+------
│ │ │ │ +00022120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00022130: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +00022140: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +00022150: 2020 2020 2020 2020 4e75 6d62 6572 546f          NumberTo
│ │ │ │ +00022160: 4227 5374 7269 6e67 286e 290a 2020 2a20  B'String(n).  * 
│ │ │ │ +00022170: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00022180: 6e2c 2061 202a 6e6f 7465 2074 6869 6e67  n, a *note thing
│ │ │ │ +00022190: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000221a0: 5468 696e 672c 2c20 6e20 6973 2061 206e  Thing,, n is a n
│ │ │ │ +000221b0: 756d 6265 722e 0a20 202a 202a 6e6f 7465  umber..  * *note
│ │ │ │ +000221c0: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ +000221d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000221e0: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ +000221f0: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ +00022200: 7075 7473 2c3a 0a20 2020 2020 202a 204d  puts,:.      * M
│ │ │ │ +00022210: 3250 7265 6369 7369 6f6e 2028 6d69 7373  2Precision (miss
│ │ │ │ +00022220: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ +00022230: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175  n) => ..., defau
│ │ │ │ +00022240: 6c74 2076 616c 7565 2035 332c 200a 0a44  lt value 53, ..D
│ │ │ │ +00022250: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00022260: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e  ======..This fun
│ │ │ │ +00022270: 6374 696f 6e20 7461 6b65 7320 6120 6e75  ction takes a nu
│ │ │ │ +00022280: 6d62 6572 2061 7320 616e 2069 6e70 7574  mber as an input
│ │ │ │ +00022290: 2074 6865 6e20 6f75 7470 7574 7320 6120   then outputs a 
│ │ │ │ +000222a0: 7374 7269 6e67 2074 6f20 7265 7072 6573  string to repres
│ │ │ │ +000222b0: 656e 740a 7468 6973 206e 756d 6265 7220  ent.this number 
│ │ │ │ +000222c0: 746f 2042 6572 7469 6e69 2e20 5468 6520  to Bertini. The 
│ │ │ │ +000222d0: 6e75 6d62 6572 7320 6172 6520 636f 6e76  numbers are conv
│ │ │ │ +000222e0: 6572 7465 6420 746f 2066 6c6f 6174 696e  erted to floatin
│ │ │ │ +000222f0: 6720 706f 696e 7420 746f 0a70 7265 6369  g point to.preci
│ │ │ │ +00022300: 7369 6f6e 2064 6574 6572 6d69 6e65 6420  sion determined 
│ │ │ │ +00022310: 6279 2074 6865 206f 7074 696f 6e20 4d32  by the option M2
│ │ │ │ +00022320: 5072 6563 6973 696f 6e2e 0a0a 2b2d 2d2d  Precision...+---
│ │ │ │  00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022360: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00022370: 4e75 6d62 6572 546f 4227 5374 7269 6e67  NumberToB'String
│ │ │ │ -00022380: 2832 2b35 2a69 6929 2020 2020 2020 2020  (2+5*ii)        
│ │ │ │ +00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00022370: 203a 204e 756d 6265 7254 6f42 2753 7472   : NumberToB'Str
│ │ │ │ +00022380: 696e 6728 322b 352a 6969 2920 2020 2020  ing(2+5*ii)     
│ │ │ │  00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000223a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000223b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -000223f0: 3d20 2e32 6531 202e 3565 3120 2020 2020  = .2e1 .5e1     
│ │ │ │ +000223e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000223f0: 6f31 203d 202e 3265 3120 2e35 6531 2020  o1 = .2e1 .5e1  
│ │ │ │  00022400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022420: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00022430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00022430: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00022440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022470: 3220 3a20 4e75 6d62 6572 546f 4227 5374  2 : NumberToB'St
│ │ │ │ -00022480: 7269 6e67 2831 2f33 2c4d 3250 7265 6369  ring(1/3,M2Preci
│ │ │ │ -00022490: 7369 6f6e 3d3e 3136 2920 2020 2020 2020  sion=>16)       
│ │ │ │ -000224a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000224b0: 5761 726e 696e 673a 2072 6174 696f 6e61  Warning: rationa
│ │ │ │ -000224c0: 6c20 6e75 6d62 6572 7320 7769 6c6c 2062  l numbers will b
│ │ │ │ -000224d0: 6520 636f 6e76 6572 7465 6420 746f 2066  e converted to f
│ │ │ │ -000224e0: 6c6f 6174 696e 6720 706f 696e 742e 7c0a  loating point.|.
│ │ │ │ -000224f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00022470: 0a7c 6932 203a 204e 756d 6265 7254 6f42  .|i2 : NumberToB
│ │ │ │ +00022480: 2753 7472 696e 6728 312f 332c 4d32 5072  'String(1/3,M2Pr
│ │ │ │ +00022490: 6563 6973 696f 6e3d 3e31 3629 2020 2020  ecision=>16)    
│ │ │ │ +000224a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224b0: 7c0a 7c57 6172 6e69 6e67 3a20 7261 7469  |.|Warning: rati
│ │ │ │ +000224c0: 6f6e 616c 206e 756d 6265 7273 2077 696c  onal numbers wil
│ │ │ │ +000224d0: 6c20 6265 2063 6f6e 7665 7274 6564 2074  l be converted t
│ │ │ │ +000224e0: 6f20 666c 6f61 7469 6e67 2070 6f69 6e74  o floating point
│ │ │ │ +000224f0: 2e7c 0a7c 2020 2020 2020 2020 2020 2020  .|.|            
│ │ │ │  00022500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022530: 0a7c 6f32 203d 202e 3333 3333 3333 3333  .|o2 = .33333333
│ │ │ │ -00022540: 3333 3333 3333 3333 3165 3020 2e30 6530  333333331e0 .0e0
│ │ │ │ -00022550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022530: 2020 7c0a 7c6f 3220 3d20 2e33 3333 3333    |.|o2 = .33333
│ │ │ │ +00022540: 3333 3333 3333 3333 3333 3331 6530 202e  333333333331e0 .
│ │ │ │ +00022550: 3065 3020 2020 2020 2020 2020 2020 2020  0e0             
│ │ │ │  00022560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022570: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00022570: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00022580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000225b0: 2d2b 0a7c 6933 203a 204e 756d 6265 7254  -+.|i3 : NumberT
│ │ │ │ -000225c0: 6f42 2753 7472 696e 6728 312f 332c 4d32  oB'String(1/3,M2
│ │ │ │ -000225d0: 5072 6563 6973 696f 6e3d 3e31 3238 2920  Precision=>128) 
│ │ │ │ -000225e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225f0: 2020 7c0a 7c57 6172 6e69 6e67 3a20 7261    |.|Warning: ra
│ │ │ │ -00022600: 7469 6f6e 616c 206e 756d 6265 7273 2077  tional numbers w
│ │ │ │ -00022610: 696c 6c20 6265 2063 6f6e 7665 7274 6564  ill be converted
│ │ │ │ -00022620: 2074 6f20 666c 6f61 7469 6e67 2070 6f69   to floating poi
│ │ │ │ -00022630: 6e74 2e7c 0a7c 2020 2020 2020 2020 2020  nt.|.|          
│ │ │ │ +000225b0: 2d2d 2d2d 2b0a 7c69 3320 3a20 4e75 6d62  ----+.|i3 : Numb
│ │ │ │ +000225c0: 6572 546f 4227 5374 7269 6e67 2831 2f33  erToB'String(1/3
│ │ │ │ +000225d0: 2c4d 3250 7265 6369 7369 6f6e 3d3e 3132  ,M2Precision=>12
│ │ │ │ +000225e0: 3829 2020 2020 2020 2020 2020 2020 2020  8)              
│ │ │ │ +000225f0: 2020 2020 207c 0a7c 5761 726e 696e 673a       |.|Warning:
│ │ │ │ +00022600: 2072 6174 696f 6e61 6c20 6e75 6d62 6572   rational number
│ │ │ │ +00022610: 7320 7769 6c6c 2062 6520 636f 6e76 6572  s will be conver
│ │ │ │ +00022620: 7465 6420 746f 2066 6c6f 6174 696e 6720  ted to floating 
│ │ │ │ +00022630: 706f 696e 742e 7c0a 7c20 2020 2020 2020  point.|.|       
│ │ │ │  00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022670: 2020 2020 7c0a 7c6f 3320 3d20 2e33 3333      |.|o3 = .333
│ │ │ │ +00022670: 2020 2020 2020 207c 0a7c 6f33 203d 202e         |.|o3 = .
│ │ │ │  00022680: 3333 3333 3333 3333 3333 3333 3333 3333  3333333333333333
│ │ │ │  00022690: 3333 3333 3333 3333 3333 3333 3333 3333  3333333333333333
│ │ │ │ -000226a0: 3333 3333 3865 3020 2e30 6530 2020 2020  33338e0 .0e0    
│ │ │ │ -000226b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000226a0: 3333 3333 3333 3338 6530 202e 3065 3020  33333338e0 .0e0 
│ │ │ │ +000226b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  000226c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000226d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000226e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226f0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f  ------+..Ways to
│ │ │ │ -00022700: 2075 7365 204e 756d 6265 7254 6f42 2753   use NumberToB'S
│ │ │ │ -00022710: 7472 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d  tring:.=========
│ │ │ │ +000226f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973  ---------+..Ways
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│ │ │ │ +00022710: 4227 5374 7269 6e67 3a0a 3d3d 3d3d 3d3d  B'String:.======
│ │ │ │  00022720: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00022730: 3d3d 3d3d 0a0a 2020 2a20 224e 756d 6265  ====..  * "Numbe
│ │ │ │ -00022740: 7254 6f42 2753 7472 696e 6728 5468 696e  rToB'String(Thin
│ │ │ │ -00022750: 6729 220a 0a46 6f72 2074 6865 2070 726f  g)"..For the pro
│ │ │ │ -00022760: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00022770: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00022780: 6f62 6a65 6374 202a 6e6f 7465 204e 756d  object *note Num
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│ │ │ │ -000227a0: 756d 6265 7254 6f42 2753 7472 696e 672c  umberToB'String,
│ │ │ │ -000227b0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -000227c0: 6f64 2066 756e 6374 696f 6e0a 7769 7468  od function.with
│ │ │ │ -000227d0: 206f 7074 696f 6e73 3a20 284d 6163 6175   options: (Macau
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│ │ │ │ -00022800: 732c 2e0a 1f0a 4669 6c65 3a20 4265 7274  s,....File: Bert
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│ │ │ │ -00022820: 5061 7468 4c69 7374 2c20 4e65 7874 3a20  PathList, Next: 
│ │ │ │ -00022830: 7261 6469 6361 6c4c 6973 742c 2050 7265  radicalList, Pre
│ │ │ │ -00022840: 763a 204e 756d 6265 7254 6f42 2753 7472  v: NumberToB'Str
│ │ │ │ -00022850: 696e 672c 2055 703a 2054 6f70 0a0a 5061  ing, Up: Top..Pa
│ │ │ │ -00022860: 7468 4c69 7374 0a2a 2a2a 2a2a 2a2a 2a0a  thList.********.
│ │ │ │ -00022870: 0a46 756e 6374 696f 6e73 2077 6974 6820  .Functions with 
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│ │ │ │ -00022890: 7420 6e61 6d65 6420 5061 7468 4c69 7374  t named PathList
│ │ │ │ -000228a0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00022730: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4e75  =======..  * "Nu
│ │ │ │ +00022740: 6d62 6572 546f 4227 5374 7269 6e67 2854  mberToB'String(T
│ │ │ │ +00022750: 6869 6e67 2922 0a0a 466f 7220 7468 6520  hing)"..For the 
│ │ │ │ +00022760: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00022770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00022780: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00022790: 4e75 6d62 6572 546f 4227 5374 7269 6e67  NumberToB'String
│ │ │ │ +000227a0: 3a20 4e75 6d62 6572 546f 4227 5374 7269  : NumberToB'Stri
│ │ │ │ +000227b0: 6e67 2c20 6973 2061 202a 6e6f 7465 206d  ng, is a *note m
│ │ │ │ +000227c0: 6574 686f 6420 6675 6e63 7469 6f6e 0a77  ethod function.w
│ │ │ │ +000227d0: 6974 6820 6f70 7469 6f6e 733a 2028 4d61  ith options: (Ma
│ │ │ │ +000227e0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +000227f0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
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│ │ │ │ +00022810: 6572 7469 6e69 2e69 6e66 6f2c 204e 6f64  ertini.info, Nod
│ │ │ │ +00022820: 653a 2050 6174 684c 6973 742c 204e 6578  e: PathList, Nex
│ │ │ │ +00022830: 743a 2072 6164 6963 616c 4c69 7374 2c20  t: radicalList, 
│ │ │ │ +00022840: 5072 6576 3a20 4e75 6d62 6572 546f 4227  Prev: NumberToB'
│ │ │ │ +00022850: 5374 7269 6e67 2c20 5570 3a20 546f 700a  String, Up: Top.
│ │ │ │ +00022860: 0a50 6174 684c 6973 740a 2a2a 2a2a 2a2a  .PathList.******
│ │ │ │ +00022870: 2a2a 0a0a 4675 6e63 7469 6f6e 7320 7769  **..Functions wi
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│ │ │ │  000228b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000228c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00022900: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
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│ │ │ │ +00022970: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a0a  a *note symbol:.
│ │ │ │ +00022980: 284d 6163 6175 6c61 7932 446f 6329 5379  (Macaulay2Doc)Sy
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│ │ │ │ +000229c0: 4e65 7874 3a20 7374 6f72 6542 4d32 4669  Next: storeBM2Fi
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│ │ │ │ +00022a50: 6365 2e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ce..************
│ │ │ │  00022a60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00022a70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00022a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00022a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00022aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00022ab0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -00022ac0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ -00022ad0: 7361 6765 3a20 0a20 2020 2020 2020 2072  sage: .        r
│ │ │ │ -00022ae0: 6164 6963 616c 4c69 7374 284c 6973 742c  adicalList(List,
│ │ │ │ -00022af0: 4e75 6d62 6572 290a 2020 2020 2020 2020  Number).        
│ │ │ │ -00022b00: 7261 6469 6361 6c4c 6973 7428 4c69 7374  radicalList(List
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│ │ │ │ -00022b20: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -00022b30: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00022b40: 3244 6f63 294c 6973 742c 2c20 4120 6c69  2Doc)List,, A li
│ │ │ │ -00022b50: 7374 206f 6620 636f 6d70 6c65 7820 6f72  st of complex or
│ │ │ │ -00022b60: 2072 6561 6c0a 2020 2020 2020 2020 6e75   real.        nu
│ │ │ │ -00022b70: 6d62 6572 732e 0a20 2020 2020 202a 204e  mbers..      * N
│ │ │ │ -00022b80: 2c20 6120 2a6e 6f74 6520 6e75 6d62 6572  , a *note number
│ │ │ │ -00022b90: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00022ba0: 4e75 6d62 6572 2c2c 2041 2073 6d61 6c6c  Number,, A small
│ │ │ │ -00022bb0: 2072 6561 6c20 6e75 6d62 6572 2e0a 0a44   real number...D
│ │ │ │ -00022bc0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00022bd0: 3d3d 3d3d 3d3d 0a0a 5468 6973 206f 7574  ======..This out
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│ │ │ │ -00022c20: 206e 6f72 6d73 0a75 7020 746f 2074 6865   norms.up to the
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│ │ │ │ -00022c40: 6661 756c 7420 6973 2031 652d 3130 292e  fault is 1e-10).
│ │ │ │ -00022c50: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +00022ab0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +00022ac0: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +00022ad0: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00022ae0: 2020 7261 6469 6361 6c4c 6973 7428 4c69    radicalList(Li
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│ │ │ │ +00022b00: 2020 2072 6164 6963 616c 4c69 7374 284c     radicalList(L
│ │ │ │ +00022b10: 6973 7429 0a20 202a 2049 6e70 7574 733a  ist).  * Inputs:
│ │ │ │ +00022b20: 0a20 2020 2020 202a 204c 2c20 6120 2a6e  .      * L, a *n
│ │ │ │ +00022b30: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00022b40: 6c61 7932 446f 6329 4c69 7374 2c2c 2041  lay2Doc)List,, A
│ │ │ │ +00022b50: 206c 6973 7420 6f66 2063 6f6d 706c 6578   list of complex
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│ │ │ │ +00022b70: 206e 756d 6265 7273 2e0a 2020 2020 2020   numbers..      
│ │ │ │ +00022b80: 2a20 4e2c 2061 202a 6e6f 7465 206e 756d  * N, a *note num
│ │ │ │ +00022b90: 6265 723a 2028 4d61 6361 756c 6179 3244  ber: (Macaulay2D
│ │ │ │ +00022ba0: 6f63 294e 756d 6265 722c 2c20 4120 736d  oc)Number,, A sm
│ │ │ │ +00022bb0: 616c 6c20 7265 616c 206e 756d 6265 722e  all real number.
│ │ │ │ +00022bc0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00022bd0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320  =========..This 
│ │ │ │ +00022be0: 6f75 7470 7574 7320 6120 7375 626c 6973  outputs a sublis
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│ │ │ │ +00022c00: 7265 616c 206e 756d 6265 7273 2074 6861  real numbers tha
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│ │ │ │ +00022c30: 7468 6520 746f 6c65 7261 6e63 6520 4e20  the tolerance N 
│ │ │ │ +00022c40: 2864 6566 6175 6c74 2069 7320 3165 2d31  (default is 1e-1
│ │ │ │ +00022c50: 3029 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  0)...+----------
│ │ │ │  00022c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00022c80: 7261 6469 6361 6c4c 6973 7428 7b32 2e30  radicalList({2.0
│ │ │ │ -00022c90: 3030 2c31 2e39 3939 7d29 2020 2020 2020  00,1.999})      
│ │ │ │ -00022ca0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00022c80: 203a 2072 6164 6963 616c 4c69 7374 287b   : radicalList({
│ │ │ │ +00022c90: 322e 3030 302c 312e 3939 397d 2920 2020  2.000,1.999})   
│ │ │ │ +00022ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cc0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -00022cd0: 7b32 2c20 312e 3939 397d 2020 2020 2020  {2, 1.999}      
│ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00022cd0: 203d 207b 322c 2031 2e39 3939 7d20 2020   = {2, 1.999}   
│ │ │ │  00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d10: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -00022d20: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +00022d10: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00022d20: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00022d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00022d40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00022d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022d60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00022d70: 7261 6469 6361 6c4c 6973 7428 7b32 2e30  radicalList({2.0
│ │ │ │ -00022d80: 3030 2c31 2e39 3939 7d2c 3165 2d31 3029  00,1.999},1e-10)
│ │ │ │ -00022d90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00022d70: 203a 2072 6164 6963 616c 4c69 7374 287b   : radicalList({
│ │ │ │ +00022d80: 322e 3030 302c 312e 3939 397d 2c31 652d  2.000,1.999},1e-
│ │ │ │ +00022d90: 3130 297c 0a7c 2020 2020 2020 2020 2020  10)|.|          
│ │ │ │  00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022db0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00022dc0: 7b32 2c20 312e 3939 397d 2020 2020 2020  {2, 1.999}      
│ │ │ │ +00022db0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00022dc0: 203d 207b 322c 2031 2e39 3939 7d20 2020   = {2, 1.999}   
│ │ │ │  00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022de0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022de0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e00: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00022e10: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +00022e00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00022e10: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00022e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e30: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00022e30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00022e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -00022e60: 7261 6469 6361 6c4c 6973 7428 7b32 2e30  radicalList({2.0
│ │ │ │ -00022e70: 3030 2c31 2e39 3939 7d2c 3165 2d32 2920  00,1.999},1e-2) 
│ │ │ │ -00022e80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +00022e60: 203a 2072 6164 6963 616c 4c69 7374 287b   : radicalList({
│ │ │ │ +00022e70: 322e 3030 302c 312e 3939 397d 2c31 652d  2.000,1.999},1e-
│ │ │ │ +00022e80: 3229 207c 0a7c 2020 2020 2020 2020 2020  2) |.|          
│ │ │ │  00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ea0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00022eb0: 7b32 7d20 2020 2020 2020 2020 2020 2020  {2}             
│ │ │ │ +00022ea0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00022eb0: 203d 207b 327d 2020 2020 2020 2020 2020   = {2}          
│ │ │ │  00022ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00022ed0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ef0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00022f00: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +00022ef0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00022f00: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00022f20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00022f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022f40: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320  --------+..Ways 
│ │ │ │ -00022f50: 746f 2075 7365 2072 6164 6963 616c 4c69  to use radicalLi
│ │ │ │ -00022f60: 7374 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  st:.============
│ │ │ │ -00022f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -00022f80: 2a20 2272 6164 6963 616c 4c69 7374 284c  * "radicalList(L
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│ │ │ │ -00022fb0: 6572 2922 0a0a 466f 7220 7468 6520 7072  er)"..For the pr
│ │ │ │ -00022fc0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -00022fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ -00022ff0: 6469 6361 6c4c 6973 743a 2072 6164 6963  dicalList: radic
│ │ │ │ -00023000: 616c 4c69 7374 2c20 6973 2061 202a 6e6f  alList, is a *no
│ │ │ │ -00023010: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -00023020: 6f6e 2077 6974 680a 6f70 7469 6f6e 733a  on with.options:
│ │ │ │ -00023030: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -00023040: 6574 686f 6446 756e 6374 696f 6e57 6974  ethodFunctionWit
│ │ │ │ -00023050: 684f 7074 696f 6e73 2c2e 0a1f 0a46 696c  hOptions,....Fil
│ │ │ │ -00023060: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c  e: Bertini.info,
│ │ │ │ -00023070: 204e 6f64 653a 2073 746f 7265 424d 3246   Node: storeBM2F
│ │ │ │ -00023080: 696c 6573 2c20 4e65 7874 3a20 7375 6250  iles, Next: subP
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│ │ │ │ +00023100: 6563 7420 7374 6f72 6542 4d32 4669 6c65  ect storeBM2File
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│ │ │ │ +000231e0: 6d61 7472 6978 2061 7420 6120 706f 696e  matrix at a poin
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│ │ │ │  00023200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00023210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00023220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00023230: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -00023240: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ -00023250: 7361 6765 3a20 0a20 2020 2020 2020 2073  sage: .        s
│ │ │ │ -00023260: 7562 506f 696e 7428 662c 762c 7029 0a20  ubPoint(f,v,p). 
│ │ │ │ -00023270: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00023280: 202a 2066 2c20 6120 2a6e 6f74 6520 7468   * f, a *note th
│ │ │ │ -00023290: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ -000232a0: 6f63 2954 6869 6e67 2c2c 2041 2070 6f6c  oc)Thing,, A pol
│ │ │ │ -000232b0: 796e 6f6d 6961 6c20 6f72 2061 206d 6174  ynomial or a mat
│ │ │ │ -000232c0: 7269 782e 0a20 2020 2020 202a 2076 2c20  rix..      * v, 
│ │ │ │ -000232d0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -000232e0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -000232f0: 2c2c 204c 6973 7420 6f66 2076 6172 6961  ,, List of varia
│ │ │ │ -00023300: 626c 6573 2074 6861 7420 7765 2077 696c  bles that we wil
│ │ │ │ -00023310: 6c20 6265 0a20 2020 2020 2020 2065 7661  l be.        eva
│ │ │ │ -00023320: 6c75 6174 6564 2061 7420 7468 6520 706f  luated at the po
│ │ │ │ -00023330: 696e 742e 0a20 2020 2020 202a 2070 2c20  int..      * p, 
│ │ │ │ -00023340: 6120 2a6e 6f74 6520 7468 696e 673a 2028  a *note thing: (
│ │ │ │ -00023350: 4d61 6361 756c 6179 3244 6f63 2954 6869  Macaulay2Doc)Thi
│ │ │ │ -00023360: 6e67 2c2c 2041 2070 6f69 6e74 206f 7220  ng,, A point or 
│ │ │ │ -00023370: 6120 6c69 7374 206f 660a 2020 2020 2020  a list of.      
│ │ │ │ -00023380: 2020 636f 6f72 6469 6e61 7465 7320 6f72    coordinates or
│ │ │ │ -00023390: 2061 206d 6174 7269 782e 0a20 202a 202a   a matrix..  * *
│ │ │ │ -000233a0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -000233b0: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -000233c0: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -000233d0: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -000233e0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -000233f0: 202a 204d 3250 7265 6369 7369 6f6e 2028   * M2Precision (
│ │ │ │ -00023400: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ -00023410: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064  ation) => ..., d
│ │ │ │ -00023420: 6566 6175 6c74 2076 616c 7565 2035 332c  efault value 53,
│ │ │ │ -00023430: 200a 2020 2020 2020 2a20 5370 6563 6966   .      * Specif
│ │ │ │ -00023440: 7956 6172 6961 626c 6573 2028 6d69 7373  yVariables (miss
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│ │ │ │ -00023470: 6c74 2076 616c 7565 2066 616c 7365 2c20  lt value false, 
│ │ │ │ -00023480: 0a20 2020 2020 202a 2053 7562 496e 746f  .      * SubInto
│ │ │ │ -00023490: 4343 2028 6d69 7373 696e 6720 646f 6375  CC (missing docu
│ │ │ │ -000234a0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e  mentation) => ..
│ │ │ │ -000234b0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -000234c0: 2066 616c 7365 2c20 0a0a 4465 7363 7269   false, ..Descri
│ │ │ │ -000234d0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -000234e0: 3d0a 0a45 7661 6c75 6174 6520 6620 6174  =..Evaluate f at
│ │ │ │ -000234f0: 2061 2070 6f69 6e74 2e0a 0a2b 2d2d 2d2d   a point...+----
│ │ │ │ +00023230: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +00023240: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +00023250: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00023260: 2020 7375 6250 6f69 6e74 2866 2c76 2c70    subPoint(f,v,p
│ │ │ │ +00023270: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +00023280: 2020 2020 2a20 662c 2061 202a 6e6f 7465      * f, a *note
│ │ │ │ +00023290: 2074 6869 6e67 3a20 284d 6163 6175 6c61   thing: (Macaula
│ │ │ │ +000232a0: 7932 446f 6329 5468 696e 672c 2c20 4120  y2Doc)Thing,, A 
│ │ │ │ +000232b0: 706f 6c79 6e6f 6d69 616c 206f 7220 6120  polynomial or a 
│ │ │ │ +000232c0: 6d61 7472 6978 2e0a 2020 2020 2020 2a20  matrix..      * 
│ │ │ │ +000232d0: 762c 2061 202a 6e6f 7465 206c 6973 743a  v, a *note list:
│ │ │ │ +000232e0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ +000232f0: 6973 742c 2c20 4c69 7374 206f 6620 7661  ist,, List of va
│ │ │ │ +00023300: 7269 6162 6c65 7320 7468 6174 2077 6520  riables that we 
│ │ │ │ +00023310: 7769 6c6c 2062 650a 2020 2020 2020 2020  will be.        
│ │ │ │ +00023320: 6576 616c 7561 7465 6420 6174 2074 6865  evaluated at the
│ │ │ │ +00023330: 2070 6f69 6e74 2e0a 2020 2020 2020 2a20   point..      * 
│ │ │ │ +00023340: 702c 2061 202a 6e6f 7465 2074 6869 6e67  p, a *note thing
│ │ │ │ +00023350: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00023360: 5468 696e 672c 2c20 4120 706f 696e 7420  Thing,, A point 
│ │ │ │ +00023370: 6f72 2061 206c 6973 7420 6f66 0a20 2020  or a list of.   
│ │ │ │ +00023380: 2020 2020 2063 6f6f 7264 696e 6174 6573       coordinates
│ │ │ │ +00023390: 206f 7220 6120 6d61 7472 6978 2e0a 2020   or a matrix..  
│ │ │ │ +000233a0: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ +000233b0: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ +000233c0: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ +000233d0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +000233e0: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ +000233f0: 2020 2020 2a20 4d32 5072 6563 6973 696f      * M2Precisio
│ │ │ │ +00023400: 6e20 286d 6973 7369 6e67 2064 6f63 756d  n (missing docum
│ │ │ │ +00023410: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e  entation) => ...
│ │ │ │ +00023420: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00023430: 3533 2c20 0a20 2020 2020 202a 2053 7065  53, .      * Spe
│ │ │ │ +00023440: 6369 6679 5661 7269 6162 6c65 7320 286d  cifyVariables (m
│ │ │ │ +00023450: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ +00023460: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ +00023470: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ +00023480: 652c 200a 2020 2020 2020 2a20 5375 6249  e, .      * SubI
│ │ │ │ +00023490: 6e74 6f43 4320 286d 6973 7369 6e67 2064  ntoCC (missing d
│ │ │ │ +000234a0: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ +000234b0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +000234c0: 6c75 6520 6661 6c73 652c 200a 0a44 6573  lue false, ..Des
│ │ │ │ +000234d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +000234e0: 3d3d 3d3d 0a0a 4576 616c 7561 7465 2066  ====..Evaluate f
│ │ │ │ +000234f0: 2061 7420 6120 706f 696e 742e 0a0a 2b2d   at a point...+-
│ │ │ │  00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023530: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 3d43  -----+.|i1 : R=C
│ │ │ │ -00023540: 435b 782c 792c 7a5d 2020 2020 2020 2020  C[x,y,z]        
│ │ │ │ +00023530: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00023540: 523d 4343 5b78 2c79 2c7a 5d20 2020 2020  R=CC[x,y,z]     
│ │ │ │  00023550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023570: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00023580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000235b0: 6f31 203d 2052 2020 2020 2020 2020 2020  o1 = R          
│ │ │ │ +000235a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000235b0: 7c0a 7c6f 3120 3d20 5220 2020 2020 2020  |.|o1 = R       
│ │ │ │  000235c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000235d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000235e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000235f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023620: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ -00023630: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +00023620: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00023630: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │  00023640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023660: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00023660: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00023670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000236a0: 6932 203a 2066 3d7a 2a78 2b79 2020 2020  i2 : f=z*x+y    
│ │ │ │ +00023690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000236a0: 2b0a 7c69 3220 3a20 663d 7a2a 782b 7920  +.|i2 : f=z*x+y 
│ │ │ │  000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000236d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000236e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023710: 2020 2020 207c 0a7c 6f32 203d 2078 2a7a       |.|o2 = x*z
│ │ │ │ -00023720: 202b 2079 2020 2020 2020 2020 2020 2020   + y            
│ │ │ │ +00023710: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +00023720: 782a 7a20 2b20 7920 2020 2020 2020 2020  x*z + y         
│ │ │ │  00023730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00023760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023790: 6f32 203a 2052 2020 2020 2020 2020 2020  o2 : R          
│ │ │ │ +00023780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023790: 7c0a 7c6f 3220 3a20 5220 2020 2020 2020  |.|o2 : R       
│ │ │ │  000237a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000237c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  000237d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000237e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000237f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023800: 2d2d 2d2d 2d2b 0a7c 6933 203a 2073 7562  -----+.|i3 : sub
│ │ │ │ -00023810: 506f 696e 7428 662c 7b78 2c79 7d2c 7b2e  Point(f,{x,y},{.
│ │ │ │ -00023820: 312c 2e32 7d29 2020 2020 2020 2020 2020  1,.2})          
│ │ │ │ +00023800: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00023810: 7375 6250 6f69 6e74 2866 2c7b 782c 797d  subPoint(f,{x,y}
│ │ │ │ +00023820: 2c7b 2e31 2c2e 327d 2920 2020 2020 2020  ,{.1,.2})       
│ │ │ │  00023830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00023850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023880: 6f33 203d 202e 317a 202b 202e 3220 2020  o3 = .1z + .2   
│ │ │ │ +00023870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023880: 7c0a 7c6f 3320 3d20 2e31 7a20 2b20 2e32  |.|o3 = .1z + .2
│ │ │ │  00023890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000238b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000238c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238f0: 2020 2020 207c 0a7c 6f33 203a 2052 2020       |.|o3 : R  
│ │ │ │ -00023900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000238f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +00023900: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00023910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023930: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00023930: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00023940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00023970: 6934 203a 2073 7562 506f 696e 7428 662c  i4 : subPoint(f,
│ │ │ │ -00023980: 7b78 2c79 2c7a 7d2c 7b2e 312c 2e32 2c2e  {x,y,z},{.1,.2,.
│ │ │ │ -00023990: 337d 2c53 7065 6369 6679 5661 7269 6162  3},SpecifyVariab
│ │ │ │ -000239a0: 6c65 733d 3e7b 797d 297c 0a7c 2020 2020  les=>{y})|.|    
│ │ │ │ +00023960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023970: 2b0a 7c69 3420 3a20 7375 6250 6f69 6e74  +.|i4 : subPoint
│ │ │ │ +00023980: 2866 2c7b 782c 792c 7a7d 2c7b 2e31 2c2e  (f,{x,y,z},{.1,.
│ │ │ │ +00023990: 322c 2e33 7d2c 5370 6563 6966 7956 6172  2,.3},SpecifyVar
│ │ │ │ +000239a0: 6961 626c 6573 3d3e 7b79 7d29 7c0a 7c20  iables=>{y})|.| 
│ │ │ │  000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239e0: 2020 2020 207c 0a7c 6f34 203d 2078 2a7a       |.|o4 = x*z
│ │ │ │ -000239f0: 202b 202e 3220 2020 2020 2020 2020 2020   + .2           
│ │ │ │ +000239e0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +000239f0: 782a 7a20 2b20 2e32 2020 2020 2020 2020  x*z + .2        
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023a60: 6f34 203a 2052 2020 2020 2020 2020 2020  o4 : R          
│ │ │ │ +00023a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a60: 7c0a 7c6f 3420 3a20 5220 2020 2020 2020  |.|o4 : R       
│ │ │ │  00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00023a90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ad0: 2d2d 2d2d 2d2b 0a2b 2d2d 2d2d 2d2d 2d2d  -----+.+--------
│ │ │ │ +00023ad0: 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d  --------+.+-----
│ │ │ │  00023ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00023b20: 0a7c 6935 203a 2052 3d43 435f 3230 305b  .|i5 : R=CC_200[
│ │ │ │ -00023b30: 782c 792c 7a5d 2020 2020 2020 2020 2020  x,y,z]          
│ │ │ │ +00023b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023b20: 2d2d 2b0a 7c69 3520 3a20 523d 4343 5f32  --+.|i5 : R=CC_2
│ │ │ │ +00023b30: 3030 5b78 2c79 2c7a 5d20 2020 2020 2020  00[x,y,z]       
│ │ │ │  00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00023b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bb0: 2020 207c 0a7c 6f35 203d 2052 2020 2020     |.|o5 = R    
│ │ │ │ +00023bb0: 2020 2020 2020 7c0a 7c6f 3520 3d20 5220        |.|o5 = R 
│ │ │ │  00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00023c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c40: 2020 2020 2020 207c 0a7c 6f35 203a 2050         |.|o5 : P
│ │ │ │ -00023c50: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00023c40: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00023c50: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │  00023c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00023c90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00023ce0: 203a 2066 3d7a 2a78 2b79 2020 2020 2020   : f=z*x+y      
│ │ │ │ +00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00023ce0: 7c69 3620 3a20 663d 7a2a 782b 7920 2020  |i6 : f=z*x+y   
│ │ │ │  00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023d20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00023d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023d70: 0a7c 6f36 203d 2078 2a7a 202b 2079 2020  .|o6 = x*z + y  
│ │ │ │ -00023d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d70: 2020 7c0a 7c6f 3620 3d20 782a 7a20 2b20    |.|o6 = x*z + 
│ │ │ │ +00023d80: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023db0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00023db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e00: 2020 207c 0a7c 6f36 203a 2052 2020 2020     |.|o6 : R    
│ │ │ │ +00023e00: 2020 2020 2020 7c0a 7c6f 3620 3a20 5220        |.|o6 : R 
│ │ │ │  00023e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e40: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00023e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e90: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2073  -------+.|i7 : s
│ │ │ │ -00023ea0: 7562 506f 696e 7428 662c 7b78 2c79 2c7a  ubPoint(f,{x,y,z
│ │ │ │ -00023eb0: 7d2c 7b2e 312c 2e32 2c2e 337d 2c53 7562  },{.1,.2,.3},Sub
│ │ │ │ -00023ec0: 496e 746f 4343 3d3e 7472 7565 2920 2020  IntoCC=>true)   
│ │ │ │ +00023e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +00023ea0: 3a20 7375 6250 6f69 6e74 2866 2c7b 782c  : subPoint(f,{x,
│ │ │ │ +00023eb0: 792c 7a7d 2c7b 2e31 2c2e 322c 2e33 7d2c  y,z},{.1,.2,.3},
│ │ │ │ +00023ec0: 5375 6249 6e74 6f43 433d 3e74 7275 6529  SubIntoCC=>true)
│ │ │ │  00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ee0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023ee0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f20: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00023f30: 203d 202e 3233 2020 2020 2020 2020 2020   = .23          
│ │ │ │ +00023f20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023f30: 7c6f 3720 3d20 2e32 3320 2020 2020 2020  |o7 = .23       
│ │ │ │  00023f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00023f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023fc0: 0a7c 6f37 203a 2043 4320 286f 6620 7072  .|o7 : CC (of pr
│ │ │ │ -00023fd0: 6563 6973 696f 6e20 3533 2920 2020 2020  ecision 53)     
│ │ │ │ +00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023fc0: 2020 7c0a 7c6f 3720 3a20 4343 2028 6f66    |.|o7 : CC (of
│ │ │ │ +00023fd0: 2070 7265 6369 7369 6f6e 2035 3329 2020   precision 53)  
│ │ │ │  00023fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024000: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00024000: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00024010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024050: 2d2d 2d2b 0a7c 6938 203a 2073 7562 506f  ---+.|i8 : subPo
│ │ │ │ -00024060: 696e 7428 662c 7b78 2c79 2c7a 7d2c 7b2e  int(f,{x,y,z},{.
│ │ │ │ -00024070: 3132 3334 3536 3738 3930 3132 3334 3536  1234567890123456
│ │ │ │ -00024080: 3738 3930 3132 3334 3536 3738 3930 3132  7890123456789012
│ │ │ │ -00024090: 3334 3536 3738 3930 7032 3030 2c7c 0a7c  34567890p200,|.|
│ │ │ │ -000240a0: 2020 2020 2020 2020 2020 2020 2030 2c31               0,1
│ │ │ │ -000240b0: 7d2c 5375 6249 6e74 6f43 433d 3e74 7275  },SubIntoCC=>tru
│ │ │ │ -000240c0: 652c 4d32 5072 6563 6973 696f 6e3d 3e32  e,M2Precision=>2
│ │ │ │ -000240d0: 3030 2920 2020 2020 2020 2020 2020 2020  00)             
│ │ │ │ -000240e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024050: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 7375  ------+.|i8 : su
│ │ │ │ +00024060: 6250 6f69 6e74 2866 2c7b 782c 792c 7a7d  bPoint(f,{x,y,z}
│ │ │ │ +00024070: 2c7b 2e31 3233 3435 3637 3839 3031 3233  ,{.1234567890123
│ │ │ │ +00024080: 3435 3637 3839 3031 3233 3435 3637 3839  4567890123456789
│ │ │ │ +00024090: 3031 3233 3435 3637 3839 3070 3230 302c  01234567890p200,
│ │ │ │ +000240a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000240b0: 302c 317d 2c53 7562 496e 746f 4343 3d3e  0,1},SubIntoCC=>
│ │ │ │ +000240c0: 7472 7565 2c4d 3250 7265 6369 7369 6f6e  true,M2Precision
│ │ │ │ +000240d0: 3d3e 3230 3029 2020 2020 2020 2020 2020  =>200)          
│ │ │ │ +000240e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000240f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024130: 207c 0a7c 6f38 203d 202e 3132 3334 3536   |.|o8 = .123456
│ │ │ │ -00024140: 3738 3930 3132 3334 3536 3738 3930 3132  7890123456789012
│ │ │ │ -00024150: 3334 3536 3738 3930 3132 3334 3536 3738  3456789012345678
│ │ │ │ -00024160: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ -00024170: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00024180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024130: 2020 2020 7c0a 7c6f 3820 3d20 2e31 3233      |.|o8 = .123
│ │ │ │ +00024140: 3435 3637 3839 3031 3233 3435 3637 3839  4567890123456789
│ │ │ │ +00024150: 3031 3233 3435 3637 3839 3031 3233 3435  0123456789012345
│ │ │ │ +00024160: 3637 3839 2020 2020 2020 2020 2020 2020  6789            
│ │ │ │ +00024170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00024180: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00024190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241c0: 2020 2020 207c 0a7c 6f38 203a 2043 4320       |.|o8 : CC 
│ │ │ │ -000241d0: 286f 6620 7072 6563 6973 696f 6e20 3230  (of precision 20
│ │ │ │ -000241e0: 3029 2020 2020 2020 2020 2020 2020 2020  0)              
│ │ │ │ +000241c0: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ +000241d0: 4343 2028 6f66 2070 7265 6369 7369 6f6e  CC (of precision
│ │ │ │ +000241e0: 2032 3030 2920 2020 2020 2020 2020 2020   200)           
│ │ │ │  000241f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024210: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00024200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024210: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00024220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024250: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665  ---------+..Cave
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│ │ │ │ -000248a0: 2020 2020 2062 6572 7469 6e69 5573 6572       bertiniUser
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│ │ │ │ +00024870: 2020 6265 7274 696e 695a 6572 6f44 696d    bertiniZeroDim
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│ │ │ │ +000248a0: 2020 2020 2020 2020 6265 7274 696e 6955          bertiniU
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│ │ │ │ +00024920: 6e65 7261 7469 6f6e 2077 6865 6e20 736f  neration when so
│ │ │ │ +00024930: 6c76 696e 6720 610a 706f 6c79 6e6f 6d69  lving a.polynomi
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│ │ │ │ +00024950: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
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│ │ │ │  000249a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00024a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
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│ │ │ │ -00024af0: 202d 2d20 5468 6973 2066 756e 6374 696f   -- This functio
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│ │ │ │ +000249f0: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00024a00: 6e74 6174 696f 6e29 0a0a 466f 7220 7468  ntation)..For th
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│ │ │ │ +00024a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00024a30: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00024a40: 6520 5573 6552 6567 656e 6572 6174 696f  e UseRegeneratio
│ │ │ │ +00024a50: 6e3a 2055 7365 5265 6765 6e65 7261 7469  n: UseRegenerati
│ │ │ │ +00024a60: 6f6e 2c20 6973 2061 202a 6e6f 7465 2073  on, is a *note s
│ │ │ │ +00024a70: 796d 626f 6c3a 0a28 4d61 6361 756c 6179  ymbol:.(Macaulay
│ │ │ │ +00024a80: 3244 6f63 2953 796d 626f 6c2c 2e0a 1f0a  2Doc)Symbol,....
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│ │ │ │ +00024aa0: 666f 2c20 4e6f 6465 3a20 7661 6c75 6542  fo, Node: valueB
│ │ │ │ +00024ab0: 4d32 2c20 4e65 7874 3a20 5661 7269 6162  M2, Next: Variab
│ │ │ │ +00024ac0: 6c65 2067 726f 7570 732c 2050 7265 763a  le groups, Prev:
│ │ │ │ +00024ad0: 2055 7365 5265 6765 6e65 7261 7469 6f6e   UseRegeneration
│ │ │ │ +00024ae0: 2c20 5570 3a20 546f 700a 0a76 616c 7565  , Up: Top..value
│ │ │ │ +00024af0: 424d 3220 2d2d 2054 6869 7320 6675 6e63  BM2 -- This func
│ │ │ │ +00024b00: 7469 6f6e 206d 616b 6573 2061 206e 756d  tion makes a num
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│ │ │ │  00024b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00024b60: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -00024b70: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ -00024b80: 653a 200a 2020 2020 2020 2020 7661 6c75  e: .        valu
│ │ │ │ -00024b90: 6542 4d32 2873 290a 2020 2a20 496e 7075  eBM2(s).  * Inpu
│ │ │ │ -00024ba0: 7473 3a0a 2020 2020 2020 2a20 732c 2061  ts:.      * s, a
│ │ │ │ -00024bb0: 202a 6e6f 7465 2073 7472 696e 673a 2028   *note string: (
│ │ │ │ -00024bc0: 4d61 6361 756c 6179 3244 6f63 2953 7472  Macaulay2Doc)Str
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│ │ │ │ -00024bf0: 2020 2020 636f 6f72 6469 6e61 7465 2e0a      coordinate..
│ │ │ │ -00024c00: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
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│ │ │ │ -00024c30: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
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│ │ │ │ -00024c50: 2020 2020 2020 2a20 4d32 5072 6563 6973        * M2Precis
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│ │ │ │ -00024c80: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -00024c90: 6520 3533 2c20 0a0a 4465 7363 7269 7074  e 53, ..Descript
│ │ │ │ -00024ca0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00024cb0: 0a54 6869 7320 6675 6e63 7469 6f6e 2074  .This function t
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│ │ │ │ -00024ce0: 6469 6e61 7465 2069 6e20 6120 4265 7274  dinate in a Bert
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│ │ │ │ -00024d00: 6c65 206f 7220 7061 7261 6d65 7465 7220  le or parameter 
│ │ │ │ -00024d10: 6669 6c65 2061 6e64 206d 616b 6573 2061  file and makes a
│ │ │ │ -00024d20: 206e 756d 6265 7220 696e 2043 432e 2057   number in CC. W
│ │ │ │ -00024d30: 6520 6361 6e20 6164 6a75 7374 2074 6865  e can adjust the
│ │ │ │ -00024d40: 2070 7265 6369 7369 6f6e 0a75 7369 6e67   precision.using
│ │ │ │ -00024d50: 2074 6865 204d 3250 7265 6369 7369 6f6e   the M2Precision
│ │ │ │ -00024d60: 206f 7074 696f 6e2e 2046 7261 6374 696f   option. Fractio
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│ │ │ │ -00024d80: 2069 6e20 7468 6520 7374 7269 6e67 2073   in the string s
│ │ │ │ -00024d90: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +00024b60: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00024b70: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +00024b80: 7361 6765 3a20 0a20 2020 2020 2020 2076  sage: .        v
│ │ │ │ +00024b90: 616c 7565 424d 3228 7329 0a20 202a 2049  alueBM2(s).  * I
│ │ │ │ +00024ba0: 6e70 7574 733a 0a20 2020 2020 202a 2073  nputs:.      * s
│ │ │ │ +00024bb0: 2c20 6120 2a6e 6f74 6520 7374 7269 6e67  , a *note string
│ │ │ │ +00024bc0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00024bd0: 5374 7269 6e67 2c2c 2041 2073 7472 696e  String,, A strin
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│ │ │ │ +00024bf0: 2020 2020 2020 2063 6f6f 7264 696e 6174         coordinat
│ │ │ │ +00024c00: 652e 0a20 202a 202a 6e6f 7465 204f 7074  e..  * *note Opt
│ │ │ │ +00024c10: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
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│ │ │ │ +00024c40: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +00024c50: 2c3a 0a20 2020 2020 202a 204d 3250 7265  ,:.      * M2Pre
│ │ │ │ +00024c60: 6369 7369 6f6e 2028 6d69 7373 696e 6720  cision (missing 
│ │ │ │ +00024c70: 646f 6375 6d65 6e74 6174 696f 6e29 203d  documentation) =
│ │ │ │ +00024c80: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00024c90: 616c 7565 2035 332c 200a 0a44 6573 6372  alue 53, ..Descr
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│ │ │ │ +00024cb0: 3d3d 0a0a 5468 6973 2066 756e 6374 696f  ==..This functio
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│ │ │ │ +00024cd0: 7265 7072 6573 656e 7469 6e67 2061 2063  representing a c
│ │ │ │ +00024ce0: 6f6f 7264 696e 6174 6520 696e 2061 2042  oordinate in a B
│ │ │ │ +00024cf0: 6572 7469 6e69 2073 6f6c 7574 696f 6e73  ertini solutions
│ │ │ │ +00024d00: 0a66 696c 6520 6f72 2070 6172 616d 6574  .file or paramet
│ │ │ │ +00024d10: 6572 2066 696c 6520 616e 6420 6d61 6b65  er file and make
│ │ │ │ +00024d20: 7320 6120 6e75 6d62 6572 2069 6e20 4343  s a number in CC
│ │ │ │ +00024d30: 2e20 5765 2063 616e 2061 646a 7573 7420  . We can adjust 
│ │ │ │ +00024d40: 7468 6520 7072 6563 6973 696f 6e0a 7573  the precision.us
│ │ │ │ +00024d50: 696e 6720 7468 6520 4d32 5072 6563 6973  ing the M2Precis
│ │ │ │ +00024d60: 696f 6e20 6f70 7469 6f6e 2e20 4672 6163  ion option. Frac
│ │ │ │ +00024d70: 7469 6f6e 7320 7368 6f75 6c64 206e 6f74  tions should not
│ │ │ │ +00024d80: 2062 6520 696e 2074 6865 2073 7472 696e   be in the strin
│ │ │ │ +00024d90: 6720 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  g s...+---------
│ │ │ │  00024da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024de0: 2d2b 0a7c 6931 203a 2076 616c 7565 424d  -+.|i1 : valueBM
│ │ │ │ -00024df0: 3228 2231 2e32 3265 2d32 2034 652d 3522  2("1.22e-2 4e-5"
│ │ │ │ -00024e00: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00024de0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7661 6c75  ----+.|i1 : valu
│ │ │ │ +00024df0: 6542 4d32 2822 312e 3232 652d 3220 3465  eBM2("1.22e-2 4e
│ │ │ │ +00024e00: 2d35 2229 2020 2020 2020 2020 2020 2020  -5")            
│ │ │ │  00024e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00024e30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00024e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e80: 207c 0a7c 6f31 203d 202e 3031 3232 2b2e   |.|o1 = .0122+.
│ │ │ │ -00024e90: 3030 3030 342a 6969 2020 2020 2020 2020  00004*ii        
│ │ │ │ +00024e80: 2020 2020 7c0a 7c6f 3120 3d20 2e30 3132      |.|o1 = .012
│ │ │ │ +00024e90: 322b 2e30 3030 3034 2a69 6920 2020 2020  2+.00004*ii     
│ │ │ │  00024ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ed0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00024ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f20: 207c 0a7c 6f31 203a 2043 4320 286f 6620   |.|o1 : CC (of 
│ │ │ │ -00024f30: 7072 6563 6973 696f 6e20 3533 2920 2020  precision 53)   
│ │ │ │ +00024f20: 2020 2020 7c0a 7c6f 3120 3a20 4343 2028      |.|o1 : CC (
│ │ │ │ +00024f30: 6f66 2070 7265 6369 7369 6f6e 2035 3329  of precision 53)
│ │ │ │  00024f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00024f70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00024f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024fc0: 2d2b 0a7c 6932 203a 2076 616c 7565 424d  -+.|i2 : valueBM
│ │ │ │ -00024fd0: 3228 2231 2e32 3220 3465 2d35 2229 2020  2("1.22 4e-5")  
│ │ │ │ -00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fc0: 2d2d 2d2d 2b0a 7c69 3220 3a20 7661 6c75  ----+.|i2 : valu
│ │ │ │ +00024fd0: 6542 4d32 2822 312e 3232 2034 652d 3522  eBM2("1.22 4e-5"
│ │ │ │ +00024fe0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025010: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025010: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025060: 207c 0a7c 6f32 203d 2031 2e32 322b 2e30   |.|o2 = 1.22+.0
│ │ │ │ -00025070: 3030 3034 2a69 6920 2020 2020 2020 2020  0004*ii         
│ │ │ │ +00025060: 2020 2020 7c0a 7c6f 3220 3d20 312e 3232      |.|o2 = 1.22
│ │ │ │ +00025070: 2b2e 3030 3030 342a 6969 2020 2020 2020  +.00004*ii      
│ │ │ │  00025080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000250b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000250c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025100: 207c 0a7c 6f32 203a 2043 4320 286f 6620   |.|o2 : CC (of 
│ │ │ │ -00025110: 7072 6563 6973 696f 6e20 3533 2920 2020  precision 53)   
│ │ │ │ +00025100: 2020 2020 7c0a 7c6f 3220 3a20 4343 2028      |.|o2 : CC (
│ │ │ │ +00025110: 6f66 2070 7265 6369 7369 6f6e 2035 3329  of precision 53)
│ │ │ │  00025120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00025150: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251a0: 2d2b 0a7c 6933 203a 2076 616c 7565 424d  -+.|i3 : valueBM
│ │ │ │ -000251b0: 3228 2231 2e32 3220 3422 2920 2020 2020  2("1.22 4")     
│ │ │ │ +000251a0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7661 6c75  ----+.|i3 : valu
│ │ │ │ +000251b0: 6542 4d32 2822 312e 3232 2034 2229 2020  eBM2("1.22 4")  
│ │ │ │  000251c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000251f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025240: 207c 0a7c 6f33 203d 2031 2e32 322b 342a   |.|o3 = 1.22+4*
│ │ │ │ -00025250: 6969 2020 2020 2020 2020 2020 2020 2020  ii              
│ │ │ │ +00025240: 2020 2020 7c0a 7c6f 3320 3d20 312e 3232      |.|o3 = 1.22
│ │ │ │ +00025250: 2b34 2a69 6920 2020 2020 2020 2020 2020  +4*ii           
│ │ │ │  00025260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025290: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025290: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000252a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252e0: 207c 0a7c 6f33 203a 2043 4320 286f 6620   |.|o3 : CC (of 
│ │ │ │ -000252f0: 7072 6563 6973 696f 6e20 3533 2920 2020  precision 53)   
│ │ │ │ +000252e0: 2020 2020 7c0a 7c6f 3320 3a20 4343 2028      |.|o3 : CC (
│ │ │ │ +000252f0: 6f66 2070 7265 6369 7369 6f6e 2035 3329  of precision 53)
│ │ │ │  00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025330: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00025330: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00025340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025380: 2d2b 0a7c 6934 203a 2076 616c 7565 424d  -+.|i4 : valueBM
│ │ │ │ -00025390: 3228 2231 2e32 3265 2b32 2034 2022 2920  2("1.22e+2 4 ") 
│ │ │ │ -000253a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025380: 2d2d 2d2d 2b0a 7c69 3420 3a20 7661 6c75  ----+.|i4 : valu
│ │ │ │ +00025390: 6542 4d32 2822 312e 3232 652b 3220 3420  eBM2("1.22e+2 4 
│ │ │ │ +000253a0: 2229 2020 2020 2020 2020 2020 2020 2020  ")              
│ │ │ │  000253b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000253d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025420: 207c 0a7c 6f34 203d 2031 3232 2b34 2a69   |.|o4 = 122+4*i
│ │ │ │ -00025430: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │ +00025420: 2020 2020 7c0a 7c6f 3420 3d20 3132 322b      |.|o4 = 122+
│ │ │ │ +00025430: 342a 6969 2020 2020 2020 2020 2020 2020  4*ii            
│ │ │ │  00025440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025470: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254c0: 207c 0a7c 6f34 203a 2043 4320 286f 6620   |.|o4 : CC (of 
│ │ │ │ -000254d0: 7072 6563 6973 696f 6e20 3533 2920 2020  precision 53)   
│ │ │ │ +000254c0: 2020 2020 7c0a 7c6f 3420 3a20 4343 2028      |.|o4 : CC (
│ │ │ │ +000254d0: 6f66 2070 7265 6369 7369 6f6e 2035 3329  of precision 53)
│ │ │ │  000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00025510: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00025520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025560: 2d2b 0a7c 6935 203a 206e 313d 7661 6c75  -+.|i5 : n1=valu
│ │ │ │ -00025570: 6542 4d32 2822 312e 3131 222c 4d32 5072  eBM2("1.11",M2Pr
│ │ │ │ -00025580: 6563 6973 696f 6e3d 3e35 3229 2020 2020  ecision=>52)    
│ │ │ │ +00025560: 2d2d 2d2d 2b0a 7c69 3520 3a20 6e31 3d76  ----+.|i5 : n1=v
│ │ │ │ +00025570: 616c 7565 424d 3228 2231 2e31 3122 2c4d  alueBM2("1.11",M
│ │ │ │ +00025580: 3250 7265 6369 7369 6f6e 3d3e 3532 2920  2Precision=>52) 
│ │ │ │  00025590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000255a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000255b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000255b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000255c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000255d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000255e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000255f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025600: 207c 0a7c 6f35 203d 2031 2e31 3120 2020   |.|o5 = 1.11   
│ │ │ │ +00025600: 2020 2020 7c0a 7c6f 3520 3d20 312e 3131      |.|o5 = 1.11
│ │ │ │  00025610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025650: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256a0: 207c 0a7c 6f35 203a 2052 5220 286f 6620   |.|o5 : RR (of 
│ │ │ │ -000256b0: 7072 6563 6973 696f 6e20 3532 2920 2020  precision 52)   
│ │ │ │ +000256a0: 2020 2020 7c0a 7c6f 3520 3a20 5252 2028      |.|o5 : RR (
│ │ │ │ +000256b0: 6f66 2070 7265 6369 7369 6f6e 2035 3229  of precision 52)
│ │ │ │  000256c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000256f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00025700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025740: 2d2b 0a7c 6936 203a 206e 323d 7661 6c75  -+.|i6 : n2=valu
│ │ │ │ -00025750: 6542 4d32 2822 312e 3131 222c 4d32 5072  eBM2("1.11",M2Pr
│ │ │ │ -00025760: 6563 6973 696f 6e3d 3e33 3030 2920 2020  ecision=>300)   
│ │ │ │ +00025740: 2d2d 2d2d 2b0a 7c69 3620 3a20 6e32 3d76  ----+.|i6 : n2=v
│ │ │ │ +00025750: 616c 7565 424d 3228 2231 2e31 3122 2c4d  alueBM2("1.11",M
│ │ │ │ +00025760: 3250 7265 6369 7369 6f6e 3d3e 3330 3029  2Precision=>300)
│ │ │ │  00025770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000257a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257e0: 207c 0a7c 6f36 203d 2031 2e31 3120 2020   |.|o6 = 1.11   
│ │ │ │ +000257e0: 2020 2020 7c0a 7c6f 3620 3d20 312e 3131      |.|o6 = 1.11
│ │ │ │  000257f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025880: 207c 0a7c 6f36 203a 2052 5220 286f 6620   |.|o6 : RR (of 
│ │ │ │ -00025890: 7072 6563 6973 696f 6e20 3330 3029 2020  precision 300)  
│ │ │ │ -000258a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025880: 2020 2020 7c0a 7c6f 3620 3a20 5252 2028      |.|o6 : RR (
│ │ │ │ +00025890: 6f66 2070 7265 6369 7369 6f6e 2033 3030  of precision 300
│ │ │ │ +000258a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000258b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000258c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000258d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000258e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000258f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025920: 2d2b 0a7c 6937 203a 2074 6f45 7874 6572  -+.|i7 : toExter
│ │ │ │ -00025930: 6e61 6c53 7472 696e 6720 6e31 2020 2020  nalString n1    
│ │ │ │ +00025920: 2d2d 2d2d 2b0a 7c69 3720 3a20 746f 4578  ----+.|i7 : toEx
│ │ │ │ +00025930: 7465 726e 616c 5374 7269 6e67 206e 3120  ternalString n1 
│ │ │ │  00025940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025970: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025970: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259c0: 207c 0a7c 6f37 203d 202e 3131 3039 3939   |.|o7 = .110999
│ │ │ │ -000259d0: 3939 3939 3939 3939 3939 3970 3532 6531  99999999999p52e1
│ │ │ │ -000259e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000259c0: 2020 2020 7c0a 7c6f 3720 3d20 2e31 3130      |.|o7 = .110
│ │ │ │ +000259d0: 3939 3939 3939 3939 3939 3939 3939 7035  99999999999999p5
│ │ │ │ +000259e0: 3265 3120 2020 2020 2020 2020 2020 2020  2e1             
│ │ │ │  000259f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00025a10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00025a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025a60: 2d2b 0a7c 6938 203a 2074 6f45 7874 6572  -+.|i8 : toExter
│ │ │ │ -00025a70: 6e61 6c53 7472 696e 6720 6e32 2020 2020  nalString n2    
│ │ │ │ +00025a60: 2d2d 2d2d 2b0a 7c69 3820 3a20 746f 4578  ----+.|i8 : toEx
│ │ │ │ +00025a70: 7465 726e 616c 5374 7269 6e67 206e 3220  ternalString n2 
│ │ │ │  00025a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ab0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00025ab0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b00: 207c 0a7c 6f38 203d 202e 3131 3130 3030   |.|o8 = .111000
│ │ │ │ +00025b00: 2020 2020 7c0a 7c6f 3820 3d20 2e31 3131      |.|o8 = .111
│ │ │ │  00025b10: 3030 3030 3030 3030 3030 3030 3030 3030  0000000000000000
│ │ │ │  00025b20: 3030 3030 3030 3030 3030 3030 3030 3030  0000000000000000
│ │ │ │  00025b30: 3030 3030 3030 3030 3030 3030 3030 3030  0000000000000000
│ │ │ │  00025b40: 3030 3030 3030 3030 3030 3030 3030 3030  0000000000000000
│ │ │ │ -00025b50: 307c 0a7c 2020 2020 2030 3030 3030 3030  0|.|     0000000
│ │ │ │ -00025b60: 3030 3030 3030 3030 3030 3030 3033 7033  00000000000003p3
│ │ │ │ -00025b70: 3030 6531 2020 2020 2020 2020 2020 2020  00e1            
│ │ │ │ +00025b50: 3030 3030 7c0a 7c20 2020 2020 3030 3030  0000|.|     0000
│ │ │ │ +00025b60: 3030 3030 3030 3030 3030 3030 3030 3030  0000000000000000
│ │ │ │ +00025b70: 3370 3330 3065 3120 2020 2020 2020 2020  3p300e1         
│ │ │ │  00025b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ba0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00025ba0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00025bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025bf0: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520  -+..Ways to use 
│ │ │ │ -00025c00: 7661 6c75 6542 4d32 3a0a 3d3d 3d3d 3d3d  valueBM2:.======
│ │ │ │ -00025c10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00025c20: 0a20 202a 2022 7661 6c75 6542 4d32 2853  .  * "valueBM2(S
│ │ │ │ -00025c30: 7472 696e 6729 220a 0a46 6f72 2074 6865  tring)"..For the
│ │ │ │ -00025c40: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00025c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00025c60: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00025c70: 2076 616c 7565 424d 323a 2076 616c 7565   valueBM2: value
│ │ │ │ -00025c80: 424d 322c 2069 7320 6120 2a6e 6f74 6520  BM2, is a *note 
│ │ │ │ -00025c90: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -00025ca0: 7769 7468 206f 7074 696f 6e73 3a0a 284d  with options:.(M
│ │ │ │ -00025cb0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00025cc0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ -00025cd0: 7469 6f6e 732c 2e0a 1f0a 4669 6c65 3a20  tions,....File: 
│ │ │ │ -00025ce0: 4265 7274 696e 692e 696e 666f 2c20 4e6f  Bertini.info, No
│ │ │ │ -00025cf0: 6465 3a20 5661 7269 6162 6c65 2067 726f  de: Variable gro
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│ │ │ │ -00025e90: 2043 435b 782c 795d 3b20 2020 2020 2020   CC[x,y];       
│ │ │ │ +00025e80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00025e90: 5220 3d20 4343 5b78 2c79 5d3b 2020 2020  R = CC[x,y];    
│ │ │ │  00025ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00025f10: 6932 203a 2046 3120 3d20 7b78 2a79 2b31  i2 : F1 = {x*y+1
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│ │ │ │ -00025f30: 7d3b 2020 2020 2020 2020 2020 2020 2020  };              
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│ │ │ │ +00025fa0: 6265 7274 696e 695a 6572 6f44 696d 536f  bertiniZeroDimSo
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│ │ │ │ -00026020: 6934 203a 2068 5220 3d43 435b 7830 2c78  i4 : hR =CC[x0,x
│ │ │ │ -00026030: 312c 7930 2c79 315d 2020 2020 2020 2020  1,y0,y1]        
│ │ │ │ +00026010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00026040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026060: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  00026070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00026090: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000260b0: 6852 2020 2020 2020 2020 2020 2020 2020  hR              
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│ │ │ │  000260d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000260e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00026100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00026120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00026130: 7c0a 7c6f 3420 3a20 506f 6c79 6e6f 6d69  |.|o4 : Polynomi
│ │ │ │ +00026140: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
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│ │ │ │ -000261c0: 3d20 7b78 312a 7931 2b78 302a 7930 2c32  = {x1*y1+x0*y0,2
│ │ │ │ -000261d0: 2a78 312a 7931 2b33 2a78 312a 7930 2b34  *x1*y1+3*x1*y0+4
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│ │ │ │ +000261c0: 4632 203d 207b 7831 2a79 312b 7830 2a79  F2 = {x1*y1+x0*y
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│ │ │ │ -00026240: 6936 203a 2062 6572 7469 6e69 5a65 726f  i6 : bertiniZero
│ │ │ │ -00026250: 4469 6d53 6f6c 7665 2846 322c 486f 6d56  DimSolve(F2,HomV
│ │ │ │ -00026260: 6172 6961 626c 6547 726f 7570 3d3e 7b7b  ariableGroup=>{{
│ │ │ │ -00026270: 7830 2c78 317d 2c7b 7930 2c79 317d 7d29  x0,x1},{y0,y1}})
│ │ │ │ -00026280: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ;|.+------------
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│ │ │ │ +00026240: 2b0a 7c69 3620 3a20 6265 7274 696e 695a  +.|i6 : bertiniZ
│ │ │ │ +00026250: 6572 6f44 696d 536f 6c76 6528 4632 2c48  eroDimSolve(F2,H
│ │ │ │ +00026260: 6f6d 5661 7269 6162 6c65 4772 6f75 703d  omVariableGroup=
│ │ │ │ +00026270: 3e7b 7b78 302c 7831 7d2c 7b79 302c 7931  >{{x0,x1},{y0,y1
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│ │ │ │  000262b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00026a60: 7574 2063 6f6e 6669 6775 7261 7469 6f6e  ut configuration
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│ │ │ │ +00026b30: 696e 6553 6f6c 737f 3533 3736 300a 4e6f  ineSols.53760.No
│ │ │ │ +00026b40: 6465 3a20 6265 7274 696e 6953 616d 706c  de: bertiniSampl
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│ │ │ │ +00026b60: 7274 696e 6954 7261 636b 486f 6d6f 746f  rtiniTrackHomoto
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│ │ │ │ +00026b80: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ +00026b90: 6f70 795f 6c70 5f70 645f 7064 5f70 645f  opy_lp_pd_pd_pd_
│ │ │ │ +00026ba0: 636d 5665 7262 6f73 653d 3e5f 7064 5f70  cmVerbose=>_pd_p
│ │ │ │ +00026bb0: 645f 7064 5f72 707f 3731 3137 370a 4e6f  d_pd_rp.71177.No
│ │ │ │ +00026bc0: 6465 3a20 6265 7274 696e 6955 7365 7248  de: bertiniUserH
│ │ │ │ +00026bd0: 6f6d 6f74 6f70 797f 3734 3933 310a 4e6f  omotopy.74931.No
│ │ │ │ +00026be0: 6465 3a20 6265 7274 696e 695a 6572 6f44  de: bertiniZeroD
│ │ │ │ +00026bf0: 696d 536f 6c76 657f 3832 3133 380a 4e6f  imSolve.82138.No
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│ │ │ │ +00026c10: 3738 3739 0a4e 6f64 653a 2069 6d70 6f72  7879.Node: impor
│ │ │ │ +00026c20: 7449 6e63 6964 656e 6365 4d61 7472 6978  tIncidenceMatrix
│ │ │ │ +00026c30: 7f38 3835 3533 0a4e 6f64 653a 2069 6d70  .88553.Node: imp
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│ │ │ │ +00026c50: 3932 3131 330a 4e6f 6465 3a20 696d 706f  92113.Node: impo
│ │ │ │ +00026c60: 7274 5061 7261 6d65 7465 7246 696c 657f  rtParameterFile.
│ │ │ │ +00026c70: 3937 3430 330a 4e6f 6465 3a20 696d 706f  97403.Node: impo
│ │ │ │ +00026c80: 7274 536f 6c75 7469 6f6e 7346 696c 657f  rtSolutionsFile.
│ │ │ │ +00026c90: 3939 3530 310a 4e6f 6465 3a20 4973 5072  99501.Node: IsPr
│ │ │ │ +00026ca0: 6f6a 6563 7469 7665 7f31 3034 3434 380a  ojective.104448.
│ │ │ │ +00026cb0: 4e6f 6465 3a20 4d61 696e 4461 7461 4469  Node: MainDataDi
│ │ │ │ +00026cc0: 7265 6374 6f72 797f 3130 3538 3030 0a4e  rectory.105800.N
│ │ │ │ +00026cd0: 6f64 653a 206d 616b 6542 2749 6e70 7574  ode: makeB'Input
│ │ │ │ +00026ce0: 4669 6c65 7f31 3036 3037 300a 4e6f 6465  File.106070.Node
│ │ │ │ +00026cf0: 3a20 6d61 6b65 4227 5365 6374 696f 6e7f  : makeB'Section.
│ │ │ │ +00026d00: 3131 3231 3131 0a4e 6f64 653a 206d 616b  112111.Node: mak
│ │ │ │ +00026d10: 6542 2753 6c69 6365 7f31 3234 3139 360a  eB'Slice.124196.
│ │ │ │ +00026d20: 4e6f 6465 3a20 6d6f 7665 4227 4669 6c65  Node: moveB'File
│ │ │ │ +00026d30: 7f31 3333 3737 310a 4e6f 6465 3a20 4e75  .133771.Node: Nu
│ │ │ │ +00026d40: 6d62 6572 546f 4227 5374 7269 6e67 7f31  mberToB'String.1
│ │ │ │ +00026d50: 3339 3332 390a 4e6f 6465 3a20 5061 7468  39329.Node: Path
│ │ │ │ +00026d60: 4c69 7374 7f31 3431 3331 390a 4e6f 6465  List.141319.Node
│ │ │ │ +00026d70: 3a20 7261 6469 6361 6c4c 6973 747f 3134  : radicalList.14
│ │ │ │ +00026d80: 3137 3139 0a4e 6f64 653a 2073 746f 7265  1719.Node: store
│ │ │ │ +00026d90: 424d 3246 696c 6573 7f31 3433 3435 340a  BM2Files.143454.
│ │ │ │ +00026da0: 4e6f 6465 3a20 7375 6250 6f69 6e74 7f31  Node: subPoint.1
│ │ │ │ +00026db0: 3433 3730 300a 4e6f 6465 3a20 546f 7044  43700.Node: TopD
│ │ │ │ +00026dc0: 6972 6563 746f 7279 7f31 3438 3431 300a  irectory.148410.
│ │ │ │ +00026dd0: 4e6f 6465 3a20 5573 6552 6567 656e 6572  Node: UseRegener
│ │ │ │ +00026de0: 6174 696f 6e7f 3134 3932 3939 0a4e 6f64  ation.149299.Nod
│ │ │ │ +00026df0: 653a 2076 616c 7565 424d 327f 3135 3031  e: valueBM2.1501
│ │ │ │ +00026e00: 3538 0a4e 6f64 653a 2056 6172 6961 626c  58.Node: Variabl
│ │ │ │ +00026e10: 6520 6772 6f75 7073 7f31 3534 3834 330a  e groups.154843.
│ │ │ │ +00026e20: 4e6f 6465 3a20 7772 6974 6553 7461 7274  Node: writeStart
│ │ │ │ +00026e30: 4669 6c65 7f31 3536 3336 320a 1f0a 456e  File.156362...En
│ │ │ │ +00026e40: 6420 5461 6720 5461 626c 650a            d Tag Table.
│ │ ├── ./usr/share/info/BettiCharacters.info.gz
│ │ │ ├── BettiCharacters.info
│ │ │ │ @@ -12411,15 +12411,15 @@
│ │ │ │  000307a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000307b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000307c0: 6939 203a 2065 6c61 7073 6564 5469 6d65  i9 : elapsedTime
│ │ │ │  000307d0: 2063 203d 2063 6861 7261 6374 6572 2041   c = character A
│ │ │ │  000307e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030800: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00030810: 202d 2d20 2e34 3434 3233 3273 2065 6c61   -- .444232s ela
│ │ │ │ +00030810: 202d 2d20 312e 3037 3431 3573 2065 6c61   -- 1.07415s ela
│ │ │ │  00030820: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -13607,15 +13607,15 @@
│ │ │ │  00035260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035280: 2d2b 0a7c 6937 203a 2065 6c61 7073 6564  -+.|i7 : elapsed
│ │ │ │  00035290: 5469 6d65 2063 3d63 6861 7261 6374 6572  Time c=character
│ │ │ │  000352a0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  000352b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000352d0: 207c 0a7c 202d 2d20 2e35 3132 3534 3773   |.| -- .512547s
│ │ │ │ +000352d0: 207c 0a7c 202d 2d20 2e38 3734 3533 3673   |.| -- .874536s
│ │ │ │  000352e0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  000352f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035320: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00035330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -15025,16 +15025,16 @@
│ │ │ │  0003ab00: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0003ab10: 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  --------------+.
│ │ │ │  0003ab40: 7c69 3230 203a 2065 6c61 7073 6564 5469  |i20 : elapsedTi
│ │ │ │  0003ab50: 6d65 2061 3120 3d20 6368 6172 6163 7465  me a1 = characte
│ │ │ │  0003ab60: 7220 4131 2020 2020 2020 2020 2020 2020  r A1            
│ │ │ │ -0003ab70: 2020 207c 0a7c 202d 2d20 2e38 3535 3735     |.| -- .85575
│ │ │ │ -0003ab80: 3373 2065 6c61 7073 6564 2020 2020 2020  3s elapsed      
│ │ │ │ +0003ab70: 2020 207c 0a7c 202d 2d20 312e 3235 3936     |.| -- 1.2596
│ │ │ │ +0003ab80: 3173 2065 6c61 7073 6564 2020 2020 2020  1s elapsed      
│ │ │ │  0003ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0003abb0: 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               |.|
│ │ │ │  0003abe0: 6f32 3020 3d20 4368 6172 6163 7465 7220  o20 = Character 
│ │ │ │  0003abf0: 6f76 6572 2052 2020 2020 2020 2020 2020  over R          
│ │ │ │ @@ -15065,15 +15065,15 @@
│ │ │ │  0003ad80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0003ad90: 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  ----------+.|i21
│ │ │ │  0003adc0: 203a 2065 6c61 7073 6564 5469 6d65 2061   : elapsedTime a
│ │ │ │  0003add0: 3220 3d20 6368 6172 6163 7465 7220 4132  2 = character A2
│ │ │ │  0003ade0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003adf0: 0a7c 202d 2d20 3337 2e38 3630 3173 2065  .| -- 37.8601s e
│ │ │ │ +0003adf0: 0a7c 202d 2d20 3538 2e31 3132 3373 2065  .| -- 58.1123s e
│ │ │ │  0003ae00: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0003ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0003ae30: 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 
│ │ │ │  0003ae60: 3d20 4368 6172 6163 7465 7220 6f76 6572  = Character over
│ │ │ │ @@ -15523,15 +15523,15 @@
│ │ │ │  0003ca20: 203a 2041 6374 696f 6e4f 6e47 7261 6465   : ActionOnGrade
│ │ │ │  0003ca30: 644d 6f64 756c 6520 2020 2020 2020 2020  dModule         
│ │ │ │  0003ca40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0003ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3332  ----------+.|i32
│ │ │ │  0003ca70: 203a 2065 6c61 7073 6564 5469 6d65 2062   : elapsedTime b
│ │ │ │  0003ca80: 203d 2063 6861 7261 6374 6572 2842 2c32   = character(B,2
│ │ │ │ -0003ca90: 3129 7c0a 7c20 2d2d 2031 372e 3233 3337  1)|.| -- 17.2337
│ │ │ │ +0003ca90: 3129 7c0a 7c20 2d2d 2032 332e 3039 3639  1)|.| -- 23.0969
│ │ │ │  0003caa0: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0003cab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0003cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cae0: 2020 7c0a 7c6f 3332 203d 2043 6861 7261    |.|o32 = Chara
│ │ │ │  0003caf0: 6374 6572 206f 7665 7220 5220 2020 2020  cter over R     
│ │ │ │  0003cb00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|
│ │ ├── ./usr/share/info/Bruns.info.gz
│ │ │ ├── Bruns.info
│ │ │ │ @@ -1019,17 +1019,17 @@
│ │ │ │  00003fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00003fc0: 6932 3320 3a20 7469 6d65 206a 3d62 7275  i23 : time j=bru
│ │ │ │  00003fd0: 6e73 2046 2e64 645f 333b 2020 2020 2020  ns F.dd_3;      
│ │ │ │  00003fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00004010: 202d 2d20 7573 6564 2030 2e31 3430 3634   -- used 0.14064
│ │ │ │ -00004020: 3473 2028 6370 7529 3b20 302e 3133 3931  4s (cpu); 0.1391
│ │ │ │ -00004030: 3938 7320 2874 6872 6561 6429 3b20 3073  98s (thread); 0s
│ │ │ │ +00004010: 202d 2d20 7573 6564 2030 2e33 3038 3030   -- used 0.30800
│ │ │ │ +00004020: 3273 2028 6370 7529 3b20 302e 3331 3039  2s (cpu); 0.3109
│ │ │ │ +00004030: 3136 7320 2874 6872 6561 6429 3b20 3073  16s (thread); 0s
│ │ │ │  00004040: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00004050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000040a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ ├── ./usr/share/info/CellularResolutions.info.gz
│ │ │ ├── CellularResolutions.info
│ │ │ │ @@ -2149,63 +2149,63 @@
│ │ │ │  00008640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008650: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00008660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008690: 207c 0a7c 6f38 203d 2052 656c 6174 696f   |.|o8 = Relatio
│ │ │ │  000086a0: 6e20 4d61 7472 6978 3a20 7c20 3120 3020  n Matrix: | 1 0 
│ │ │ │ -000086b0: 3020 3020 3020 3120 3020 3120 3020 3020  0 0 0 1 0 1 0 0 
│ │ │ │ -000086c0: 3120 3020 3020 3020 3020 3120 3120 7c7c  1 0 0 0 0 1 1 ||
│ │ │ │ +000086b0: 3020 3020 3020 3120 3020 3020 3120 3020  0 0 0 1 0 0 1 0 
│ │ │ │ +000086c0: 3120 3020 3020 3020 3120 3020 3120 7c7c  1 0 0 0 1 0 1 ||
│ │ │ │  000086d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000086e0: 2020 2020 2020 2020 7c20 3020 3120 3020          | 0 1 0 
│ │ │ │ -000086f0: 3020 3020 3020 3120 3020 3120 3120 3120  0 0 0 1 0 1 1 1 
│ │ │ │ -00008700: 3020 3020 3120 3120 3120 3120 7c7c 0a7c  0 0 1 1 1 1 ||.|
│ │ │ │ +000086f0: 3020 3020 3020 3120 3020 3120 3020 3020  0 0 0 1 0 1 0 0 
│ │ │ │ +00008700: 3120 3020 3020 3020 3120 3120 7c7c 0a7c  1 0 0 0 1 1 ||.|
│ │ │ │  00008710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008720: 2020 2020 2020 7c20 3020 3020 3120 3020        | 0 0 1 0 
│ │ │ │ -00008730: 3020 3020 3120 3020 3020 3020 3020 3120  0 0 1 0 0 0 0 1 
│ │ │ │ -00008740: 3120 3120 3120 3020 3020 7c7c 0a7c 2020  1 1 1 0 0 ||.|  
│ │ │ │ +00008730: 3020 3020 3020 3120 3020 3120 3120 3120  0 0 0 1 0 1 1 1 
│ │ │ │ +00008740: 3020 3120 3120 3120 3120 7c7c 0a7c 2020  0 1 1 1 1 ||.|  
│ │ │ │  00008750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008760: 2020 2020 7c20 3020 3020 3020 3120 3020      | 0 0 0 1 0 
│ │ │ │ -00008770: 3120 3020 3020 3120 3020 3020 3120 3020  1 0 0 1 0 0 1 0 
│ │ │ │ -00008780: 3120 3020 3120 3020 7c7c 0a7c 2020 2020  1 0 1 0 ||.|    
│ │ │ │ +00008770: 3120 3020 3120 3020 3020 3020 3020 3120  1 0 1 0 0 0 0 1 
│ │ │ │ +00008780: 3120 3120 3020 3020 7c7c 0a7c 2020 2020  1 1 0 0 ||.|    
│ │ │ │  00008790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000087a0: 2020 7c20 3020 3020 3020 3020 3120 3020    | 0 0 0 0 1 0 
│ │ │ │ -000087b0: 3020 3120 3020 3120 3020 3020 3120 3020  0 1 0 1 0 0 1 0 
│ │ │ │ -000087c0: 3120 3020 3120 7c7c 0a7c 2020 2020 2020  1 0 1 ||.|      
│ │ │ │ +000087b0: 3120 3020 3020 3120 3020 3020 3120 3120  1 0 0 1 0 0 1 1 
│ │ │ │ +000087c0: 3020 3120 3020 7c7c 0a7c 2020 2020 2020  0 1 0 ||.|      
│ │ │ │  000087d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000087e0: 7c20 3020 3020 3020 3020 3020 3120 3020  | 0 0 0 0 0 1 0 
│ │ │ │ -000087f0: 3020 3020 3020 3020 3020 3020 3020 3020  0 0 0 0 0 0 0 0 
│ │ │ │ -00008800: 3120 3020 7c7c 0a7c 2020 2020 2020 2020  1 0 ||.|        
│ │ │ │ +000087f0: 3020 3020 3020 3020 3020 3020 3020 3120  0 0 0 0 0 0 0 1 
│ │ │ │ +00008800: 3020 3020 7c7c 0a7c 2020 2020 2020 2020  0 0 ||.|        
│ │ │ │  00008810: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  00008820: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -00008830: 3020 3020 3020 3020 3020 3120 3120 3020  0 0 0 0 0 1 1 0 
│ │ │ │ +00008830: 3020 3020 3020 3020 3020 3020 3020 3120  0 0 0 0 0 0 0 1 
│ │ │ │  00008840: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │  00008850: 2020 2020 2020 2020 2020 2020 7c20 3020              | 0 
│ │ │ │  00008860: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -00008870: 3020 3020 3020 3020 3020 3020 3020 3120  0 0 0 0 0 0 0 1 
│ │ │ │ +00008870: 3020 3020 3020 3020 3120 3120 3020 3020  0 0 0 0 1 1 0 0 
│ │ │ │  00008880: 7c7c 0a7c 2020 2020 2020 2020 2020 2020  ||.|            
│ │ │ │  00008890: 2020 2020 2020 2020 2020 7c20 3020 3020            | 0 0 
│ │ │ │  000088a0: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -000088b0: 3020 3020 3020 3120 3020 3120 3020 7c7c  0 0 0 1 0 1 0 ||
│ │ │ │ +000088b0: 3020 3020 3020 3020 3020 3020 3120 7c7c  0 0 0 0 0 0 1 ||
│ │ │ │  000088c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000088d0: 2020 2020 2020 2020 7c20 3020 3020 3020          | 0 0 0 
│ │ │ │  000088e0: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -000088f0: 3020 3020 3020 3120 3020 3120 7c7c 0a7c  0 0 0 1 0 1 ||.|
│ │ │ │ +000088f0: 3020 3020 3120 3020 3120 3020 7c7c 0a7c  0 0 1 0 1 0 ||.|
│ │ │ │  00008900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008910: 2020 2020 2020 7c20 3020 3020 3020 3020        | 0 0 0 0 
│ │ │ │  00008920: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -00008930: 3020 3020 3020 3120 3120 7c7c 0a7c 2020  0 0 0 1 1 ||.|  
│ │ │ │ +00008930: 3020 3020 3120 3020 3120 7c7c 0a7c 2020  0 0 1 0 1 ||.|  
│ │ │ │  00008940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008950: 2020 2020 7c20 3020 3020 3020 3020 3020      | 0 0 0 0 0 
│ │ │ │  00008960: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -00008970: 3120 3020 3020 3020 7c7c 0a7c 2020 2020  1 0 0 0 ||.|    
│ │ │ │ +00008970: 3020 3020 3120 3120 7c7c 0a7c 2020 2020  0 0 1 1 ||.|    
│ │ │ │  00008980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008990: 2020 7c20 3020 3020 3020 3020 3020 3020    | 0 0 0 0 0 0 
│ │ │ │ -000089a0: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ -000089b0: 3120 3020 3020 7c7c 0a7c 2020 2020 2020  1 0 0 ||.|      
│ │ │ │ +000089a0: 3020 3020 3020 3020 3020 3020 3120 3120  0 0 0 0 0 0 1 1 
│ │ │ │ +000089b0: 3020 3020 3020 7c7c 0a7c 2020 2020 2020  0 0 0 ||.|      
│ │ │ │  000089c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000089d0: 7c20 3020 3020 3020 3020 3020 3020 3020  | 0 0 0 0 0 0 0 
│ │ │ │  000089e0: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │  000089f0: 3020 3020 7c7c 0a7c 2020 2020 2020 2020  0 0 ||.|        
│ │ │ │  00008a00: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  00008a10: 3020 3020 3020 3020 3020 3020 3020 3020  0 0 0 0 0 0 0 0 
│ │ │ │  00008a20: 3020 3020 3020 3020 3020 3020 3120 3020  0 0 0 0 0 0 1 0 
│ │ │ │ @@ -2559,17 +2559,17 @@
│ │ │ │  00009fe0: 6443 292f 6365 6c6c 4c61 6265 6c20 2020  dC)/cellLabel   
│ │ │ │  00009ff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0000a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0000a040: 7c6f 3130 203d 207b 7b78 2a77 2c20 782a  |o10 = {{x*w, x*
│ │ │ │ -0000a050: 792c 2079 2a7a 2c20 792a 7a7d 2c20 7b79  y, y*z, y*z}, {y
│ │ │ │ -0000a060: 2a7a 2c20 782a 792a 7a2c 2078 2a79 2a7a  *z, x*y*z, x*y*z
│ │ │ │ -0000a070: 2a77 2c20 782a 792a 777d 2c20 7b78 2a79  *w, x*y*w}, {x*y
│ │ │ │ +0000a050: 792c 2079 2a7a 2c20 792a 7a7d 2c20 7b78  y, y*z, y*z}, {x
│ │ │ │ +0000a060: 2a79 2a77 2c20 792a 7a2c 2078 2a79 2a7a  *y*w, y*z, x*y*z
│ │ │ │ +0000a070: 2c20 782a 792a 7a2a 777d 2c20 7b78 2a79  , x*y*z*w}, {x*y
│ │ │ │  0000a080: 2a7a 2a77 7d7d 7c0a 7c20 2020 2020 2020  *z*w}}|.|       
│ │ │ │  0000a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a0c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0000a0d0: 7c6f 3130 203a 204c 6973 7420 2020 2020  |o10 : List     
│ │ │ │  0000a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2790,33 +2790,33 @@
│ │ │ │  0000ae50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000aea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0000aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aec0: 2020 2033 2032 2020 2032 2033 2020 2020     3 2   2 3    
│ │ │ │ -0000aed0: 2034 2020 2035 2020 2035 2020 2034 2020   4   5   5   4  
│ │ │ │ +0000aec0: 2020 2035 2020 2034 2020 2020 3320 3220     5   4    3 2 
│ │ │ │ +0000aed0: 2020 3220 3320 2020 2020 3420 2020 3520    2 3     4   5 
│ │ │ │  0000aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aef0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │  0000af00: 3d20 4861 7368 5461 626c 657b 3020 3d3e  = HashTable{0 =>
│ │ │ │ -0000af10: 207b 7820 7920 2c20 7820 7920 2c20 782a   {x y , x y , x*
│ │ │ │ -0000af20: 7920 2c20 7820 2c20 7820 2c20 7820 797d  y , x , x , x y}
│ │ │ │ +0000af10: 207b 7820 2c20 7820 792c 2078 2079 202c   {x , x y, x y ,
│ │ │ │ +0000af20: 2078 2079 202c 2078 2a79 202c 2078 207d   x y , x*y , x }
│ │ │ │  0000af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000af40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0000af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000af60: 2020 2035 2034 2020 2035 2020 2020 3520     5 4   5    5 
│ │ │ │ -0000af70: 3220 2020 3520 3320 2020 3520 3420 2020  2   5 3   5 4   
│ │ │ │ -0000af80: 3420 3220 2020 3420 3420 2020 3520 2020  4 2   4 4   5   
│ │ │ │ -0000af90: 2033 2033 2020 2020 2020 7c0a 7c20 2020   3 3      |.|   
│ │ │ │ +0000af60: 2020 2035 2032 2020 2035 2033 2020 2035     5 2   5 3   5
│ │ │ │ +0000af70: 2034 2020 2034 2032 2020 2034 2034 2020   4   4 2   4 4  
│ │ │ │ +0000af80: 2035 2020 2020 3320 3320 2020 3520 3220   5    3 3   5 2 
│ │ │ │ +0000af90: 2020 3220 3420 2020 2020 7c0a 7c20 2020    2 4     |.|   
│ │ │ │  0000afa0: 2020 2020 2020 2020 2020 2020 3120 3d3e              1 =>
│ │ │ │ -0000afb0: 207b 7820 7920 2c20 7820 792c 2078 2079   {x y , x y, x y
│ │ │ │ -0000afc0: 202c 2078 2079 202c 2078 2079 202c 2078   , x y , x y , x
│ │ │ │ -0000afd0: 2079 202c 2078 2079 202c 2078 2079 2c20   y , x y , x y, 
│ │ │ │ -0000afe0: 7820 7920 2c20 2020 2020 7c0a 7c20 2020  x y ,     |.|   
│ │ │ │ +0000afb0: 207b 7820 7920 2c20 7820 7920 2c20 7820   {x y , x y , x 
│ │ │ │ +0000afc0: 7920 2c20 7820 7920 2c20 7820 7920 2c20  y , x y , x y , 
│ │ │ │ +0000afd0: 7820 792c 2078 2079 202c 2078 2079 202c  x y, x y , x y ,
│ │ │ │ +0000afe0: 2078 2079 202c 2020 2020 7c0a 7c20 2020   x y ,    |.|   
│ │ │ │  0000aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b000: 2020 2035 2032 2020 2035 2034 2020 2035     5 2   5 4   5
│ │ │ │  0000b010: 2033 2020 2035 2034 2020 2035 2032 2020   3   5 4   5 2  
│ │ │ │  0000b020: 2035 2034 2020 2035 2033 2020 2035 2034   5 4   5 3   5 4
│ │ │ │  0000b030: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0000b040: 2020 2020 2020 2020 2020 2020 3220 3d3e              2 =>
│ │ │ │  0000b050: 207b 7820 7920 2c20 7820 7920 2c20 7820   {x y , x y , x 
│ │ │ │ @@ -2834,25 +2834,25 @@
│ │ │ │  0000b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b120: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  0000b130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b170: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ -0000b180: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +0000b180: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │  0000b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1c0: 2020 2020 2020 2020 2020 7c0a 7c20 3520            |.| 5 
│ │ │ │ -0000b1d0: 3220 2020 3220 3420 2020 3520 3320 2020  2   2 4   5 3   
│ │ │ │ +0000b1d0: 3320 2020 3520 3420 2020 3520 2020 2020  3   5 4   5     
│ │ │ │  0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b210: 2020 2020 2020 2020 2020 7c0a 7c78 2079            |.|x y
│ │ │ │ -0000b220: 202c 2078 2079 202c 2078 2079 207d 2020   , x y , x y }  
│ │ │ │ +0000b220: 202c 2078 2079 202c 2078 2079 7d20 2020   , x y , x y}   
│ │ │ │  0000b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b260: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0000b270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -4936,25 +4936,25 @@
│ │ │ │  00013470: 3220 3a20 6661 6365 506f 7365 7420 4320  2 : facePoset C 
│ │ │ │  00013480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013490: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000134a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134c0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │  000134d0: 3d20 5265 6c61 7469 6f6e 204d 6174 7269  = Relation Matri
│ │ │ │ -000134e0: 783a 207c 2031 2030 2030 2030 2030 2031  x: | 1 0 0 0 0 1
│ │ │ │ +000134e0: 783a 207c 2031 2030 2030 2030 2031 2030  x: | 1 0 0 0 1 0
│ │ │ │  000134f0: 2031 2030 2031 207c 7c0a 7c20 2020 2020   1 0 1 ||.|     
│ │ │ │  00013500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013510: 2020 7c20 3020 3120 3020 3020 3120 3020    | 0 1 0 0 1 0 
│ │ │ │ -00013520: 3120 3020 3120 7c7c 0a7c 2020 2020 2020  1 0 1 ||.|      
│ │ │ │ +00013510: 2020 7c20 3020 3120 3020 3020 3020 3020    | 0 1 0 0 0 0 
│ │ │ │ +00013520: 3120 3120 3120 7c7c 0a7c 2020 2020 2020  1 1 1 ||.|      
│ │ │ │  00013530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013540: 207c 2030 2030 2031 2030 2031 2030 2030   | 0 0 1 0 1 0 0
│ │ │ │ +00013540: 207c 2030 2030 2031 2030 2030 2031 2030   | 0 0 1 0 0 1 0
│ │ │ │  00013550: 2031 2031 207c 7c0a 7c20 2020 2020 2020   1 1 ||.|       
│ │ │ │  00013560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013570: 7c20 3020 3020 3020 3120 3020 3120 3020  | 0 0 0 1 0 1 0 
│ │ │ │ -00013580: 3120 3120 7c7c 0a7c 2020 2020 2020 2020  1 1 ||.|        
│ │ │ │ +00013570: 7c20 3020 3020 3020 3120 3120 3120 3020  | 0 0 0 1 1 1 0 
│ │ │ │ +00013580: 3020 3120 7c7c 0a7c 2020 2020 2020 2020  0 1 ||.|        
│ │ │ │  00013590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000135a0: 2030 2030 2030 2030 2031 2030 2030 2030   0 0 0 0 1 0 0 0
│ │ │ │  000135b0: 2031 207c 7c0a 7c20 2020 2020 2020 2020   1 ||.|         
│ │ │ │  000135c0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  000135d0: 3020 3020 3020 3020 3020 3120 3020 3020  0 0 0 0 0 1 0 0 
│ │ │ │  000135e0: 3120 7c7c 0a7c 2020 2020 2020 2020 2020  1 ||.|          
│ │ │ │  000135f0: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ ├── ./usr/share/info/ChainComplexExtras.info.gz
│ │ │ ├── ChainComplexExtras.info
│ │ │ │ @@ -4516,17 +4516,17 @@
│ │ │ │  00011a30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00011a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011a60: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │  00011a70: 7469 6d65 206d 203d 206d 696e 696d 697a  time m = minimiz
│ │ │ │  00011a80: 6520 2845 5b31 5d29 3b20 2020 2020 2020  e (E[1]);       
│ │ │ │  00011a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00011aa0: 0a7c 202d 2d20 7573 6564 2030 2e32 3831  .| -- used 0.281
│ │ │ │ -00011ab0: 3335 3173 2028 6370 7529 3b20 302e 3230  351s (cpu); 0.20
│ │ │ │ -00011ac0: 3431 3937 7320 2874 6872 6561 6429 3b20  4197s (thread); 
│ │ │ │ +00011aa0: 0a7c 202d 2d20 7573 6564 2030 2e33 3330  .| -- used 0.330
│ │ │ │ +00011ab0: 3032 3873 2028 6370 7529 3b20 302e 3237  028s (cpu); 0.27
│ │ │ │ +00011ac0: 3530 3034 7320 2874 6872 6561 6429 3b20  5004s (thread); 
│ │ │ │  00011ad0: 3073 2028 6763 297c 0a2b 2d2d 2d2d 2d2d  0s (gc)|.+------
│ │ │ │  00011ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00011b10: 0a7c 6931 3420 3a20 6973 5175 6173 6949  .|i14 : isQuasiI
│ │ │ │  00011b20: 736f 6d6f 7270 6869 736d 206d 2020 2020  somorphism m    
│ │ │ │  00011b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6196,31 +6196,31 @@
│ │ │ │  00018330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018340: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 7469  ------+.|i8 : ti
│ │ │ │  00018350: 6d65 206d 203d 2072 6573 6f6c 7574 696f  me m = resolutio
│ │ │ │  00018360: 6e4f 6643 6861 696e 436f 6d70 6c65 7820  nOfChainComplex 
│ │ │ │  00018370: 433b 2020 2020 2020 2020 2020 2020 2020  C;              
│ │ │ │  00018380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018390: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000183a0: 6420 302e 3039 3632 3538 7320 2863 7075  d 0.096258s (cpu
│ │ │ │ -000183b0: 293b 2030 2e30 3935 3834 3632 7320 2874  ); 0.0958462s (t
│ │ │ │ -000183c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +000183a0: 6420 302e 3132 3430 3538 7320 2863 7075  d 0.124058s (cpu
│ │ │ │ +000183b0: 293b 2030 2e31 3233 3037 3773 2028 7468  ); 0.123077s (th
│ │ │ │ +000183c0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000183d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000183f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018430: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7469  ------+.|i9 : ti
│ │ │ │  00018440: 6d65 206e 203d 2063 6172 7461 6e45 696c  me n = cartanEil
│ │ │ │  00018450: 656e 6265 7267 5265 736f 6c75 7469 6f6e  enbergResolution
│ │ │ │  00018460: 2043 3b20 2020 2020 2020 2020 2020 2020   C;             
│ │ │ │  00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018480: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00018490: 6420 302e 3230 3231 3136 7320 2863 7075  d 0.202116s (cpu
│ │ │ │ -000184a0: 293b 2030 2e31 3439 3937 3873 2028 7468  ); 0.149978s (th
│ │ │ │ +00018490: 6420 302e 3330 3436 3139 7320 2863 7075  d 0.304619s (cpu
│ │ │ │ +000184a0: 293b 2030 2e32 3534 3338 3273 2028 7468  ); 0.254382s (th
│ │ │ │  000184b0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000184c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000184d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000184f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/CharacteristicClasses.info.gz
│ │ │ ├── CharacteristicClasses.info
│ │ │ │ @@ -1066,16 +1066,16 @@
│ │ │ │  00004290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000042a0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │  000042b0: 6d65 2043 534d 2055 2020 2020 2020 2020  me CSM U        
│ │ │ │  000042c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000042f0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00004300: 6420 302e 3236 3137 3033 7320 2863 7075  d 0.261703s (cpu
│ │ │ │ -00004310: 293b 2030 2e31 3139 3731 3373 2028 7468  ); 0.119713s (th
│ │ │ │ +00004300: 6420 302e 3535 3632 3336 7320 2863 7075  d 0.556236s (cpu
│ │ │ │ +00004310: 293b 2030 2e31 3338 3134 3373 2028 7468  ); 0.138143s (th
│ │ │ │  00004320: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00004330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004340: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00004350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1151,16 +1151,16 @@
│ │ │ │  000047e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000047f0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7469  ------+.|i4 : ti
│ │ │ │  00004800: 6d65 2043 534d 2855 2c43 6865 636b 536d  me CSM(U,CheckSm
│ │ │ │  00004810: 6f6f 7468 3d3e 6661 6c73 6529 2020 2020  ooth=>false)    
│ │ │ │  00004820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004840: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00004850: 6420 302e 3534 3434 3939 7320 2863 7075  d 0.544499s (cpu
│ │ │ │ -00004860: 293b 2030 2e32 3635 3936 3173 2028 7468  ); 0.265961s (th
│ │ │ │ +00004850: 6420 302e 3637 3038 3131 7320 2863 7075  d 0.670811s (cpu
│ │ │ │ +00004860: 293b 2030 2e34 3038 3230 3473 2028 7468  ); 0.408204s (th
│ │ │ │  00004870: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00004880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004890: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000048a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000048b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000048c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000048d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4113,16 +4113,16 @@
│ │ │ │  00010100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010120: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │  00010130: 7469 6d65 2043 534d 2849 2c43 6f6d 704d  time CSM(I,CompM
│ │ │ │  00010140: 6574 686f 643d 3e50 726f 6a65 6374 6976  ethod=>Projectiv
│ │ │ │  00010150: 6544 6567 7265 6529 2020 2020 2020 2020  eDegree)        
│ │ │ │  00010160: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00010170: 2d20 7573 6564 2031 2e32 3739 3634 7320  - used 1.27964s 
│ │ │ │ -00010180: 2863 7075 293b 2030 2e33 3535 3036 3573  (cpu); 0.355065s
│ │ │ │ +00010170: 2d20 7573 6564 2032 2e30 3833 3033 7320  - used 2.08303s 
│ │ │ │ +00010180: 2863 7075 293b 2030 2e36 3734 3038 3573  (cpu); 0.674085s
│ │ │ │  00010190: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  000101a0: 6329 2020 2020 2020 2020 2020 2020 7c0a  c)            |.
│ │ │ │  000101b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000101c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101f0: 207c 0a7c 2020 2020 2020 2035 2020 2020   |.|       5    
│ │ │ │ @@ -4172,16 +4172,16 @@
│ │ │ │  000104b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000104c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000104d0: 2d2d 2b0a 7c69 3620 3a20 7469 6d65 2043  --+.|i6 : time C
│ │ │ │  000104e0: 534d 2849 2c43 6f6d 704d 6574 686f 643d  SM(I,CompMethod=
│ │ │ │  000104f0: 3e50 6e52 6573 6964 7561 6c29 2020 2020  >PnResidual)    
│ │ │ │  00010500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010510: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00010520: 2032 2e31 3534 3231 7320 2863 7075 293b   2.15421s (cpu);
│ │ │ │ -00010530: 2031 2e37 3333 3534 7320 2874 6872 6561   1.73354s (threa
│ │ │ │ +00010520: 2033 2e37 3237 3937 7320 2863 7075 293b   3.72797s (cpu);
│ │ │ │ +00010530: 2033 2e31 3835 3832 7320 2874 6872 6561   3.18582s (threa
│ │ │ │  00010540: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00010550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00010560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000105a0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ @@ -4260,16 +4260,16 @@
│ │ │ │  00010a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010a50: 2d2b 0a7c 6931 3020 3a20 7469 6d65 2043  -+.|i10 : time C
│ │ │ │  00010a60: 534d 284b 2c43 6f6d 704d 6574 686f 643d  SM(K,CompMethod=
│ │ │ │  00010a70: 3e50 726f 6a65 6374 6976 6544 6567 7265  >ProjectiveDegre
│ │ │ │  00010a80: 6529 2020 2020 2020 2020 2020 2020 2020  e)              
│ │ │ │  00010a90: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00010aa0: 302e 3536 3037 3135 7320 2863 7075 293b  0.560715s (cpu);
│ │ │ │ -00010ab0: 2030 2e32 3633 3530 3773 2028 7468 7265   0.263507s (thre
│ │ │ │ +00010aa0: 302e 3534 3631 3133 7320 2863 7075 293b  0.546113s (cpu);
│ │ │ │ +00010ab0: 2030 2e33 3630 3232 3473 2028 7468 7265   0.360224s (thre
│ │ │ │  00010ac0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00010ad0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00010ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010b10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00010b20: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ @@ -4318,18 +4318,18 @@
│ │ │ │  00010dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  00010e00: 3120 3a20 7469 6d65 2043 534d 284b 2c43  1 : time CSM(K,C
│ │ │ │  00010e10: 6f6d 704d 6574 686f 643d 3e50 6e52 6573  ompMethod=>PnRes
│ │ │ │  00010e20: 6964 7561 6c29 2020 2020 2020 2020 2020  idual)          
│ │ │ │  00010e30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00010e40: 7c20 2d2d 2075 7365 6420 302e 3331 3938  | -- used 0.3198
│ │ │ │ -00010e50: 3831 7320 2863 7075 293b 2030 2e31 3633  81s (cpu); 0.163
│ │ │ │ -00010e60: 3239 3273 2028 7468 7265 6164 293b 2030  292s (thread); 0
│ │ │ │ -00010e70: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00010e40: 7c20 2d2d 2075 7365 6420 302e 3436 3632  | -- used 0.4662
│ │ │ │ +00010e50: 3535 7320 2863 7075 293b 2030 2e30 3934  55s (cpu); 0.094
│ │ │ │ +00010e60: 3535 3734 7320 2874 6872 6561 6429 3b20  5574s (thread); 
│ │ │ │ +00010e70: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00010e80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00010e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010ec0: 2020 2020 7c0a 7c20 2020 2020 2020 2033      |.|        3
│ │ │ │  00010ed0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00010ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5289,16 +5289,16 @@
│ │ │ │  00014a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ab0: 2d2d 2d2d 2b0a 7c69 3135 203a 2074 696d  ----+.|i15 : tim
│ │ │ │  00014ac0: 6520 6373 6d4b 3d43 534d 2841 2c4b 2920  e csmK=CSM(A,K) 
│ │ │ │  00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ae0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00014af0: 2d20 7573 6564 2031 2e35 3139 3632 7320  - used 1.51962s 
│ │ │ │ -00014b00: 2863 7075 293b 2030 2e35 3239 3531 3573  (cpu); 0.529515s
│ │ │ │ +00014af0: 2d20 7573 6564 2034 2e31 3638 3433 7320  - used 4.16843s 
│ │ │ │ +00014b00: 2863 7075 293b 2030 2e38 3537 3037 3573  (cpu); 0.857075s
│ │ │ │  00014b10: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  00014b20: 6329 7c0a 7c20 2020 2020 2020 2020 2020  c)|.|           
│ │ │ │  00014b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00014b60: 2020 2020 3220 3220 2020 2020 3220 2020      2 2     2   
│ │ │ │  00014b70: 2020 2020 2020 3220 2020 2032 2020 2020        2    2    
│ │ │ │ @@ -5467,16 +5467,16 @@
│ │ │ │  000155a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155c0: 2d2d 2d2d 2b0a 7c69 3232 203a 2074 696d  ----+.|i22 : tim
│ │ │ │  000155d0: 6520 4353 4d28 412c 4b2c 6d29 2020 2020  e CSM(A,K,m)    
│ │ │ │  000155e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000155f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015600: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -00015610: 3236 3836 3031 7320 2863 7075 293b 2030  268601s (cpu); 0
│ │ │ │ -00015620: 2e31 3030 3334 3173 2028 7468 7265 6164  .100341s (thread
│ │ │ │ +00015610: 3330 3635 3331 7320 2863 7075 293b 2030  306531s (cpu); 0
│ │ │ │ +00015620: 2e31 3233 3936 3873 2028 7468 7265 6164  .123968s (thread
│ │ │ │  00015630: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00015640: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00015680: 7c20 2020 2020 2020 2032 2032 2020 2020  |        2 2    
│ │ │ │  00015690: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ @@ -6424,18 +6424,18 @@
│ │ │ │  00019170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019180: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │  00019190: 3a20 7469 6d65 2045 756c 6572 2849 2c49  : time Euler(I,I
│ │ │ │  000191a0: 6e70 7574 4973 536d 6f6f 7468 3d3e 7472  nputIsSmooth=>tr
│ │ │ │  000191b0: 7565 2920 2020 2020 2020 2020 2020 2020  ue)             
│ │ │ │  000191c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000191d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -000191e0: 2075 7365 6420 302e 3133 3435 3731 7320   used 0.134571s 
│ │ │ │ -000191f0: 2863 7075 293b 2030 2e30 3630 3534 3731  (cpu); 0.0605471
│ │ │ │ -00019200: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00019210: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +000191e0: 2075 7365 6420 302e 3334 3736 3673 2028   used 0.34766s (
│ │ │ │ +000191f0: 6370 7529 3b20 302e 3135 3130 3131 7320  cpu); 0.151011s 
│ │ │ │ +00019200: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00019210: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00019220: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00019230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019270: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │  00019280: 3d20 3420 2020 2020 2020 2020 2020 2020  = 4             
│ │ │ │ @@ -6449,18 +6449,18 @@
│ │ │ │  00019300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019310: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │  00019320: 3a20 7469 6d65 2045 756c 6572 2049 2020  : time Euler I  
│ │ │ │  00019330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019360: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00019370: 2075 7365 6420 302e 3331 3631 3138 7320   used 0.316118s 
│ │ │ │ -00019380: 2863 7075 293b 2030 2e31 3637 3235 3673  (cpu); 0.167256s
│ │ │ │ -00019390: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -000193a0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00019370: 2075 7365 6420 302e 3433 3837 3873 2028   used 0.43878s (
│ │ │ │ +00019380: 6370 7529 3b20 302e 3138 3136 7320 2874  cpu); 0.1816s (t
│ │ │ │ +00019390: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +000193a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000193b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000193c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000193d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000193e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000193f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019400: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │  00019410: 3d20 3420 2020 2020 2020 2020 2020 2020  = 4             
│ │ │ │ @@ -6653,17 +6653,17 @@
│ │ │ │  00019fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00019fe0: 7c69 3130 203a 2074 696d 6520 4575 6c65  |i10 : time Eule
│ │ │ │  00019ff0: 7228 4a2c 4d65 7468 6f64 3d3e 4469 7265  r(J,Method=>Dire
│ │ │ │  0001a000: 6374 436f 6d70 6c65 7465 496e 7429 2020  ctCompleteInt)  
│ │ │ │  0001a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a020: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -0001a030: 3330 3937 3873 2028 6370 7529 3b20 302e  30978s (cpu); 0.
│ │ │ │ -0001a040: 3039 3332 3431 3773 2028 7468 7265 6164  0932417s (thread
│ │ │ │ -0001a050: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0001a030: 3534 3734 3835 7320 2863 7075 293b 2030  547485s (cpu); 0
│ │ │ │ +0001a040: 2e31 3338 3439 7320 2874 6872 6561 6429  .13849s (thread)
│ │ │ │ +0001a050: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0001a060: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a0a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │  0001a0b0: 203d 2032 2020 2020 2020 2020 2020 2020   = 2            
│ │ │ │  0001a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6674,17 +6674,17 @@
│ │ │ │  0001a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a130: 2d2d 2b0a 7c69 3131 203a 2074 696d 6520  --+.|i11 : time 
│ │ │ │  0001a140: 4575 6c65 7228 4a2c 4d65 7468 6f64 3d3e  Euler(J,Method=>
│ │ │ │  0001a150: 4469 7265 6374 436f 6d70 6c65 7465 496e  DirectCompleteIn
│ │ │ │  0001a160: 742c 496e 6473 4f66 536d 6f6f 7468 3d3e  t,IndsOfSmooth=>
│ │ │ │  0001a170: 7b30 2c31 7d29 7c0a 7c20 2d2d 2075 7365  {0,1})|.| -- use
│ │ │ │ -0001a180: 6420 302e 3230 3337 3734 7320 2863 7075  d 0.203774s (cpu
│ │ │ │ -0001a190: 293b 2030 2e30 3930 3034 3638 7320 2874  ); 0.0900468s (t
│ │ │ │ -0001a1a0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001a180: 6420 302e 3237 3436 3034 7320 2863 7075  d 0.274604s (cpu
│ │ │ │ +0001a190: 293b 2030 2e31 3135 3837 3873 2028 7468  ); 0.115878s (th
│ │ │ │ +0001a1a0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  0001a1b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0001a200: 7c6f 3131 203d 2032 2020 2020 2020 2020  |o11 = 2        
│ │ │ │  0001a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7280,17 +7280,17 @@
│ │ │ │  0001c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c710: 2b0a 7c69 3320 3a20 7469 6d65 2043 534d  +.|i3 : time CSM
│ │ │ │  0001c720: 2849 2c4d 6574 686f 643d 3e44 6972 6563  (I,Method=>Direc
│ │ │ │  0001c730: 7443 6f6d 706c 6574 496e 7429 2020 2020  tCompletInt)    
│ │ │ │  0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c750: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0001c760: 6564 2034 2e35 3036 3638 7320 2863 7075  ed 4.50668s (cpu
│ │ │ │ -0001c770: 293b 2031 2e32 3930 3232 7320 2874 6872  ); 1.29022s (thr
│ │ │ │ -0001c780: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +0001c760: 6564 2031 372e 3130 3831 7320 2863 7075  ed 17.1081s (cpu
│ │ │ │ +0001c770: 293b 2032 2e34 3139 3173 2028 7468 7265  ); 2.4191s (thre
│ │ │ │ +0001c780: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  0001c790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0001c7a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7e0: 2020 2020 207c 0a7c 2020 2020 2020 2032       |.|       2
│ │ │ │  0001c7f0: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ @@ -7342,16 +7342,16 @@
│ │ │ │  0001cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001caf0: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2043  --+.|i4 : time C
│ │ │ │  0001cb00: 534d 2849 2c4d 6574 686f 643d 3e44 6972  SM(I,Method=>Dir
│ │ │ │  0001cb10: 6563 7443 6f6d 706c 6574 496e 742c 496e  ectCompletInt,In
│ │ │ │  0001cb20: 6473 4f66 536d 6f6f 7468 3d3e 7b31 2c32  dsOfSmooth=>{1,2
│ │ │ │  0001cb30: 7d29 2020 2020 2020 207c 0a7c 202d 2d20  })       |.| -- 
│ │ │ │ -0001cb40: 7573 6564 2034 2e31 3533 3132 7320 2863  used 4.15312s (c
│ │ │ │ -0001cb50: 7075 293b 2031 2e32 3132 3535 7320 2874  pu); 1.21255s (t
│ │ │ │ +0001cb40: 7573 6564 2031 372e 3331 3637 7320 2863  used 17.3167s (c
│ │ │ │ +0001cb50: 7075 293b 2032 2e32 3330 3432 7320 2874  pu); 2.23042s (t
│ │ │ │  0001cb60: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ @@ -7478,16 +7478,16 @@
│ │ │ │  0001d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d380: 2d2d 2b0a 7c69 3320 3a20 7469 6d65 2043  --+.|i3 : time C
│ │ │ │  0001d390: 534d 2049 2020 2020 2020 2020 2020 2020  SM I            
│ │ │ │  0001d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d3b0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0001d3c0: 2d20 7573 6564 2031 2e33 3839 3737 7320  - used 1.38977s 
│ │ │ │ -0001d3d0: 2863 7075 293b 2030 2e35 3035 3435 3873  (cpu); 0.505458s
│ │ │ │ +0001d3c0: 2d20 7573 6564 2032 2e39 3636 3338 7320  - used 2.96638s 
│ │ │ │ +0001d3d0: 2863 7075 293b 2030 2e37 3931 3136 3773  (cpu); 0.791167s
│ │ │ │  0001d3e0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  0001d3f0: 6329 2020 7c0a 7c20 2020 2020 2020 2020  c)  |.|         
│ │ │ │  0001d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001d430: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │  0001d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7528,16 +7528,16 @@
│ │ │ │  0001d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d6a0: 2b0a 7c69 3420 3a20 7469 6d65 2043 534d  +.|i4 : time CSM
│ │ │ │  0001d6b0: 2849 2c49 6e70 7574 4973 536d 6f6f 7468  (I,InputIsSmooth
│ │ │ │  0001d6c0: 3d3e 7472 7565 2920 2020 2020 2020 2020  =>true)         
│ │ │ │  0001d6d0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0001d6e0: 7573 6564 2030 2e31 3738 3232 3773 2028  used 0.178227s (
│ │ │ │ -0001d6f0: 6370 7529 3b20 302e 3035 3035 3337 3473  cpu); 0.0505374s
│ │ │ │ +0001d6e0: 7573 6564 2030 2e32 3438 3931 3373 2028  used 0.248913s (
│ │ │ │ +0001d6f0: 6370 7529 3b20 302e 3034 3131 3834 3973  cpu); 0.0411849s
│ │ │ │  0001d700: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  0001d710: 6329 7c0a 7c20 2020 2020 2020 2020 2020  c)|.|           
│ │ │ │  0001d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d740: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001d750: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │  0001d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7583,4088 +7583,4087 @@
│ │ │ │  0001d9e0: 6c79 2c20 7573 6520 7468 6520 636f 6d6d  ly, use the comm
│ │ │ │  0001d9f0: 616e 6420 2a6e 6f74 6520 4368 6572 6e3a  and *note Chern:
│ │ │ │  0001da00: 2043 6865 726e 2c20 696e 7374 6561 640a   Chern, instead.
│ │ │ │  0001da10: 696e 2074 6869 7320 6361 7365 2e0a 0a2b  in this case...+
│ │ │ │  0001da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001da50: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074  -------+.|i5 : t
│ │ │ │ -0001da60: 696d 6520 4368 6572 6e20 4920 2020 2020  ime Chern I     
│ │ │ │ +0001da50: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7469  ------+.|i5 : ti
│ │ │ │ +0001da60: 6d65 2043 6865 726e 2049 2020 2020 2020  me Chern I      
│ │ │ │  0001da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da90: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -0001daa0: 3434 3833 3535 7320 2863 7075 293b 2030  448355s (cpu); 0
│ │ │ │ -0001dab0: 2e30 3331 3731 3037 7320 2874 6872 6561  .0317107s (threa
│ │ │ │ -0001dac0: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │ +0001da80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001da90: 0a7c 202d 2d20 7573 6564 2030 2e34 3036  .| -- used 0.406
│ │ │ │ +0001daa0: 3334 3673 2028 6370 7529 3b20 302e 3033  346s (cpu); 0.03
│ │ │ │ +0001dab0: 3136 3735 3673 2028 7468 7265 6164 293b  16756s (thread);
│ │ │ │ +0001dac0: 2030 7320 2867 6329 7c0a 7c20 2020 2020   0s (gc)|.|     
│ │ │ │  0001dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db00: 2020 2020 207c 0a7c 2020 2020 2020 2033       |.|       3
│ │ │ │ +0001db00: 207c 0a7c 2020 2020 2020 2033 2020 2020   |.|       3    
│ │ │ │  0001db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001db40: 0a7c 6f35 203d 2034 6820 2020 2020 2020  .|o5 = 4h       
│ │ │ │ +0001db30: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +0001db40: 3d20 3468 2020 2020 2020 2020 2020 2020  = 4h            
│ │ │ │  0001db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001db80: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +0001db70: 2020 207c 0a7c 2020 2020 2020 2031 2020     |.|       1  
│ │ │ │ +0001db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dbb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001dba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dbe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001dbf0: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │ +0001dbe0: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ +0001dbf0: 6820 5d20 2020 2020 2020 2020 2020 2020  h ]             
│ │ │ │  0001dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001dc30: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +0001dc10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001dc20: 7c20 2020 2020 2020 2020 3120 2020 2020  |         1     
│ │ │ │ +0001dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc60: 207c 0a7c 6f35 203a 202d 2d2d 2d2d 2d20   |.|o5 : ------ 
│ │ │ │ +0001dc50: 2020 2020 2020 207c 0a7c 6f35 203a 202d         |.|o5 : -
│ │ │ │ +0001dc60: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │  0001dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001dca0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +0001dc90: 7c0a 7c20 2020 2020 2020 2035 2020 2020  |.|        5    
│ │ │ │ +0001dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcd0: 2020 2020 207c 0a7c 2020 2020 2020 2068       |.|       h
│ │ │ │ +0001dcc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001dcd0: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │  0001dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001dd10: 0a7c 2020 2020 2020 2020 3120 2020 2020  .|        1     
│ │ │ │ +0001dd00: 2020 7c0a 7c20 2020 2020 2020 2031 2020    |.|        1  
│ │ │ │ +0001dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001dd30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dd80: 2d2d 2d2b 0a0a 4675 6e63 7469 6f6e 7320  ---+..Functions 
│ │ │ │ -0001dd90: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
│ │ │ │ -0001dda0: 6775 6d65 6e74 206e 616d 6564 2049 6e70  gument named Inp
│ │ │ │ -0001ddb0: 7574 4973 536d 6f6f 7468 3a0a 3d3d 3d3d  utIsSmooth:.====
│ │ │ │ +0001dd70: 2d2d 2d2d 2b0a 0a46 756e 6374 696f 6e73  ----+..Functions
│ │ │ │ +0001dd80: 2077 6974 6820 6f70 7469 6f6e 616c 2061   with optional a
│ │ │ │ +0001dd90: 7267 756d 656e 7420 6e61 6d65 6420 496e  rgument named In
│ │ │ │ +0001dda0: 7075 7449 7353 6d6f 6f74 683a 0a3d 3d3d  putIsSmooth:.===
│ │ │ │ +0001ddb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0001ddc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0001ddd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001dde0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001ddf0: 3d0a 0a20 202a 2022 4353 4d28 2e2e 2e2c  =..  * "CSM(...,
│ │ │ │ -0001de00: 496e 7075 7449 7353 6d6f 6f74 683d 3e2e  InputIsSmooth=>.
│ │ │ │ -0001de10: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ -0001de20: 6520 4353 4d3a 2043 534d 2c20 2d2d 2054  e CSM: CSM, -- T
│ │ │ │ -0001de30: 6865 0a20 2020 2043 6865 726e 2d53 6368  he.    Chern-Sch
│ │ │ │ -0001de40: 7761 7274 7a2d 4d61 6350 6865 7273 6f6e  wartz-MacPherson
│ │ │ │ -0001de50: 2063 6c61 7373 0a20 202a 2045 756c 6572   class.  * Euler
│ │ │ │ -0001de60: 282e 2e2e 2c49 6e70 7574 4973 536d 6f6f  (...,InputIsSmoo
│ │ │ │ -0001de70: 7468 3d3e 2e2e 2e29 2028 6d69 7373 696e  th=>...) (missin
│ │ │ │ -0001de80: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ -0001de90: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0001dea0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0001deb0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0001dec0: 6563 7420 2a6e 6f74 6520 496e 7075 7449  ect *note InputI
│ │ │ │ -0001ded0: 7353 6d6f 6f74 683a 2049 6e70 7574 4973  sSmooth: InputIs
│ │ │ │ -0001dee0: 536d 6f6f 7468 2c20 6973 2061 202a 6e6f  Smooth, is a *no
│ │ │ │ -0001def0: 7465 2073 796d 626f 6c3a 0a28 4d61 6361  te symbol:.(Maca
│ │ │ │ -0001df00: 756c 6179 3244 6f63 2953 796d 626f 6c2c  ulay2Doc)Symbol,
│ │ │ │ -0001df10: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163  ....File: Charac
│ │ │ │ -0001df20: 7465 7269 7374 6963 436c 6173 7365 732e  teristicClasses.
│ │ │ │ -0001df30: 696e 666f 2c20 4e6f 6465 3a20 6973 4d75  info, Node: isMu
│ │ │ │ -0001df40: 6c74 6948 6f6d 6f67 656e 656f 7573 2c20  ltiHomogeneous, 
│ │ │ │ -0001df50: 4e65 7874 3a20 4d65 7468 6f64 2c20 5072  Next: Method, Pr
│ │ │ │ -0001df60: 6576 3a20 496e 7075 7449 7353 6d6f 6f74  ev: InputIsSmoot
│ │ │ │ -0001df70: 682c 2055 703a 2054 6f70 0a0a 6973 4d75  h, Up: Top..isMu
│ │ │ │ -0001df80: 6c74 6948 6f6d 6f67 656e 656f 7573 202d  ltiHomogeneous -
│ │ │ │ -0001df90: 2d20 4368 6563 6b73 2069 6620 616e 2069  - Checks if an i
│ │ │ │ -0001dfa0: 6465 616c 2069 7320 686f 6d6f 6765 6e65  deal is homogene
│ │ │ │ -0001dfb0: 6f75 7320 7769 7468 2072 6573 7065 6374  ous with respect
│ │ │ │ -0001dfc0: 2074 6f20 7468 6520 6772 6164 696e 6720   to the grading 
│ │ │ │ -0001dfd0: 6f6e 2069 7473 2072 696e 6720 2869 2e65  on its ring (i.e
│ │ │ │ -0001dfe0: 2e20 6d75 6c74 692d 686f 6d6f 6765 6e65  . multi-homogene
│ │ │ │ -0001dff0: 6f75 7320 696e 2074 6865 206d 756c 7469  ous in the multi
│ │ │ │ -0001e000: 2d67 7261 6465 6420 6361 7365 290a 2a2a  -graded case).**
│ │ │ │ +0001dde0: 3d3d 0a0a 2020 2a20 2243 534d 282e 2e2e  ==..  * "CSM(...
│ │ │ │ +0001ddf0: 2c49 6e70 7574 4973 536d 6f6f 7468 3d3e  ,InputIsSmooth=>
│ │ │ │ +0001de00: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ +0001de10: 7465 2043 534d 3a20 4353 4d2c 202d 2d20  te CSM: CSM, -- 
│ │ │ │ +0001de20: 5468 650a 2020 2020 4368 6572 6e2d 5363  The.    Chern-Sc
│ │ │ │ +0001de30: 6877 6172 747a 2d4d 6163 5068 6572 736f  hwartz-MacPherso
│ │ │ │ +0001de40: 6e20 636c 6173 730a 2020 2a20 4575 6c65  n class.  * Eule
│ │ │ │ +0001de50: 7228 2e2e 2e2c 496e 7075 7449 7353 6d6f  r(...,InputIsSmo
│ │ │ │ +0001de60: 6f74 683d 3e2e 2e2e 2920 286d 6973 7369  oth=>...) (missi
│ │ │ │ +0001de70: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ +0001de80: 290a 0a46 6f72 2074 6865 2070 726f 6772  )..For the progr
│ │ │ │ +0001de90: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +0001dea0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +0001deb0: 6a65 6374 202a 6e6f 7465 2049 6e70 7574  ject *note Input
│ │ │ │ +0001dec0: 4973 536d 6f6f 7468 3a20 496e 7075 7449  IsSmooth: InputI
│ │ │ │ +0001ded0: 7353 6d6f 6f74 682c 2069 7320 6120 2a6e  sSmooth, is a *n
│ │ │ │ +0001dee0: 6f74 6520 7379 6d62 6f6c 3a0a 284d 6163  ote symbol:.(Mac
│ │ │ │ +0001def0: 6175 6c61 7932 446f 6329 5379 6d62 6f6c  aulay2Doc)Symbol
│ │ │ │ +0001df00: 2c2e 0a1f 0a46 696c 653a 2043 6861 7261  ,....File: Chara
│ │ │ │ +0001df10: 6374 6572 6973 7469 6343 6c61 7373 6573  cteristicClasses
│ │ │ │ +0001df20: 2e69 6e66 6f2c 204e 6f64 653a 2069 734d  .info, Node: isM
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│ │ │ │ -0001e0e0: 2020 2020 2069 734d 756c 7469 486f 6d6f       isMultiHomo
│ │ │ │ -0001e0f0: 6765 6e65 6f75 7320 660a 2020 2a20 496e  geneous f.  * In
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│ │ │ │ -0001e230: 3d3d 3d3d 3d3d 0a0a 5465 7374 7320 6966  ======..Tests if
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│ │ │ │ -0001e250: 206f 7220 5269 6e67 456c 656d 656e 7420   or RingElement 
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│ │ │ │ +0001e0d0: 2020 2020 2020 6973 4d75 6c74 6948 6f6d        isMultiHom
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│ │ │ │ +0001e250: 2069 7320 486f 6d6f 6765 6e65 6f75 7320   is Homogeneous 
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│ │ │ │ +0001e300: 7261 6465 6420 636f 6f72 6469 6e61 7465  raded coordinate
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│ │ │ │ +0001e320: 7661 7269 6574 6965 732c 0a61 6e64 2068  varieties,.and h
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│ │ │ │ +0001e380: 6d65 7468 6f64 7320 7468 6520 2a6e 6f74  methods the *not
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│ │ │ │ +0001e3a0: 5269 6e67 3a20 4d75 6c74 6950 726f 6a43  Ring: MultiProjC
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│ │ │ │ +0001e3f0: 6c54 6f72 6963 5661 7269 6574 6965 7320  lToricVarieties 
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│ │ │ │ -0001e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001e460: 7264 5269 6e67 287b 312c 322c 317d 2920  rdRing({1,2,1}) 
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│ │ │ │ -0001e4e0: 6f31 203d 2052 2020 2020 2020 2020 2020  o1 = R          
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│ │ │ │ +0001e720: 7c0a 7c6f 3220 3a20 4c69 7374 2020 2020  |.|o2 : List    
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│ │ │ │ -0001e7b0: 2d2d 2d2b 0a7c 6933 203a 2049 3d69 6465  ---+.|i3 : I=ide
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│ │ │ │ -0001e7d0: 2a78 5f30 2a78 5f34 2c78 5f36 5e33 2920  *x_0*x_4,x_6^3) 
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│ │ │ │ -0001e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001e890: 2078 2078 202c 2078 2029 2020 2020 2020   x x , x )      
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│ │ │ │ -0001e8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001e8c0: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ -0001e8d0: 2020 3020 3120 3420 2020 3620 2020 2020    0 1 4   6     
│ │ │ │ -0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001ea40: 7472 7565 2020 2020 2020 2020 2020 2020  true            
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│ │ │ │ +0001ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0001eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001eb10: 496e 7075 7420 7465 726d 2062 656c 6f77  Input term below
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│ │ │ │ -0001eb50: 0a7c 2d20 7820 7820 7820 202b 2078 2078  .|- x x x  + x x
│ │ │ │ +0001eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001eac0: 3520 3a20 6973 4d75 6c74 6948 6f6d 6f67  5 : isMultiHomog
│ │ │ │ +0001ead0: 656e 656f 7573 2069 6465 616c 2878 5f30  eneous ideal(x_0
│ │ │ │ +0001eae0: 2a78 5f33 2d78 5f31 2a78 5f30 2a78 5f34  *x_3-x_1*x_0*x_4
│ │ │ │ +0001eaf0: 2c78 5f36 5e33 2920 2020 2020 2020 7c0a  ,x_6^3)       |.
│ │ │ │ +0001eb00: 7c49 6e70 7574 2074 6572 6d20 6265 6c6f  |Input term belo
│ │ │ │ +0001eb10: 7720 6973 206e 6f74 2068 6f6d 6f67 656e  w is not homogen
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│ │ │ │ +0001eb30: 7420 746f 2074 6865 2067 7261 6469 6e67  t to the grading
│ │ │ │ +0001eb40: 7c0a 7c2d 2078 2078 2078 2020 2b20 7820  |.|- x x x  + x 
│ │ │ │ +0001eb50: 7820 2020 2020 2020 2020 2020 2020 2020  x               
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│ │ │ │ -0001eb90: 207c 0a7c 2020 2030 2031 2034 2020 2020   |.|   0 1 4    
│ │ │ │ -0001eba0: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ +0001eb80: 2020 7c0a 7c20 2020 3020 3120 3420 2020    |.|   0 1 4   
│ │ │ │ +0001eb90: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
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│ │ │ │ +0001ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec10: 2020 2020 207c 0a7c 6f35 203d 2066 616c       |.|o5 = fal
│ │ │ │ -0001ec20: 7365 2020 2020 2020 2020 2020 2020 2020  se              
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│ │ │ │ +0001ec10: 6c73 6520 2020 2020 2020 2020 2020 2020  lse             
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│ │ │ │ -0001ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ec40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e6f 7465  ---------+..Note
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│ │ │ │ -0001ecc0: 6f6d 6f67 656e 656f 7573 2074 6865 2064  omogeneous the d
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│ │ │ │ -0001ecf0: 2061 2067 6976 656e 2067 656e 6572 6174   a given generat
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│ │ │ │ -0001ed20: 6520 6973 4d75 6c74 6948 6f6d 6f67 656e  e isMultiHomogen
│ │ │ │ -0001ed30: 656f 7573 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  eous:.==========
│ │ │ │ -0001ed40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001ed50: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 4d75  =====..  * "isMu
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│ │ │ │ -0001ee70: 204d 756c 7469 5072 6f6a 436f 6f72 6452   MultiProjCoordR
│ │ │ │ -0001ee80: 696e 672c 2050 7265 763a 2069 734d 756c  ing, Prev: isMul
│ │ │ │ -0001ee90: 7469 486f 6d6f 6765 6e65 6f75 732c 2055  tiHomogeneous, U
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│ │ │ │ -0001eeb0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469  *****..Descripti
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│ │ │ │ -0001eed0: 5468 6520 6f70 7469 6f6e 204d 6574 686f  The option Metho
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│ │ │ │ -0001efe0: 2057 6865 6e20 7468 6520 696e 7075 740a   When the input.
│ │ │ │ -0001eff0: 6964 6561 6c20 6973 2061 2063 6f6d 706c  ideal is a compl
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│ │ │ │ -0001f010: 206f 6e65 206d 6179 2c20 706f 7465 6e74   one may, potent
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│ │ │ │ -0001f030: 7468 6520 636f 6d70 7574 6174 696f 6e0a  the computation.
│ │ │ │ -0001f040: 6279 2073 6574 7469 6e67 204d 6574 686f  by setting Metho
│ │ │ │ -0001f050: 643d 3e20 4469 7265 6374 436f 6d70 6c65  d=> DirectComple
│ │ │ │ -0001f060: 7465 496e 742e 2054 6865 206f 7074 696f  teInt. The optio
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│ │ │ │ -0001f080: 2075 7365 6420 6279 2074 6865 0a63 6f6d   used by the.com
│ │ │ │ -0001f090: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a  mands *note CSM:
│ │ │ │ -0001f0a0: 2043 534d 2c20 616e 6420 2a6e 6f74 6520   CSM, and *note 
│ │ │ │ -0001f0b0: 4575 6c65 723a 2045 756c 6572 2c20 616e  Euler: Euler, an
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│ │ │ │ -0001f0e0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d   CompMethod: Com
│ │ │ │ -0001f0f0: 704d 6574 686f 642c 3d3e 5072 6f6a 6563  pMethod,=>Projec
│ │ │ │ -0001f100: 7469 7665 4465 6772 6565 2e20 5468 6520  tiveDegree. The 
│ │ │ │ -0001f110: 4d65 7468 6f64 2049 6e63 6c75 7369 6f6e  Method Inclusion
│ │ │ │ -0001f120: 4578 636c 7573 696f 6e0a 7769 6c6c 2061  Exclusion.will a
│ │ │ │ -0001f130: 6c77 6179 7320 6265 2075 7365 6420 7769  lways be used wi
│ │ │ │ -0001f140: 7468 202a 6e6f 7465 2043 6f6d 704d 6574  th *note CompMet
│ │ │ │ -0001f150: 686f 643a 2043 6f6d 704d 6574 686f 642c  hod: CompMethod,
│ │ │ │ -0001f160: 2050 6e52 6573 6964 7561 6c20 6f72 2062   PnResidual or b
│ │ │ │ -0001f170: 6572 7469 6e69 2e0a 0a2b 2d2d 2d2d 2d2d  ertini...+------
│ │ │ │ +0001ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e 6f74  ----------+..Not
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│ │ │ │ +0001ed00: 7361 6d65 2e0a 0a57 6179 7320 746f 2075  same...Ways to u
│ │ │ │ +0001ed10: 7365 2069 734d 756c 7469 486f 6d6f 6765  se isMultiHomoge
│ │ │ │ +0001ed20: 6e65 6f75 733a 0a3d 3d3d 3d3d 3d3d 3d3d  neous:.=========
│ │ │ │ +0001ed30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001ed40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 734d  ======..  * "isM
│ │ │ │ +0001ed50: 756c 7469 486f 6d6f 6765 6e65 6f75 7328  ultiHomogeneous(
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│ │ │ │ +0001eef0: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20  *note CSM: CSM, 
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│ │ │ │ +0001ef70: 2049 6e63 6c75 7369 6f6e 4578 636c 7573   InclusionExclus
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│ │ │ │ +0001efd0: 2e20 5768 656e 2074 6865 2069 6e70 7574  . When the input
│ │ │ │ +0001efe0: 0a69 6465 616c 2069 7320 6120 636f 6d70  .ideal is a comp
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│ │ │ │ +0001f020: 2074 6865 2063 6f6d 7075 7461 7469 6f6e   the computation
│ │ │ │ +0001f030: 0a62 7920 7365 7474 696e 6720 4d65 7468  .by setting Meth
│ │ │ │ +0001f040: 6f64 3d3e 2044 6972 6563 7443 6f6d 706c  od=> DirectCompl
│ │ │ │ +0001f050: 6574 6549 6e74 2e20 5468 6520 6f70 7469  eteInt. The opti
│ │ │ │ +0001f060: 6f6e 204d 6574 686f 6420 6973 206f 6e6c  on Method is onl
│ │ │ │ +0001f070: 7920 7573 6564 2062 7920 7468 650a 636f  y used by the.co
│ │ │ │ +0001f080: 6d6d 616e 6473 202a 6e6f 7465 2043 534d  mmands *note CSM
│ │ │ │ +0001f090: 3a20 4353 4d2c 2061 6e64 202a 6e6f 7465  : CSM, and *note
│ │ │ │ +0001f0a0: 2045 756c 6572 3a20 4575 6c65 722c 2061   Euler: Euler, a
│ │ │ │ +0001f0b0: 6e64 206f 6e6c 7920 696e 2063 6f6d 6269  nd only in combi
│ │ │ │ +0001f0c0: 6e61 7469 6f6e 2077 6974 680a 2a6e 6f74  nation with.*not
│ │ │ │ +0001f0d0: 6520 436f 6d70 4d65 7468 6f64 3a20 436f  e CompMethod: Co
│ │ │ │ +0001f0e0: 6d70 4d65 7468 6f64 2c3d 3e50 726f 6a65  mpMethod,=>Proje
│ │ │ │ +0001f0f0: 6374 6976 6544 6567 7265 652e 2054 6865  ctiveDegree. The
│ │ │ │ +0001f100: 204d 6574 686f 6420 496e 636c 7573 696f   Method Inclusio
│ │ │ │ +0001f110: 6e45 7863 6c75 7369 6f6e 0a77 696c 6c20  nExclusion.will 
│ │ │ │ +0001f120: 616c 7761 7973 2062 6520 7573 6564 2077  always be used w
│ │ │ │ +0001f130: 6974 6820 2a6e 6f74 6520 436f 6d70 4d65  ith *note CompMe
│ │ │ │ +0001f140: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64  thod: CompMethod
│ │ │ │ +0001f150: 2c20 506e 5265 7369 6475 616c 206f 7220  , PnResidual or 
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│ │ │ │ +0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f1b0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5a5a  --+.|i1 : R = ZZ
│ │ │ │ -0001f1c0: 2f33 3237 3439 5b78 5f30 2e2e 785f 365d  /32749[x_0..x_6]
│ │ │ │ -0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f1a0: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a  ---+.|i1 : R = Z
│ │ │ │ +0001f1b0: 5a2f 3332 3734 395b 785f 302e 2e78 5f36  Z/32749[x_0..x_6
│ │ │ │ +0001f1c0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0001f1d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f1e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f220: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -0001f230: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0001f210: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ +0001f220: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0001f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001f250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f2a0: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ -0001f2b0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ -0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f290: 0a7c 6f31 203a 2050 6f6c 796e 6f6d 6961  .|o1 : Polynomia
│ │ │ │ +0001f2a0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +0001f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f310: 2d2d 2d2d 2b0a 7c69 3220 3a20 493d 6964  ----+.|i2 : I=id
│ │ │ │ -0001f320: 6561 6c28 7261 6e64 6f6d 2832 2c52 292c  eal(random(2,R),
│ │ │ │ -0001f330: 7261 6e64 6f6d 2831 2c52 292c 525f 302a  random(1,R),R_0*
│ │ │ │ -0001f340: 525f 312a 525f 362d 525f 305e 3329 3b7c  R_1*R_6-R_0^3);|
│ │ │ │ -0001f350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001f300: 2d2d 2d2d 2d2b 0a7c 6932 203a 2049 3d69  -----+.|i2 : I=i
│ │ │ │ +0001f310: 6465 616c 2872 616e 646f 6d28 322c 5229  deal(random(2,R)
│ │ │ │ +0001f320: 2c72 616e 646f 6d28 312c 5229 2c52 5f30  ,random(1,R),R_0
│ │ │ │ +0001f330: 2a52 5f31 2a52 5f36 2d52 5f30 5e33 293b  *R_1*R_6-R_0^3);
│ │ │ │ +0001f340: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f380: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -0001f390: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │ +0001f370: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001f380: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ +0001f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001f3b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f400: 2b0a 7c69 3320 3a20 7469 6d65 2043 534d  +.|i3 : time CSM
│ │ │ │ -0001f410: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ -0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f430: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0001f440: 2d20 7573 6564 2032 2e39 3539 3331 7320  - used 2.95931s 
│ │ │ │ -0001f450: 2863 7075 293b 2030 2e39 3739 3137 3773  (cpu); 0.979177s
│ │ │ │ -0001f460: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -0001f470: 6329 2020 2020 7c0a 7c20 2020 2020 2020  c)    |.|       
│ │ │ │ +0001f3f0: 2d2b 0a7c 6933 203a 2074 696d 6520 4353  -+.|i3 : time CS
│ │ │ │ +0001f400: 4d20 4920 2020 2020 2020 2020 2020 2020  M I             
│ │ │ │ +0001f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f430: 2d2d 2075 7365 6420 392e 3832 3435 3273  -- used 9.82452s
│ │ │ │ +0001f440: 2028 6370 7529 3b20 312e 3633 3638 3573   (cpu); 1.63685s
│ │ │ │ +0001f450: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +0001f460: 6329 2020 2020 207c 0a7c 2020 2020 2020  c)     |.|      
│ │ │ │ +0001f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4b0: 207c 0a7c 2020 2020 2020 2020 3520 2020   |.|        5   
│ │ │ │ -0001f4c0: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │ -0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001f4f0: 3320 3d20 3132 6820 202b 2031 3068 2020  3 = 12h  + 10h  
│ │ │ │ -0001f500: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f520: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001f530: 2020 3120 2020 2020 2031 2020 2020 2031    1      1     1
│ │ │ │ +0001f4a0: 2020 7c0a 7c20 2020 2020 2020 2035 2020    |.|        5  
│ │ │ │ +0001f4b0: 2020 2020 3420 2020 2020 3320 2020 2020      4     3     
│ │ │ │ +0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f4d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f4e0: 6f33 203d 2031 3268 2020 2b20 3130 6820  o3 = 12h  + 10h 
│ │ │ │ +0001f4f0: 202b 2036 6820 2020 2020 2020 2020 2020   + 6h           
│ │ │ │ +0001f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001f520: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +0001f530: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  0001f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f560: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f5a0: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │ +0001f580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f590: 7c20 2020 2020 5a5a 5b68 205d 2020 2020  |     ZZ[h ]    
│ │ │ │ +0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f5e0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +0001f5c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f5d0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f610: 2020 207c 0a7c 6f33 203a 202d 2d2d 2d2d     |.|o3 : -----
│ │ │ │ -0001f620: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ -0001f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f650: 7c20 2020 2020 2020 2037 2020 2020 2020  |        7      
│ │ │ │ +0001f600: 2020 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d      |.|o3 : ----
│ │ │ │ +0001f610: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +0001f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f640: 0a7c 2020 2020 2020 2020 3720 2020 2020  .|        7     
│ │ │ │ +0001f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f680: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001f690: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │ +0001f670: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f680: 2020 2020 6820 2020 2020 2020 2020 2020      h           
│ │ │ │ +0001f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6c0: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ +0001f6b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f6c0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  0001f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001f700: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001f6f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f730: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -0001f740: 3a20 7469 6d65 2043 534d 2849 2c4d 6574  : time CSM(I,Met
│ │ │ │ -0001f750: 686f 643d 3e44 6972 6563 7443 6f6d 706c  hod=>DirectCompl
│ │ │ │ -0001f760: 6574 6549 6e74 2920 2020 2020 2020 2020  eteInt)         
│ │ │ │ -0001f770: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0001f780: 2030 2e39 3839 3436 3373 2028 6370 7529   0.989463s (cpu)
│ │ │ │ -0001f790: 3b20 302e 3334 3738 3135 7320 2874 6872  ; 0.347815s (thr
│ │ │ │ -0001f7a0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -0001f7b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001f720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +0001f730: 203a 2074 696d 6520 4353 4d28 492c 4d65   : time CSM(I,Me
│ │ │ │ +0001f740: 7468 6f64 3d3e 4469 7265 6374 436f 6d70  thod=>DirectComp
│ │ │ │ +0001f750: 6c65 7465 496e 7429 2020 2020 2020 2020  leteInt)        
│ │ │ │ +0001f760: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +0001f770: 6420 322e 3634 3736 3973 2028 6370 7529  d 2.64769s (cpu)
│ │ │ │ +0001f780: 3b20 302e 3434 3631 3137 7320 2874 6872  ; 0.446117s (thr
│ │ │ │ +0001f790: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
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│ │ │ │ -0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001f7e0: 2020 2020 2020 2035 2020 2020 2020 3420         5      4 
│ │ │ │ +0001f7f0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
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│ │ │ │ +0001f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8d0: 2020 2020 2020 207c 0a7c 2020 2020 205a         |.|     Z
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│ │ │ │ +0001f8c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001f8d0: 5a5a 5b68 205d 2020 2020 2020 2020 2020  ZZ[h ]          
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│ │ │ │ -0001f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f950: 6f34 203a 202d 2d2d 2d2d 2d20 2020 2020  o4 : ------     
│ │ │ │ +0001f930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f940: 7c6f 3420 3a20 2d2d 2d2d 2d2d 2020 2020  |o4 : ------    
│ │ │ │ +0001f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001f970: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -0001f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9c0: 2020 207c 0a7c 2020 2020 2020 2068 2020     |.|       h  
│ │ │ │ +0001f9b0: 2020 2020 7c0a 7c20 2020 2020 2020 6820      |.|       h 
│ │ │ │ +0001f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fa00: 7c20 2020 2020 2020 2031 2020 2020 2020  |        1      
│ │ │ │ +0001f9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f9f0: 0a7c 2020 2020 2020 2020 3120 2020 2020  .|        1     
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│ │ │ │ +0001fb80: 6420 4d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  d Method:.======
│ │ │ │ +0001fb90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0001fba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001fbb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001fbc0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4353  =======..  * "CS
│ │ │ │ -0001fbd0: 4d28 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e  M(...,Method=>..
│ │ │ │ -0001fbe0: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ -0001fbf0: 2043 534d 3a20 4353 4d2c 202d 2d20 5468   CSM: CSM, -- Th
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│ │ │ │ -0001fc10: 6172 747a 2d4d 6163 5068 6572 736f 6e20  artz-MacPherson 
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│ │ │ │  0001fd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fdc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fdd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001fde0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -000200d0: 202a 6e6f 7465 2072 696e 673a 2028 4d61   *note ring: (Ma
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│ │ │ │ -000200f0: 2c20 7468 6520 6772 6164 6564 2063 6f6f  , the graded coo
│ │ │ │ -00020100: 7264 696e 6174 6520 7269 6e67 206f 6620  rdinate ring of 
│ │ │ │ -00020110: 7468 650a 2020 2020 2020 2020 5c50 505e  the.        \PP^
│ │ │ │ -00020120: 7b6e 5f31 7d20 782e 2e2e 2e20 7820 5c50  {n_1} x.... x \P
│ │ │ │ -00020130: 505e 7b6e 5f6d 7d20 7768 6572 6520 7b6e  P^{n_m} where {n
│ │ │ │ -00020140: 5f31 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074  _1,...,n_m} is t
│ │ │ │ -00020150: 6865 2069 6e70 7574 206c 6973 7420 6f66  he input list of
│ │ │ │ -00020160: 0a20 2020 2020 2020 2064 696d 656e 7369  .        dimensi
│ │ │ │ -00020170: 6f6e 730a 0a44 6573 6372 6970 7469 6f6e  ons..Description
│ │ │ │ -00020180: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f  .===========..Co
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│ │ │ │ -000201a0: 6420 636f 6f72 6469 6e61 7465 2072 696e  d coordinate rin
│ │ │ │ -000201b0: 6720 6f66 2074 6865 205c 5050 5e7b 6e5f  g of the \PP^{n_
│ │ │ │ -000201c0: 317d 2078 2e2e 2e2e 2078 205c 5050 5e7b  1} x.... x \PP^{
│ │ │ │ -000201d0: 6e5f 6d7d 2077 6865 7265 0a7b 6e5f 312c  n_m} where.{n_1,
│ │ │ │ -000201e0: 2e2e 2e2c 6e5f 6d7d 2069 7320 7468 6520  ...,n_m} is the 
│ │ │ │ -000201f0: 696e 7075 7420 6c69 7374 206f 6620 6469  input list of di
│ │ │ │ -00020200: 6d65 6e73 696f 6e73 2e20 5468 6973 206d  mensions. This m
│ │ │ │ -00020210: 6574 686f 6420 6973 2075 7365 6420 746f  ethod is used to
│ │ │ │ -00020220: 2071 7569 636b 6c79 0a62 7569 6c64 2074   quickly.build t
│ │ │ │ -00020230: 6865 2063 6f6f 7264 696e 6174 6520 7269  he coordinate ri
│ │ │ │ -00020240: 6e67 206f 6620 6120 7072 6f64 7563 7420  ng of a product 
│ │ │ │ -00020250: 6f66 2070 726f 6a65 6374 6976 6520 7370  of projective sp
│ │ │ │ -00020260: 6163 6573 2066 6f72 2075 7365 2069 6e0a  aces for use in.
│ │ │ │ -00020270: 636f 6d70 7574 6174 696f 6e73 2e0a 0a2b  computations...+
│ │ │ │ +0001fde0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ +0001fdf0: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ +0001fe00: 200a 2020 2020 2020 2020 4d75 6c74 6950   .        MultiP
│ │ │ │ +0001fe10: 726f 6a43 6f6f 7264 5269 6e67 2044 696d  rojCoordRing Dim
│ │ │ │ +0001fe20: 730a 2020 2020 2020 2020 4d75 6c74 6950  s.        MultiP
│ │ │ │ +0001fe30: 726f 6a43 6f6f 7264 5269 6e67 2028 436f  rojCoordRing (Co
│ │ │ │ +0001fe40: 6566 6652 696e 672c 4469 6d73 290a 2020  effRing,Dims).  
│ │ │ │ +0001fe50: 2020 2020 2020 4d75 6c74 6950 726f 6a43        MultiProjC
│ │ │ │ +0001fe60: 6f6f 7264 5269 6e67 2028 7661 722c 4469  oordRing (var,Di
│ │ │ │ +0001fe70: 6d73 290a 2020 2020 2020 2020 4d75 6c74  ms).        Mult
│ │ │ │ +0001fe80: 6950 726f 6a43 6f6f 7264 5269 6e67 2028  iProjCoordRing (
│ │ │ │ +0001fe90: 436f 6566 6652 696e 672c 7661 722c 4469  CoeffRing,var,Di
│ │ │ │ +0001fea0: 6d73 290a 2020 2a20 496e 7075 7473 3a0a  ms).  * Inputs:.
│ │ │ │ +0001feb0: 2020 2020 2020 2a20 4469 6d73 2c20 6120        * Dims, a 
│ │ │ │ +0001fec0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +0001fed0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +0001fee0: 2072 6570 7265 7365 6e74 696e 6720 7468   representing th
│ │ │ │ +0001fef0: 6520 6469 6d65 6e73 696f 6e73 206f 660a  e dimensions of.
│ │ │ │ +0001ff00: 2020 2020 2020 2020 7468 6520 7072 6f6a          the proj
│ │ │ │ +0001ff10: 6563 7469 7665 2073 7061 6365 732c 2069  ective spaces, i
│ │ │ │ +0001ff20: 2e65 2e20 7b6e 5f31 2c2e 2e2e 2c6e 5f6d  .e. {n_1,...,n_m
│ │ │ │ +0001ff30: 7d20 636f 7272 6573 706f 6e64 7320 746f  } corresponds to
│ │ │ │ +0001ff40: 205c 5050 5e7b 6e5f 317d 0a20 2020 2020   \PP^{n_1}.     
│ │ │ │ +0001ff50: 2020 2078 2e2e 2e2e 2078 205c 5050 5e7b     x.... x \PP^{
│ │ │ │ +0001ff60: 6e5f 6d7d 0a20 2020 2020 202a 2043 6f65  n_m}.      * Coe
│ │ │ │ +0001ff70: 6666 5269 6e67 2c20 6120 2a6e 6f74 6520  ffRing, a *note 
│ │ │ │ +0001ff80: 7269 6e67 3a20 284d 6163 6175 6c61 7932  ring: (Macaulay2
│ │ │ │ +0001ff90: 446f 6329 5269 6e67 2c2c 2074 6865 2063  Doc)Ring,, the c
│ │ │ │ +0001ffa0: 6f65 6666 6963 6965 6e74 2072 696e 6720  oefficient ring 
│ │ │ │ +0001ffb0: 6f66 0a20 2020 2020 2020 2074 6865 2067  of.        the g
│ │ │ │ +0001ffc0: 7261 6465 6420 706f 6c79 6e6f 6d69 616c  raded polynomial
│ │ │ │ +0001ffd0: 2072 696e 6720 746f 2062 6520 6275 696c   ring to be buil
│ │ │ │ +0001ffe0: 7420 6279 2074 6865 206d 6574 686f 642c  t by the method,
│ │ │ │ +0001fff0: 2062 7920 6465 6661 756c 7420 7468 6973   by default this
│ │ │ │ +00020000: 0a20 2020 2020 2020 2069 7320 5c5a 5a2f  .        is \ZZ/
│ │ │ │ +00020010: 3332 3734 390a 2020 2020 2020 2a20 7661  32749.      * va
│ │ │ │ +00020020: 722c 2061 202a 6e6f 7465 2073 796d 626f  r, a *note symbo
│ │ │ │ +00020030: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +00020040: 2953 796d 626f 6c2c 2c20 746f 2062 6520  )Symbol,, to be 
│ │ │ │ +00020050: 7573 6564 2066 6f72 2074 6865 0a20 2020  used for the.   
│ │ │ │ +00020060: 2020 2020 2069 6e74 6572 6d65 6469 6174       intermediat
│ │ │ │ +00020070: 6573 206f 6620 7468 6520 6772 6164 6564  es of the graded
│ │ │ │ +00020080: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67   polynomial ring
│ │ │ │ +00020090: 2074 6f20 6265 2062 7569 6c74 2062 7920   to be built by 
│ │ │ │ +000200a0: 7468 6520 6d65 7468 6f64 0a20 202a 204f  the method.  * O
│ │ │ │ +000200b0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +000200c0: 6120 2a6e 6f74 6520 7269 6e67 3a20 284d  a *note ring: (M
│ │ │ │ +000200d0: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ +000200e0: 2c2c 2074 6865 2067 7261 6465 6420 636f  ,, the graded co
│ │ │ │ +000200f0: 6f72 6469 6e61 7465 2072 696e 6720 6f66  ordinate ring of
│ │ │ │ +00020100: 2074 6865 0a20 2020 2020 2020 205c 5050   the.        \PP
│ │ │ │ +00020110: 5e7b 6e5f 317d 2078 2e2e 2e2e 2078 205c  ^{n_1} x.... x \
│ │ │ │ +00020120: 5050 5e7b 6e5f 6d7d 2077 6865 7265 207b  PP^{n_m} where {
│ │ │ │ +00020130: 6e5f 312c 2e2e 2e2c 6e5f 6d7d 2069 7320  n_1,...,n_m} is 
│ │ │ │ +00020140: 7468 6520 696e 7075 7420 6c69 7374 206f  the input list o
│ │ │ │ +00020150: 660a 2020 2020 2020 2020 6469 6d65 6e73  f.        dimens
│ │ │ │ +00020160: 696f 6e73 0a0a 4465 7363 7269 7074 696f  ions..Descriptio
│ │ │ │ +00020170: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43  n.===========..C
│ │ │ │ +00020180: 6f6d 7075 7465 7320 7468 6520 6772 6164  omputes the grad
│ │ │ │ +00020190: 6564 2063 6f6f 7264 696e 6174 6520 7269  ed coordinate ri
│ │ │ │ +000201a0: 6e67 206f 6620 7468 6520 5c50 505e 7b6e  ng of the \PP^{n
│ │ │ │ +000201b0: 5f31 7d20 782e 2e2e 2e20 7820 5c50 505e  _1} x.... x \PP^
│ │ │ │ +000201c0: 7b6e 5f6d 7d20 7768 6572 650a 7b6e 5f31  {n_m} where.{n_1
│ │ │ │ +000201d0: 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074 6865  ,...,n_m} is the
│ │ │ │ +000201e0: 2069 6e70 7574 206c 6973 7420 6f66 2064   input list of d
│ │ │ │ +000201f0: 696d 656e 7369 6f6e 732e 2054 6869 7320  imensions. This 
│ │ │ │ +00020200: 6d65 7468 6f64 2069 7320 7573 6564 2074  method is used t
│ │ │ │ +00020210: 6f20 7175 6963 6b6c 790a 6275 696c 6420  o quickly.build 
│ │ │ │ +00020220: 7468 6520 636f 6f72 6469 6e61 7465 2072  the coordinate r
│ │ │ │ +00020230: 696e 6720 6f66 2061 2070 726f 6475 6374  ing of a product
│ │ │ │ +00020240: 206f 6620 7072 6f6a 6563 7469 7665 2073   of projective s
│ │ │ │ +00020250: 7061 6365 7320 666f 7220 7573 6520 696e  paces for use in
│ │ │ │ +00020260: 0a63 6f6d 7075 7461 7469 6f6e 732e 0a0a  .computations...
│ │ │ │ +00020270: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00020280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000202d0: 6931 203a 2053 3d4d 756c 7469 5072 6f6a  i1 : S=MultiProj
│ │ │ │ -000202e0: 436f 6f72 6452 696e 6728 5151 2c73 796d  CoordRing(QQ,sym
│ │ │ │ -000202f0: 626f 6c20 7a2c 7b31 2c33 2c33 7d29 2020  bol z,{1,3,3})  
│ │ │ │ -00020300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000202c0: 7c69 3120 3a20 533d 4d75 6c74 6950 726f  |i1 : S=MultiPro
│ │ │ │ +000202d0: 6a43 6f6f 7264 5269 6e67 2851 512c 7379  jCoordRing(QQ,sy
│ │ │ │ +000202e0: 6d62 6f6c 207a 2c7b 312c 332c 337d 2920  mbol z,{1,3,3}) 
│ │ │ │ +000202f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020370: 6f31 203d 2053 2020 2020 2020 2020 2020  o1 = S          
│ │ │ │ +00020350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020360: 7c6f 3120 3d20 5320 2020 2020 2020 2020  |o1 = S         
│ │ │ │ +00020370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000203a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000203b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000203c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020410: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ -00020420: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +000203f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020400: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +00020410: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020450: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00020460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000204b0: 6932 203a 2064 6567 7265 6573 2053 2020  i2 : degrees S  
│ │ │ │ +00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000204a0: 7c69 3220 3a20 6465 6772 6565 7320 5320  |i2 : degrees S 
│ │ │ │ +000204b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000204c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000204d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000204e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000204f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000204e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000204f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020550: 6f32 203d 207b 7b31 2c20 302c 2030 7d2c  o2 = {{1, 0, 0},
│ │ │ │ -00020560: 207b 312c 2030 2c20 307d 2c20 7b30 2c20   {1, 0, 0}, {0, 
│ │ │ │ -00020570: 312c 2030 7d2c 207b 302c 2031 2c20 307d  1, 0}, {0, 1, 0}
│ │ │ │ -00020580: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c  , {0, 1, 0}, {0,
│ │ │ │ -00020590: 2031 2c20 307d 2c20 7b30 2c20 207c 0a7c   1, 0}, {0,  |.|
│ │ │ │ -000205a0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │ +00020530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020540: 7c6f 3220 3d20 7b7b 312c 2030 2c20 307d  |o2 = {{1, 0, 0}
│ │ │ │ +00020550: 2c20 7b31 2c20 302c 2030 7d2c 207b 302c  , {1, 0, 0}, {0,
│ │ │ │ +00020560: 2031 2c20 307d 2c20 7b30 2c20 312c 2030   1, 0}, {0, 1, 0
│ │ │ │ +00020570: 7d2c 207b 302c 2031 2c20 307d 2c20 7b30  }, {0, 1, 0}, {0
│ │ │ │ +00020580: 2c20 312c 2030 7d2c 207b 302c 2020 7c0a  , 1, 0}, {0,  |.
│ │ │ │ +00020590: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +000205a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000205b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000205c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000205d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000205e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -000205f0: 2020 2020 2030 2c20 317d 2c20 7b30 2c20       0, 1}, {0, 
│ │ │ │ -00020600: 302c 2031 7d2c 207b 302c 2030 2c20 317d  0, 1}, {0, 0, 1}
│ │ │ │ -00020610: 2c20 7b30 2c20 302c 2031 7d7d 2020 2020  , {0, 0, 1}}    
│ │ │ │ -00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020630: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000205d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000205e0: 7c20 2020 2020 302c 2031 7d2c 207b 302c  |     0, 1}, {0,
│ │ │ │ +000205f0: 2030 2c20 317d 2c20 7b30 2c20 302c 2031   0, 1}, {0, 0, 1
│ │ │ │ +00020600: 7d2c 207b 302c 2030 2c20 317d 7d20 2020  }, {0, 0, 1}}   
│ │ │ │ +00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020620: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020630: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020690: 6f32 203a 204c 6973 7420 2020 2020 2020  o2 : List       
│ │ │ │ +00020670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020680: 7c6f 3220 3a20 4c69 7374 2020 2020 2020  |o2 : List      
│ │ │ │ +00020690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000206a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000206c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000206d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000206c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000206d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000206f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020730: 6933 203a 2052 3d4d 756c 7469 5072 6f6a  i3 : R=MultiProj
│ │ │ │ -00020740: 436f 6f72 6452 696e 6720 7b32 2c33 7d20  CoordRing {2,3} 
│ │ │ │ +00020710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020720: 7c69 3320 3a20 523d 4d75 6c74 6950 726f  |i3 : R=MultiPro
│ │ │ │ +00020730: 6a43 6f6f 7264 5269 6e67 207b 322c 337d  jCoordRing {2,3}
│ │ │ │ +00020740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020770: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020770: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000207a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000207b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000207c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000207d0: 6f33 203d 2052 2020 2020 2020 2020 2020  o3 = R          
│ │ │ │ +000207b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000207c0: 7c6f 3320 3d20 5220 2020 2020 2020 2020  |o3 = R         
│ │ │ │ +000207d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000207e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000207f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020810: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020870: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -00020880: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00020850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020860: 7c6f 3320 3a20 506f 6c79 6e6f 6d69 616c  |o3 : Polynomial
│ │ │ │ +00020870: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00020880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000208a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000208b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000208c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000208d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000208e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000208f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020910: 6934 203a 2063 6f65 6666 6963 6965 6e74  i4 : coefficient
│ │ │ │ -00020920: 5269 6e67 2052 2020 2020 2020 2020 2020  Ring R          
│ │ │ │ +000208f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020900: 7c69 3420 3a20 636f 6566 6669 6369 656e  |i4 : coefficien
│ │ │ │ +00020910: 7452 696e 6720 5220 2020 2020 2020 2020  tRing R         
│ │ │ │ +00020920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000209b0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00020990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000209a0: 7c20 2020 2020 2020 5a5a 2020 2020 2020  |       ZZ      
│ │ │ │ +000209b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000209c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000209d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020a00: 6f34 203d 202d 2d2d 2d2d 2020 2020 2020  o4 = -----      
│ │ │ │ +000209e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000209f0: 7c6f 3420 3d20 2d2d 2d2d 2d20 2020 2020  |o4 = -----     
│ │ │ │ +00020a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020a50: 2020 2020 2033 3237 3439 2020 2020 2020       32749      
│ │ │ │ +00020a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020a40: 7c20 2020 2020 3332 3734 3920 2020 2020  |     32749     
│ │ │ │ +00020a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020a90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020af0: 6f34 203a 2051 756f 7469 656e 7452 696e  o4 : QuotientRin
│ │ │ │ -00020b00: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00020ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020ae0: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269  |o4 : QuotientRi
│ │ │ │ +00020af0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b30: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020b20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020b30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00020b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020b90: 6935 203a 2064 6573 6372 6962 6520 5220  i5 : describe R 
│ │ │ │ +00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020b80: 7c69 3520 3a20 6465 7363 7269 6265 2052  |i5 : describe R
│ │ │ │ +00020b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020bc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020bd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020c30: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00020c10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020c20: 7c20 2020 2020 2020 5a5a 2020 2020 2020  |       ZZ      
│ │ │ │ +00020c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020c80: 6f35 203d 202d 2d2d 2d2d 5b78 202e 2e78  o5 = -----[x ..x
│ │ │ │ -00020c90: 202c 2044 6567 7265 6573 203d 3e20 7b33   , Degrees => {3
│ │ │ │ -00020ca0: 3a7b 317d 2c20 343a 7b30 7d7d 2c20 4865  :{1}, 4:{0}}, He
│ │ │ │ -00020cb0: 6674 203d 3e20 7b32 3a31 7d5d 2020 2020  ft => {2:1}]    
│ │ │ │ -00020cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020cd0: 2020 2020 2033 3237 3439 2020 3020 2020       32749  0   
│ │ │ │ -00020ce0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ -00020cf0: 207b 307d 2020 2020 7b31 7d20 2020 2020   {0}    {1}     
│ │ │ │ -00020d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d10: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020c60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020c70: 7c6f 3520 3d20 2d2d 2d2d 2d5b 7820 2e2e  |o5 = -----[x ..
│ │ │ │ +00020c80: 7820 2c20 4465 6772 6565 7320 3d3e 207b  x , Degrees => {
│ │ │ │ +00020c90: 333a 7b31 7d2c 2034 3a7b 307d 7d2c 2048  3:{1}, 4:{0}}, H
│ │ │ │ +00020ca0: 6566 7420 3d3e 207b 323a 317d 5d20 2020  eft => {2:1}]   
│ │ │ │ +00020cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020cc0: 7c20 2020 2020 3332 3734 3920 2030 2020  |     32749  0  
│ │ │ │ +00020cd0: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00020ce0: 2020 7b30 7d20 2020 207b 317d 2020 2020    {0}    {1}    
│ │ │ │ +00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020d10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00020d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020d70: 6936 203a 2041 3d43 686f 7752 696e 6720  i6 : A=ChowRing 
│ │ │ │ -00020d80: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020d60: 7c69 3620 3a20 413d 4368 6f77 5269 6e67  |i6 : A=ChowRing
│ │ │ │ +00020d70: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00020d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020da0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020db0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020e10: 6f36 203d 2041 2020 2020 2020 2020 2020  o6 = A          
│ │ │ │ +00020df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020e00: 7c6f 3620 3d20 4120 2020 2020 2020 2020  |o6 = A         
│ │ │ │ +00020e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020e40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020e50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020eb0: 6f36 203a 2051 756f 7469 656e 7452 696e  o6 : QuotientRin
│ │ │ │ -00020ec0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00020e90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020ea0: 7c6f 3620 3a20 5175 6f74 6965 6e74 5269  |o6 : QuotientRi
│ │ │ │ +00020eb0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00020ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ef0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020ee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020ef0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00020f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020f50: 6937 203a 2064 6573 6372 6962 6520 4120  i7 : describe A 
│ │ │ │ +00020f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020f40: 7c69 3720 3a20 6465 7363 7269 6265 2041  |i7 : describe A
│ │ │ │ +00020f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020f80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020f90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020ff0: 2020 2020 205a 5a5b 6820 2e2e 6820 5d20       ZZ[h ..h ] 
│ │ │ │ +00020fd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020fe0: 7c20 2020 2020 5a5a 5b68 202e 2e68 205d  |     ZZ[h ..h ]
│ │ │ │ +00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021040: 2020 2020 2020 2020 2031 2020 2032 2020           1   2  
│ │ │ │ +00021020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021030: 7c20 2020 2020 2020 2020 3120 2020 3220  |         1   2 
│ │ │ │ +00021040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021090: 6f37 203d 202d 2d2d 2d2d 2d2d 2d2d 2d20  o7 = ---------- 
│ │ │ │ +00021070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021080: 7c6f 3720 3d20 2d2d 2d2d 2d2d 2d2d 2d2d  |o7 = ----------
│ │ │ │ +00021090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000210e0: 2020 2020 2020 2020 3320 2020 3420 2020          3   4   
│ │ │ │ +000210c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000210d0: 7c20 2020 2020 2020 2033 2020 2034 2020  |        3   4  
│ │ │ │ +000210e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021130: 2020 2020 2020 2868 202c 2068 2029 2020        (h , h )  
│ │ │ │ +00021110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021120: 7c20 2020 2020 2028 6820 2c20 6820 2920  |      (h , h ) 
│ │ │ │ +00021130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021180: 2020 2020 2020 2020 3120 2020 3220 2020          1   2   
│ │ │ │ +00021160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021170: 7c20 2020 2020 2020 2031 2020 2032 2020  |        1   2  
│ │ │ │ +00021180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000211a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000211b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000211c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000211f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00021220: 6938 203a 2053 6567 7265 2841 2c69 6465  i8 : Segre(A,ide
│ │ │ │ -00021230: 616c 2072 616e 646f 6d28 7b31 2c31 7d2c  al random({1,1},
│ │ │ │ -00021240: 5229 2920 2020 2020 2020 2020 2020 2020  R))             
│ │ │ │ -00021250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00021210: 7c69 3820 3a20 5365 6772 6528 412c 6964  |i8 : Segre(A,id
│ │ │ │ +00021220: 6561 6c20 7261 6e64 6f6d 287b 312c 317d  eal random({1,1}
│ │ │ │ +00021230: 2c52 2929 2020 2020 2020 2020 2020 2020  ,R))            
│ │ │ │ +00021240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021260: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00021270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000212c0: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ -000212d0: 3220 3220 2020 2020 2020 3320 2020 2020  2 2       3     
│ │ │ │ -000212e0: 3220 2020 2020 2020 2020 3220 2020 2033  2         2    3
│ │ │ │ -000212f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00021300: 2032 2020 2020 2020 2020 2020 207c 0a7c   2           |.|
│ │ │ │ -00021310: 6f38 203d 2031 3068 2068 2020 2d20 3668  o8 = 10h h  - 6h
│ │ │ │ -00021320: 2068 2020 2d20 3468 2068 2020 2b20 3368   h  - 4h h  + 3h
│ │ │ │ -00021330: 2068 2020 2b20 3368 2068 2020 2b20 6820   h  + 3h h  + h 
│ │ │ │ -00021340: 202d 2068 2020 2d20 3268 2068 2020 2d20   - h  - 2h h  - 
│ │ │ │ -00021350: 6820 202b 2068 2020 2b20 6820 207c 0a7c  h  + h  + h  |.|
│ │ │ │ -00021360: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -00021370: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │ -00021380: 3120 3220 2020 2020 3120 3220 2020 2032  1 2     1 2    2
│ │ │ │ -00021390: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -000213a0: 2032 2020 2020 3120 2020 2032 207c 0a7c   2    1    2 |.|
│ │ │ │ +000212a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000212b0: 7c20 2020 2020 2020 2032 2033 2020 2020  |        2 3    
│ │ │ │ +000212c0: 2032 2032 2020 2020 2020 2033 2020 2020   2 2       3    
│ │ │ │ +000212d0: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ +000212e0: 3320 2020 2032 2020 2020 2020 2020 2020  3    2          
│ │ │ │ +000212f0: 2020 3220 2020 2020 2020 2020 2020 7c0a    2           |.
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│ │ │ │ +00021310: 6820 6820 202d 2034 6820 6820 202b 2033  h h  - 4h h  + 3
│ │ │ │ +00021320: 6820 6820 202b 2033 6820 6820 202b 2068  h h  + 3h h  + h
│ │ │ │ +00021330: 2020 2d20 6820 202d 2032 6820 6820 202d    - h  - 2h h  -
│ │ │ │ +00021340: 2068 2020 2b20 6820 202b 2068 2020 7c0a   h  + h  + h  |.
│ │ │ │ +00021350: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +00021360: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +00021370: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +00021380: 3220 2020 2031 2020 2020 2031 2032 2020  2    1     1 2  
│ │ │ │ +00021390: 2020 3220 2020 2031 2020 2020 3220 7c0a    2    1    2 |.
│ │ │ │ +000213a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000213b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000213c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000213d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021400: 6f38 203a 2041 2020 2020 2020 2020 2020  o8 : A          
│ │ │ │ +000213e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000213f0: 7c6f 3820 3a20 4120 2020 2020 2020 2020  |o8 : A         
│ │ │ │ +00021400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021440: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021440: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00021450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
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│ │ │ │ +00021480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ +000214a0: 7469 5072 6f6a 436f 6f72 6452 696e 673a  tiProjCoordRing:
│ │ │ │ +000214b0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │  000214c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000214d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -000214e0: 0a20 202a 2022 4d75 6c74 6950 726f 6a43  .  * "MultiProjC
│ │ │ │ -000214f0: 6f6f 7264 5269 6e67 284c 6973 7429 220a  oordRing(List)".
│ │ │ │ -00021500: 2020 2a20 224d 756c 7469 5072 6f6a 436f    * "MultiProjCo
│ │ │ │ -00021510: 6f72 6452 696e 6728 5269 6e67 2c4c 6973  ordRing(Ring,Lis
│ │ │ │ -00021520: 7429 220a 2020 2a20 224d 756c 7469 5072  t)".  * "MultiPr
│ │ │ │ -00021530: 6f6a 436f 6f72 6452 696e 6728 5269 6e67  ojCoordRing(Ring
│ │ │ │ -00021540: 2c53 796d 626f 6c2c 4c69 7374 2922 0a20  ,Symbol,List)". 
│ │ │ │ -00021550: 202a 2022 4d75 6c74 6950 726f 6a43 6f6f   * "MultiProjCoo
│ │ │ │ -00021560: 7264 5269 6e67 2853 796d 626f 6c2c 4c69  rdRing(Symbol,Li
│ │ │ │ -00021570: 7374 2922 0a0a 466f 7220 7468 6520 7072  st)"..For the pr
│ │ │ │ -00021580: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -00021590: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -000215a0: 206f 626a 6563 7420 2a6e 6f74 6520 4d75   object *note Mu
│ │ │ │ -000215b0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -000215c0: 3a20 4d75 6c74 6950 726f 6a43 6f6f 7264  : MultiProjCoord
│ │ │ │ -000215d0: 5269 6e67 2c20 6973 2061 202a 6e6f 7465  Ring, is a *note
│ │ │ │ -000215e0: 206d 6574 686f 640a 6675 6e63 7469 6f6e   method.function
│ │ │ │ -000215f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00021600: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ -00021610: 0a1f 0a46 696c 653a 2043 6861 7261 6374  ...File: Charact
│ │ │ │ -00021620: 6572 6973 7469 6343 6c61 7373 6573 2e69  eristicClasses.i
│ │ │ │ -00021630: 6e66 6f2c 204e 6f64 653a 204f 7574 7075  nfo, Node: Outpu
│ │ │ │ -00021640: 742c 204e 6578 743a 2070 726f 6261 6269  t, Next: probabi
│ │ │ │ -00021650: 6c69 7374 6963 2061 6c67 6f72 6974 686d  listic algorithm
│ │ │ │ -00021660: 2c20 5072 6576 3a20 4d75 6c74 6950 726f  , Prev: MultiPro
│ │ │ │ -00021670: 6a43 6f6f 7264 5269 6e67 2c20 5570 3a20  jCoordRing, Up: 
│ │ │ │ -00021680: 546f 700a 0a4f 7574 7075 740a 2a2a 2a2a  Top..Output.****
│ │ │ │ -00021690: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a  **..Description.
│ │ │ │ -000216a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -000216b0: 206f 7074 696f 6e20 4f75 7470 7574 2069   option Output i
│ │ │ │ -000216c0: 7320 6f6e 6c79 2075 7365 6420 6279 2074  s only used by t
│ │ │ │ -000216d0: 6865 2063 6f6d 6d61 6e64 7320 2a6e 6f74  he commands *not
│ │ │ │ -000216e0: 6520 4353 4d3a 2043 534d 2c2c 202a 6e6f  e CSM: CSM,, *no
│ │ │ │ -000216f0: 7465 2053 6567 7265 3a0a 5365 6772 652c  te Segre:.Segre,
│ │ │ │ -00021700: 2c20 2a6e 6f74 6520 4368 6572 6e3a 2043  , *note Chern: C
│ │ │ │ -00021710: 6865 726e 2c20 616e 6420 2a6e 6f74 6520  hern, and *note 
│ │ │ │ -00021720: 4575 6c65 723a 2045 756c 6572 2c20 746f  Euler: Euler, to
│ │ │ │ -00021730: 2073 7065 6369 6679 2074 6865 2074 7970   specify the typ
│ │ │ │ -00021740: 6520 6f66 0a6f 7574 7075 7420 746f 2062  e of.output to b
│ │ │ │ -00021750: 6520 7265 7475 726e 6564 2074 6f20 7468  e returned to th
│ │ │ │ -00021760: 6520 7573 6564 2e20 5468 6973 206f 7074  e used. This opt
│ │ │ │ -00021770: 696f 6e20 7769 6c6c 2062 6520 6967 6e6f  ion will be igno
│ │ │ │ -00021780: 7265 6420 7768 656e 2075 7365 6420 7769  red when used wi
│ │ │ │ -00021790: 7468 0a2a 6e6f 7465 2043 6f6d 704d 6574  th.*note CompMet
│ │ │ │ -000217a0: 686f 643a 2043 6f6d 704d 6574 686f 642c  hod: CompMethod,
│ │ │ │ -000217b0: 2050 6e52 6573 6964 7561 6c20 6f72 2062   PnResidual or b
│ │ │ │ -000217c0: 6572 7469 6e69 2e20 5468 6520 6f70 7469  ertini. The opti
│ │ │ │ -000217d0: 6f6e 2077 696c 6c20 616c 736f 2062 650a  on will also be.
│ │ │ │ -000217e0: 6967 6e6f 7265 2077 6865 6e20 2a6e 6f74  ignore when *not
│ │ │ │ -000217f0: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64  e Method: Method
│ │ │ │ -00021800: 2c3d 3e44 6972 6563 7443 6f6d 706c 6574  ,=>DirectComplet
│ │ │ │ -00021810: 6549 6e74 2069 7320 7573 6564 2e20 5468  eInt is used. Th
│ │ │ │ -00021820: 6520 6465 6661 756c 740a 6f75 7470 7574  e default.output
│ │ │ │ -00021830: 2066 6f72 2061 6c6c 2074 6865 7365 206d   for all these m
│ │ │ │ -00021840: 6574 686f 6473 2069 7320 4368 6f77 5269  ethods is ChowRi
│ │ │ │ -00021850: 6e67 456c 656c 6d65 6e74 2077 6869 6368  ngElelment which
│ │ │ │ -00021860: 2077 696c 6c20 7265 7475 726e 2061 6e20   will return an 
│ │ │ │ -00021870: 656c 656d 656e 740a 6f66 2074 6865 2061  element.of the a
│ │ │ │ -00021880: 7070 726f 7072 6961 7465 2043 686f 7720  ppropriate Chow 
│ │ │ │ -00021890: 7269 6e67 2e20 416c 6c20 6d65 7468 6f64  ring. All method
│ │ │ │ -000218a0: 7320 616c 736f 2068 6176 6520 616e 206f  s also have an o
│ │ │ │ -000218b0: 7074 696f 6e20 4861 7368 466f 726d 2077  ption HashForm w
│ │ │ │ -000218c0: 6869 6368 0a72 6574 7572 6e73 2061 6464  hich.returns add
│ │ │ │ -000218d0: 6974 696f 6e61 6c20 696e 666f 726d 6174  itional informat
│ │ │ │ -000218e0: 696f 6e20 636f 6d70 7574 6564 2062 7920  ion computed by 
│ │ │ │ -000218f0: 7468 6520 6d65 7468 6f64 7320 6475 7269  the methods duri
│ │ │ │ -00021900: 6e67 2074 6865 6972 2073 7461 6e64 6172  ng their standar
│ │ │ │ -00021910: 640a 6f70 6572 6174 696f 6e2e 0a0a 2b2d  d.operation...+-
│ │ │ │ +000214d0: 0a0a 2020 2a20 224d 756c 7469 5072 6f6a  ..  * "MultiProj
│ │ │ │ +000214e0: 436f 6f72 6452 696e 6728 4c69 7374 2922  CoordRing(List)"
│ │ │ │ +000214f0: 0a20 202a 2022 4d75 6c74 6950 726f 6a43  .  * "MultiProjC
│ │ │ │ +00021500: 6f6f 7264 5269 6e67 2852 696e 672c 4c69  oordRing(Ring,Li
│ │ │ │ +00021510: 7374 2922 0a20 202a 2022 4d75 6c74 6950  st)".  * "MultiP
│ │ │ │ +00021520: 726f 6a43 6f6f 7264 5269 6e67 2852 696e  rojCoordRing(Rin
│ │ │ │ +00021530: 672c 5379 6d62 6f6c 2c4c 6973 7429 220a  g,Symbol,List)".
│ │ │ │ +00021540: 2020 2a20 224d 756c 7469 5072 6f6a 436f    * "MultiProjCo
│ │ │ │ +00021550: 6f72 6452 696e 6728 5379 6d62 6f6c 2c4c  ordRing(Symbol,L
│ │ │ │ +00021560: 6973 7429 220a 0a46 6f72 2074 6865 2070  ist)"..For the p
│ │ │ │ +00021570: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00021580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00021590: 6520 6f62 6a65 6374 202a 6e6f 7465 204d  e object *note M
│ │ │ │ +000215a0: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ +000215b0: 673a 204d 756c 7469 5072 6f6a 436f 6f72  g: MultiProjCoor
│ │ │ │ +000215c0: 6452 696e 672c 2069 7320 6120 2a6e 6f74  dRing, is a *not
│ │ │ │ +000215d0: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ +000215e0: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
│ │ │ │ +000215f0: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +00021600: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163  ....File: Charac
│ │ │ │ +00021610: 7465 7269 7374 6963 436c 6173 7365 732e  teristicClasses.
│ │ │ │ +00021620: 696e 666f 2c20 4e6f 6465 3a20 4f75 7470  info, Node: Outp
│ │ │ │ +00021630: 7574 2c20 4e65 7874 3a20 7072 6f62 6162  ut, Next: probab
│ │ │ │ +00021640: 696c 6973 7469 6320 616c 676f 7269 7468  ilistic algorith
│ │ │ │ +00021650: 6d2c 2050 7265 763a 204d 756c 7469 5072  m, Prev: MultiPr
│ │ │ │ +00021660: 6f6a 436f 6f72 6452 696e 672c 2055 703a  ojCoordRing, Up:
│ │ │ │ +00021670: 2054 6f70 0a0a 4f75 7470 7574 0a2a 2a2a   Top..Output.***
│ │ │ │ +00021680: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e  ***..Description
│ │ │ │ +00021690: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ +000216a0: 6520 6f70 7469 6f6e 204f 7574 7075 7420  e option Output 
│ │ │ │ +000216b0: 6973 206f 6e6c 7920 7573 6564 2062 7920  is only used by 
│ │ │ │ +000216c0: 7468 6520 636f 6d6d 616e 6473 202a 6e6f  the commands *no
│ │ │ │ +000216d0: 7465 2043 534d 3a20 4353 4d2c 2c20 2a6e  te CSM: CSM,, *n
│ │ │ │ +000216e0: 6f74 6520 5365 6772 653a 0a53 6567 7265  ote Segre:.Segre
│ │ │ │ +000216f0: 2c2c 202a 6e6f 7465 2043 6865 726e 3a20  ,, *note Chern: 
│ │ │ │ +00021700: 4368 6572 6e2c 2061 6e64 202a 6e6f 7465  Chern, and *note
│ │ │ │ +00021710: 2045 756c 6572 3a20 4575 6c65 722c 2074   Euler: Euler, t
│ │ │ │ +00021720: 6f20 7370 6563 6966 7920 7468 6520 7479  o specify the ty
│ │ │ │ +00021730: 7065 206f 660a 6f75 7470 7574 2074 6f20  pe of.output to 
│ │ │ │ +00021740: 6265 2072 6574 7572 6e65 6420 746f 2074  be returned to t
│ │ │ │ +00021750: 6865 2075 7365 642e 2054 6869 7320 6f70  he used. This op
│ │ │ │ +00021760: 7469 6f6e 2077 696c 6c20 6265 2069 676e  tion will be ign
│ │ │ │ +00021770: 6f72 6564 2077 6865 6e20 7573 6564 2077  ored when used w
│ │ │ │ +00021780: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65  ith.*note CompMe
│ │ │ │ +00021790: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64  thod: CompMethod
│ │ │ │ +000217a0: 2c20 506e 5265 7369 6475 616c 206f 7220  , PnResidual or 
│ │ │ │ +000217b0: 6265 7274 696e 692e 2054 6865 206f 7074  bertini. The opt
│ │ │ │ +000217c0: 696f 6e20 7769 6c6c 2061 6c73 6f20 6265  ion will also be
│ │ │ │ +000217d0: 0a69 676e 6f72 6520 7768 656e 202a 6e6f  .ignore when *no
│ │ │ │ +000217e0: 7465 204d 6574 686f 643a 204d 6574 686f  te Method: Metho
│ │ │ │ +000217f0: 642c 3d3e 4469 7265 6374 436f 6d70 6c65  d,=>DirectComple
│ │ │ │ +00021800: 7465 496e 7420 6973 2075 7365 642e 2054  teInt is used. T
│ │ │ │ +00021810: 6865 2064 6566 6175 6c74 0a6f 7574 7075  he default.outpu
│ │ │ │ +00021820: 7420 666f 7220 616c 6c20 7468 6573 6520  t for all these 
│ │ │ │ +00021830: 6d65 7468 6f64 7320 6973 2043 686f 7752  methods is ChowR
│ │ │ │ +00021840: 696e 6745 6c65 6c6d 656e 7420 7768 6963  ingElelment whic
│ │ │ │ +00021850: 6820 7769 6c6c 2072 6574 7572 6e20 616e  h will return an
│ │ │ │ +00021860: 2065 6c65 6d65 6e74 0a6f 6620 7468 6520   element.of the 
│ │ │ │ +00021870: 6170 7072 6f70 7269 6174 6520 4368 6f77  appropriate Chow
│ │ │ │ +00021880: 2072 696e 672e 2041 6c6c 206d 6574 686f   ring. All metho
│ │ │ │ +00021890: 6473 2061 6c73 6f20 6861 7665 2061 6e20  ds also have an 
│ │ │ │ +000218a0: 6f70 7469 6f6e 2048 6173 6846 6f72 6d20  option HashForm 
│ │ │ │ +000218b0: 7768 6963 680a 7265 7475 726e 7320 6164  which.returns ad
│ │ │ │ +000218c0: 6469 7469 6f6e 616c 2069 6e66 6f72 6d61  ditional informa
│ │ │ │ +000218d0: 7469 6f6e 2063 6f6d 7075 7465 6420 6279  tion computed by
│ │ │ │ +000218e0: 2074 6865 206d 6574 686f 6473 2064 7572   the methods dur
│ │ │ │ +000218f0: 696e 6720 7468 6569 7220 7374 616e 6461  ing their standa
│ │ │ │ +00021900: 7264 0a6f 7065 7261 7469 6f6e 2e0a 0a2b  rd.operation...+
│ │ │ │ +00021910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00021970: 3120 3a20 5220 3d20 5a5a 2f33 3237 3439  1 : R = ZZ/32749
│ │ │ │ -00021980: 5b78 5f30 2e2e 785f 365d 2020 2020 2020  [x_0..x_6]      
│ │ │ │ +00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00021960: 6931 203a 2052 203d 205a 5a2f 3332 3734  i1 : R = ZZ/3274
│ │ │ │ +00021970: 395b 785f 302e 2e78 5f36 5d20 2020 2020  9[x_0..x_6]     
│ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000219a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000219b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021a10: 3120 3d20 5220 2020 2020 2020 2020 2020  1 = R           
│ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021a00: 6f31 203d 2052 2020 2020 2020 2020 2020  o1 = R          
│ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021a40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021ab0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ -00021ac0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00021a90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021aa0: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ +00021ab0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00021ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021af0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021ae0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00021b50: 3220 3a20 413d 4368 6f77 5269 6e67 2852  2 : A=ChowRing(R
│ │ │ │ -00021b60: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00021b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00021b40: 6932 203a 2041 3d43 686f 7752 696e 6728  i2 : A=ChowRing(
│ │ │ │ +00021b50: 5229 2020 2020 2020 2020 2020 2020 2020  R)              
│ │ │ │ +00021b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021be0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021bf0: 3220 3d20 4120 2020 2020 2020 2020 2020  2 = A           
│ │ │ │ +00021bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021be0: 6f32 203d 2041 2020 2020 2020 2020 2020  o2 = A          
│ │ │ │ +00021bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021c20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021c90: 3220 3a20 5175 6f74 6965 6e74 5269 6e67  2 : QuotientRing
│ │ │ │ +00021c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021c80: 6f32 203a 2051 756f 7469 656e 7452 696e  o2 : QuotientRin
│ │ │ │ +00021c90: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00021ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021cd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021cc0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00021cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00021d30: 3320 3a20 493d 6964 6561 6c28 7261 6e64  3 : I=ideal(rand
│ │ │ │ -00021d40: 6f6d 2832 2c52 292c 525f 302a 525f 312a  om(2,R),R_0*R_1*
│ │ │ │ -00021d50: 525f 362d 525f 305e 3329 3b20 2020 2020  R_6-R_0^3);     
│ │ │ │ -00021d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00021d20: 6933 203a 2049 3d69 6465 616c 2872 616e  i3 : I=ideal(ran
│ │ │ │ +00021d30: 646f 6d28 322c 5229 2c52 5f30 2a52 5f31  dom(2,R),R_0*R_1
│ │ │ │ +00021d40: 2a52 5f36 2d52 5f30 5e33 293b 2020 2020  *R_6-R_0^3);    
│ │ │ │ +00021d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021dd0: 3320 3a20 4964 6561 6c20 6f66 2052 2020  3 : Ideal of R  
│ │ │ │ +00021db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021dc0: 6f33 203a 2049 6465 616c 206f 6620 5220  o3 : Ideal of R 
│ │ │ │ +00021dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021e00: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00021e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00021e70: 3420 3a20 6373 6d3d 4353 4d28 412c 492c  4 : csm=CSM(A,I,
│ │ │ │ -00021e80: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ -00021e90: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00021ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00021e60: 6934 203a 2063 736d 3d43 534d 2841 2c49  i4 : csm=CSM(A,I
│ │ │ │ +00021e70: 2c4f 7574 7075 743d 3e48 6173 6846 6f72  ,Output=>HashFor
│ │ │ │ +00021e80: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ +00021e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021f10: 3420 3d20 4d75 7461 626c 6548 6173 6854  4 = MutableHashT
│ │ │ │ -00021f20: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020  able{...4...}   
│ │ │ │ +00021ef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021f00: 6f34 203d 204d 7574 6162 6c65 4861 7368  o4 = MutableHash
│ │ │ │ +00021f10: 5461 626c 657b 2e2e 2e34 2e2e 2e7d 2020  Table{...4...}  
│ │ │ │ +00021f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021f40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021fb0: 3420 3a20 4d75 7461 626c 6548 6173 6854  4 : MutableHashT
│ │ │ │ -00021fc0: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +00021f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021fa0: 6f34 203a 204d 7574 6162 6c65 4861 7368  o4 : MutableHash
│ │ │ │ +00021fb0: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +00021fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ff0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021fe0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00021ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022050: 3520 3a20 7065 656b 2063 736d 2020 2020  5 : peek csm    
│ │ │ │ +00022030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00022040: 6935 203a 2070 6565 6b20 6373 6d20 2020  i5 : peek csm   
│ │ │ │ +00022050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000220d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000220e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022110: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ -00022120: 2020 2020 2020 3320 2020 2020 2032 2020        3      2  
│ │ │ │ -00022130: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00022140: 3520 3d20 4d75 7461 626c 6548 6173 6854  5 = MutableHashT
│ │ │ │ -00022150: 6162 6c65 7b7b 302c 2031 7d20 3d3e 2032  able{{0, 1} => 2
│ │ │ │ -00022160: 6820 202b 2032 3368 2020 2b20 3332 6820  h  + 23h  + 32h 
│ │ │ │ -00022170: 202b 2033 3368 2020 2b20 3138 6820 202b   + 33h  + 18h  +
│ │ │ │ -00022180: 2035 6820 7d20 2020 2020 2020 7c0a 7c20   5h }       |.| 
│ │ │ │ +00022100: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ +00022110: 3420 2020 2020 2033 2020 2020 2020 3220  4      3      2 
│ │ │ │ +00022120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022130: 6f35 203d 204d 7574 6162 6c65 4861 7368  o5 = MutableHash
│ │ │ │ +00022140: 5461 626c 657b 7b30 2c20 317d 203d 3e20  Table{{0, 1} => 
│ │ │ │ +00022150: 3268 2020 2b20 3233 6820 202b 2033 3268  2h  + 23h  + 32h
│ │ │ │ +00022160: 2020 2b20 3333 6820 202b 2031 3868 2020    + 33h  + 18h  
│ │ │ │ +00022170: 2b20 3568 207d 2020 2020 2020 207c 0a7c  + 5h }       |.|
│ │ │ │ +00022180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221b0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -000221c0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -000221d0: 2020 2031 2020 2020 2020 2020 7c0a 7c20     1        |.| 
│ │ │ │ +000221a0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +000221b0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +000221c0: 2020 2020 3120 2020 2020 2020 207c 0a7c      1        |.|
│ │ │ │ +000221d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221f0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00022200: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00022210: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ -00022220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022240: 2020 2020 2043 534d 203d 3e20 3130 6820       CSM => 10h 
│ │ │ │ -00022250: 202b 2031 3268 2020 2b20 3232 6820 202b   + 12h  + 22h  +
│ │ │ │ -00022260: 2031 3668 2020 2b20 3668 2020 2020 2020   16h  + 6h      
│ │ │ │ -00022270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000221f0: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ +00022200: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ +00022210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022230: 2020 2020 2020 4353 4d20 3d3e 2031 3068        CSM => 10h
│ │ │ │ +00022240: 2020 2b20 3132 6820 202b 2032 3268 2020    + 12h  + 22h  
│ │ │ │ +00022250: 2b20 3136 6820 202b 2036 6820 2020 2020  + 16h  + 6h     
│ │ │ │ +00022260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022290: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -000222a0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -000222b0: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -000222c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000222d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222e0: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -000222f0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -00022300: 2020 2033 2020 2020 2020 3220 2020 2020     3      2     
│ │ │ │ -00022310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022330: 2020 2020 207b 307d 203d 3e20 3668 2020       {0} => 6h  
│ │ │ │ -00022340: 2b20 3138 6820 202b 2032 3668 2020 2b20  + 18h  + 26h  + 
│ │ │ │ -00022350: 3232 6820 202b 2031 3068 2020 2b20 3268  22h  + 10h  + 2h
│ │ │ │ -00022360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022380: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00022390: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -000223a0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -000223b0: 3120 2020 2020 2020 2020 2020 7c0a 7c20  1           |.| 
│ │ │ │ -000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223d0: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -000223e0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -000223f0: 2020 2033 2020 2020 2020 3220 2020 2020     3      2     
│ │ │ │ -00022400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022420: 2020 2020 207b 317d 203d 3e20 3668 2020       {1} => 6h  
│ │ │ │ -00022430: 2b20 3137 6820 202b 2032 3868 2020 2b20  + 17h  + 28h  + 
│ │ │ │ -00022440: 3237 6820 202b 2031 3468 2020 2b20 3368  27h  + 14h  + 3h
│ │ │ │ -00022450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022470: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00022480: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00022490: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -000224a0: 3120 2020 2020 2020 2020 2020 7c0a 2b2d  1           |.+-
│ │ │ │ +00022290: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +000222a0: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +000222b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000222d0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ +000222e0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ +000222f0: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ +00022300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022320: 2020 2020 2020 7b30 7d20 3d3e 2036 6820        {0} => 6h 
│ │ │ │ +00022330: 202b 2031 3868 2020 2b20 3236 6820 202b   + 18h  + 26h  +
│ │ │ │ +00022340: 2032 3268 2020 2b20 3130 6820 202b 2032   22h  + 10h  + 2
│ │ │ │ +00022350: 6820 2020 2020 2020 2020 2020 207c 0a7c  h            |.|
│ │ │ │ +00022360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00022380: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00022390: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +000223a0: 2031 2020 2020 2020 2020 2020 207c 0a7c   1           |.|
│ │ │ │ +000223b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000223c0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ +000223d0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ +000223e0: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ +000223f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022410: 2020 2020 2020 7b31 7d20 3d3e 2036 6820        {1} => 6h 
│ │ │ │ +00022420: 202b 2031 3768 2020 2b20 3238 6820 202b   + 17h  + 28h  +
│ │ │ │ +00022430: 2032 3768 2020 2b20 3134 6820 202b 2033   27h  + 14h  + 3
│ │ │ │ +00022440: 6820 2020 2020 2020 2020 2020 207c 0a7c  h            |.|
│ │ │ │ +00022450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00022470: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00022480: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00022490: 2031 2020 2020 2020 2020 2020 207c 0a2b   1           |.+
│ │ │ │ +000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000224b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000224c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000224d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000224e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000224f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022500: 3620 3a20 4353 4d28 412c 6964 6561 6c20  6 : CSM(A,ideal 
│ │ │ │ -00022510: 495f 3029 3d3d 6373 6d23 7b30 7d20 2020  I_0)==csm#{0}   
│ │ │ │ +000224e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000224f0: 6936 203a 2043 534d 2841 2c69 6465 616c  i6 : CSM(A,ideal
│ │ │ │ +00022500: 2049 5f30 293d 3d63 736d 237b 307d 2020   I_0)==csm#{0}  
│ │ │ │ +00022510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000225a0: 3620 3d20 7472 7565 2020 2020 2020 2020  6 = true        
│ │ │ │ +00022580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022590: 6f36 203d 2074 7275 6520 2020 2020 2020  o6 = true       
│ │ │ │ +000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000225d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022640: 3720 3a20 4353 4d28 412c 6964 6561 6c28  7 : CSM(A,ideal(
│ │ │ │ -00022650: 495f 302a 495f 3129 293d 3d63 736d 237b  I_0*I_1))==csm#{
│ │ │ │ -00022660: 302c 317d 2020 2020 2020 2020 2020 2020  0,1}            
│ │ │ │ -00022670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00022630: 6937 203a 2043 534d 2841 2c69 6465 616c  i7 : CSM(A,ideal
│ │ │ │ +00022640: 2849 5f30 2a49 5f31 2929 3d3d 6373 6d23  (I_0*I_1))==csm#
│ │ │ │ +00022650: 7b30 2c31 7d20 2020 2020 2020 2020 2020  {0,1}           
│ │ │ │ +00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000226e0: 3720 3d20 7472 7565 2020 2020 2020 2020  7 = true        
│ │ │ │ +000226c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000226d0: 6f37 203d 2074 7275 6520 2020 2020 2020  o7 = true       
│ │ │ │ +000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022720: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00022710: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022780: 3820 3a20 633d 4368 6572 6e28 2049 2c20  8 : c=Chern( I, 
│ │ │ │ -00022790: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ -000227a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -000227b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00022770: 6938 203a 2063 3d43 6865 726e 2820 492c  i8 : c=Chern( I,
│ │ │ │ +00022780: 204f 7574 7075 743d 3e48 6173 6846 6f72   Output=>HashFor
│ │ │ │ +00022790: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ +000227a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000227b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000227c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022810: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00022820: 3820 3d20 4d75 7461 626c 6548 6173 6854  8 = MutableHashT
│ │ │ │ -00022830: 6162 6c65 7b2e 2e2e 362e 2e2e 7d20 2020  able{...6...}   
│ │ │ │ +00022800: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022810: 6f38 203d 204d 7574 6162 6c65 4861 7368  o8 = MutableHash
│ │ │ │ +00022820: 5461 626c 657b 2e2e 2e36 2e2e 2e7d 2020  Table{...6...}  
│ │ │ │ +00022830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000228c0: 3820 3a20 4d75 7461 626c 6548 6173 6854  8 : MutableHashT
│ │ │ │ -000228d0: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +000228a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000228b0: 6f38 203a 204d 7574 6162 6c65 4861 7368  o8 : MutableHash
│ │ │ │ +000228c0: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +000228d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000228f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022960: 3920 3a20 7065 656b 2063 2020 2020 2020  9 : peek c      
│ │ │ │ +00022940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00022950: 6939 203a 2070 6565 6b20 6320 2020 2020  i9 : peek c     
│ │ │ │ +00022960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022990: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000229a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000229e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000229f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a20: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00022a30: 2020 3320 2020 2020 2034 2020 2020 2020    3      4      
│ │ │ │ -00022a40: 2035 2020 2020 2020 2036 2020 7c0a 7c6f   5       6  |.|o
│ │ │ │ -00022a50: 3920 3d20 4d75 7461 626c 6548 6173 6854  9 = MutableHashT
│ │ │ │ -00022a60: 6162 6c65 7b53 6567 7265 4c69 7374 203d  able{SegreList =
│ │ │ │ -00022a70: 3e20 7b30 2c20 302c 2036 6820 2c20 2d33  > {0, 0, 6h , -3
│ │ │ │ -00022a80: 3068 202c 2031 3134 6820 2c20 2d33 3930  0h , 114h , -390
│ │ │ │ -00022a90: 6820 2c20 3132 3636 6820 7d7d 7c0a 7c20  h , 1266h }}|.| 
│ │ │ │ +00022a10: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00022a20: 2020 2033 2020 2020 2020 3420 2020 2020     3      4     
│ │ │ │ +00022a30: 2020 3520 2020 2020 2020 3620 207c 0a7c    5       6  |.|
│ │ │ │ +00022a40: 6f39 203d 204d 7574 6162 6c65 4861 7368  o9 = MutableHash
│ │ │ │ +00022a50: 5461 626c 657b 5365 6772 654c 6973 7420  Table{SegreList 
│ │ │ │ +00022a60: 3d3e 207b 302c 2030 2c20 3668 202c 202d  => {0, 0, 6h , -
│ │ │ │ +00022a70: 3330 6820 2c20 3131 3468 202c 202d 3339  30h , 114h , -39
│ │ │ │ +00022a80: 3068 202c 2031 3236 3668 207d 7d7c 0a7c  0h , 1266h }}|.|
│ │ │ │ +00022a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ac0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00022ad0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00022ae0: 2031 2020 2020 2020 2031 2020 7c0a 7c20   1       1  |.| 
│ │ │ │ +00022ab0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00022ac0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00022ad0: 2020 3120 2020 2020 2020 3120 207c 0a7c    1       1  |.|
│ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b10: 2020 2020 2020 2020 2032 2020 2020 3320           2    3 
│ │ │ │ -00022b20: 2020 2034 2020 2020 3520 2020 2036 2020     4    5    6  
│ │ │ │ -00022b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b50: 2020 2020 2047 6c69 7374 203d 3e20 7b31       Glist => {1
│ │ │ │ -00022b60: 2c20 3368 202c 2033 6820 2c20 3368 202c  , 3h , 3h , 3h ,
│ │ │ │ -00022b70: 2033 6820 2c20 3368 202c 2033 6820 7d20   3h , 3h , 3h } 
│ │ │ │ -00022b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022b00: 2020 2020 2020 2020 2020 3220 2020 2033            2    3
│ │ │ │ +00022b10: 2020 2020 3420 2020 2035 2020 2020 3620      4    5    6 
│ │ │ │ +00022b20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b40: 2020 2020 2020 476c 6973 7420 3d3e 207b        Glist => {
│ │ │ │ +00022b50: 312c 2033 6820 2c20 3368 202c 2033 6820  1, 3h , 3h , 3h 
│ │ │ │ +00022b60: 2c20 3368 202c 2033 6820 2c20 3368 207d  , 3h , 3h , 3h }
│ │ │ │ +00022b70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ba0: 2020 2020 2031 2020 2020 3120 2020 2031       1    1    1
│ │ │ │  00022bb0: 2020 2020 3120 2020 2031 2020 2020 3120      1    1    1 
│ │ │ │ -00022bc0: 2020 2031 2020 2020 3120 2020 2031 2020     1    1    1  
│ │ │ │ -00022bd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c00: 2020 2036 2020 2020 2020 2035 2020 2020     6       5    
│ │ │ │ -00022c10: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ -00022c20: 3220 2020 2020 2020 2020 2020 7c0a 7c20  2           |.| 
│ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c40: 2020 2020 2053 6567 7265 203d 3e20 3132       Segre => 12
│ │ │ │ -00022c50: 3636 6820 202d 2033 3930 6820 202b 2031  66h  - 390h  + 1
│ │ │ │ -00022c60: 3134 6820 202d 2033 3068 2020 2b20 3668  14h  - 30h  + 6h
│ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022bf0: 2020 2020 3620 2020 2020 2020 3520 2020      6       5   
│ │ │ │ +00022c00: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00022c10: 2032 2020 2020 2020 2020 2020 207c 0a7c   2           |.|
│ │ │ │ +00022c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c30: 2020 2020 2020 5365 6772 6520 3d3e 2031        Segre => 1
│ │ │ │ +00022c40: 3236 3668 2020 2d20 3339 3068 2020 2b20  266h  - 390h  + 
│ │ │ │ +00022c50: 3131 3468 2020 2d20 3330 6820 202b 2036  114h  - 30h  + 6
│ │ │ │ +00022c60: 6820 2020 2020 2020 2020 2020 207c 0a7c  h            |.|
│ │ │ │ +00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ca0: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ -00022cb0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00022cc0: 3120 2020 2020 2020 2020 2020 7c0a 7c20  1           |.| 
│ │ │ │ +00022c90: 2020 2020 3120 2020 2020 2020 3120 2020      1       1   
│ │ │ │ +00022ca0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00022cb0: 2031 2020 2020 2020 2020 2020 207c 0a7c   1           |.|
│ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cf0: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ -00022d00: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ -00022d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d30: 2020 2020 2043 6865 726e 203d 3e20 3930       Chern => 90
│ │ │ │ -00022d40: 6820 202d 2031 3268 2020 2b20 3330 6820  h  - 12h  + 30h 
│ │ │ │ -00022d50: 202b 2031 3268 2020 2b20 3668 2020 2020   + 12h  + 6h    
│ │ │ │ -00022d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022ce0: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ +00022cf0: 3420 2020 2020 2033 2020 2020 2032 2020  4      3     2  
│ │ │ │ +00022d00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d20: 2020 2020 2020 4368 6572 6e20 3d3e 2039        Chern => 9
│ │ │ │ +00022d30: 3068 2020 2d20 3132 6820 202b 2033 3068  0h  - 12h  + 30h
│ │ │ │ +00022d40: 2020 2b20 3132 6820 202b 2036 6820 2020    + 12h  + 6h   
│ │ │ │ +00022d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d90: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00022da0: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ -00022db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022dd0: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00022de0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -00022df0: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │ -00022e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e20: 2020 2020 2043 4620 3d3e 2039 3068 2020       CF => 90h  
│ │ │ │ -00022e30: 2d20 3132 6820 202b 2033 3068 2020 2b20  - 12h  + 30h  + 
│ │ │ │ -00022e40: 3132 6820 202b 2036 6820 2020 2020 2020  12h  + 6h       
│ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e70: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00022e80: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00022e90: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ -00022ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ec0: 2020 2020 2020 2020 2020 2020 3620 2020              6   
│ │ │ │ -00022ed0: 2020 3520 2020 2020 3420 2020 2020 3320    5     4     3 
│ │ │ │ -00022ee0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00022ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f10: 2020 2020 2047 203d 3e20 3368 2020 2b20       G => 3h  + 
│ │ │ │ -00022f20: 3368 2020 2b20 3368 2020 2b20 3368 2020  3h  + 3h  + 3h  
│ │ │ │ -00022f30: 2b20 3368 2020 2b20 3368 2020 2b20 3120  + 3h  + 3h  + 1 
│ │ │ │ -00022f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f60: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00022f70: 2020 3120 2020 2020 3120 2020 2020 3120    1     1     1 
│ │ │ │ -00022f80: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00022f90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00022d80: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00022d90: 3120 2020 2020 2031 2020 2020 2031 2020  1      1     1  
│ │ │ │ +00022da0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022dc0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ +00022dd0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ +00022de0: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ +00022df0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e10: 2020 2020 2020 4346 203d 3e20 3930 6820        CF => 90h 
│ │ │ │ +00022e20: 202d 2031 3268 2020 2b20 3330 6820 202b   - 12h  + 30h  +
│ │ │ │ +00022e30: 2031 3268 2020 2b20 3668 2020 2020 2020   12h  + 6h      
│ │ │ │ +00022e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e60: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00022e70: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00022e80: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00022e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022eb0: 2020 2020 2020 2020 2020 2020 2036 2020               6  
│ │ │ │ +00022ec0: 2020 2035 2020 2020 2034 2020 2020 2033     5     4     3
│ │ │ │ +00022ed0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00022ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f00: 2020 2020 2020 4720 3d3e 2033 6820 202b        G => 3h  +
│ │ │ │ +00022f10: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ +00022f20: 202b 2033 6820 202b 2033 6820 202b 2031   + 3h  + 3h  + 1
│ │ │ │ +00022f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f50: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00022f60: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ +00022f70: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00022f80: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022ff0: 3130 203a 2073 6567 3d53 6567 7265 2820  10 : seg=Segre( 
│ │ │ │ -00023000: 492c 204f 7574 7075 743d 3e48 6173 6846  I, Output=>HashF
│ │ │ │ -00023010: 6f72 6d29 2020 2020 2020 2020 2020 2020  orm)            
│ │ │ │ -00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00022fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00022fe0: 6931 3020 3a20 7365 673d 5365 6772 6528  i10 : seg=Segre(
│ │ │ │ +00022ff0: 2049 2c20 4f75 7470 7574 3d3e 4861 7368   I, Output=>Hash
│ │ │ │ +00023000: 466f 726d 2920 2020 2020 2020 2020 2020  Form)           
│ │ │ │ +00023010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023080: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00023090: 3130 203d 204d 7574 6162 6c65 4861 7368  10 = MutableHash
│ │ │ │ -000230a0: 5461 626c 657b 2e2e 2e34 2e2e 2e7d 2020  Table{...4...}  
│ │ │ │ +00023070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023080: 6f31 3020 3d20 4d75 7461 626c 6548 6173  o10 = MutableHas
│ │ │ │ +00023090: 6854 6162 6c65 7b2e 2e2e 342e 2e2e 7d20  hTable{...4...} 
│ │ │ │ +000230a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000230c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000230d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023120: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00023130: 3130 203a 204d 7574 6162 6c65 4861 7368  10 : MutableHash
│ │ │ │ -00023140: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +00023110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023120: 6f31 3020 3a20 4d75 7461 626c 6548 6173  o10 : MutableHas
│ │ │ │ +00023130: 6854 6162 6c65 2020 2020 2020 2020 2020  hTable          
│ │ │ │ +00023140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023170: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00023160: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00023170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000231a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000231b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000231d0: 3131 203a 2070 6565 6b20 7365 6720 2020  11 : peek seg   
│ │ │ │ +000231b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000231c0: 6931 3120 3a20 7065 656b 2073 6567 2020  i11 : peek seg  
│ │ │ │ +000231d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023290: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000232a0: 2020 2033 2020 2020 2020 3420 2020 2020     3      4     
│ │ │ │ -000232b0: 2020 3520 2020 2020 2020 2020 7c0a 7c6f    5         |.|o
│ │ │ │ -000232c0: 3131 203d 204d 7574 6162 6c65 4861 7368  11 = MutableHash
│ │ │ │ -000232d0: 5461 626c 657b 5365 6772 654c 6973 7420  Table{SegreList 
│ │ │ │ -000232e0: 3d3e 207b 302c 2030 2c20 3668 202c 202d  => {0, 0, 6h , -
│ │ │ │ -000232f0: 3330 6820 2c20 3131 3468 202c 202d 3339  30h , 114h , -39
│ │ │ │ -00023300: 3068 202c 2020 2020 2020 2020 7c0a 7c20  0h ,        |.| 
│ │ │ │ +00023280: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00023290: 2020 2020 3320 2020 2020 2034 2020 2020      3      4    
│ │ │ │ +000232a0: 2020 2035 2020 2020 2020 2020 207c 0a7c     5         |.|
│ │ │ │ +000232b0: 6f31 3120 3d20 4d75 7461 626c 6548 6173  o11 = MutableHas
│ │ │ │ +000232c0: 6854 6162 6c65 7b53 6567 7265 4c69 7374  hTable{SegreList
│ │ │ │ +000232d0: 203d 3e20 7b30 2c20 302c 2036 6820 2c20   => {0, 0, 6h , 
│ │ │ │ +000232e0: 2d33 3068 202c 2031 3134 6820 2c20 2d33  -30h , 114h , -3
│ │ │ │ +000232f0: 3930 6820 2c20 2020 2020 2020 207c 0a7c  90h ,        |.|
│ │ │ │ +00023300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023330: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00023340: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00023350: 2020 3120 2020 2020 2020 2020 7c0a 7c20    1         |.| 
│ │ │ │ +00023320: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00023330: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00023340: 2020 2031 2020 2020 2020 2020 207c 0a7c     1         |.|
│ │ │ │ +00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023380: 2020 2020 2020 2020 2020 3220 2020 2033            2    3
│ │ │ │ -00023390: 2020 2020 3420 2020 2035 2020 2020 3620      4    5    6 
│ │ │ │ -000233a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000233b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000233c0: 2020 2020 2020 476c 6973 7420 3d3e 207b        Glist => {
│ │ │ │ -000233d0: 312c 2033 6820 2c20 3368 202c 2033 6820  1, 3h , 3h , 3h 
│ │ │ │ -000233e0: 2c20 3368 202c 2033 6820 2c20 3368 207d  , 3h , 3h , 3h }
│ │ │ │ -000233f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023370: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00023380: 3320 2020 2034 2020 2020 3520 2020 2036  3    4    5    6
│ │ │ │ +00023390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000233a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233b0: 2020 2020 2020 2047 6c69 7374 203d 3e20         Glist => 
│ │ │ │ +000233c0: 7b31 2c20 3368 202c 2033 6820 2c20 3368  {1, 3h , 3h , 3h
│ │ │ │ +000233d0: 202c 2033 6820 2c20 3368 202c 2033 6820   , 3h , 3h , 3h 
│ │ │ │ +000233e0: 7d20 2020 2020 2020 2020 2020 207c 0a7c  }            |.|
│ │ │ │ +000233f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023420: 2020 2020 2031 2020 2020 3120 2020 2031       1    1    1
│ │ │ │ -00023430: 2020 2020 3120 2020 2031 2020 2020 3120      1    1    1 
│ │ │ │ -00023440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023410: 2020 2020 2020 3120 2020 2031 2020 2020        1    1    
│ │ │ │ +00023420: 3120 2020 2031 2020 2020 3120 2020 2031  1    1    1    1
│ │ │ │ +00023430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023470: 2020 2020 3620 2020 2020 2020 3520 2020      6       5   
│ │ │ │ -00023480: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ -00023490: 2032 2020 2020 2020 2020 2020 7c0a 7c20   2          |.| 
│ │ │ │ -000234a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234b0: 2020 2020 2020 5365 6772 6520 3d3e 2031        Segre => 1
│ │ │ │ -000234c0: 3236 3668 2020 2d20 3339 3068 2020 2b20  266h  - 390h  + 
│ │ │ │ -000234d0: 3131 3468 2020 2d20 3330 6820 202b 2036  114h  - 30h  + 6
│ │ │ │ -000234e0: 6820 2020 2020 2020 2020 2020 7c0a 7c20  h           |.| 
│ │ │ │ +00023460: 2020 2020 2036 2020 2020 2020 2035 2020       6       5  
│ │ │ │ +00023470: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00023480: 2020 3220 2020 2020 2020 2020 207c 0a7c    2          |.|
│ │ │ │ +00023490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000234a0: 2020 2020 2020 2053 6567 7265 203d 3e20         Segre => 
│ │ │ │ +000234b0: 3132 3636 6820 202d 2033 3930 6820 202b  1266h  - 390h  +
│ │ │ │ +000234c0: 2031 3134 6820 202d 2033 3068 2020 2b20   114h  - 30h  + 
│ │ │ │ +000234d0: 3668 2020 2020 2020 2020 2020 207c 0a7c  6h           |.|
│ │ │ │ +000234e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023510: 2020 2020 3120 2020 2020 2020 3120 2020      1       1   
│ │ │ │ -00023520: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00023530: 2031 2020 2020 2020 2020 2020 7c0a 7c20   1          |.| 
│ │ │ │ -00023540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023550: 2020 2020 2020 2020 2020 2020 2036 2020               6  
│ │ │ │ -00023560: 2020 2035 2020 2020 2034 2020 2020 2033     5     4     3
│ │ │ │ -00023570: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00023580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00023590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235a0: 2020 2020 2020 4720 3d3e 2033 6820 202b        G => 3h  +
│ │ │ │ -000235b0: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ -000235c0: 202b 2033 6820 202b 2033 6820 202b 2031   + 3h  + 3h  + 1
│ │ │ │ -000235d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000235e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235f0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -00023600: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ -00023610: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00023620: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00023500: 2020 2020 2031 2020 2020 2020 2031 2020       1       1  
│ │ │ │ +00023510: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00023520: 2020 3120 2020 2020 2020 2020 207c 0a7c    1          |.|
│ │ │ │ +00023530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023540: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ +00023550: 2020 2020 3520 2020 2020 3420 2020 2020      5     4     
│ │ │ │ +00023560: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │ +00023570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023590: 2020 2020 2020 2047 203d 3e20 3368 2020         G => 3h  
│ │ │ │ +000235a0: 2b20 3368 2020 2b20 3368 2020 2b20 3368  + 3h  + 3h  + 3h
│ │ │ │ +000235b0: 2020 2b20 3368 2020 2b20 3368 2020 2b20    + 3h  + 3h  + 
│ │ │ │ +000235c0: 3120 2020 2020 2020 2020 2020 207c 0a7c  1            |.|
│ │ │ │ +000235d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000235e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000235f0: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00023600: 3120 2020 2020 3120 2020 2020 3120 2020  1     1     1   
│ │ │ │ +00023610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -00023680: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ +00023660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00023670: 2020 2020 2036 2020 2020 2020 2020 2020       6          
│ │ │ │ +00023680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236c0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ -000236d0: 3236 3668 207d 7d20 2020 2020 2020 2020  266h }}         
│ │ │ │ +000236b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000236c0: 3132 3636 6820 7d7d 2020 2020 2020 2020  1266h }}        
│ │ │ │ +000236d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00023720: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00023700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023710: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00023720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023760: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00023750: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00023760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000237a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000237b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000237c0: 3132 203a 2065 753d 4575 6c65 7228 2049  12 : eu=Euler( I
│ │ │ │ -000237d0: 2c20 4f75 7470 7574 3d3e 4861 7368 466f  , Output=>HashFo
│ │ │ │ -000237e0: 726d 2920 2020 2020 2020 2020 2020 2020  rm)             
│ │ │ │ -000237f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000237a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000237b0: 6931 3220 3a20 6575 3d45 756c 6572 2820  i12 : eu=Euler( 
│ │ │ │ +000237c0: 492c 204f 7574 7075 743d 3e48 6173 6846  I, Output=>HashF
│ │ │ │ +000237d0: 6f72 6d29 2020 2020 2020 2020 2020 2020  orm)            
│ │ │ │ +000237e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000237f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023850: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00023860: 3132 203d 204d 7574 6162 6c65 4861 7368  12 = MutableHash
│ │ │ │ -00023870: 5461 626c 657b 2e2e 2e35 2e2e 2e7d 2020  Table{...5...}  
│ │ │ │ +00023840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023850: 6f31 3220 3d20 4d75 7461 626c 6548 6173  o12 = MutableHas
│ │ │ │ +00023860: 6854 6162 6c65 7b2e 2e2e 352e 2e2e 7d20  hTable{...5...} 
│ │ │ │ +00023870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000238a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00023900: 3132 203a 204d 7574 6162 6c65 4861 7368  12 : MutableHash
│ │ │ │ -00023910: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +000238e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000238f0: 6f31 3220 3a20 4d75 7461 626c 6548 6173  o12 : MutableHas
│ │ │ │ +00023900: 6854 6162 6c65 2020 2020 2020 2020 2020  hTable          
│ │ │ │ +00023910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00023930: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00023940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000239a0: 3133 203a 2070 6565 6b20 6575 2020 2020  13 : peek eu    
│ │ │ │ +00023980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00023990: 6931 3320 3a20 7065 656b 2065 7520 2020  i13 : peek eu   
│ │ │ │ +000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000239d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000239e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00023a40: 3133 203d 204d 7574 6162 6c65 4861 7368  13 = MutableHash
│ │ │ │ -00023a50: 5461 626c 657b 4575 6c65 7220 3d3e 2031  Table{Euler => 1
│ │ │ │ -00023a60: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a80: 2020 2020 207d 2020 2020 2020 7c0a 7c20       }      |.| 
│ │ │ │ +00023a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023a30: 6f31 3320 3d20 4d75 7461 626c 6548 6173  o13 = MutableHas
│ │ │ │ +00023a40: 6854 6162 6c65 7b45 756c 6572 203d 3e20  hTable{Euler => 
│ │ │ │ +00023a50: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │ +00023a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a70: 2020 2020 2020 7d20 2020 2020 207c 0a7c        }      |.|
│ │ │ │ +00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ab0: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ -00023ac0: 3420 2020 2020 2033 2020 2020 2020 3220  4      3      2 
│ │ │ │ -00023ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00023ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023af0: 2020 2020 2020 7b30 2c20 317d 203d 3e20        {0, 1} => 
│ │ │ │ -00023b00: 3268 2020 2b20 3233 6820 202b 2033 3268  2h  + 23h  + 32h
│ │ │ │ -00023b10: 2020 2b20 3333 6820 202b 2031 3868 2020    + 33h  + 18h  
│ │ │ │ -00023b20: 2b20 3568 2020 2020 2020 2020 7c0a 7c20  + 5h        |.| 
│ │ │ │ +00023aa0: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ +00023ab0: 2034 2020 2020 2020 3320 2020 2020 2032   4      3      2
│ │ │ │ +00023ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ae0: 2020 2020 2020 207b 302c 2031 7d20 3d3e         {0, 1} =>
│ │ │ │ +00023af0: 2032 6820 202b 2032 3368 2020 2b20 3332   2h  + 23h  + 32
│ │ │ │ +00023b00: 6820 202b 2033 3368 2020 2b20 3138 6820  h  + 33h  + 18h 
│ │ │ │ +00023b10: 202b 2035 6820 2020 2020 2020 207c 0a7c   + 5h        |.|
│ │ │ │ +00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b50: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00023b60: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00023b70: 2020 2020 3120 2020 2020 2020 7c0a 7c20      1       |.| 
│ │ │ │ +00023b40: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00023b50: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00023b60: 2020 2020 2031 2020 2020 2020 207c 0a7c       1       |.|
│ │ │ │ +00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ba0: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ -00023bb0: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023be0: 2020 2020 2020 4353 4d20 3d3e 2031 3068        CSM => 10h
│ │ │ │ -00023bf0: 2020 2b20 3132 6820 202b 2032 3268 2020    + 12h  + 22h  
│ │ │ │ -00023c00: 2b20 3136 6820 202b 2036 6820 2020 2020  + 16h  + 6h     
│ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023b90: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ +00023ba0: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ +00023bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bd0: 2020 2020 2020 2043 534d 203d 3e20 3130         CSM => 10
│ │ │ │ +00023be0: 6820 202b 2031 3268 2020 2b20 3232 6820  h  + 12h  + 22h 
│ │ │ │ +00023bf0: 202b 2031 3668 2020 2b20 3668 2020 2020   + 16h  + 6h    
│ │ │ │ +00023c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c40: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00023c50: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00023c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023c30: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00023c40: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ +00023c50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c80: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00023c90: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00023ca0: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ -00023cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00023cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cd0: 2020 2020 2020 7b30 7d20 3d3e 2036 6820        {0} => 6h 
│ │ │ │ -00023ce0: 202b 2031 3868 2020 2b20 3236 6820 202b   + 18h  + 26h  +
│ │ │ │ -00023cf0: 2032 3268 2020 2b20 3130 6820 202b 2032   22h  + 10h  + 2
│ │ │ │ -00023d00: 6820 2020 2020 2020 2020 2020 7c0a 7c20  h           |.| 
│ │ │ │ +00023c80: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ +00023c90: 2020 2020 2033 2020 2020 2020 3220 2020       3      2   
│ │ │ │ +00023ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023cc0: 2020 2020 2020 207b 307d 203d 3e20 3668         {0} => 6h
│ │ │ │ +00023cd0: 2020 2b20 3138 6820 202b 2032 3668 2020    + 18h  + 26h  
│ │ │ │ +00023ce0: 2b20 3232 6820 202b 2031 3068 2020 2b20  + 22h  + 10h  + 
│ │ │ │ +00023cf0: 3268 2020 2020 2020 2020 2020 207c 0a7c  2h           |.|
│ │ │ │ +00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d20: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00023d30: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00023d40: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00023d50: 2031 2020 2020 2020 2020 2020 7c0a 7c20   1          |.| 
│ │ │ │ +00023d20: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00023d30: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00023d40: 2020 3120 2020 2020 2020 2020 207c 0a7c    1          |.|
│ │ │ │ +00023d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d70: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00023d80: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00023d90: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ -00023da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dc0: 2020 2020 2020 7b31 7d20 3d3e 2036 6820        {1} => 6h 
│ │ │ │ -00023dd0: 202b 2031 3768 2020 2b20 3238 6820 202b   + 17h  + 28h  +
│ │ │ │ -00023de0: 2032 3768 2020 2b20 3134 6820 202b 2033   27h  + 14h  + 3
│ │ │ │ -00023df0: 6820 2020 2020 2020 2020 2020 7c0a 7c20  h           |.| 
│ │ │ │ +00023d70: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ +00023d80: 2020 2020 2033 2020 2020 2020 3220 2020       3      2   
│ │ │ │ +00023d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023db0: 2020 2020 2020 207b 317d 203d 3e20 3668         {1} => 6h
│ │ │ │ +00023dc0: 2020 2b20 3137 6820 202b 2032 3868 2020    + 17h  + 28h  
│ │ │ │ +00023dd0: 2b20 3237 6820 202b 2031 3468 2020 2b20  + 27h  + 14h  + 
│ │ │ │ +00023de0: 3368 2020 2020 2020 2020 2020 207c 0a7c  3h           |.|
│ │ │ │ +00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e10: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00023e20: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00023e30: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00023e40: 2031 2020 2020 2020 2020 2020 7c0a 2b2d   1          |.+-
│ │ │ │ +00023e10: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00023e20: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00023e30: 2020 3120 2020 2020 2020 2020 207c 0a2b    1          |.+
│ │ │ │ +00023e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54  ------------+..T
│ │ │ │ -00023ea0: 6865 204d 7574 6162 6c65 4861 7368 5461  he MutableHashTa
│ │ │ │ -00023eb0: 626c 6520 7265 7475 726e 6564 2077 6974  ble returned wit
│ │ │ │ -00023ec0: 6820 7468 6520 6f70 7469 6f6e 204f 7574  h the option Out
│ │ │ │ -00023ed0: 7075 743d 3e48 6173 6846 6f72 6d20 636f  put=>HashForm co
│ │ │ │ -00023ee0: 6e74 6169 6e73 0a64 6966 6665 7265 6e74  ntains.different
│ │ │ │ -00023ef0: 2069 6e66 6f72 6d61 7469 6f6e 2064 6570   information dep
│ │ │ │ -00023f00: 656e 6469 6e67 206f 6e20 7468 6520 6d65  ending on the me
│ │ │ │ -00023f10: 7468 6f64 2077 6974 6820 7768 6963 6820  thod with which 
│ │ │ │ -00023f20: 6974 2069 7320 7573 6564 2e0a 4164 6469  it is used..Addi
│ │ │ │ -00023f30: 7469 6f6e 616c 6c79 2069 6620 7468 6520  tionally if the 
│ │ │ │ -00023f40: 6f70 7469 6f6e 202a 6e6f 7465 2049 6e70  option *note Inp
│ │ │ │ -00023f50: 7574 4973 536d 6f6f 7468 3a20 496e 7075  utIsSmooth: Inpu
│ │ │ │ -00023f60: 7449 7353 6d6f 6f74 682c 2069 7320 7573  tIsSmooth, is us
│ │ │ │ -00023f70: 6564 2074 6865 6e20 7468 650a 6861 7368  ed then the.hash
│ │ │ │ -00023f80: 2074 6162 6c65 2072 6574 7572 6e65 6420   table returned 
│ │ │ │ -00023f90: 6279 2074 6865 206d 6574 686f 6473 2045  by the methods E
│ │ │ │ -00023fa0: 756c 6572 2061 6e64 2043 534d 2077 696c  uler and CSM wil
│ │ │ │ -00023fb0: 6c20 6265 2074 6865 2073 616d 6520 6173  l be the same as
│ │ │ │ -00023fc0: 2074 6861 740a 7265 7475 726e 6564 2062   that.returned b
│ │ │ │ -00023fd0: 7920 4368 6572 6e2e 2057 6865 6e20 7573  y Chern. When us
│ │ │ │ -00023fe0: 696e 6720 7468 6520 2a6e 6f74 6520 4353  ing the *note CS
│ │ │ │ -00023ff0: 4d3a 2043 534d 2c20 2063 6f6d 6d61 6e64  M: CSM,  command
│ │ │ │ -00024000: 2069 6e20 7468 6520 6465 6661 756c 740a   in the default.
│ │ │ │ -00024010: 636f 6e66 6967 7572 6174 696f 6e73 2028  configurations (
│ │ │ │ -00024020: 7468 6174 2069 7320 2a6e 6f74 6520 4d65  that is *note Me
│ │ │ │ -00024030: 7468 6f64 3a20 4d65 7468 6f64 2c3d 3e49  thod: Method,=>I
│ │ │ │ -00024040: 6e63 6c75 7369 6f6e 4578 636c 7573 696f  nclusionExclusio
│ │ │ │ -00024050: 6e2c 202a 6e6f 7465 0a43 6f6d 704d 6574  n, *note.CompMet
│ │ │ │ -00024060: 686f 643a 2043 6f6d 704d 6574 686f 642c  hod: CompMethod,
│ │ │ │ -00024070: 3d3e 5072 6f6a 6563 7469 7665 4465 6772  =>ProjectiveDegr
│ │ │ │ -00024080: 6565 2920 7468 6572 6520 6973 2074 6865  ee) there is the
│ │ │ │ -00024090: 2061 6464 6974 696f 6e61 6c20 6f70 7469   additional opti
│ │ │ │ -000240a0: 6f6e 2074 6f0a 7365 7420 4f75 7470 7574  on to.set Output
│ │ │ │ -000240b0: 3d3e 4861 7368 466f 726d 584c 2e20 5468  =>HashFormXL. Th
│ │ │ │ -000240c0: 6973 2072 6574 7572 6e73 2061 6c6c 2074  is returns all t
│ │ │ │ -000240d0: 6865 2075 7375 616c 2069 6e66 6f72 6d61  he usual informa
│ │ │ │ -000240e0: 7469 6f6e 2074 6861 740a 4f75 7470 7574  tion that.Output
│ │ │ │ -000240f0: 3d3e 4861 7368 466f 726d 2077 6f75 6c64  =>HashForm would
│ │ │ │ -00024100: 2066 6f72 2074 6869 7320 636f 6e66 6967   for this config
│ │ │ │ -00024110: 7572 6174 696f 6e20 7769 7468 2074 6865  uration with the
│ │ │ │ -00024120: 2061 6464 6974 696f 6e20 6f66 2074 6865   addition of the
│ │ │ │ -00024130: 0a70 726f 6a65 6374 6976 6520 6465 6772  .projective degr
│ │ │ │ -00024140: 6565 7320 616e 6420 5365 6772 6520 636c  ees and Segre cl
│ │ │ │ -00024150: 6173 7365 7320 6f66 2073 696e 6775 6c61  asses of singula
│ │ │ │ -00024160: 7269 7479 2073 7562 7363 6865 6d65 7320  rity subschemes 
│ │ │ │ -00024170: 6765 6e65 7261 7465 6420 6279 2074 6865  generated by the
│ │ │ │ -00024180: 0a68 7970 6572 7375 7266 6163 6573 2063  .hypersurfaces c
│ │ │ │ -00024190: 6f6e 7369 6465 7265 6420 696e 2074 6865  onsidered in the
│ │ │ │ -000241a0: 2069 6e63 6c75 7369 6f6e 2f65 7863 6c75   inclusion/exclu
│ │ │ │ -000241b0: 7369 6f6e 2070 726f 6365 6475 7265 2c20  sion procedure, 
│ │ │ │ -000241c0: 7468 6174 2069 7320 696e 0a66 696e 6469  that is in.findi
│ │ │ │ -000241d0: 6e67 2074 6865 2043 534d 2063 6c61 7373  ng the CSM class
│ │ │ │ -000241e0: 206f 6620 616c 6c20 6879 7065 7273 7572   of all hypersur
│ │ │ │ -000241f0: 6661 6365 7320 6765 6e65 7261 7465 6420  faces generated 
│ │ │ │ -00024200: 6279 2074 616b 696e 6720 6120 7072 6f64  by taking a prod
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│ │ │ │ -00024240: 6465 616c 2e20 4e6f 7465 2074 6861 742c  deal. Note that,
│ │ │ │ -00024250: 2073 696e 6365 2074 6865 2043 534d 2063   since the CSM c
│ │ │ │ -00024260: 6c61 7373 0a6f 6620 6120 7375 6273 6368  lass.of a subsch
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│ │ │ │ -000242a0: 6f72 2065 7175 6976 616c 656e 746c 7920  or equivalently 
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│ │ │ │ -000242f0: 636c 6173 7320 6f66 2074 6865 0a72 6164  class of the.rad
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│ │ │ │ -00024340: 6f72 0a65 6666 6963 6965 6e63 7920 7265  or.efficiency re
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│ │ │ │ -00024360: 6520 7072 6f6a 6563 7469 7665 2064 6567  e projective deg
│ │ │ │ -00024370: 7265 6573 2061 6e64 2053 6567 7265 2063  rees and Segre c
│ │ │ │ -00024380: 6c61 7373 6573 2063 6f6d 7075 7465 640a  lasses computed.
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│ │ │ │ -000243a0: 6265 2074 686f 7365 206f 6620 7468 6520  be those of the 
│ │ │ │ -000243b0: 7261 6469 6361 6c20 6f66 2061 6e20 6964  radical of an id
│ │ │ │ -000243c0: 6561 6c20 6465 6669 6e65 6420 6279 2061  eal defined by a
│ │ │ │ -000243d0: 2070 6f6c 796e 6f6d 6961 6c0a 7768 6963   polynomial.whic
│ │ │ │ -000243e0: 6820 6973 2061 2070 726f 6475 6374 206f  h is a product o
│ │ │ │ -000243f0: 6620 736f 6d65 2073 7562 7365 7420 6f66  f some subset of
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│ │ │ │ -00024430: 706c 6520 6265 6c6f 772e 0a0a 2b2d 2d2d  ple below...+---
│ │ │ │ +00023e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00023e90: 5468 6520 4d75 7461 626c 6548 6173 6854  The MutableHashT
│ │ │ │ +00023ea0: 6162 6c65 2072 6574 7572 6e65 6420 7769  able returned wi
│ │ │ │ +00023eb0: 7468 2074 6865 206f 7074 696f 6e20 4f75  th the option Ou
│ │ │ │ +00023ec0: 7470 7574 3d3e 4861 7368 466f 726d 2063  tput=>HashForm c
│ │ │ │ +00023ed0: 6f6e 7461 696e 730a 6469 6666 6572 656e  ontains.differen
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│ │ │ │ +00023f30: 206f 7074 696f 6e20 2a6e 6f74 6520 496e   option *note In
│ │ │ │ +00023f40: 7075 7449 7353 6d6f 6f74 683a 2049 6e70  putIsSmooth: Inp
│ │ │ │ +00023f50: 7574 4973 536d 6f6f 7468 2c20 6973 2075  utIsSmooth, is u
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│ │ │ │ +00023f80: 2062 7920 7468 6520 6d65 7468 6f64 7320   by the methods 
│ │ │ │ +00023f90: 4575 6c65 7220 616e 6420 4353 4d20 7769  Euler and CSM wi
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│ │ │ │ +00023fc0: 6279 2043 6865 726e 2e20 5768 656e 2075  by Chern. When u
│ │ │ │ +00023fd0: 7369 6e67 2074 6865 202a 6e6f 7465 2043  sing the *note C
│ │ │ │ +00023fe0: 534d 3a20 4353 4d2c 2020 636f 6d6d 616e  SM: CSM,  comman
│ │ │ │ +00023ff0: 6420 696e 2074 6865 2064 6566 6175 6c74  d in the default
│ │ │ │ +00024000: 0a63 6f6e 6669 6775 7261 7469 6f6e 7320  .configurations 
│ │ │ │ +00024010: 2874 6861 7420 6973 202a 6e6f 7465 204d  (that is *note M
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│ │ │ │ +00024030: 496e 636c 7573 696f 6e45 7863 6c75 7369  InclusionExclusi
│ │ │ │ +00024040: 6f6e 2c20 2a6e 6f74 650a 436f 6d70 4d65  on, *note.CompMe
│ │ │ │ +00024050: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64  thod: CompMethod
│ │ │ │ +00024060: 2c3d 3e50 726f 6a65 6374 6976 6544 6567  ,=>ProjectiveDeg
│ │ │ │ +00024070: 7265 6529 2074 6865 7265 2069 7320 7468  ree) there is th
│ │ │ │ +00024080: 6520 6164 6469 7469 6f6e 616c 206f 7074  e additional opt
│ │ │ │ +00024090: 696f 6e20 746f 0a73 6574 204f 7574 7075  ion to.set Outpu
│ │ │ │ +000240a0: 743d 3e48 6173 6846 6f72 6d58 4c2e 2054  t=>HashFormXL. T
│ │ │ │ +000240b0: 6869 7320 7265 7475 726e 7320 616c 6c20  his returns all 
│ │ │ │ +000240c0: 7468 6520 7573 7561 6c20 696e 666f 726d  the usual inform
│ │ │ │ +000240d0: 6174 696f 6e20 7468 6174 0a4f 7574 7075  ation that.Outpu
│ │ │ │ +000240e0: 743d 3e48 6173 6846 6f72 6d20 776f 756c  t=>HashForm woul
│ │ │ │ +000240f0: 6420 666f 7220 7468 6973 2063 6f6e 6669  d for this confi
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│ │ │ │ +00024110: 6520 6164 6469 7469 6f6e 206f 6620 7468  e addition of th
│ │ │ │ +00024120: 650a 7072 6f6a 6563 7469 7665 2064 6567  e.projective deg
│ │ │ │ +00024130: 7265 6573 2061 6e64 2053 6567 7265 2063  rees and Segre c
│ │ │ │ +00024140: 6c61 7373 6573 206f 6620 7369 6e67 756c  lasses of singul
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│ │ │ │ +00024160: 2067 656e 6572 6174 6564 2062 7920 7468   generated by th
│ │ │ │ +00024170: 650a 6879 7065 7273 7572 6661 6365 7320  e.hypersurfaces 
│ │ │ │ +00024180: 636f 6e73 6964 6572 6564 2069 6e20 7468  considered in th
│ │ │ │ +00024190: 6520 696e 636c 7573 696f 6e2f 6578 636c  e inclusion/excl
│ │ │ │ +000241a0: 7573 696f 6e20 7072 6f63 6564 7572 652c  usion procedure,
│ │ │ │ +000241b0: 2074 6861 7420 6973 2069 6e0a 6669 6e64   that is in.find
│ │ │ │ +000241c0: 696e 6720 7468 6520 4353 4d20 636c 6173  ing the CSM clas
│ │ │ │ +000241d0: 7320 6f66 2061 6c6c 2068 7970 6572 7375  s of all hypersu
│ │ │ │ +000241e0: 7266 6163 6573 2067 656e 6572 6174 6564  rfaces generated
│ │ │ │ +000241f0: 2062 7920 7461 6b69 6e67 2061 2070 726f   by taking a pro
│ │ │ │ +00024200: 6475 6374 206f 660a 736f 6d65 2073 7562  duct of.some sub
│ │ │ │ +00024210: 7365 7473 206f 6620 6765 6e65 7261 746f  sets of generato
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│ │ │ │ +00024230: 6964 6561 6c2e 204e 6f74 6520 7468 6174  ideal. Note that
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│ │ │ │ +00024250: 636c 6173 730a 6f66 2061 2073 7562 7363  class.of a subsc
│ │ │ │ +00024260: 6865 6d65 2065 7175 616c 7320 7468 6520  heme equals the 
│ │ │ │ +00024270: 4353 4d20 636c 6173 7320 6f66 2069 7473  CSM class of its
│ │ │ │ +00024280: 2072 6564 7563 6564 2073 6368 656d 652c   reduced scheme,
│ │ │ │ +00024290: 206f 7220 6571 7569 7661 6c65 6e74 6c79   or equivalently
│ │ │ │ +000242a0: 2066 6f72 0a75 7320 7468 6520 4353 4d20   for.us the CSM 
│ │ │ │ +000242b0: 636c 6173 7320 636f 7272 6573 706f 6e64  class correspond
│ │ │ │ +000242c0: 696e 6720 746f 2061 6e20 6964 6561 6c20  ing to an ideal 
│ │ │ │ +000242d0: 4920 6571 7561 6c73 2074 6865 2043 534d  I equals the CSM
│ │ │ │ +000242e0: 2063 6c61 7373 206f 6620 7468 650a 7261   class of the.ra
│ │ │ │ +000242f0: 6469 6361 6c20 6f66 2049 2c20 7468 656e  dical of I, then
│ │ │ │ +00024300: 2069 6e74 6572 6e61 6c6c 7920 7765 2061   internally we a
│ │ │ │ +00024310: 6c77 6179 7320 776f 726b 2077 6974 6820  lways work with 
│ │ │ │ +00024320: 7261 6469 6361 6c20 6964 6561 6c73 2028  radical ideals (
│ │ │ │ +00024330: 666f 720a 6566 6669 6369 656e 6379 2072  for.efficiency r
│ │ │ │ +00024340: 6561 736f 6e73 292e 2048 656e 6365 2074  easons). Hence t
│ │ │ │ +00024350: 6865 2070 726f 6a65 6374 6976 6520 6465  he projective de
│ │ │ │ +00024360: 6772 6565 7320 616e 6420 5365 6772 6520  grees and Segre 
│ │ │ │ +00024370: 636c 6173 7365 7320 636f 6d70 7574 6564  classes computed
│ │ │ │ +00024380: 0a69 6e74 6572 6e61 6c6c 7920 7769 6c6c  .internally will
│ │ │ │ +00024390: 2062 6520 7468 6f73 6520 6f66 2074 6865   be those of the
│ │ │ │ +000243a0: 2072 6164 6963 616c 206f 6620 616e 2069   radical of an i
│ │ │ │ +000243b0: 6465 616c 2064 6566 696e 6564 2062 7920  deal defined by 
│ │ │ │ +000243c0: 6120 706f 6c79 6e6f 6d69 616c 0a77 6869  a polynomial.whi
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│ │ │ │ +000243e0: 6f66 2073 6f6d 6520 7375 6273 6574 206f  of some subset o
│ │ │ │ +000243f0: 6620 7468 6520 6765 6e65 7261 746f 7273  f the generators
│ │ │ │ +00024400: 2e20 5765 2069 6c6c 7573 7472 6174 6520  . We illustrate 
│ │ │ │ +00024410: 7468 6973 2077 6974 6820 616e 0a65 7861  this with an.exa
│ │ │ │ +00024420: 6d70 6c65 2062 656c 6f77 2e0a 0a2b 2d2d  mple below...+--
│ │ │ │ +00024430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024480: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ -00024490: 203a 2063 736d 584c 6861 7368 3d43 534d   : csmXLhash=CSM
│ │ │ │ -000244a0: 2841 2c49 2c4f 7574 7075 743d 3e48 6173  (A,I,Output=>Has
│ │ │ │ -000244b0: 6846 6f72 6d58 4c29 2020 2020 2020 2020  hFormXL)        
│ │ │ │ -000244c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00024470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00024480: 3420 3a20 6373 6d58 4c68 6173 683d 4353  4 : csmXLhash=CS
│ │ │ │ +00024490: 4d28 412c 492c 4f75 7470 7574 3d3e 4861  M(A,I,Output=>Ha
│ │ │ │ +000244a0: 7368 466f 726d 584c 2920 2020 2020 2020  shFormXL)       
│ │ │ │ +000244b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024520: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -00024530: 203d 204d 7574 6162 6c65 4861 7368 5461   = MutableHashTa
│ │ │ │ -00024540: 626c 657b 2e2e 2e31 302e 2e2e 7d20 2020  ble{...10...}   
│ │ │ │ +00024510: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00024520: 3420 3d20 4d75 7461 626c 6548 6173 6854  4 = MutableHashT
│ │ │ │ +00024530: 6162 6c65 7b2e 2e2e 3130 2e2e 2e7d 2020  able{...10...}  
│ │ │ │ +00024540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024570: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00024560: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -000245d0: 203a 204d 7574 6162 6c65 4861 7368 5461   : MutableHashTa
│ │ │ │ -000245e0: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +000245b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000245c0: 3420 3a20 4d75 7461 626c 6548 6173 6854  4 : MutableHashT
│ │ │ │ +000245d0: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +000245e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024610: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00024600: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00024610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024660: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135  ----------+.|i15
│ │ │ │ -00024670: 203a 2070 6565 6b20 6373 6d58 4c68 6173   : peek csmXLhas
│ │ │ │ -00024680: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +00024650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00024660: 3520 3a20 7065 656b 2063 736d 584c 6861  5 : peek csmXLha
│ │ │ │ +00024670: 7368 2020 2020 2020 2020 2020 2020 2020  sh              
│ │ │ │ +00024680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000246a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024700: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -00024710: 203d 204d 7574 6162 6c65 4861 7368 5461   = MutableHashTa
│ │ │ │ -00024720: 626c 657b 4728 4a61 636f 6269 616e 297b  ble{G(Jacobian){
│ │ │ │ -00024730: 307d 203d 3e20 3020 2020 2020 2020 2020  0} => 0         
│ │ │ │ -00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024770: 2020 2020 5365 6772 6528 4a61 636f 6269      Segre(Jacobi
│ │ │ │ -00024780: 616e 297b 307d 203d 3e20 3020 2020 2020  an){0} => 0     
│ │ │ │ -00024790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000246f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00024700: 3520 3d20 4d75 7461 626c 6548 6173 6854  5 = MutableHashT
│ │ │ │ +00024710: 6162 6c65 7b47 284a 6163 6f62 6961 6e29  able{G(Jacobian)
│ │ │ │ +00024720: 7b30 7d20 3d3e 2030 2020 2020 2020 2020  {0} => 0        
│ │ │ │ +00024730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024740: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024760: 2020 2020 2053 6567 7265 284a 6163 6f62       Segre(Jacob
│ │ │ │ +00024770: 6961 6e29 7b30 7d20 3d3e 2030 2020 2020  ian){0} => 0    
│ │ │ │ +00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247e0: 2036 2020 2020 2020 2035 2020 2020 2020   6       5      
│ │ │ │ -000247f0: 2034 2020 2020 2020 2020 7c0a 7c20 2020   4        |.|   
│ │ │ │ -00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024810: 2020 2020 5365 6772 6528 4a61 636f 6269      Segre(Jacobi
│ │ │ │ -00024820: 616e 297b 302c 2031 7d20 3d3e 2033 3930  an){0, 1} => 390
│ │ │ │ -00024830: 6820 202d 2033 3836 6820 202b 2031 3530  h  - 386h  + 150
│ │ │ │ -00024840: 6820 202d 2020 2020 2020 7c0a 7c20 2020  h  -      |.|   
│ │ │ │ +000247d0: 2020 3620 2020 2020 2020 3520 2020 2020    6       5     
│ │ │ │ +000247e0: 2020 3420 2020 2020 2020 207c 0a7c 2020    4        |.|  
│ │ │ │ +000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024800: 2020 2020 2053 6567 7265 284a 6163 6f62       Segre(Jacob
│ │ │ │ +00024810: 6961 6e29 7b30 2c20 317d 203d 3e20 3339  ian){0, 1} => 39
│ │ │ │ +00024820: 3068 2020 2d20 3338 3668 2020 2b20 3135  0h  - 386h  + 15
│ │ │ │ +00024830: 3068 2020 2d20 2020 2020 207c 0a7c 2020  0h  -      |.|  
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│ │ │ │  00024f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00025280: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025290: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +000252a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000252d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000252e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000252f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025330: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │ -00025340: 203a 204b 3d69 6465 616c 2049 5f30 2a49   : K=ideal I_0*I
│ │ │ │ -00025350: 5f31 3b20 2020 2020 2020 2020 2020 2020  _1;             
│ │ │ │ +00025320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00025330: 3620 3a20 4b3d 6964 6561 6c20 495f 302a  6 : K=ideal I_0*
│ │ │ │ +00025340: 495f 313b 2020 2020 2020 2020 2020 2020  I_1;            
│ │ │ │ +00025350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025380: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00025370: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -000253e0: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ +000253c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000253d0: 3620 3a20 4964 6561 6c20 6f66 2052 2020  6 : Ideal of R  
│ │ │ │ +000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025420: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025410: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025470: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │ -00025480: 203a 2043 534d 2841 2c72 6164 6963 616c   : CSM(A,radical
│ │ │ │ -00025490: 204b 293d 3d43 534d 2841 2c4b 2920 2020   K)==CSM(A,K)   
│ │ │ │ +00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00025470: 3720 3a20 4353 4d28 412c 7261 6469 6361  7 : CSM(A,radica
│ │ │ │ +00025480: 6c20 4b29 3d3d 4353 4d28 412c 4b29 2020  l K)==CSM(A,K)  
│ │ │ │ +00025490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000254b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000254c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025510: 2020 2020 2020 2020 2020 7c0a 7c6f 3137            |.|o17
│ │ │ │ -00025520: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00025500: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00025510: 3720 3d20 7472 7565 2020 2020 2020 2020  7 = true        
│ │ │ │ +00025520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025560: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025550: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -000255c0: 203a 204a 3d69 6465 616c 206a 6163 6f62   : J=ideal jacob
│ │ │ │ -000255d0: 6961 6e20 7261 6469 6361 6c20 4b3b 2020  ian radical K;  
│ │ │ │ +000255a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000255b0: 3820 3a20 4a3d 6964 6561 6c20 6a61 636f  8 : J=ideal jaco
│ │ │ │ +000255c0: 6269 616e 2072 6164 6963 616c 204b 3b20  bian radical K; 
│ │ │ │ +000255d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000255e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000255f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000255f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025650: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -00025660: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ +00025640: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00025650: 3820 3a20 4964 6561 6c20 6f66 2052 2020  8 : Ideal of R  
│ │ │ │ +00025660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025690: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000256a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000256b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000256c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000256d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000256e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000256f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139  ----------+.|i19
│ │ │ │ -00025700: 203a 2073 6567 4a3d 5365 6772 6528 412c   : segJ=Segre(A,
│ │ │ │ -00025710: 4a2c 4f75 7470 7574 3d3e 4861 7368 466f  J,Output=>HashFo
│ │ │ │ -00025720: 726d 2920 2020 2020 2020 2020 2020 2020  rm)             
│ │ │ │ -00025730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025740: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000256e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000256f0: 3920 3a20 7365 674a 3d53 6567 7265 2841  9 : segJ=Segre(A
│ │ │ │ +00025700: 2c4a 2c4f 7574 7075 743d 3e48 6173 6846  ,J,Output=>HashF
│ │ │ │ +00025710: 6f72 6d29 2020 2020 2020 2020 2020 2020  orm)            
│ │ │ │ +00025720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025790: 2020 2020 2020 2020 2020 7c0a 7c6f 3139            |.|o19
│ │ │ │ -000257a0: 203d 204d 7574 6162 6c65 4861 7368 5461   = MutableHashTa
│ │ │ │ -000257b0: 626c 657b 2e2e 2e34 2e2e 2e7d 2020 2020  ble{...4...}    
│ │ │ │ +00025780: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00025790: 3920 3d20 4d75 7461 626c 6548 6173 6854  9 = MutableHashT
│ │ │ │ +000257a0: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020  able{...4...}   
│ │ │ │ +000257b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000257d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000257e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025830: 2020 2020 2020 2020 2020 7c0a 7c6f 3139            |.|o19
│ │ │ │ -00025840: 203a 204d 7574 6162 6c65 4861 7368 5461   : MutableHashTa
│ │ │ │ -00025850: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +00025820: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00025830: 3920 3a20 4d75 7461 626c 6548 6173 6854  9 : MutableHashT
│ │ │ │ +00025840: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +00025850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025880: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025870: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000258a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000258b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000258c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000258d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230  ----------+.|i20
│ │ │ │ -000258e0: 203a 2063 736d 584c 6861 7368 2328 2247   : csmXLhash#("G
│ │ │ │ -000258f0: 284a 6163 6f62 6961 6e29 227c 746f 5374  (Jacobian)"|toSt
│ │ │ │ -00025900: 7269 6e67 287b 302c 317d 2929 3d3d 7365  ring({0,1}))==se
│ │ │ │ -00025910: 674a 2322 4722 2020 2020 2020 2020 2020  gJ#"G"          
│ │ │ │ -00025920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000258c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +000258d0: 3020 3a20 6373 6d58 4c68 6173 6823 2822  0 : csmXLhash#("
│ │ │ │ +000258e0: 4728 4a61 636f 6269 616e 2922 7c74 6f53  G(Jacobian)"|toS
│ │ │ │ +000258f0: 7472 696e 6728 7b30 2c31 7d29 293d 3d73  tring({0,1}))==s
│ │ │ │ +00025900: 6567 4a23 2247 2220 2020 2020 2020 2020  egJ#"G"         
│ │ │ │ +00025910: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025970: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ -00025980: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00025960: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00025970: 3020 3d20 7472 7565 2020 2020 2020 2020  0 = true        
│ │ │ │ +00025980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000259b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000259c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000259d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000259e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000259f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │ -00025a20: 203a 2063 736d 584c 6861 7368 2328 2253   : csmXLhash#("S
│ │ │ │ -00025a30: 6567 7265 284a 6163 6f62 6961 6e29 227c  egre(Jacobian)"|
│ │ │ │ -00025a40: 746f 5374 7269 6e67 287b 302c 317d 2929  toString({0,1}))
│ │ │ │ -00025a50: 3d3d 7365 674a 2322 5365 6772 6522 2020  ==segJ#"Segre"  
│ │ │ │ -00025a60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00025a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00025a10: 3120 3a20 6373 6d58 4c68 6173 6823 2822  1 : csmXLhash#("
│ │ │ │ +00025a20: 5365 6772 6528 4a61 636f 6269 616e 2922  Segre(Jacobian)"
│ │ │ │ +00025a30: 7c74 6f53 7472 696e 6728 7b30 2c31 7d29  |toString({0,1})
│ │ │ │ +00025a40: 293d 3d73 6567 4a23 2253 6567 7265 2220  )==segJ#"Segre" 
│ │ │ │ +00025a50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ab0: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ -00025ac0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00025aa0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00025ab0: 3120 3d20 7472 7565 2020 2020 2020 2020  1 = true        
│ │ │ │ +00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025af0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 756e  ----------+..Fun
│ │ │ │ -00025b60: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ -00025b70: 6f6e 616c 2061 7267 756d 656e 7420 6e61  onal argument na
│ │ │ │ -00025b80: 6d65 6420 4f75 7470 7574 3a0a 3d3d 3d3d  med Output:.====
│ │ │ │ +00025b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675  -----------+..Fu
│ │ │ │ +00025b50: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00025b60: 696f 6e61 6c20 6172 6775 6d65 6e74 206e  ional argument n
│ │ │ │ +00025b70: 616d 6564 204f 7574 7075 743a 0a3d 3d3d  amed Output:.===
│ │ │ │ +00025b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00025b90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00025ba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00025bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00025bc0: 2243 6865 726e 282e 2e2e 2c4f 7574 7075  "Chern(...,Outpu
│ │ │ │ -00025bd0: 743d 3e2e 2e2e 2922 202d 2d20 7365 6520  t=>...)" -- see 
│ │ │ │ -00025be0: 2a6e 6f74 6520 4368 6572 6e3a 2043 6865  *note Chern: Che
│ │ │ │ -00025bf0: 726e 2c20 2d2d 2054 6865 2043 6865 726e  rn, -- The Chern
│ │ │ │ -00025c00: 2063 6c61 7373 0a20 202a 2022 4353 4d28   class.  * "CSM(
│ │ │ │ -00025c10: 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e 2e29  ...,Output=>...)
│ │ │ │ -00025c20: 2220 2d2d 2073 6565 202a 6e6f 7465 2043  " -- see *note C
│ │ │ │ -00025c30: 534d 3a20 4353 4d2c 202d 2d20 5468 650a  SM: CSM, -- The.
│ │ │ │ -00025c40: 2020 2020 4368 6572 6e2d 5363 6877 6172      Chern-Schwar
│ │ │ │ -00025c50: 747a 2d4d 6163 5068 6572 736f 6e20 636c  tz-MacPherson cl
│ │ │ │ -00025c60: 6173 730a 2020 2a20 2245 756c 6572 282e  ass.  * "Euler(.
│ │ │ │ -00025c70: 2e2e 2c4f 7574 7075 743d 3e2e 2e2e 2922  ..,Output=>...)"
│ │ │ │ -00025c80: 202d 2d20 7365 6520 2a6e 6f74 6520 4575   -- see *note Eu
│ │ │ │ -00025c90: 6c65 723a 2045 756c 6572 2c20 2d2d 2054  ler: Euler, -- T
│ │ │ │ -00025ca0: 6865 2045 756c 6572 0a20 2020 2043 6861  he Euler.    Cha
│ │ │ │ -00025cb0: 7261 6374 6572 6973 7469 630a 2020 2a20  racteristic.  * 
│ │ │ │ -00025cc0: 2253 6567 7265 282e 2e2e 2c4f 7574 7075  "Segre(...,Outpu
│ │ │ │ -00025cd0: 743d 3e2e 2e2e 2922 202d 2d20 7365 6520  t=>...)" -- see 
│ │ │ │ -00025ce0: 2a6e 6f74 6520 5365 6772 653a 2053 6567  *note Segre: Seg
│ │ │ │ -00025cf0: 7265 2c20 2d2d 2054 6865 2053 6567 7265  re, -- The Segre
│ │ │ │ -00025d00: 2063 6c61 7373 206f 6620 610a 2020 2020   class of a.    
│ │ │ │ -00025d10: 7375 6273 6368 656d 650a 0a46 6f72 2074  subscheme..For t
│ │ │ │ -00025d20: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -00025d30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00025d40: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -00025d50: 7465 204f 7574 7075 743a 204f 7574 7075  te Output: Outpu
│ │ │ │ -00025d60: 742c 2069 7320 6120 2a6e 6f74 6520 7379  t, is a *note sy
│ │ │ │ -00025d70: 6d62 6f6c 3a20 284d 6163 6175 6c61 7932  mbol: (Macaulay2
│ │ │ │ -00025d80: 446f 6329 5379 6d62 6f6c 2c2e 0a1f 0a46  Doc)Symbol,....F
│ │ │ │ -00025d90: 696c 653a 2043 6861 7261 6374 6572 6973  ile: Characteris
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│ │ │ │ -00025dd0: 4e65 7874 3a20 5365 6772 652c 2050 7265  Next: Segre, Pre
│ │ │ │ -00025de0: 763a 204f 7574 7075 742c 2055 703a 2054  v: Output, Up: T
│ │ │ │ -00025df0: 6f70 0a0a 7072 6f62 6162 696c 6973 7469  op..probabilisti
│ │ │ │ -00025e00: 6320 616c 676f 7269 7468 6d0a 2a2a 2a2a  c algorithm.****
│ │ │ │ -00025e10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025e20: 2a2a 2a0a 0a54 6865 2061 6c67 6f72 6974  ***..The algorit
│ │ │ │ -00025e30: 686d 7320 7573 6564 2066 6f72 2074 6865  hms used for the
│ │ │ │ -00025e40: 2063 6f6d 7075 7461 7469 6f6e 206f 6620   computation of 
│ │ │ │ -00025e50: 6368 6172 6163 7465 7269 7374 6963 2063  characteristic c
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│ │ │ │ -00025ea0: 7365 7320 636f 7272 6563 746c 7920 666f  ses correctly fo
│ │ │ │ -00025eb0: 7220 610a 6765 6e65 7261 6c20 6368 6f69  r a.general choi
│ │ │ │ -00025ec0: 6365 206f 6620 6365 7274 6169 6e20 706f  ce of certain po
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│ │ │ │ -00025ee0: 6973 2c20 7468 6572 6520 6973 2061 6e20  is, there is an 
│ │ │ │ -00025ef0: 6f70 656e 2064 656e 7365 205a 6172 6973  open dense Zaris
│ │ │ │ -00025f00: 6b69 0a73 6574 2066 6f72 2077 6869 6368  ki.set for which
│ │ │ │ -00025f10: 2074 6865 2061 6c67 6f72 6974 686d 2079   the algorithm y
│ │ │ │ -00025f20: 6965 6c64 7320 7468 6520 636f 7272 6563  ields the correc
│ │ │ │ -00025f30: 7420 636c 6173 732c 2069 2e65 2e2c 2074  t class, i.e., t
│ │ │ │ -00025f40: 6865 2063 6f72 7265 6374 2063 6c61 7373  he correct class
│ │ │ │ -00025f50: 0a69 7320 6361 6c63 756c 6174 6564 2077  .is calculated w
│ │ │ │ -00025f60: 6974 6820 7072 6f62 6162 696c 6974 7920  ith probability 
│ │ │ │ -00025f70: 312e 2048 6f77 6576 6572 2c20 7369 6e63  1. However, sinc
│ │ │ │ -00025f80: 6520 7468 6520 696d 706c 656d 656e 7461  e the implementa
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│ │ │ │ -00025fa0: 6120 6469 7363 7265 7465 2070 726f 6261  a discrete proba
│ │ │ │ -00025fb0: 6269 6c69 7479 2073 7061 6365 2074 6865  bility space the
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│ │ │ │ -00025fd0: 6c6c 2c20 6275 7420 6e6f 6e2d 7a65 726f  ll, but non-zero
│ │ │ │ -00025fe0: 2c20 7072 6f62 6162 696c 6974 790a 6f66  , probability.of
│ │ │ │ -00025ff0: 206e 6f74 2063 6f6d 7075 7469 6e67 2074   not computing t
│ │ │ │ -00026000: 6865 2063 6f72 7265 6374 2063 6c61 7373  he correct class
│ │ │ │ -00026010: 2e20 536b 6570 7469 6361 6c20 7573 6572  . Skeptical user
│ │ │ │ -00026020: 7320 7368 6f75 6c64 2072 6570 6561 7420  s should repeat 
│ │ │ │ -00026030: 6361 6c63 756c 6174 696f 6e73 0a73 6576  calculations.sev
│ │ │ │ -00026040: 6572 616c 2074 696d 6573 2074 6f20 696e  eral times to in
│ │ │ │ -00026050: 6372 6561 7365 2074 6865 2070 726f 6261  crease the proba
│ │ │ │ -00026060: 6269 6c69 7479 206f 6620 636f 6d70 7574  bility of comput
│ │ │ │ -00026070: 696e 6720 7468 6520 636f 7272 6563 7420  ing the correct 
│ │ │ │ -00026080: 636c 6173 732e 0a0a 496e 2074 6865 2063  class...In the c
│ │ │ │ -00026090: 6173 6520 6f66 2074 6865 2073 796d 626f  ase of the symbo
│ │ │ │ -000260a0: 6c69 6320 696d 706c 656d 656e 7461 7469  lic implementati
│ │ │ │ -000260b0: 6f6e 206f 6620 7468 6520 5072 6f6a 6563  on of the Projec
│ │ │ │ -000260c0: 7469 7665 4465 6772 6565 206d 6574 686f  tiveDegree metho
│ │ │ │ -000260d0: 640a 7072 6163 7469 6361 6c20 6578 7065  d.practical expe
│ │ │ │ -000260e0: 7269 656e 6365 2061 6e64 2061 6c67 6f72  rience and algor
│ │ │ │ -000260f0: 6974 686d 2074 6573 7469 6e67 2069 6e64  ithm testing ind
│ │ │ │ -00026100: 6963 6174 6520 7468 6174 2061 2066 696e  icate that a fin
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│ │ │ │ -00026120: 7665 7220 3235 3030 3020 656c 656d 656e  ver 25000 elemen
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│ │ │ │ -00026140: 7375 6666 6963 6965 6e74 2074 6f20 6578  sufficient to ex
│ │ │ │ -00026150: 7065 6374 2061 2063 6f72 7265 6374 2072  pect a correct r
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│ │ │ │ -00026170: 7072 6f62 6162 696c 6974 792c 2069 2e65  probability, i.e
│ │ │ │ -00026180: 2e20 7573 696e 6720 7468 6520 6669 6e69  . using the fini
│ │ │ │ -00026190: 7465 2066 6965 6c64 206b 6b3d 5a5a 2f32  te field kk=ZZ/2
│ │ │ │ -000261a0: 3530 3733 2074 6865 2065 7870 6572 696d  5073 the experim
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│ │ │ │ -000261c0: 6661 696c 7572 6520 7769 7468 2074 6865  failure with the
│ │ │ │ -000261d0: 2050 726f 6a65 6374 6976 6544 6567 7265   ProjectiveDegre
│ │ │ │ -000261e0: 6520 616c 676f 7269 7468 6d20 6f6e 2061  e algorithm on a
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│ │ │ │ -00026200: 706c 6573 0a77 6173 206c 6573 7320 7468  ples.was less th
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│ │ │ │ -00026220: 2074 6865 2066 696e 6974 6520 6669 656c   the finite fiel
│ │ │ │ -00026230: 6420 6b6b 3d5a 5a2f 3332 3734 3920 7265  d kk=ZZ/32749 re
│ │ │ │ -00026240: 7375 6c74 6564 2069 6e20 6e6f 0a66 6169  sulted in no.fai
│ │ │ │ -00026250: 6c75 7265 7320 696e 206f 7665 7220 3130  lures in over 10
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│ │ │ │ -00026270: 7365 7665 7261 6c20 6469 6666 6572 656e  several differen
│ │ │ │ -00026280: 7420 6578 616d 706c 6573 2e0a 0a57 6520  t examples...We 
│ │ │ │ -00026290: 696c 6c75 7374 7261 7465 2074 6865 2070  illustrate the p
│ │ │ │ -000262a0: 726f 6261 6269 6c69 7374 6963 2062 6568  robabilistic beh
│ │ │ │ -000262b0: 6176 696f 7572 2077 6974 6820 616e 2065  aviour with an e
│ │ │ │ -000262c0: 7861 6d70 6c65 2077 6865 7265 2074 6865  xample where the
│ │ │ │ -000262d0: 2063 686f 7365 6e0a 7261 6e64 6f6d 2073   chosen.random s
│ │ │ │ -000262e0: 6565 6420 6c65 6164 7320 746f 2061 2077  eed leads to a w
│ │ │ │ -000262f0: 726f 6e67 2072 6573 756c 7420 696e 2074  rong result in t
│ │ │ │ -00026300: 6865 2066 6972 7374 2063 616c 6375 6c61  he first calcula
│ │ │ │ -00026310: 7469 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  tion...+--------
│ │ │ │ +00025ba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00025bb0: 2022 4368 6572 6e28 2e2e 2e2c 4f75 7470   "Chern(...,Outp
│ │ │ │ +00025bc0: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565  ut=>...)" -- see
│ │ │ │ +00025bd0: 202a 6e6f 7465 2043 6865 726e 3a20 4368   *note Chern: Ch
│ │ │ │ +00025be0: 6572 6e2c 202d 2d20 5468 6520 4368 6572  ern, -- The Cher
│ │ │ │ +00025bf0: 6e20 636c 6173 730a 2020 2a20 2243 534d  n class.  * "CSM
│ │ │ │ +00025c00: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e  (...,Output=>...
│ │ │ │ +00025c10: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ +00025c20: 4353 4d3a 2043 534d 2c20 2d2d 2054 6865  CSM: CSM, -- The
│ │ │ │ +00025c30: 0a20 2020 2043 6865 726e 2d53 6368 7761  .    Chern-Schwa
│ │ │ │ +00025c40: 7274 7a2d 4d61 6350 6865 7273 6f6e 2063  rtz-MacPherson c
│ │ │ │ +00025c50: 6c61 7373 0a20 202a 2022 4575 6c65 7228  lass.  * "Euler(
│ │ │ │ +00025c60: 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e 2e29  ...,Output=>...)
│ │ │ │ +00025c70: 2220 2d2d 2073 6565 202a 6e6f 7465 2045  " -- see *note E
│ │ │ │ +00025c80: 756c 6572 3a20 4575 6c65 722c 202d 2d20  uler: Euler, -- 
│ │ │ │ +00025c90: 5468 6520 4575 6c65 720a 2020 2020 4368  The Euler.    Ch
│ │ │ │ +00025ca0: 6172 6163 7465 7269 7374 6963 0a20 202a  aracteristic.  *
│ │ │ │ +00025cb0: 2022 5365 6772 6528 2e2e 2e2c 4f75 7470   "Segre(...,Outp
│ │ │ │ +00025cc0: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565  ut=>...)" -- see
│ │ │ │ +00025cd0: 202a 6e6f 7465 2053 6567 7265 3a20 5365   *note Segre: Se
│ │ │ │ +00025ce0: 6772 652c 202d 2d20 5468 6520 5365 6772  gre, -- The Segr
│ │ │ │ +00025cf0: 6520 636c 6173 7320 6f66 2061 0a20 2020  e class of a.   
│ │ │ │ +00025d00: 2073 7562 7363 6865 6d65 0a0a 466f 7220   subscheme..For 
│ │ │ │ +00025d10: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00025d20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00025d30: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +00025d40: 6f74 6520 4f75 7470 7574 3a20 4f75 7470  ote Output: Outp
│ │ │ │ +00025d50: 7574 2c20 6973 2061 202a 6e6f 7465 2073  ut, is a *note s
│ │ │ │ +00025d60: 796d 626f 6c3a 2028 4d61 6361 756c 6179  ymbol: (Macaulay
│ │ │ │ +00025d70: 3244 6f63 2953 796d 626f 6c2c 2e0a 1f0a  2Doc)Symbol,....
│ │ │ │ +00025d80: 4669 6c65 3a20 4368 6172 6163 7465 7269  File: Characteri
│ │ │ │ +00025d90: 7374 6963 436c 6173 7365 732e 696e 666f  sticClasses.info
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│ │ │ │ +00025db0: 6973 7469 6320 616c 676f 7269 7468 6d2c  istic algorithm,
│ │ │ │ +00025dc0: 204e 6578 743a 2053 6567 7265 2c20 5072   Next: Segre, Pr
│ │ │ │ +00025dd0: 6576 3a20 4f75 7470 7574 2c20 5570 3a20  ev: Output, Up: 
│ │ │ │ +00025de0: 546f 700a 0a70 726f 6261 6269 6c69 7374  Top..probabilist
│ │ │ │ +00025df0: 6963 2061 6c67 6f72 6974 686d 0a2a 2a2a  ic algorithm.***
│ │ │ │ +00025e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00025e10: 2a2a 2a2a 0a0a 5468 6520 616c 676f 7269  ****..The algori
│ │ │ │ +00025e20: 7468 6d73 2075 7365 6420 666f 7220 7468  thms used for th
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│ │ │ │ +00025e80: 616c 6375 6c61 7465 2074 6865 2063 6c61  alculate the cla
│ │ │ │ +00025e90: 7373 6573 2063 6f72 7265 6374 6c79 2066  sses correctly f
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│ │ │ │ +00025ec0: 6f6c 796e 6f6d 6961 6c73 2e20 5468 6174  olynomials. That
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│ │ │ │ +00025ee0: 206f 7065 6e20 6465 6e73 6520 5a61 7269   open dense Zari
│ │ │ │ +00025ef0: 736b 690a 7365 7420 666f 7220 7768 6963  ski.set for whic
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│ │ │ │ +00025f30: 7468 6520 636f 7272 6563 7420 636c 6173  the correct clas
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│ │ │ │ +00025f50: 7769 7468 2070 726f 6261 6269 6c69 7479  with probability
│ │ │ │ +00025f60: 2031 2e20 486f 7765 7665 722c 2073 696e   1. However, sin
│ │ │ │ +00025f70: 6365 2074 6865 2069 6d70 6c65 6d65 6e74  ce the implement
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│ │ │ │ +00026020: 2063 616c 6375 6c61 7469 6f6e 730a 7365   calculations.se
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│ │ │ │ +000260b0: 6374 6976 6544 6567 7265 6520 6d65 7468  ctiveDegree meth
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│ │ │ │ +00026240: 696c 7572 6573 2069 6e20 6f76 6572 2031  ilures in over 1
│ │ │ │ +00026250: 3030 3030 2061 7474 656d 7074 7320 6f66  0000 attempts of
│ │ │ │ +00026260: 2073 6576 6572 616c 2064 6966 6665 7265   several differe
│ │ │ │ +00026270: 6e74 2065 7861 6d70 6c65 732e 0a0a 5765  nt examples...We
│ │ │ │ +00026280: 2069 6c6c 7573 7472 6174 6520 7468 6520   illustrate the 
│ │ │ │ +00026290: 7072 6f62 6162 696c 6973 7469 6320 6265  probabilistic be
│ │ │ │ +000262a0: 6861 7669 6f75 7220 7769 7468 2061 6e20  haviour with an 
│ │ │ │ +000262b0: 6578 616d 706c 6520 7768 6572 6520 7468  example where th
│ │ │ │ +000262c0: 6520 6368 6f73 656e 0a72 616e 646f 6d20  e chosen.random 
│ │ │ │ +000262d0: 7365 6564 206c 6561 6473 2074 6f20 6120  seed leads to a 
│ │ │ │ +000262e0: 7772 6f6e 6720 7265 7375 6c74 2069 6e20  wrong result in 
│ │ │ │ +000262f0: 7468 6520 6669 7273 7420 6361 6c63 756c  the first calcul
│ │ │ │ +00026300: 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d  ation...+-------
│ │ │ │ +00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026340: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00026350: 7365 7452 616e 646f 6d53 6565 6420 3132  setRandomSeed 12
│ │ │ │ -00026360: 313b 2020 2020 2020 2020 2020 2020 2020  1;              
│ │ │ │ -00026370: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00026330: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00026340: 2073 6574 5261 6e64 6f6d 5365 6564 2031   setRandomSeed 1
│ │ │ │ +00026350: 3231 3b20 2020 2020 2020 2020 2020 2020  21;             
│ │ │ │ +00026360: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00026370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000263a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000263b0: 7c69 3220 3a20 5220 3d20 5151 5b78 2c79  |i2 : R = QQ[x,y
│ │ │ │ -000263c0: 2c7a 2c77 5d20 2020 2020 2020 2020 2020  ,z,w]           
│ │ │ │ -000263d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000263a0: 0a7c 6932 203a 2052 203d 2051 515b 782c  .|i2 : R = QQ[x,
│ │ │ │ +000263b0: 792c 7a2c 775d 2020 2020 2020 2020 2020  y,z,w]          
│ │ │ │ +000263c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000263d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000263e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026410: 2020 2020 7c0a 7c6f 3220 3d20 5220 2020      |.|o2 = R   
│ │ │ │ +00026400: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ +00026410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00026430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026470: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00026480: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ -00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00026460: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00026470: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +00026480: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00026490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000264a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000264b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000264c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000264d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000264e0: 2b0a 7c69 3320 3a20 4920 3d20 6d69 6e6f  +.|i3 : I = mino
│ │ │ │ -000264f0: 7273 2832 2c6d 6174 7269 787b 7b78 2c79  rs(2,matrix{{x,y
│ │ │ │ -00026500: 2c7a 7d2c 7b79 2c7a 2c77 7d7d 2920 2020  ,z},{y,z,w}})   
│ │ │ │ -00026510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000264d0: 2d2b 0a7c 6933 203a 2049 203d 206d 696e  -+.|i3 : I = min
│ │ │ │ +000264e0: 6f72 7328 322c 6d61 7472 6978 7b7b 782c  ors(2,matrix{{x,
│ │ │ │ +000264f0: 792c 7a7d 2c7b 792c 7a2c 777d 7d29 2020  y,z},{y,z,w}})  
│ │ │ │ +00026500: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026540: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00026550: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00026560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026570: 2032 2020 2020 2020 207c 0a7c 6f33 203d   2       |.|o3 =
│ │ │ │ -00026580: 2069 6465 616c 2028 2d20 7920 202b 2078   ideal (- y  + x
│ │ │ │ -00026590: 2a7a 2c20 2d20 792a 7a20 2b20 782a 772c  *z, - y*z + x*w,
│ │ │ │ -000265a0: 202d 207a 2020 2b20 792a 7729 7c0a 7c20   - z  + y*w)|.| 
│ │ │ │ +00026530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00026540: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00026550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026560: 2020 3220 2020 2020 2020 7c0a 7c6f 3320    2       |.|o3 
│ │ │ │ +00026570: 3d20 6964 6561 6c20 282d 2079 2020 2b20  = ideal (- y  + 
│ │ │ │ +00026580: 782a 7a2c 202d 2079 2a7a 202b 2078 2a77  x*z, - y*z + x*w
│ │ │ │ +00026590: 2c20 2d20 7a20 202b 2079 2a77 297c 0a7c  , - z  + y*w)|.|
│ │ │ │ +000265a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000265e0: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -000265f0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -00026600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026610: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000265d0: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66  |.|o3 : Ideal of
│ │ │ │ +000265e0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +000265f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026600: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026640: 2d2d 2d2d 2d2b 0a7c 6934 203a 2043 6865  -----+.|i4 : Che
│ │ │ │ -00026650: 726e 2028 492c 436f 6d70 4d65 7468 6f64  rn (I,CompMethod
│ │ │ │ -00026660: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │ -00026670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026630: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 4368  ------+.|i4 : Ch
│ │ │ │ +00026640: 6572 6e20 2849 2c43 6f6d 704d 6574 686f  ern (I,CompMetho
│ │ │ │ +00026650: 643d 3e50 6e52 6573 6964 7561 6c29 2020  d=>PnResidual)  
│ │ │ │ +00026660: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00026670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000266b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -000266c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000266e0: 7c6f 3420 3d20 3448 2020 2020 2020 2020  |o4 = 4H        
│ │ │ │ +00026690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000266a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000266b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000266d0: 0a7c 6f34 203d 2034 4820 2020 2020 2020  .|o4 = 4H       
│ │ │ │ +000266e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000266f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026710: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026740: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b48      |.|     ZZ[H
│ │ │ │ -00026750: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00026760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026770: 2020 2020 2020 207c 0a7c 6f34 203a 202d         |.|o4 : -
│ │ │ │ -00026780: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ -00026790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000267a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000267b0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -000267c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000267d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000267e0: 2020 2020 2020 2048 2020 2020 2020 2020         H        
│ │ │ │ +00026730: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ +00026740: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │ +00026750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026760: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +00026770: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ +00026780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000267a0: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +000267b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000267c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000267d0: 7c20 2020 2020 2020 4820 2020 2020 2020  |       H       
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026810: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00026800: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00026810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026840: 2d2d 2d2b 0a7c 6935 203a 2043 6865 726e  ---+.|i5 : Chern
│ │ │ │ -00026850: 2028 492c 436f 6d70 4d65 7468 6f64 3d3e   (I,CompMethod=>
│ │ │ │ -00026860: 506e 5265 7369 6475 616c 2920 2020 2020  PnResidual)     
│ │ │ │ -00026870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00026830: 2d2d 2d2d 2b0a 7c69 3520 3a20 4368 6572  ----+.|i5 : Cher
│ │ │ │ +00026840: 6e20 2849 2c43 6f6d 704d 6574 686f 643d  n (I,CompMethod=
│ │ │ │ +00026850: 3e50 6e52 6573 6964 7561 6c29 2020 2020  >PnResidual)    
│ │ │ │ +00026860: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00026870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000268b0: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │ -000268c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000268e0: 3520 3d20 3248 2020 2b20 3348 2020 2020  5 = 2H  + 3H    
│ │ │ │ +00026890: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000268a0: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ +000268b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000268c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000268d0: 6f35 203d 2032 4820 202b 2033 4820 2020  o5 = 2H  + 3H   
│ │ │ │ +000268e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026910: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026900: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00026910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026940: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │ +00026930: 2020 207c 0a7c 2020 2020 205a 5a5b 485d     |.|     ZZ[H]
│ │ │ │ +00026940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026970: 2020 2020 207c 0a7c 6f35 203a 202d 2d2d       |.|o5 : ---
│ │ │ │ -00026980: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ -00026990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000269b0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -000269c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000269e0: 2020 2020 2048 2020 2020 2020 2020 2020       H          
│ │ │ │ -000269f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00026a10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00026960: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d        |.|o5 : --
│ │ │ │ +00026970: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00026980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026990: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000269a0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +000269b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000269d0: 2020 2020 2020 4820 2020 2020 2020 2020        H         
│ │ │ │ +000269e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00026a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026a40: 2d2b 0a7c 6936 203a 2043 6865 726e 2028  -+.|i6 : Chern (
│ │ │ │ -00026a50: 492c 436f 6d70 4d65 7468 6f64 3d3e 506e  I,CompMethod=>Pn
│ │ │ │ -00026a60: 5265 7369 6475 616c 2920 2020 2020 2020  Residual)       
│ │ │ │ -00026a70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026a30: 2d2d 2b0a 7c69 3620 3a20 4368 6572 6e20  --+.|i6 : Chern 
│ │ │ │ +00026a40: 2849 2c43 6f6d 704d 6574 686f 643d 3e50  (I,CompMethod=>P
│ │ │ │ +00026a50: 6e52 6573 6964 7561 6c29 2020 2020 2020  nResidual)      
│ │ │ │ +00026a60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00026ab0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ -00026ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ad0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00026ae0: 3d20 3248 2020 2b20 3348 2020 2020 2020  = 2H  + 3H      
│ │ │ │ -00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00026a90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026aa0: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ac0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00026ad0: 203d 2032 4820 202b 2033 4820 2020 2020   = 2H  + 3H     
│ │ │ │ +00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026af0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026b00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b40: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020  |.|     ZZ[H]   
│ │ │ │ +00026b30: 207c 0a7c 2020 2020 205a 5a5b 485d 2020   |.|     ZZ[H]  
│ │ │ │ +00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b70: 2020 207c 0a7c 6f36 203a 202d 2d2d 2d2d     |.|o6 : -----
│ │ │ │ +00026b60: 2020 2020 7c0a 7c6f 3620 3a20 2d2d 2d2d      |.|o6 : ----
│ │ │ │ +00026b70: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │  00026b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00026bb0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
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│ │ │ │ -00026ff0: 2020 2020 5365 6772 6528 4368 2c58 2c4a      Segre(Ch,X,J
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│ │ │ │ -00027350: 2020 2020 206f 6620 7468 6520 746f 7269       of the tori
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│ │ │ │ -00027370: 2872 696e 6720 4a29 2f28 5352 2b4c 5229  (ring J)/(SR+LR)
│ │ │ │ -00027380: 2077 6865 7265 2053 5220 6973 2074 6865   where SR is the
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│ │ │ │ -000273a0: 2d52 6569 736e 6572 2069 6465 616c 206f  -Reisner ideal o
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│ │ │ │ -000273d0: 6865 206c 696e 6561 720a 2020 2020 2020  he linear.      
│ │ │ │ -000273e0: 2020 7265 6c61 7469 6f6e 7320 6964 6561    relations idea
│ │ │ │ -000273f0: 6c2c 2074 6869 7320 7269 6e67 2073 686f  l, this ring sho
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│ │ │ │ -00027420: 2020 2020 2054 6f72 6963 4368 6f77 5269       ToricChowRi
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│ │ │ │ -00027440: 672c 2063 6f6d 6d61 6e64 0a20 202a 202a  g, command.  * *
│ │ │ │ -00027450: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -00027460: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -00027470: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -00027480: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -00027490: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -000274a0: 202a 2043 6f6d 704d 6574 686f 6420 286d   * CompMethod (m
│ │ │ │ -000274b0: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ -000274c0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ -000274d0: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -000274e0: 2020 2020 5072 6f6a 6563 7469 7665 4465      ProjectiveDe
│ │ │ │ -000274f0: 6772 6565 2c20 5072 6f6a 6563 7469 7665  gree, Projective
│ │ │ │ -00027500: 4465 6772 6565 2c20 7468 6973 2061 6c67  Degree, this alg
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│ │ │ │ -00027520: 6564 2066 6f72 0a20 2020 2020 2020 2073  ed for.        s
│ │ │ │ -00027530: 7562 7363 6865 6d65 7320 6f66 2061 6e79  ubschemes of any
│ │ │ │ -00027540: 2061 7070 6c69 6361 626c 6520 746f 7269   applicable tori
│ │ │ │ -00027550: 6320 7661 7269 6574 7920 2874 6869 7320  c variety (this 
│ │ │ │ -00027560: 6d61 7920 6265 2063 6865 636b 6564 2075  may be checked u
│ │ │ │ -00027570: 7369 6e67 0a20 2020 2020 2020 2074 6865  sing.        the
│ │ │ │ -00027580: 202a 6e6f 7465 2043 6865 636b 546f 7269   *note CheckTori
│ │ │ │ -00027590: 6356 6172 6965 7479 5661 6c69 643a 2043  cVarietyValid: C
│ │ │ │ -000275a0: 6865 636b 546f 7269 6356 6172 6965 7479  heckToricVariety
│ │ │ │ -000275b0: 5661 6c69 642c 2063 6f6d 6d61 6e64 290a  Valid, command).
│ │ │ │ -000275c0: 2020 2020 2020 2a20 436f 6d70 4d65 7468        * CompMeth
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│ │ │ │ -00027660: 6e20 6f6e 6c79 0a20 2020 2020 202a 204f  n only.      * O
│ │ │ │ -00027670: 7574 7075 7420 3d3e 202e 2e2e 2c20 6465  utput => ..., de
│ │ │ │ -00027680: 6661 756c 7420 7661 6c75 6520 4368 6f77  fault value Chow
│ │ │ │ -00027690: 5269 6e67 456c 656d 656e 742c 2043 686f  RingElement, Cho
│ │ │ │ -000276a0: 7752 696e 6745 6c65 6d65 6e74 2c20 7265  wRingElement, re
│ │ │ │ -000276b0: 7475 726e 730a 2020 2020 2020 2020 6120  turns.        a 
│ │ │ │ -000276c0: 5269 6e67 456c 656d 656e 7420 696e 2074  RingElement in t
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│ │ │ │ -000276e0: 7468 6520 6170 7072 6f70 7269 6174 6520  the appropriate 
│ │ │ │ -000276f0: 616d 6269 656e 7420 7370 6163 650a 2020  ambient space.  
│ │ │ │ -00027700: 2020 2020 2a20 4f75 7470 7574 203d 3e20      * Output => 
│ │ │ │ -00027710: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -00027720: 7565 2043 686f 7752 696e 6745 6c65 6d65  ue ChowRingEleme
│ │ │ │ -00027730: 6e74 2c20 4861 7368 466f 726d 2c20 4861  nt, HashForm, Ha
│ │ │ │ -00027740: 7368 466f 726d 0a20 2020 2020 2020 2072  shForm.        r
│ │ │ │ -00027750: 6574 7572 6e73 2061 204d 7574 6162 6c65  eturns a Mutable
│ │ │ │ -00027760: 4861 7368 5461 626c 6520 636f 6e74 6169  HashTable contai
│ │ │ │ -00027770: 6e69 6e67 2074 6865 2066 6f6c 6c6f 7769  ning the followi
│ │ │ │ -00027780: 6e67 206b 6579 733a 2022 4722 2028 7468  ng keys: "G" (th
│ │ │ │ -00027790: 650a 2020 2020 2020 2020 706f 6c79 6e6f  e.        polyno
│ │ │ │ -000277a0: 6d69 616c 2077 6974 6820 636f 6566 6669  mial with coeffi
│ │ │ │ -000277b0: 6369 656e 7473 206f 6620 7468 6520 6879  cients of the hy
│ │ │ │ -000277c0: 7065 7270 6c61 6e65 2063 6c61 7373 6573  perplane classes
│ │ │ │ -000277d0: 2072 6570 7265 7365 6e74 696e 6720 7468   representing th
│ │ │ │ -000277e0: 650a 2020 2020 2020 2020 7072 6f6a 6563  e.        projec
│ │ │ │ -000277f0: 7469 7665 2064 6567 7265 6573 292c 2022  tive degrees), "
│ │ │ │ -00027800: 476c 6973 7422 2028 7468 6520 6c69 7374  Glist" (the list
│ │ │ │ -00027810: 2066 6f72 6d20 6f66 2022 4722 2920 2c20   form of "G") , 
│ │ │ │ -00027820: 2253 6567 7265 2220 2874 6865 0a20 2020  "Segre" (the.   
│ │ │ │ -00027830: 2020 2020 2074 6f74 616c 2053 6567 7265       total Segre
│ │ │ │ -00027840: 2063 6c61 7373 206f 6620 7468 6520 696e   class of the in
│ │ │ │ -00027850: 7075 7429 2c22 5365 6772 654c 6973 7422  put),"SegreList"
│ │ │ │ -00027860: 2028 7468 6520 6c69 7374 2066 6f72 6d20   (the list form 
│ │ │ │ -00027870: 6f66 2022 5365 6772 6522 290a 2020 2a20  of "Segre").  * 
│ │ │ │ -00027880: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00027890: 2061 202a 6e6f 7465 2072 696e 6720 656c   a *note ring el
│ │ │ │ -000278a0: 656d 656e 743a 2028 4d61 6361 756c 6179  ement: (Macaulay
│ │ │ │ -000278b0: 3244 6f63 2952 696e 6745 6c65 6d65 6e74  2Doc)RingElement
│ │ │ │ -000278c0: 2c2c 2074 6865 2070 7573 6866 6f72 7761  ,, the pushforwa
│ │ │ │ -000278d0: 7264 206f 660a 2020 2020 2020 2020 7468  rd of.        th
│ │ │ │ -000278e0: 6520 746f 7461 6c20 5365 6772 6520 636c  e total Segre cl
│ │ │ │ -000278f0: 6173 7320 6f66 2074 6865 2073 6368 656d  ass of the schem
│ │ │ │ -00027900: 6520 5620 6465 6669 6e65 6420 6279 2074  e V defined by t
│ │ │ │ -00027910: 6865 2069 6e70 7574 2069 6465 616c 2074  he input ideal t
│ │ │ │ -00027920: 6f20 7468 650a 2020 2020 2020 2020 6170  o the.        ap
│ │ │ │ -00027930: 7072 6f70 7269 6174 6520 4368 6f77 2072  propriate Chow r
│ │ │ │ -00027940: 696e 670a 0a44 6573 6372 6970 7469 6f6e  ing..Description
│ │ │ │ -00027950: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f  .===========..Fo
│ │ │ │ -00027960: 7220 6120 7375 6273 6368 656d 6520 5620  r a subscheme V 
│ │ │ │ -00027970: 6f66 2061 6e20 6170 706c 6963 6162 6c65  of an applicable
│ │ │ │ -00027980: 2074 6f72 6963 2076 6172 6965 7479 2058   toric variety X
│ │ │ │ -00027990: 2074 6869 7320 636f 6d6d 616e 6420 636f   this command co
│ │ │ │ -000279a0: 6d70 7574 6573 2074 6865 0a70 7573 682d  mputes the.push-
│ │ │ │ -000279b0: 666f 7277 6172 6420 6f66 2074 6865 2074  forward of the t
│ │ │ │ -000279c0: 6f74 616c 2053 6567 7265 2063 6c61 7373  otal Segre class
│ │ │ │ -000279d0: 2073 2856 2c58 2920 6f66 2056 2069 6e20   s(V,X) of V in 
│ │ │ │ -000279e0: 5820 746f 2074 6865 2043 686f 7720 7269  X to the Chow ri
│ │ │ │ -000279f0: 6e67 206f 6620 582e 0a0a 2b2d 2d2d 2d2d  ng of X...+-----
│ │ │ │ +00026f80: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00026f90: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +00026fa0: 7361 6765 3a20 0a20 2020 2020 2020 2053  sage: .        S
│ │ │ │ +00026fb0: 6567 7265 2049 0a20 2020 2020 2020 2053  egre I.        S
│ │ │ │ +00026fc0: 6567 7265 2841 2c49 290a 2020 2020 2020  egre(A,I).      
│ │ │ │ +00026fd0: 2020 5365 6772 6528 582c 4a29 0a20 2020    Segre(X,J).   
│ │ │ │ +00026fe0: 2020 2020 2053 6567 7265 2843 682c 582c       Segre(Ch,X,
│ │ │ │ +00026ff0: 4a29 0a20 202a 2049 6e70 7574 733a 0a20  J).  * Inputs:. 
│ │ │ │ +00027000: 2020 2020 202a 2049 2c20 616e 202a 6e6f       * I, an *no
│ │ │ │ +00027010: 7465 2069 6465 616c 3a20 284d 6163 6175  te ideal: (Macau
│ │ │ │ +00027020: 6c61 7932 446f 6329 4964 6561 6c2c 2c20  lay2Doc)Ideal,, 
│ │ │ │ +00027030: 6120 6d75 6c74 692d 686f 6d6f 6765 6e65  a multi-homogene
│ │ │ │ +00027040: 6f75 7320 6964 6561 6c20 696e 2061 0a20  ous ideal in a. 
│ │ │ │ +00027050: 2020 2020 2020 2067 7261 6465 6420 706f         graded po
│ │ │ │ +00027060: 6c79 6e6f 6d69 616c 2072 696e 6720 6f76  lynomial ring ov
│ │ │ │ +00027070: 6572 2061 2066 6965 6c64 2064 6566 696e  er a field defin
│ │ │ │ +00027080: 696e 6720 6120 636c 6f73 6564 2073 7562  ing a closed sub
│ │ │ │ +00027090: 7363 6865 6d65 2056 206f 660a 2020 2020  scheme V of.    
│ │ │ │ +000270a0: 2020 2020 5c50 505e 7b6e 5f31 7d78 2e2e      \PP^{n_1}x..
│ │ │ │ +000270b0: 2e78 5c50 505e 7b6e 5f6d 7d0a 2020 2020  .x\PP^{n_m}.    
│ │ │ │ +000270c0: 2020 2a20 412c 2061 202a 6e6f 7465 2071    * A, a *note q
│ │ │ │ +000270d0: 756f 7469 656e 7420 7269 6e67 3a20 284d  uotient ring: (M
│ │ │ │ +000270e0: 6163 6175 6c61 7932 446f 6329 5175 6f74  acaulay2Doc)Quot
│ │ │ │ +000270f0: 6965 6e74 5269 6e67 2c2c 0a20 2020 2020  ientRing,,.     
│ │ │ │ +00027100: 2020 2041 3d5c 5a5a 5b68 5f31 2c2e 2e2e     A=\ZZ[h_1,...
│ │ │ │ +00027110: 2c68 5f6d 5d2f 2868 5f31 5e7b 6e5f 312b  ,h_m]/(h_1^{n_1+
│ │ │ │ +00027120: 317d 2c2e 2e2e 2c68 5f6d 5e7b 6e5f 6d2b  1},...,h_m^{n_m+
│ │ │ │ +00027130: 317d 2920 7175 6f74 6965 6e74 2072 696e  1}) quotient rin
│ │ │ │ +00027140: 670a 2020 2020 2020 2020 7265 7072 6573  g.        repres
│ │ │ │ +00027150: 656e 7469 6e67 2074 6865 2043 686f 7720  enting the Chow 
│ │ │ │ +00027160: 7269 6e67 206f 6620 5c50 505e 7b6e 5f31  ring of \PP^{n_1
│ │ │ │ +00027170: 7d78 2e2e 2e78 5c50 505e 7b6e 5f6d 7d2c  }x...x\PP^{n_m},
│ │ │ │ +00027180: 2074 6869 7320 7269 6e67 2073 686f 756c   this ring shoul
│ │ │ │ +00027190: 640a 2020 2020 2020 2020 6265 2062 7569  d.        be bui
│ │ │ │ +000271a0: 6c74 2075 7369 6e67 2074 6865 202a 6e6f  lt using the *no
│ │ │ │ +000271b0: 7465 2043 686f 7752 696e 673a 2043 686f  te ChowRing: Cho
│ │ │ │ +000271c0: 7752 696e 672c 2063 6f6d 6d61 6e64 0a20  wRing, command. 
│ │ │ │ +000271d0: 2020 2020 202a 204a 2c20 616e 202a 6e6f       * J, an *no
│ │ │ │ +000271e0: 7465 2069 6465 616c 3a20 284d 6163 6175  te ideal: (Macau
│ │ │ │ +000271f0: 6c61 7932 446f 6329 4964 6561 6c2c 2c20  lay2Doc)Ideal,, 
│ │ │ │ +00027200: 696e 2074 6865 2067 7261 6465 6420 706f  in the graded po
│ │ │ │ +00027210: 6c79 6e6f 6d69 616c 2072 696e 670a 2020  lynomial ring.  
│ │ │ │ +00027220: 2020 2020 2020 7768 6963 6820 6973 2063        which is c
│ │ │ │ +00027230: 6f6f 7264 696e 6174 6520 7269 6e67 206f  oordinate ring o
│ │ │ │ +00027240: 6620 7468 6520 4e6f 726d 616c 2054 6f72  f the Normal Tor
│ │ │ │ +00027250: 6963 2056 6172 6965 7479 2058 0a20 2020  ic Variety X.   
│ │ │ │ +00027260: 2020 202a 2058 2c20 6120 2a6e 6f74 6520     * X, a *note 
│ │ │ │ +00027270: 6e6f 726d 616c 2074 6f72 6963 2076 6172  normal toric var
│ │ │ │ +00027280: 6965 7479 3a0a 2020 2020 2020 2020 284e  iety:.        (N
│ │ │ │ +00027290: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +000272a0: 6965 7329 4e6f 726d 616c 546f 7269 6356  ies)NormalToricV
│ │ │ │ +000272b0: 6172 6965 7479 2c2c 2077 6869 6368 2069  ariety,, which i
│ │ │ │ +000272c0: 7320 7468 6520 616d 6269 656e 7420 7370  s the ambient sp
│ │ │ │ +000272d0: 6163 650a 2020 2020 2020 2020 7768 6963  ace.        whic
│ │ │ │ +000272e0: 6820 636f 6e74 6169 6e73 2056 284a 290a  h contains V(J).
│ │ │ │ +000272f0: 2020 2020 2020 2a20 4368 2c20 6120 2a6e        * Ch, a *n
│ │ │ │ +00027300: 6f74 6520 7175 6f74 6965 6e74 2072 696e  ote quotient rin
│ │ │ │ +00027310: 673a 2028 4d61 6361 756c 6179 3244 6f63  g: (Macaulay2Doc
│ │ │ │ +00027320: 2951 756f 7469 656e 7452 696e 672c 2c20  )QuotientRing,, 
│ │ │ │ +00027330: 7468 6520 4368 6f77 2072 696e 670a 2020  the Chow ring.  
│ │ │ │ +00027340: 2020 2020 2020 6f66 2074 6865 2074 6f72        of the tor
│ │ │ │ +00027350: 6963 2076 6172 6965 7479 2058 2c20 4368  ic variety X, Ch
│ │ │ │ +00027360: 3d28 7269 6e67 204a 292f 2853 522b 4c52  =(ring J)/(SR+LR
│ │ │ │ +00027370: 2920 7768 6572 6520 5352 2069 7320 7468  ) where SR is th
│ │ │ │ +00027380: 650a 2020 2020 2020 2020 5374 616e 6c65  e.        Stanle
│ │ │ │ +00027390: 792d 5265 6973 6e65 7220 6964 6561 6c20  y-Reisner ideal 
│ │ │ │ +000273a0: 6f66 2074 6865 2066 616e 2064 6566 696e  of the fan defin
│ │ │ │ +000273b0: 696e 6720 5820 616e 6420 4c52 2069 7320  ing X and LR is 
│ │ │ │ +000273c0: 7468 6520 6c69 6e65 6172 0a20 2020 2020  the linear.     
│ │ │ │ +000273d0: 2020 2072 656c 6174 696f 6e73 2069 6465     relations ide
│ │ │ │ +000273e0: 616c 2c20 7468 6973 2072 696e 6720 7368  al, this ring sh
│ │ │ │ +000273f0: 6f75 6c64 2062 6520 6275 696c 7420 7573  ould be built us
│ │ │ │ +00027400: 696e 6720 7468 6520 2a6e 6f74 650a 2020  ing the *note.  
│ │ │ │ +00027410: 2020 2020 2020 546f 7269 6343 686f 7752        ToricChowR
│ │ │ │ +00027420: 696e 673a 2054 6f72 6963 4368 6f77 5269  ing: ToricChowRi
│ │ │ │ +00027430: 6e67 2c20 636f 6d6d 616e 640a 2020 2a20  ng, command.  * 
│ │ │ │ +00027440: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ +00027450: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ +00027460: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ +00027470: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00027480: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ +00027490: 2020 2a20 436f 6d70 4d65 7468 6f64 2028    * CompMethod (
│ │ │ │ +000274a0: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ +000274b0: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064  ation) => ..., d
│ │ │ │ +000274c0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ +000274d0: 2020 2020 2050 726f 6a65 6374 6976 6544       ProjectiveD
│ │ │ │ +000274e0: 6567 7265 652c 2050 726f 6a65 6374 6976  egree, Projectiv
│ │ │ │ +000274f0: 6544 6567 7265 652c 2074 6869 7320 616c  eDegree, this al
│ │ │ │ +00027500: 676f 7269 7468 6d20 6d61 7920 6265 2075  gorithm may be u
│ │ │ │ +00027510: 7365 6420 666f 720a 2020 2020 2020 2020  sed for.        
│ │ │ │ +00027520: 7375 6273 6368 656d 6573 206f 6620 616e  subschemes of an
│ │ │ │ +00027530: 7920 6170 706c 6963 6162 6c65 2074 6f72  y applicable tor
│ │ │ │ +00027540: 6963 2076 6172 6965 7479 2028 7468 6973  ic variety (this
│ │ │ │ +00027550: 206d 6179 2062 6520 6368 6563 6b65 6420   may be checked 
│ │ │ │ +00027560: 7573 696e 670a 2020 2020 2020 2020 7468  using.        th
│ │ │ │ +00027570: 6520 2a6e 6f74 6520 4368 6563 6b54 6f72  e *note CheckTor
│ │ │ │ +00027580: 6963 5661 7269 6574 7956 616c 6964 3a20  icVarietyValid: 
│ │ │ │ +00027590: 4368 6563 6b54 6f72 6963 5661 7269 6574  CheckToricVariet
│ │ │ │ +000275a0: 7956 616c 6964 2c20 636f 6d6d 616e 6429  yValid, command)
│ │ │ │ +000275b0: 0a20 2020 2020 202a 2043 6f6d 704d 6574  .      * CompMet
│ │ │ │ +000275c0: 686f 6420 286d 6973 7369 6e67 2064 6f63  hod (missing doc
│ │ │ │ +000275d0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ +000275e0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +000275f0: 650a 2020 2020 2020 2020 5072 6f6a 6563  e.        Projec
│ │ │ │ +00027600: 7469 7665 4465 6772 6565 2c20 506e 5265  tiveDegree, PnRe
│ │ │ │ +00027610: 7369 6475 616c 2c20 7468 6973 2061 6c67  sidual, this alg
│ │ │ │ +00027620: 6f72 6974 686d 206d 6179 2062 6520 7573  orithm may be us
│ │ │ │ +00027630: 6564 2066 6f72 2073 7562 7363 6865 6d65  ed for subscheme
│ │ │ │ +00027640: 730a 2020 2020 2020 2020 6f66 205c 5050  s.        of \PP
│ │ │ │ +00027650: 5e6e 206f 6e6c 790a 2020 2020 2020 2a20  ^n only.      * 
│ │ │ │ +00027660: 4f75 7470 7574 203d 3e20 2e2e 2e2c 2064  Output => ..., d
│ │ │ │ +00027670: 6566 6175 6c74 2076 616c 7565 2043 686f  efault value Cho
│ │ │ │ +00027680: 7752 696e 6745 6c65 6d65 6e74 2c20 4368  wRingElement, Ch
│ │ │ │ +00027690: 6f77 5269 6e67 456c 656d 656e 742c 2072  owRingElement, r
│ │ │ │ +000276a0: 6574 7572 6e73 0a20 2020 2020 2020 2061  eturns.        a
│ │ │ │ +000276b0: 2052 696e 6745 6c65 6d65 6e74 2069 6e20   RingElement in 
│ │ │ │ +000276c0: 7468 6520 4368 6f77 2072 696e 6720 6f66  the Chow ring of
│ │ │ │ +000276d0: 2074 6865 2061 7070 726f 7072 6961 7465   the appropriate
│ │ │ │ +000276e0: 2061 6d62 6965 6e74 2073 7061 6365 0a20   ambient space. 
│ │ │ │ +000276f0: 2020 2020 202a 204f 7574 7075 7420 3d3e       * Output =>
│ │ │ │ +00027700: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +00027710: 6c75 6520 4368 6f77 5269 6e67 456c 656d  lue ChowRingElem
│ │ │ │ +00027720: 656e 742c 2048 6173 6846 6f72 6d2c 2048  ent, HashForm, H
│ │ │ │ +00027730: 6173 6846 6f72 6d0a 2020 2020 2020 2020  ashForm.        
│ │ │ │ +00027740: 7265 7475 726e 7320 6120 4d75 7461 626c  returns a Mutabl
│ │ │ │ +00027750: 6548 6173 6854 6162 6c65 2063 6f6e 7461  eHashTable conta
│ │ │ │ +00027760: 696e 696e 6720 7468 6520 666f 6c6c 6f77  ining the follow
│ │ │ │ +00027770: 696e 6720 6b65 7973 3a20 2247 2220 2874  ing keys: "G" (t
│ │ │ │ +00027780: 6865 0a20 2020 2020 2020 2070 6f6c 796e  he.        polyn
│ │ │ │ +00027790: 6f6d 6961 6c20 7769 7468 2063 6f65 6666  omial with coeff
│ │ │ │ +000277a0: 6963 6965 6e74 7320 6f66 2074 6865 2068  icients of the h
│ │ │ │ +000277b0: 7970 6572 706c 616e 6520 636c 6173 7365  yperplane classe
│ │ │ │ +000277c0: 7320 7265 7072 6573 656e 7469 6e67 2074  s representing t
│ │ │ │ +000277d0: 6865 0a20 2020 2020 2020 2070 726f 6a65  he.        proje
│ │ │ │ +000277e0: 6374 6976 6520 6465 6772 6565 7329 2c20  ctive degrees), 
│ │ │ │ +000277f0: 2247 6c69 7374 2220 2874 6865 206c 6973  "Glist" (the lis
│ │ │ │ +00027800: 7420 666f 726d 206f 6620 2247 2229 202c  t form of "G") ,
│ │ │ │ +00027810: 2022 5365 6772 6522 2028 7468 650a 2020   "Segre" (the.  
│ │ │ │ +00027820: 2020 2020 2020 746f 7461 6c20 5365 6772        total Segr
│ │ │ │ +00027830: 6520 636c 6173 7320 6f66 2074 6865 2069  e class of the i
│ │ │ │ +00027840: 6e70 7574 292c 2253 6567 7265 4c69 7374  nput),"SegreList
│ │ │ │ +00027850: 2220 2874 6865 206c 6973 7420 666f 726d  " (the list form
│ │ │ │ +00027860: 206f 6620 2253 6567 7265 2229 0a20 202a   of "Segre").  *
│ │ │ │ +00027870: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +00027880: 2a20 6120 2a6e 6f74 6520 7269 6e67 2065  * a *note ring e
│ │ │ │ +00027890: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61  lement: (Macaula
│ │ │ │ +000278a0: 7932 446f 6329 5269 6e67 456c 656d 656e  y2Doc)RingElemen
│ │ │ │ +000278b0: 742c 2c20 7468 6520 7075 7368 666f 7277  t,, the pushforw
│ │ │ │ +000278c0: 6172 6420 6f66 0a20 2020 2020 2020 2074  ard of.        t
│ │ │ │ +000278d0: 6865 2074 6f74 616c 2053 6567 7265 2063  he total Segre c
│ │ │ │ +000278e0: 6c61 7373 206f 6620 7468 6520 7363 6865  lass of the sche
│ │ │ │ +000278f0: 6d65 2056 2064 6566 696e 6564 2062 7920  me V defined by 
│ │ │ │ +00027900: 7468 6520 696e 7075 7420 6964 6561 6c20  the input ideal 
│ │ │ │ +00027910: 746f 2074 6865 0a20 2020 2020 2020 2061  to the.        a
│ │ │ │ +00027920: 7070 726f 7072 6961 7465 2043 686f 7720  ppropriate Chow 
│ │ │ │ +00027930: 7269 6e67 0a0a 4465 7363 7269 7074 696f  ring..Descriptio
│ │ │ │ +00027940: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a46  n.===========..F
│ │ │ │ +00027950: 6f72 2061 2073 7562 7363 6865 6d65 2056  or a subscheme V
│ │ │ │ +00027960: 206f 6620 616e 2061 7070 6c69 6361 626c   of an applicabl
│ │ │ │ +00027970: 6520 746f 7269 6320 7661 7269 6574 7920  e toric variety 
│ │ │ │ +00027980: 5820 7468 6973 2063 6f6d 6d61 6e64 2063  X this command c
│ │ │ │ +00027990: 6f6d 7075 7465 7320 7468 650a 7075 7368  omputes the.push
│ │ │ │ +000279a0: 2d66 6f72 7761 7264 206f 6620 7468 6520  -forward of the 
│ │ │ │ +000279b0: 746f 7461 6c20 5365 6772 6520 636c 6173  total Segre clas
│ │ │ │ +000279c0: 7320 7328 562c 5829 206f 6620 5620 696e  s s(V,X) of V in
│ │ │ │ +000279d0: 2058 2074 6f20 7468 6520 4368 6f77 2072   X to the Chow r
│ │ │ │ +000279e0: 696e 6720 6f66 2058 2e0a 0a2b 2d2d 2d2d  ing of X...+----
│ │ │ │ +000279f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00027a30: 7365 7452 616e 646f 6d53 6565 6420 3732  setRandomSeed 72
│ │ │ │ -00027a40: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -00027a50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027a10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00027a20: 2073 6574 5261 6e64 6f6d 5365 6564 2037   setRandomSeed 7
│ │ │ │ +00027a30: 323b 2020 2020 2020 2020 2020 2020 2020  2;              
│ │ │ │ +00027a40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00027a90: 5220 3d20 5a5a 2f33 3237 3439 5b77 2c79  R = ZZ/32749[w,y
│ │ │ │ -00027aa0: 2c7a 5d20 2020 2020 2020 2020 2020 2020  ,z]             
│ │ │ │ -00027ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027a70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +00027a80: 2052 203d 205a 5a2f 3332 3734 395b 772c   R = ZZ/32749[w,
│ │ │ │ +00027a90: 792c 7a5d 2020 2020 2020 2020 2020 2020  y,z]            
│ │ │ │ +00027aa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ae0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00027af0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -00027b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027ad0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ +00027ae0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00027af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b40: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00027b50: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ -00027b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027b30: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ +00027b40: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ +00027b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ba0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -00027bb0: 5365 6772 6528 6964 6561 6c28 772a 7929  Segre(ideal(w*y)
│ │ │ │ -00027bc0: 2c43 6f6d 704d 6574 686f 643d 3e50 6e52  ,CompMethod=>PnR
│ │ │ │ -00027bd0: 6573 6964 7561 6c29 7c0a 7c20 2020 2020  esidual)|.|     
│ │ │ │ +00027b90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00027ba0: 2053 6567 7265 2869 6465 616c 2877 2a79   Segre(ideal(w*y
│ │ │ │ +00027bb0: 292c 436f 6d70 4d65 7468 6f64 3d3e 506e  ),CompMethod=>Pn
│ │ │ │ +00027bc0: 5265 7369 6475 616c 297c 0a7c 2020 2020  Residual)|.|    
│ │ │ │ +00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027c10: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00027c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c30: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00027c40: 2d20 3448 2020 2b20 3248 2020 2020 2020  - 4H  + 2H      
│ │ │ │ -00027c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027bf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027c00: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00027c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027c20: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +00027c30: 202d 2034 4820 202b 2032 4820 2020 2020   - 4H  + 2H     
│ │ │ │ +00027c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027c50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027ca0: 5a5a 5b48 5d20 2020 2020 2020 2020 2020  ZZ[H]           
│ │ │ │ -00027cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027cc0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00027cd0: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ -00027ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027cf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027d00: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00027d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027d20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027d30: 2020 4820 2020 2020 2020 2020 2020 2020    H             
│ │ │ │ -00027d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027d50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027c80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027c90: 205a 5a5b 485d 2020 2020 2020 2020 2020   ZZ[H]          
│ │ │ │ +00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027cb0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00027cc0: 202d 2d2d 2d2d 2020 2020 2020 2020 2020   -----          
│ │ │ │ +00027cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027ce0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027cf0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +00027d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027d20: 2020 2048 2020 2020 2020 2020 2020 2020     H            
│ │ │ │ +00027d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00027d90: 413d 4368 6f77 5269 6e67 2852 2920 2020  A=ChowRing(R)   
│ │ │ │ -00027da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027d70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00027d80: 2041 3d43 686f 7752 696e 6728 5229 2020   A=ChowRing(R)  
│ │ │ │ +00027d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027da0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027de0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -00027df0: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │ -00027e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027e10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027dd0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +00027de0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │ +00027df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027e40: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -00027e50: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ -00027e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027e70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027e30: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +00027e40: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00027e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ea0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -00027eb0: 5365 6772 6528 412c 6964 6561 6c28 775e  Segre(A,ideal(w^
│ │ │ │ -00027ec0: 322a 792c 772a 795e 3229 2920 2020 2020  2*y,w*y^2))     
│ │ │ │ -00027ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027e90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +00027ea0: 2053 6567 7265 2841 2c69 6465 616c 2877   Segre(A,ideal(w
│ │ │ │ +00027eb0: 5e32 2a79 2c77 2a79 5e32 2929 2020 2020  ^2*y,w*y^2))    
│ │ │ │ +00027ec0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027f10: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00027f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f30: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -00027f40: 2d20 3368 2020 2b20 3268 2020 2020 2020  - 3h  + 2h      
│ │ │ │ -00027f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027f70: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00027f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027ef0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027f00: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00027f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f20: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +00027f30: 202d 2033 6820 202b 2032 6820 2020 2020   - 3h  + 2h     
│ │ │ │ +00027f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027f60: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00027f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fc0: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -00027fd0: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │ -00027fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ff0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00027fb0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +00027fc0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │ +00027fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027fe0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028020: 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e 6f77 2063  --------+..Now c
│ │ │ │ -00028030: 6f6e 7369 6465 7220 616e 2065 7861 6d70  onsider an examp
│ │ │ │ -00028040: 6c65 2069 6e20 5c50 505e 3220 5c74 696d  le in \PP^2 \tim
│ │ │ │ -00028050: 6573 205c 5050 5e32 2c20 6966 2077 6520  es \PP^2, if we 
│ │ │ │ -00028060: 696e 7075 7420 7468 6520 4368 6f77 2072  input the Chow r
│ │ │ │ -00028070: 696e 6720 4120 7468 650a 6f75 7470 7574  ing A the.output
│ │ │ │ -00028080: 2077 696c 6c20 6265 2072 6574 7572 6e65   will be returne
│ │ │ │ -00028090: 6420 696e 2074 6865 2073 616d 6520 7269  d in the same ri
│ │ │ │ -000280a0: 6e67 2e20 546f 2065 6e73 7572 6520 7072  ng. To ensure pr
│ │ │ │ -000280b0: 6f70 6572 2066 756e 6374 696f 6e20 6f66  oper function of
│ │ │ │ -000280c0: 2074 6865 0a6d 6574 686f 6473 2077 6520   the.methods we 
│ │ │ │ -000280d0: 6275 696c 6420 7468 6520 4368 6f77 2072  build the Chow r
│ │ │ │ -000280e0: 696e 6720 7573 696e 6720 7468 6520 2a6e  ing using the *n
│ │ │ │ -000280f0: 6f74 6520 4368 6f77 5269 6e67 3a20 4368  ote ChowRing: Ch
│ │ │ │ -00028100: 6f77 5269 6e67 2c20 636f 6d6d 616e 642e  owRing, command.
│ │ │ │ -00028110: 2057 650a 6d61 7920 616c 736f 2072 6574   We.may also ret
│ │ │ │ -00028120: 7572 6e20 6120 4d75 7461 626c 6548 6173  urn a MutableHas
│ │ │ │ -00028130: 6854 6162 6c65 2e0a 0a2b 2d2d 2d2d 2d2d  hTable...+------
│ │ │ │ +00028010: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e6f 7720  ---------+..Now 
│ │ │ │ +00028020: 636f 6e73 6964 6572 2061 6e20 6578 616d  consider an exam
│ │ │ │ +00028030: 706c 6520 696e 205c 5050 5e32 205c 7469  ple in \PP^2 \ti
│ │ │ │ +00028040: 6d65 7320 5c50 505e 322c 2069 6620 7765  mes \PP^2, if we
│ │ │ │ +00028050: 2069 6e70 7574 2074 6865 2043 686f 7720   input the Chow 
│ │ │ │ +00028060: 7269 6e67 2041 2074 6865 0a6f 7574 7075  ring A the.outpu
│ │ │ │ +00028070: 7420 7769 6c6c 2062 6520 7265 7475 726e  t will be return
│ │ │ │ +00028080: 6564 2069 6e20 7468 6520 7361 6d65 2072  ed in the same r
│ │ │ │ +00028090: 696e 672e 2054 6f20 656e 7375 7265 2070  ing. To ensure p
│ │ │ │ +000280a0: 726f 7065 7220 6675 6e63 7469 6f6e 206f  roper function o
│ │ │ │ +000280b0: 6620 7468 650a 6d65 7468 6f64 7320 7765  f the.methods we
│ │ │ │ +000280c0: 2062 7569 6c64 2074 6865 2043 686f 7720   build the Chow 
│ │ │ │ +000280d0: 7269 6e67 2075 7369 6e67 2074 6865 202a  ring using the *
│ │ │ │ +000280e0: 6e6f 7465 2043 686f 7752 696e 673a 2043  note ChowRing: C
│ │ │ │ +000280f0: 686f 7752 696e 672c 2063 6f6d 6d61 6e64  howRing, command
│ │ │ │ +00028100: 2e20 5765 0a6d 6179 2061 6c73 6f20 7265  . We.may also re
│ │ │ │ +00028110: 7475 726e 2061 204d 7574 6162 6c65 4861  turn a MutableHa
│ │ │ │ +00028120: 7368 5461 626c 652e 0a0a 2b2d 2d2d 2d2d  shTable...+-----
│ │ │ │ +00028130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028180: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2052  -------+.|i6 : R
│ │ │ │ -00028190: 3d4d 756c 7469 5072 6f6a 436f 6f72 6452  =MultiProjCoordR
│ │ │ │ -000281a0: 696e 6728 7b32 2c32 7d29 2020 2020 2020  ing({2,2})      
│ │ │ │ +00028170: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +00028180: 523d 4d75 6c74 6950 726f 6a43 6f6f 7264  R=MultiProjCoord
│ │ │ │ +00028190: 5269 6e67 287b 322c 327d 2920 2020 2020  Ring({2,2})     
│ │ │ │ +000281a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000281c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000281d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028220: 2020 2020 2020 207c 0a7c 6f36 203d 2052         |.|o6 = R
│ │ │ │ +00028210: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +00028220: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00028230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282c0: 2020 2020 2020 207c 0a7c 6f36 203a 2050         |.|o6 : P
│ │ │ │ -000282d0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +000282b0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +000282c0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +000282d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028310: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00028300: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00028310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028360: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2072  -------+.|i7 : r
│ │ │ │ -00028370: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +00028350: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +00028360: 723d 6765 6e73 2052 2020 2020 2020 2020  r=gens R        
│ │ │ │ +00028370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000283a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000283b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000283c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000283d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000283e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028400: 2020 2020 2020 207c 0a7c 6f37 203d 207b         |.|o7 = {
│ │ │ │ -00028410: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -00028420: 7820 2c20 7820 7d20 2020 2020 2020 2020  x , x }         
│ │ │ │ +000283f0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +00028400: 7b78 202c 2078 202c 2078 202c 2078 202c  {x , x , x , x ,
│ │ │ │ +00028410: 2078 202c 2078 207d 2020 2020 2020 2020   x , x }        
│ │ │ │ +00028420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028450: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028460: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ -00028470: 2034 2020 2035 2020 2020 2020 2020 2020   4   5          
│ │ │ │ +00028440: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028450: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ +00028460: 2020 3420 2020 3520 2020 2020 2020 2020    4   5         
│ │ │ │ +00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028490: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000284a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284f0: 2020 2020 2020 207c 0a7c 6f37 203a 204c         |.|o7 : L
│ │ │ │ -00028500: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +000284e0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ +000284f0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +00028500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028540: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00028530: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028590: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2041  -------+.|i8 : A
│ │ │ │ -000285a0: 3d43 686f 7752 696e 6728 5229 2020 2020  =ChowRing(R)    
│ │ │ │ +00028580: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +00028590: 413d 4368 6f77 5269 6e67 2852 2920 2020  A=ChowRing(R)   
│ │ │ │ +000285a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000285b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000285c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000285d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000285e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000285f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028630: 2020 2020 2020 207c 0a7c 6f38 203d 2041         |.|o8 = A
│ │ │ │ +00028620: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ +00028630: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │  00028640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000286a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000286b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286d0: 2020 2020 2020 207c 0a7c 6f38 203a 2051         |.|o8 : Q
│ │ │ │ -000286e0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +000286c0: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ +000286d0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +000286e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000286f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028720: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00028710: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028770: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2049  -------+.|i9 : I
│ │ │ │ -00028780: 3d69 6465 616c 2872 5f30 5e32 2a72 5f33  =ideal(r_0^2*r_3
│ │ │ │ -00028790: 2d72 5f34 2a72 5f31 2a72 5f32 2c72 5f32  -r_4*r_1*r_2,r_2
│ │ │ │ -000287a0: 5e32 2a72 5f35 2920 2020 2020 2020 2020  ^2*r_5)         
│ │ │ │ -000287b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028760: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +00028770: 493d 6964 6561 6c28 725f 305e 322a 725f  I=ideal(r_0^2*r_
│ │ │ │ +00028780: 332d 725f 342a 725f 312a 725f 322c 725f  3-r_4*r_1*r_2,r_
│ │ │ │ +00028790: 325e 322a 725f 3529 2020 2020 2020 2020  2^2*r_5)        
│ │ │ │ +000287a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000287c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028810: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028810: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00028820: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00028830: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00028830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028860: 2020 2020 2020 207c 0a7c 6f39 203d 2069         |.|o9 = i
│ │ │ │ -00028870: 6465 616c 2028 7820 7820 202d 2078 2078  deal (x x  - x x
│ │ │ │ -00028880: 2078 202c 2078 2078 2029 2020 2020 2020   x , x x )      
│ │ │ │ +00028850: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ +00028860: 6964 6561 6c20 2878 2078 2020 2d20 7820  ideal (x x  - x 
│ │ │ │ +00028870: 7820 7820 2c20 7820 7820 2920 2020 2020  x x , x x )     
│ │ │ │ +00028880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000288c0: 2020 2020 2020 2030 2033 2020 2020 3120         0 3    1 
│ │ │ │ -000288d0: 3220 3420 2020 3220 3520 2020 2020 2020  2 4   2 5       
│ │ │ │ +000288a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000288b0: 2020 2020 2020 2020 3020 3320 2020 2031          0 3    1
│ │ │ │ +000288c0: 2032 2034 2020 2032 2035 2020 2020 2020   2 4   2 5      
│ │ │ │ +000288d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000288e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028900: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000288f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028950: 2020 2020 2020 207c 0a7c 6f39 203a 2049         |.|o9 : I
│ │ │ │ -00028960: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +00028940: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ +00028950: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │ +00028960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00028990: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000289a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000289b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000289c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000289d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000289e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000289f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -00028a00: 5365 6772 6520 4920 2020 2020 2020 2020  Segre I         
│ │ │ │ +000289e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ +000289f0: 2053 6567 7265 2049 2020 2020 2020 2020   Segre I        
│ │ │ │ +00028a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028a30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028aa0: 2020 2032 2032 2020 2020 2020 3220 2020     2 2      2   
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│ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ae0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ -00028af0: 3732 6820 6820 202d 2032 3468 2068 2020  72h h  - 24h h  
│ │ │ │ -00028b00: 2d20 3132 6820 6820 202b 2034 6820 202b  - 12h h  + 4h  +
│ │ │ │ -00028b10: 2034 6820 6820 202b 2068 2020 2020 2020   4h h  + h      
│ │ │ │ -00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028b40: 2020 2031 2032 2020 2020 2020 3120 3220     1 2      1 2 
│ │ │ │ -00028b50: 2020 2020 2031 2032 2020 2020 2031 2020       1 2     1  
│ │ │ │ -00028b60: 2020 2031 2032 2020 2020 3220 2020 2020     1 2    2     
│ │ │ │ -00028b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028a90: 2020 2020 3220 3220 2020 2020 2032 2020      2 2      2  
│ │ │ │ +00028aa0: 2020 2020 2020 2020 3220 2020 2020 3220          2     2 
│ │ │ │ +00028ab0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ad0: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ +00028ae0: 2037 3268 2068 2020 2d20 3234 6820 6820   72h h  - 24h h 
│ │ │ │ +00028af0: 202d 2031 3268 2068 2020 2b20 3468 2020   - 12h h  + 4h  
│ │ │ │ +00028b00: 2b20 3468 2068 2020 2b20 6820 2020 2020  + 4h h  + h     
│ │ │ │ +00028b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028b30: 2020 2020 3120 3220 2020 2020 2031 2032      1 2      1 2
│ │ │ │ +00028b40: 2020 2020 2020 3120 3220 2020 2020 3120        1 2     1 
│ │ │ │ +00028b50: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028be0: 5a5a 5b68 202e 2e68 205d 2020 2020 2020  ZZ[h ..h ]      
│ │ │ │ +00028bc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028bd0: 205a 5a5b 6820 2e2e 6820 5d20 2020 2020   ZZ[h ..h ]     
│ │ │ │ +00028be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028c30: 2020 2020 3120 2020 3220 2020 2020 2020      1   2       
│ │ │ │ +00028c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028c20: 2020 2020 2031 2020 2032 2020 2020 2020       1   2      
│ │ │ │ +00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c70: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ -00028c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 2020  ----------      
│ │ │ │ +00028c60: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ +00028c70: 202d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020   ----------     
│ │ │ │ +00028c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028cd0: 2020 2033 2020 2033 2020 2020 2020 2020     3   3        
│ │ │ │ +00028cb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028cc0: 2020 2020 3320 2020 3320 2020 2020 2020      3   3       
│ │ │ │ +00028cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028d20: 2028 6820 2c20 6820 2920 2020 2020 2020   (h , h )       
│ │ │ │ +00028d00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028d10: 2020 2868 202c 2068 2029 2020 2020 2020    (h , h )      
│ │ │ │ +00028d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028d70: 2020 2031 2020 2032 2020 2020 2020 2020     1   2        
│ │ │ │ +00028d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028d60: 2020 2020 3120 2020 3220 2020 2020 2020      1   2       
│ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028db0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00028da0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00028db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028e00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ -00028e10: 7331 3d53 6567 7265 2841 2c49 2920 2020  s1=Segre(A,I)   
│ │ │ │ +00028df0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +00028e00: 2073 313d 5365 6772 6528 412c 4929 2020   s1=Segre(A,I)  
│ │ │ │ +00028e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ea0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028eb0: 2020 2032 2032 2020 2020 2020 3220 2020     2 2      2   
│ │ │ │ -00028ec0: 2020 2020 2020 2032 2020 2020 2032 2020         2     2  
│ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ef0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ -00028f00: 3732 6820 6820 202d 2032 3468 2068 2020  72h h  - 24h h  
│ │ │ │ -00028f10: 2d20 3132 6820 6820 202b 2034 6820 202b  - 12h h  + 4h  +
│ │ │ │ -00028f20: 2034 6820 6820 202b 2068 2020 2020 2020   4h h  + h      
│ │ │ │ -00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028f50: 2020 2031 2032 2020 2020 2020 3120 3220     1 2      1 2 
│ │ │ │ -00028f60: 2020 2020 2031 2032 2020 2020 2031 2020       1 2     1  
│ │ │ │ -00028f70: 2020 2031 2032 2020 2020 3220 2020 2020     1 2    2     
│ │ │ │ -00028f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028ea0: 2020 2020 3220 3220 2020 2020 2032 2020      2 2      2  
│ │ │ │ +00028eb0: 2020 2020 2020 2020 3220 2020 2020 3220          2     2 
│ │ │ │ +00028ec0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ee0: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +00028ef0: 2037 3268 2068 2020 2d20 3234 6820 6820   72h h  - 24h h 
│ │ │ │ +00028f00: 202d 2031 3268 2068 2020 2b20 3468 2020   - 12h h  + 4h  
│ │ │ │ +00028f10: 2b20 3468 2068 2020 2b20 6820 2020 2020  + 4h h  + h     
│ │ │ │ +00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028f40: 2020 2020 3120 3220 2020 2020 2031 2032      1 2      1 2
│ │ │ │ +00028f50: 2020 2020 2020 3120 3220 2020 2020 3120        1 2     1 
│ │ │ │ +00028f60: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +00028f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fe0: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ -00028ff0: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │ +00028fd0: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ +00028fe0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │ +00028ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029030: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00029020: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00029030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029080: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -00029090: 5365 6748 6173 683d 5365 6772 6528 412c  SegHash=Segre(A,
│ │ │ │ -000290a0: 492c 4f75 7470 7574 3d3e 4861 7368 466f  I,Output=>HashFo
│ │ │ │ -000290b0: 726d 2920 2020 2020 2020 2020 2020 2020  rm)             
│ │ │ │ -000290c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000290d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029070: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │ +00029080: 2053 6567 4861 7368 3d53 6567 7265 2841   SegHash=Segre(A
│ │ │ │ +00029090: 2c49 2c4f 7574 7075 743d 3e48 6173 6846  ,I,Output=>HashF
│ │ │ │ +000290a0: 6f72 6d29 2020 2020 2020 2020 2020 2020  orm)            
│ │ │ │ +000290b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000290c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000290d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029120: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ -00029130: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ -00029140: 7b2e 2e2e 342e 2e2e 7d20 2020 2020 2020  {...4...}       
│ │ │ │ +00029110: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ +00029120: 204d 7574 6162 6c65 4861 7368 5461 626c   MutableHashTabl
│ │ │ │ +00029130: 657b 2e2e 2e34 2e2e 2e7d 2020 2020 2020  e{...4...}      
│ │ │ │ +00029140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029170: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291c0: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ -000291d0: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ +000291b0: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ +000291c0: 204d 7574 6162 6c65 4861 7368 5461 626c   MutableHashTabl
│ │ │ │ +000291d0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  000291e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029210: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00029200: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00029210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029260: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -00029270: 7065 656b 2053 6567 4861 7368 2020 2020  peek SegHash    
│ │ │ │ +00029250: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ +00029260: 2070 6565 6b20 5365 6748 6173 6820 2020   peek SegHash   
│ │ │ │ +00029270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000292a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000292b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000292a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000292b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000292c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000292d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000292e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000292f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029300: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -00029310: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ -00029320: 7b47 203d 3e20 3268 2020 2b20 6820 202b  {G => 2h  + h  +
│ │ │ │ -00029330: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00029340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029350: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00029360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029370: 2020 2020 2020 2020 3120 2020 2032 2020          1    2  
│ │ │ │ +000292f0: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +00029300: 204d 7574 6162 6c65 4861 7368 5461 626c   MutableHashTabl
│ │ │ │ +00029310: 657b 4720 3d3e 2032 6820 202b 2068 2020  e{G => 2h  + h  
│ │ │ │ +00029320: 2b20 3120 2020 2020 2020 2020 2020 2020  + 1             
│ │ │ │ +00029330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029360: 2020 2020 2020 2020 2031 2020 2020 3220           1    2 
│ │ │ │ +00029370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000293b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293c0: 2047 6c69 7374 203d 3e20 7b31 2c20 3268   Glist => {1, 2h
│ │ │ │ -000293d0: 2020 2b20 6820 2c20 302c 2030 2c20 307d    + h , 0, 0, 0}
│ │ │ │ -000293e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029390: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000293a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000293b0: 2020 476c 6973 7420 3d3e 207b 312c 2032    Glist => {1, 2
│ │ │ │ +000293c0: 6820 202b 2068 202c 2030 2c20 302c 2030  h  + h , 0, 0, 0
│ │ │ │ +000293d0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000293e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000293f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029420: 3120 2020 2032 2020 2020 2020 2020 2020  1    2          
│ │ │ │ -00029430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029410: 2031 2020 2020 3220 2020 2020 2020 2020   1    2         
│ │ │ │ +00029420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029470: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00029480: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00029490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000294a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000294b0: 2053 6567 7265 4c69 7374 203d 3e20 7b30   SegreList => {0
│ │ │ │ -000294c0: 2c20 302c 2034 6820 202b 2034 6820 6820  , 0, 4h  + 4h h 
│ │ │ │ -000294d0: 202b 2068 202c 202d 2020 2020 2020 2020   + h , -        
│ │ │ │ -000294e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029460: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00029470: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00029480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000294a0: 2020 5365 6772 654c 6973 7420 3d3e 207b    SegreList => {
│ │ │ │ +000294b0: 302c 2030 2c20 3468 2020 2b20 3468 2068  0, 0, 4h  + 4h h
│ │ │ │ +000294c0: 2020 2b20 6820 2c20 2d20 2020 2020 2020    + h , -       
│ │ │ │ +000294d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000294e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000294f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029510: 2020 2020 2020 2031 2020 2020 2031 2032         1     1 2
│ │ │ │ -00029520: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00029530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00029540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029550: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ -00029560: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00029570: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ -00029580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00029590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000295a0: 2053 6567 7265 203d 3e20 3732 6820 6820   Segre => 72h h 
│ │ │ │ -000295b0: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ -000295c0: 6820 202b 2034 6820 2020 2020 2020 2020  h  + 4h         
│ │ │ │ -000295d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000295e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000295f0: 2020 2020 2020 2020 2020 2020 2031 2032               1 2
│ │ │ │ -00029600: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ -00029610: 2032 2020 2020 2031 2020 2020 2020 2020   2     1        
│ │ │ │ -00029620: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +00029500: 2020 2020 2020 2020 3120 2020 2020 3120          1     1 
│ │ │ │ +00029510: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ +00029520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029540: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00029550: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ +00029560: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ +00029570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029590: 2020 5365 6772 6520 3d3e 2037 3268 2068    Segre => 72h h
│ │ │ │ +000295a0: 2020 2d20 3234 6820 6820 202d 2031 3268    - 24h h  - 12h
│ │ │ │ +000295b0: 2068 2020 2b20 3468 2020 2020 2020 2020   h  + 4h        
│ │ │ │ +000295c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000295d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000295e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000295f0: 3220 2020 2020 2031 2032 2020 2020 2020  2      1 2      
│ │ │ │ +00029600: 3120 3220 2020 2020 3120 2020 2020 2020  1 2     1       
│ │ │ │ +00029610: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +00029620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029670: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ -00029680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029690: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ +00029660: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00029670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029680: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │ +00029690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000296b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000296c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000296b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000296c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029710: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029760: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297b0: 2020 2020 2020 207c 0a7c 2020 2032 2020         |.|   2  
│ │ │ │ -000297c0: 2020 2020 2020 2020 3220 2020 2020 3220          2     2 
│ │ │ │ -000297d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000297a0: 2020 2020 2020 2020 7c0a 7c20 2020 3220          |.|   2 
│ │ │ │ +000297b0: 2020 2020 2020 2020 2032 2020 2020 2032           2     2
│ │ │ │ +000297c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000297d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029800: 2020 2020 2020 207c 0a7c 3234 6820 6820         |.|24h h 
│ │ │ │ -00029810: 202d 2031 3268 2068 202c 2037 3268 2068   - 12h h , 72h h
│ │ │ │ -00029820: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +000297f0: 2020 2020 2020 2020 7c0a 7c32 3468 2068          |.|24h h
│ │ │ │ +00029800: 2020 2d20 3132 6820 6820 2c20 3732 6820    - 12h h , 72h 
│ │ │ │ +00029810: 6820 7d20 2020 2020 2020 2020 2020 2020  h }             
│ │ │ │ +00029820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029850: 2020 2020 2020 207c 0a7c 2020 2031 2032         |.|   1 2
│ │ │ │ -00029860: 2020 2020 2020 3120 3220 2020 2020 3120        1 2     1 
│ │ │ │ -00029870: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00029840: 2020 2020 2020 2020 7c0a 7c20 2020 3120          |.|   1 
│ │ │ │ +00029850: 3220 2020 2020 2031 2032 2020 2020 2031  2      1 2     1
│ │ │ │ +00029860: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00029870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000298b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00029890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000298a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000298b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298f0: 2020 2020 2020 207c 0a7c 2b20 3468 2068         |.|+ 4h h
│ │ │ │ -00029900: 2020 2b20 6820 2020 2020 2020 2020 2020    + h           
│ │ │ │ +000298e0: 2020 2020 2020 2020 7c0a 7c2b 2034 6820          |.|+ 4h 
│ │ │ │ +000298f0: 6820 202b 2068 2020 2020 2020 2020 2020  h  + h          
│ │ │ │ +00029900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029940: 2020 2020 2020 207c 0a7c 2020 2020 3120         |.|    1 
│ │ │ │ -00029950: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ +00029930: 2020 2020 2020 2020 7c0a 7c20 2020 2031          |.|    1
│ │ │ │ +00029940: 2032 2020 2020 3220 2020 2020 2020 2020   2    2         
│ │ │ │ +00029950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00029980: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00029990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000299a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000299b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000299c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000299d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000299e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ -000299f0: 7331 3d3d 5365 6748 6173 6823 2253 6567  s1==SegHash#"Seg
│ │ │ │ -00029a00: 7265 2220 2020 2020 2020 2020 2020 2020  re"             
│ │ │ │ +000299d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │ +000299e0: 2073 313d 3d53 6567 4861 7368 2322 5365   s1==SegHash#"Se
│ │ │ │ +000299f0: 6772 6522 2020 2020 2020 2020 2020 2020  gre"            
│ │ │ │ +00029a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029a20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a80: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ -00029a90: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00029a70: 2020 2020 2020 2020 7c0a 7c6f 3134 203d          |.|o14 =
│ │ │ │ +00029a80: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00029a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ad0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00029ac0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00029ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b20: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865  -------+..In the
│ │ │ │ -00029b30: 2063 6173 6520 7768 6572 6520 7468 6520   case where the 
│ │ │ │ -00029b40: 616d 6269 656e 7420 7370 6163 6520 6973  ambient space is
│ │ │ │ -00029b50: 2061 2074 6f72 6963 2076 6172 6965 7479   a toric variety
│ │ │ │ -00029b60: 2077 6869 6368 2069 7320 6e6f 7420 6120   which is not a 
│ │ │ │ -00029b70: 7072 6f64 7563 740a 6f66 2070 726f 6a65  product.of proje
│ │ │ │ -00029b80: 6374 6976 6520 7370 6163 6573 2077 6520  ctive spaces we 
│ │ │ │ -00029b90: 6d75 7374 206c 6f61 6420 7468 6520 4e6f  must load the No
│ │ │ │ -00029ba0: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ -00029bb0: 6573 2070 6163 6b61 6765 2061 6e64 206d  es package and m
│ │ │ │ -00029bc0: 7573 740a 616c 736f 2069 6e70 7574 2074  ust.also input t
│ │ │ │ -00029bd0: 6865 2074 6f72 6963 2076 6172 6965 7479  he toric variety
│ │ │ │ -00029be0: 2e20 4966 2074 6865 2074 6f72 6963 2076  . If the toric v
│ │ │ │ -00029bf0: 6172 6965 7479 2069 7320 6120 7072 6f64  ariety is a prod
│ │ │ │ -00029c00: 7563 7420 6f66 2070 726f 6a65 6374 6976  uct of projectiv
│ │ │ │ -00029c10: 650a 7370 6163 6520 6974 2069 7320 7265  e.space it is re
│ │ │ │ -00029c20: 636f 6d6d 656e 6465 6420 746f 2075 7365  commended to use
│ │ │ │ -00029c30: 2074 6865 2066 6f72 6d20 6162 6f76 6520   the form above 
│ │ │ │ -00029c40: 7261 7468 6572 2074 6861 6e20 696e 7075  rather than inpu
│ │ │ │ -00029c50: 7474 696e 6720 7468 6520 746f 7269 630a  tting the toric.
│ │ │ │ -00029c60: 7661 7269 6574 7920 666f 7220 6566 6669  variety for effi
│ │ │ │ -00029c70: 6369 656e 6379 2072 6561 736f 6e73 2e0a  ciency reasons..
│ │ │ │ -00029c80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00029b10: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468  --------+..In th
│ │ │ │ +00029b20: 6520 6361 7365 2077 6865 7265 2074 6865  e case where the
│ │ │ │ +00029b30: 2061 6d62 6965 6e74 2073 7061 6365 2069   ambient space i
│ │ │ │ +00029b40: 7320 6120 746f 7269 6320 7661 7269 6574  s a toric variet
│ │ │ │ +00029b50: 7920 7768 6963 6820 6973 206e 6f74 2061  y which is not a
│ │ │ │ +00029b60: 2070 726f 6475 6374 0a6f 6620 7072 6f6a   product.of proj
│ │ │ │ +00029b70: 6563 7469 7665 2073 7061 6365 7320 7765  ective spaces we
│ │ │ │ +00029b80: 206d 7573 7420 6c6f 6164 2074 6865 204e   must load the N
│ │ │ │ +00029b90: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +00029ba0: 6965 7320 7061 636b 6167 6520 616e 6420  ies package and 
│ │ │ │ +00029bb0: 6d75 7374 0a61 6c73 6f20 696e 7075 7420  must.also input 
│ │ │ │ +00029bc0: 7468 6520 746f 7269 6320 7661 7269 6574  the toric variet
│ │ │ │ +00029bd0: 792e 2049 6620 7468 6520 746f 7269 6320  y. If the toric 
│ │ │ │ +00029be0: 7661 7269 6574 7920 6973 2061 2070 726f  variety is a pro
│ │ │ │ +00029bf0: 6475 6374 206f 6620 7072 6f6a 6563 7469  duct of projecti
│ │ │ │ +00029c00: 7665 0a73 7061 6365 2069 7420 6973 2072  ve.space it is r
│ │ │ │ +00029c10: 6563 6f6d 6d65 6e64 6564 2074 6f20 7573  ecommended to us
│ │ │ │ +00029c20: 6520 7468 6520 666f 726d 2061 626f 7665  e the form above
│ │ │ │ +00029c30: 2072 6174 6865 7220 7468 616e 2069 6e70   rather than inp
│ │ │ │ +00029c40: 7574 7469 6e67 2074 6865 2074 6f72 6963  utting the toric
│ │ │ │ +00029c50: 0a76 6172 6965 7479 2066 6f72 2065 6666  .variety for eff
│ │ │ │ +00029c60: 6963 6965 6e63 7920 7265 6173 6f6e 732e  iciency reasons.
│ │ │ │ +00029c70: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +00029c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135  ----------+.|i15
│ │ │ │ -00029cd0: 203a 206e 6565 6473 5061 636b 6167 6520   : needsPackage 
│ │ │ │ -00029ce0: 224e 6f72 6d61 6c54 6f72 6963 5661 7269  "NormalToricVari
│ │ │ │ -00029cf0: 6574 6965 7322 2020 2020 2020 2020 2020  eties"          
│ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00029cc0: 3520 3a20 6e65 6564 7350 6163 6b61 6765  5 : needsPackage
│ │ │ │ +00029cd0: 2022 4e6f 726d 616c 546f 7269 6356 6172   "NormalToricVar
│ │ │ │ +00029ce0: 6965 7469 6573 2220 2020 2020 2020 2020  ieties"         
│ │ │ │ +00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00029d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d60: 7c0a 7c6f 3135 203d 204e 6f72 6d61 6c54  |.|o15 = NormalT
│ │ │ │ -00029d70: 6f72 6963 5661 7269 6574 6965 7320 2020  oricVarieties   
│ │ │ │ +00029d50: 207c 0a7c 6f31 3520 3d20 4e6f 726d 616c   |.|o15 = Normal
│ │ │ │ +00029d60: 546f 7269 6356 6172 6965 7469 6573 2020  ToricVarieties  
│ │ │ │ +00029d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029da0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 7c0a 7c6f 3135 203a 2050        |.|o15 : P
│ │ │ │ -00029e00: 6163 6b61 6765 2020 2020 2020 2020 2020  ackage          
│ │ │ │ +00029de0: 2020 2020 2020 207c 0a7c 6f31 3520 3a20         |.|o15 : 
│ │ │ │ +00029df0: 5061 636b 6167 6520 2020 2020 2020 2020  Package         
│ │ │ │ +00029e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00029e30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00029e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00029e90: 3136 203a 2052 686f 203d 207b 7b31 2c30  16 : Rho = {{1,0
│ │ │ │ -00029ea0: 2c30 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30  ,0},{0,1,0},{0,0
│ │ │ │ -00029eb0: 2c31 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30  ,1},{-1,-1,0},{0
│ │ │ │ -00029ec0: 2c30 2c2d 317d 7d20 2020 2020 2020 2020  ,0,-1}}         
│ │ │ │ -00029ed0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00029e80: 6931 3620 3a20 5268 6f20 3d20 7b7b 312c  i16 : Rho = {{1,
│ │ │ │ +00029e90: 302c 307d 2c7b 302c 312c 307d 2c7b 302c  0,0},{0,1,0},{0,
│ │ │ │ +00029ea0: 302c 317d 2c7b 2d31 2c2d 312c 307d 2c7b  0,1},{-1,-1,0},{
│ │ │ │ +00029eb0: 302c 302c 2d31 7d7d 2020 2020 2020 2020  0,0,-1}}        
│ │ │ │ +00029ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f20: 2020 7c0a 7c6f 3136 203d 207b 7b31 2c20    |.|o16 = {{1, 
│ │ │ │ -00029f30: 302c 2030 7d2c 207b 302c 2031 2c20 307d  0, 0}, {0, 1, 0}
│ │ │ │ -00029f40: 2c20 7b30 2c20 302c 2031 7d2c 207b 2d31  , {0, 0, 1}, {-1
│ │ │ │ -00029f50: 2c20 2d31 2c20 307d 2c20 7b30 2c20 302c  , -1, 0}, {0, 0,
│ │ │ │ -00029f60: 202d 317d 7d20 2020 2020 2020 207c 0a7c   -1}}        |.|
│ │ │ │ +00029f10: 2020 207c 0a7c 6f31 3620 3d20 7b7b 312c     |.|o16 = {{1,
│ │ │ │ +00029f20: 2030 2c20 307d 2c20 7b30 2c20 312c 2030   0, 0}, {0, 1, 0
│ │ │ │ +00029f30: 7d2c 207b 302c 2030 2c20 317d 2c20 7b2d  }, {0, 0, 1}, {-
│ │ │ │ +00029f40: 312c 202d 312c 2030 7d2c 207b 302c 2030  1, -1, 0}, {0, 0
│ │ │ │ +00029f50: 2c20 2d31 7d7d 2020 2020 2020 2020 7c0a  , -1}}        |.
│ │ │ │ +00029f60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00029f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fb0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a          |.|o16 :
│ │ │ │ -00029fc0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +00029fa0: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ +00029fb0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a000: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00029ff0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0002a050: 7c69 3137 203a 2053 6967 6d61 203d 207b  |i17 : Sigma = {
│ │ │ │ -0002a060: 7b30 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c  {0,1,2},{1,2,3},
│ │ │ │ -0002a070: 7b30 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c  {0,2,3},{0,1,4},
│ │ │ │ -0002a080: 7b31 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d  {1,3,4},{0,3,4}}
│ │ │ │ -0002a090: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002a040: 0a7c 6931 3720 3a20 5369 676d 6120 3d20  .|i17 : Sigma = 
│ │ │ │ +0002a050: 7b7b 302c 312c 327d 2c7b 312c 322c 337d  {{0,1,2},{1,2,3}
│ │ │ │ +0002a060: 2c7b 302c 322c 337d 2c7b 302c 312c 347d  ,{0,2,3},{0,1,4}
│ │ │ │ +0002a070: 2c7b 312c 332c 347d 2c7b 302c 332c 347d  ,{1,3,4},{0,3,4}
│ │ │ │ +0002a080: 7d20 2020 2020 2020 2020 7c0a 7c20 2020  }         |.|   
│ │ │ │ +0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0e0: 2020 2020 7c0a 7c6f 3137 203d 207b 7b30      |.|o17 = {{0
│ │ │ │ -0002a0f0: 2c20 312c 2032 7d2c 207b 312c 2032 2c20  , 1, 2}, {1, 2, 
│ │ │ │ -0002a100: 337d 2c20 7b30 2c20 322c 2033 7d2c 207b  3}, {0, 2, 3}, {
│ │ │ │ -0002a110: 302c 2031 2c20 347d 2c20 7b31 2c20 332c  0, 1, 4}, {1, 3,
│ │ │ │ -0002a120: 2034 7d2c 207b 302c 2033 2c20 347d 7d7c   4}, {0, 3, 4}}|
│ │ │ │ -0002a130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a0d0: 2020 2020 207c 0a7c 6f31 3720 3d20 7b7b       |.|o17 = {{
│ │ │ │ +0002a0e0: 302c 2031 2c20 327d 2c20 7b31 2c20 322c  0, 1, 2}, {1, 2,
│ │ │ │ +0002a0f0: 2033 7d2c 207b 302c 2032 2c20 337d 2c20   3}, {0, 2, 3}, 
│ │ │ │ +0002a100: 7b30 2c20 312c 2034 7d2c 207b 312c 2033  {0, 1, 4}, {1, 3
│ │ │ │ +0002a110: 2c20 347d 2c20 7b30 2c20 332c 2034 7d7d  , 4}, {0, 3, 4}}
│ │ │ │ +0002a120: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a170: 2020 2020 2020 2020 2020 7c0a 7c6f 3137            |.|o17
│ │ │ │ -0002a180: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0002a160: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0002a170: 3720 3a20 4c69 7374 2020 2020 2020 2020  7 : List        
│ │ │ │ +0002a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002a1b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a210: 2b0a 7c69 3138 203a 2058 203d 206e 6f72  +.|i18 : X = nor
│ │ │ │ -0002a220: 6d61 6c54 6f72 6963 5661 7269 6574 7928  malToricVariety(
│ │ │ │ -0002a230: 5268 6f2c 5369 676d 612c 436f 6566 6669  Rho,Sigma,Coeffi
│ │ │ │ -0002a240: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33  cientRing =>ZZ/3
│ │ │ │ -0002a250: 3237 3439 2920 2020 2020 207c 0a7c 2020  2749)      |.|  
│ │ │ │ +0002a200: 2d2b 0a7c 6931 3820 3a20 5820 3d20 6e6f  -+.|i18 : X = no
│ │ │ │ +0002a210: 726d 616c 546f 7269 6356 6172 6965 7479  rmalToricVariety
│ │ │ │ +0002a220: 2852 686f 2c53 6967 6d61 2c43 6f65 6666  (Rho,Sigma,Coeff
│ │ │ │ +0002a230: 6963 6965 6e74 5269 6e67 203d 3e5a 5a2f  icientRing =>ZZ/
│ │ │ │ +0002a240: 3332 3734 3929 2020 2020 2020 7c0a 7c20  32749)      |.| 
│ │ │ │ +0002a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a2a0: 2020 2020 2020 7c0a 7c6f 3138 203d 2058        |.|o18 = X
│ │ │ │ +0002a290: 2020 2020 2020 207c 0a7c 6f31 3820 3d20         |.|o18 = 
│ │ │ │ +0002a2a0: 5820 2020 2020 2020 2020 2020 2020 2020  X               
│ │ │ │  0002a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a2f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a2e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a330: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002a340: 3138 203a 204e 6f72 6d61 6c54 6f72 6963  18 : NormalToric
│ │ │ │ -0002a350: 5661 7269 6574 7920 2020 2020 2020 2020  Variety         
│ │ │ │ +0002a320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002a330: 6f31 3820 3a20 4e6f 726d 616c 546f 7269  o18 : NormalTori
│ │ │ │ +0002a340: 6356 6172 6965 7479 2020 2020 2020 2020  cVariety        
│ │ │ │ +0002a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a380: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002a370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a3d0: 2d2d 2b0a 7c69 3139 203a 2043 6865 636b  --+.|i19 : Check
│ │ │ │ -0002a3e0: 546f 7269 6356 6172 6965 7479 5661 6c69  ToricVarietyVali
│ │ │ │ -0002a3f0: 6428 5829 2020 2020 2020 2020 2020 2020  d(X)            
│ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002a3c0: 2d2d 2d2b 0a7c 6931 3920 3a20 4368 6563  ---+.|i19 : Chec
│ │ │ │ +0002a3d0: 6b54 6f72 6963 5661 7269 6574 7956 616c  kToricVarietyVal
│ │ │ │ +0002a3e0: 6964 2858 2920 2020 2020 2020 2020 2020  id(X)           
│ │ │ │ +0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a410: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a460: 2020 2020 2020 2020 7c0a 7c6f 3139 203d          |.|o19 =
│ │ │ │ -0002a470: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +0002a450: 2020 2020 2020 2020 207c 0a7c 6f31 3920           |.|o19 
│ │ │ │ +0002a460: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │ +0002a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a4a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0002a500: 7c69 3230 203a 2052 3d72 696e 6728 5829  |i20 : R=ring(X)
│ │ │ │ +0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002a4f0: 0a7c 6932 3020 3a20 523d 7269 6e67 2858  .|i20 : R=ring(X
│ │ │ │ +0002a500: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0002a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002a530: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a590: 2020 2020 7c0a 7c6f 3230 203d 2052 2020      |.|o20 = R  
│ │ │ │ +0002a580: 2020 2020 207c 0a7c 6f32 3020 3d20 5220       |.|o20 = R 
│ │ │ │ +0002a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a5e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a5d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a620: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ -0002a630: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -0002a640: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0002a610: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0002a620: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269  0 : PolynomialRi
│ │ │ │ +0002a630: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0002a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a670: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002a660: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a6c0: 2b0a 7c69 3231 203a 2049 3d69 6465 616c  +.|i21 : I=ideal
│ │ │ │ -0002a6d0: 2852 5f30 5e34 2a52 5f31 2c52 5f30 2a52  (R_0^4*R_1,R_0*R
│ │ │ │ -0002a6e0: 5f33 2a52 5f34 2a52 5f32 2d52 5f32 5e32  _3*R_4*R_2-R_2^2
│ │ │ │ -0002a6f0: 2a52 5f30 5e32 2920 2020 2020 2020 2020  *R_0^2)         
│ │ │ │ -0002a700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a6b0: 2d2b 0a7c 6932 3120 3a20 493d 6964 6561  -+.|i21 : I=idea
│ │ │ │ +0002a6c0: 6c28 525f 305e 342a 525f 312c 525f 302a  l(R_0^4*R_1,R_0*
│ │ │ │ +0002a6d0: 525f 332a 525f 342a 525f 322d 525f 325e  R_3*R_4*R_2-R_2^
│ │ │ │ +0002a6e0: 322a 525f 305e 3229 2020 2020 2020 2020  2*R_0^2)        
│ │ │ │ +0002a6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a750: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002a760: 2020 2020 2020 2034 2020 2020 2020 2032         4       2
│ │ │ │ -0002a770: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002a740: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002a750: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +0002a760: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ +0002a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7a0: 207c 0a7c 6f32 3120 3d20 6964 6561 6c20   |.|o21 = ideal 
│ │ │ │ -0002a7b0: 2878 2078 202c 202d 2078 2078 2020 2b20  (x x , - x x  + 
│ │ │ │ -0002a7c0: 7820 7820 7820 7820 2920 2020 2020 2020  x x x x )       
│ │ │ │ -0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002a7f0: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -0002a800: 2020 2020 2030 2032 2020 2020 3020 3220       0 2    0 2 
│ │ │ │ -0002a810: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ -0002a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a830: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002a790: 2020 7c0a 7c6f 3231 203d 2069 6465 616c    |.|o21 = ideal
│ │ │ │ +0002a7a0: 2028 7820 7820 2c20 2d20 7820 7820 202b   (x x , - x x  +
│ │ │ │ +0002a7b0: 2078 2078 2078 2078 2029 2020 2020 2020   x x x x )      
│ │ │ │ +0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a7d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002a7e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0002a7f0: 3120 2020 2020 3020 3220 2020 2030 2032  1     0 2    0 2
│ │ │ │ +0002a800: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +0002a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a880: 2020 7c0a 7c6f 3231 203a 2049 6465 616c    |.|o21 : Ideal
│ │ │ │ -0002a890: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +0002a870: 2020 207c 0a7c 6f32 3120 3a20 4964 6561     |.|o21 : Idea
│ │ │ │ +0002a880: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ +0002a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002a8b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a8c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a910: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a  --------+.|i22 :
│ │ │ │ -0002a920: 2053 6567 7265 2858 2c49 2920 2020 2020   Segre(X,I)     
│ │ │ │ +0002a900: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220  ---------+.|i22 
│ │ │ │ +0002a910: 3a20 5365 6772 6528 582c 4929 2020 2020  : Segre(X,I)    
│ │ │ │ +0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a960: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002a950: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002a9b0: 7c20 2020 2020 2020 2020 2020 3220 2020  |           2   
│ │ │ │ -0002a9c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002a990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a9a0: 0a7c 2020 2020 2020 2020 2020 2032 2020  .|           2  
│ │ │ │ +0002a9b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9f0: 2020 2020 2020 2020 207c 0a7c 6f32 3220           |.|o22 
│ │ │ │ -0002aa00: 3d20 2d20 3732 7820 7820 202b 2033 7820  = - 72x x  + 3x 
│ │ │ │ -0002aa10: 202b 2038 7820 7820 202b 2078 2020 2020   + 8x x  + x    
│ │ │ │ +0002a9e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3232            |.|o22
│ │ │ │ +0002a9f0: 203d 202d 2037 3278 2078 2020 2b20 3378   = - 72x x  + 3x
│ │ │ │ +0002aa00: 2020 2b20 3878 2078 2020 2b20 7820 2020    + 8x x  + x   
│ │ │ │ +0002aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002aa50: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ -0002aa60: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0002aa30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002aa40: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ +0002aa50: 2033 2034 2020 2020 3320 2020 2020 2020   3 4    3       
│ │ │ │ +0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002aa90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002aa80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aad0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aaf0: 2020 205a 5a5b 7820 2e2e 7820 5d20 2020     ZZ[x ..x ]   
│ │ │ │ +0002aac0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002aef0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
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│ │ │ │ -0002af50: 202b 2033 7820 202b 2038 7820 7820 202b   + 3x  + 8x x  +
│ │ │ │ -0002af60: 2078 2020 2020 2020 2020 2020 2020 2020   x              
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│ │ │ │ +0002af40: 2020 2b20 3378 2020 2b20 3878 2078 2020    + 3x  + 8x x  
│ │ │ │ +0002af50: 2b20 7820 2020 2020 2020 2020 2020 2020  + x             
│ │ │ │ +0002af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002af80: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ +0002af90: 2033 2020 2020 2033 2034 2020 2020 3320   3     3 4    3 
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│ │ │ │ -0002b2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -0002b2f0: 2022 5365 6772 6528 4964 6561 6c29 220a   "Segre(Ideal)".
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│ │ │ │ -0002b350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +0002b280: 7465 0a70 726f 6261 6269 6c69 7374 6963  te.probabilistic
│ │ │ │ +0002b290: 2061 6c67 6f72 6974 686d 3a20 7072 6f62   algorithm: prob
│ │ │ │ +0002b2a0: 6162 696c 6973 7469 6320 616c 676f 7269  abilistic algori
│ │ │ │ +0002b2b0: 7468 6d2c 2e0a 0a57 6179 7320 746f 2075  thm,...Ways to u
│ │ │ │ +0002b2c0: 7365 2053 6567 7265 3a0a 3d3d 3d3d 3d3d  se Segre:.======
│ │ │ │ +0002b2d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +0002b2e0: 2a20 2253 6567 7265 2849 6465 616c 2922  * "Segre(Ideal)"
│ │ │ │ +0002b2f0: 0a20 202a 2022 5365 6772 6528 4964 6561  .  * "Segre(Idea
│ │ │ │ +0002b300: 6c2c 5379 6d62 6f6c 2922 0a20 202a 2022  l,Symbol)".  * "
│ │ │ │ +0002b310: 5365 6772 6528 5175 6f74 6965 6e74 5269  Segre(QuotientRi
│ │ │ │ +0002b320: 6e67 2c49 6465 616c 2922 0a0a 466f 7220  ng,Ideal)"..For 
│ │ │ │ +0002b330: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +0002b340: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002b350: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +0002b360: 6f74 6520 5365 6772 653a 2053 6567 7265  ote Segre: Segre
│ │ │ │ +0002b370: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +0002b380: 686f 6420 6675 6e63 7469 6f6e 2077 6974  hod function wit
│ │ │ │ +0002b390: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361  h options:.(Maca
│ │ │ │ +0002b3a0: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +0002b3b0: 756e 6374 696f 6e57 6974 684f 7074 696f  unctionWithOptio
│ │ │ │ +0002b3c0: 6e73 2c2e 0a1f 0a46 696c 653a 2043 6861  ns,....File: Cha
│ │ │ │ +0002b3d0: 7261 6374 6572 6973 7469 6343 6c61 7373  racteristicClass
│ │ │ │ +0002b3e0: 6573 2e69 6e66 6f2c 204e 6f64 653a 2054  es.info, Node: T
│ │ │ │ +0002b3f0: 6f72 6963 4368 6f77 5269 6e67 2c20 5072  oricChowRing, Pr
│ │ │ │ +0002b400: 6576 3a20 5365 6772 652c 2055 703a 2054  ev: Segre, Up: T
│ │ │ │ +0002b410: 6f70 0a0a 546f 7269 6343 686f 7752 696e  op..ToricChowRin
│ │ │ │ +0002b420: 6720 2d2d 2043 6f6d 7075 7465 7320 7468  g -- Computes th
│ │ │ │ +0002b430: 6520 4368 6f77 2072 696e 6720 6f66 2061  e Chow ring of a
│ │ │ │ +0002b440: 206e 6f72 6d61 6c20 746f 7269 6320 7661   normal toric va
│ │ │ │ +0002b450: 7269 6574 790a 2a2a 2a2a 2a2a 2a2a 2a2a  riety.**********
│ │ │ │ +0002b460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002b470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002b480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002b490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002b4a0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -0002b4b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ -0002b4c0: 7361 6765 3a20 0a20 2020 2020 2020 2054  sage: .        T
│ │ │ │ -0002b4d0: 6f72 6963 4368 6f77 5269 6e67 2058 0a20  oricChowRing X. 
│ │ │ │ -0002b4e0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -0002b4f0: 202a 2052 2c20 6120 2a6e 6f74 6520 6e6f   * R, a *note no
│ │ │ │ -0002b500: 726d 616c 2074 6f72 6963 2076 6172 6965  rmal toric varie
│ │ │ │ -0002b510: 7479 3a0a 2020 2020 2020 2020 284e 6f72  ty:.        (Nor
│ │ │ │ -0002b520: 6d61 6c54 6f72 6963 5661 7269 6574 6965  malToricVarietie
│ │ │ │ -0002b530: 7329 4e6f 726d 616c 546f 7269 6356 6172  s)NormalToricVar
│ │ │ │ -0002b540: 6965 7479 2c2c 2041 206e 6f72 6d61 6c20  iety,, A normal 
│ │ │ │ -0002b550: 746f 7269 6320 7661 7269 6574 790a 2020  toric variety.  
│ │ │ │ -0002b560: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0002b570: 202a 2061 202a 6e6f 7465 2071 756f 7469   * a *note quoti
│ │ │ │ -0002b580: 656e 7420 7269 6e67 3a20 284d 6163 6175  ent ring: (Macau
│ │ │ │ -0002b590: 6c61 7932 446f 6329 5175 6f74 6965 6e74  lay2Doc)Quotient
│ │ │ │ -0002b5a0: 5269 6e67 2c2c 200a 0a44 6573 6372 6970  Ring,, ..Descrip
│ │ │ │ -0002b5b0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -0002b5c0: 0a0a 4c65 7420 5820 6265 2061 2074 6f72  ..Let X be a tor
│ │ │ │ -0002b5d0: 6963 2076 6172 6965 7479 2077 6974 6820  ic variety with 
│ │ │ │ -0002b5e0: 746f 7461 6c20 636f 6f72 6469 6e61 7465  total coordinate
│ │ │ │ -0002b5f0: 2072 696e 6720 2843 6f78 2072 696e 6729   ring (Cox ring)
│ │ │ │ -0002b600: 2052 2e20 5468 6973 206d 6574 686f 640a   R. This method.
│ │ │ │ -0002b610: 636f 6d70 7574 6573 2074 6865 2043 686f  computes the Cho
│ │ │ │ -0002b620: 7720 7269 6e67 2020 4368 6f77 2072 696e  w ring  Chow rin
│ │ │ │ -0002b630: 6720 4368 3d52 2f28 5352 2b4c 5229 206f  g Ch=R/(SR+LR) o
│ │ │ │ -0002b640: 6620 583b 2068 6572 6520 5352 2069 7320  f X; here SR is 
│ │ │ │ -0002b650: 7468 650a 5374 616e 6c65 792d 5265 6973  the.Stanley-Reis
│ │ │ │ -0002b660: 6e65 7220 6964 6561 6c20 6f66 2074 6865  ner ideal of the
│ │ │ │ -0002b670: 2063 6f72 7265 7370 6f6e 6469 6e67 2066   corresponding f
│ │ │ │ -0002b680: 616e 2061 6e64 204c 5220 6973 2074 6865  an and LR is the
│ │ │ │ -0002b690: 2069 6465 616c 206f 6620 6c69 6e65 6172   ideal of linear
│ │ │ │ -0002b6a0: 0a72 656c 6174 696f 6e73 2061 6d6f 756e  .relations amoun
│ │ │ │ -0002b6b0: 7420 7468 6520 7261 7973 2e20 4974 2069  t the rays. It i
│ │ │ │ -0002b6c0: 7320 6e65 6564 6564 2066 6f72 2069 6e70  s needed for inp
│ │ │ │ -0002b6d0: 7574 2069 6e74 6f20 7468 6520 6d65 7468  ut into the meth
│ │ │ │ -0002b6e0: 6f64 7320 2a6e 6f74 6520 5365 6772 653a  ods *note Segre:
│ │ │ │ -0002b6f0: 0a53 6567 7265 2c2c 202a 6e6f 7465 2043  .Segre,, *note C
│ │ │ │ -0002b700: 6865 726e 3a20 4368 6572 6e2c 2061 6e64  hern: Chern, and
│ │ │ │ -0002b710: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c   *note CSM: CSM,
│ │ │ │ -0002b720: 2069 6e20 7468 6520 6361 7365 7320 7768   in the cases wh
│ │ │ │ -0002b730: 6572 6520 6120 746f 7269 630a 7661 7269  ere a toric.vari
│ │ │ │ -0002b740: 6574 7920 6973 2061 6c73 6f20 696e 7075  ety is also inpu
│ │ │ │ -0002b750: 7420 746f 2065 6e73 7572 6520 7468 6174  t to ensure that
│ │ │ │ -0002b760: 2074 6865 7365 206d 6574 686f 6473 2072   these methods r
│ │ │ │ -0002b770: 6574 7572 6e20 7265 7375 6c74 7320 696e  eturn results in
│ │ │ │ -0002b780: 2074 6865 2073 616d 650a 7269 6e67 2e20   the same.ring. 
│ │ │ │ -0002b790: 5765 2067 6976 6520 616e 2065 7861 6d70  We give an examp
│ │ │ │ -0002b7a0: 6c65 206f 6620 7468 6520 7573 6520 6f66  le of the use of
│ │ │ │ -0002b7b0: 2074 6869 7320 6d65 7468 6f64 2074 6f20   this method to 
│ │ │ │ -0002b7c0: 776f 726b 2077 6974 6820 656c 656d 656e  work with elemen
│ │ │ │ -0002b7d0: 7473 206f 6620 7468 650a 4368 6f77 2072  ts of the.Chow r
│ │ │ │ -0002b7e0: 696e 6720 6f66 2061 2074 6f72 6963 2076  ing of a toric v
│ │ │ │ -0002b7f0: 6172 6965 7479 0a0a 2b2d 2d2d 2d2d 2d2d  ariety..+-------
│ │ │ │ +0002b490: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +0002b4a0: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +0002b4b0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0002b4c0: 546f 7269 6343 686f 7752 696e 6720 580a  ToricChowRing X.
│ │ │ │ +0002b4d0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0002b4e0: 2020 2a20 522c 2061 202a 6e6f 7465 206e    * R, a *note n
│ │ │ │ +0002b4f0: 6f72 6d61 6c20 746f 7269 6320 7661 7269  ormal toric vari
│ │ │ │ +0002b500: 6574 793a 0a20 2020 2020 2020 2028 4e6f  ety:.        (No
│ │ │ │ +0002b510: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ +0002b520: 6573 294e 6f72 6d61 6c54 6f72 6963 5661  es)NormalToricVa
│ │ │ │ +0002b530: 7269 6574 792c 2c20 4120 6e6f 726d 616c  riety,, A normal
│ │ │ │ +0002b540: 2074 6f72 6963 2076 6172 6965 7479 0a20   toric variety. 
│ │ │ │ +0002b550: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0002b560: 2020 2a20 6120 2a6e 6f74 6520 7175 6f74    * a *note quot
│ │ │ │ +0002b570: 6965 6e74 2072 696e 673a 2028 4d61 6361  ient ring: (Maca
│ │ │ │ +0002b580: 756c 6179 3244 6f63 2951 756f 7469 656e  ulay2Doc)Quotien
│ │ │ │ +0002b590: 7452 696e 672c 2c20 0a0a 4465 7363 7269  tRing,, ..Descri
│ │ │ │ +0002b5a0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0002b5b0: 3d0a 0a4c 6574 2058 2062 6520 6120 746f  =..Let X be a to
│ │ │ │ +0002b5c0: 7269 6320 7661 7269 6574 7920 7769 7468  ric variety with
│ │ │ │ +0002b5d0: 2074 6f74 616c 2063 6f6f 7264 696e 6174   total coordinat
│ │ │ │ +0002b5e0: 6520 7269 6e67 2028 436f 7820 7269 6e67  e ring (Cox ring
│ │ │ │ +0002b5f0: 2920 522e 2054 6869 7320 6d65 7468 6f64  ) R. This method
│ │ │ │ +0002b600: 0a63 6f6d 7075 7465 7320 7468 6520 4368  .computes the Ch
│ │ │ │ +0002b610: 6f77 2072 696e 6720 2043 686f 7720 7269  ow ring  Chow ri
│ │ │ │ +0002b620: 6e67 2043 683d 522f 2853 522b 4c52 2920  ng Ch=R/(SR+LR) 
│ │ │ │ +0002b630: 6f66 2058 3b20 6865 7265 2053 5220 6973  of X; here SR is
│ │ │ │ +0002b640: 2074 6865 0a53 7461 6e6c 6579 2d52 6569   the.Stanley-Rei
│ │ │ │ +0002b650: 736e 6572 2069 6465 616c 206f 6620 7468  sner ideal of th
│ │ │ │ +0002b660: 6520 636f 7272 6573 706f 6e64 696e 6720  e corresponding 
│ │ │ │ +0002b670: 6661 6e20 616e 6420 4c52 2069 7320 7468  fan and LR is th
│ │ │ │ +0002b680: 6520 6964 6561 6c20 6f66 206c 696e 6561  e ideal of linea
│ │ │ │ +0002b690: 720a 7265 6c61 7469 6f6e 7320 616d 6f75  r.relations amou
│ │ │ │ +0002b6a0: 6e74 2074 6865 2072 6179 732e 2049 7420  nt the rays. It 
│ │ │ │ +0002b6b0: 6973 206e 6565 6465 6420 666f 7220 696e  is needed for in
│ │ │ │ +0002b6c0: 7075 7420 696e 746f 2074 6865 206d 6574  put into the met
│ │ │ │ +0002b6d0: 686f 6473 202a 6e6f 7465 2053 6567 7265  hods *note Segre
│ │ │ │ +0002b6e0: 3a0a 5365 6772 652c 2c20 2a6e 6f74 6520  :.Segre,, *note 
│ │ │ │ +0002b6f0: 4368 6572 6e3a 2043 6865 726e 2c20 616e  Chern: Chern, an
│ │ │ │ +0002b700: 6420 2a6e 6f74 6520 4353 4d3a 2043 534d  d *note CSM: CSM
│ │ │ │ +0002b710: 2c20 696e 2074 6865 2063 6173 6573 2077  , in the cases w
│ │ │ │ +0002b720: 6865 7265 2061 2074 6f72 6963 0a76 6172  here a toric.var
│ │ │ │ +0002b730: 6965 7479 2069 7320 616c 736f 2069 6e70  iety is also inp
│ │ │ │ +0002b740: 7574 2074 6f20 656e 7375 7265 2074 6861  ut to ensure tha
│ │ │ │ +0002b750: 7420 7468 6573 6520 6d65 7468 6f64 7320  t these methods 
│ │ │ │ +0002b760: 7265 7475 726e 2072 6573 756c 7473 2069  return results i
│ │ │ │ +0002b770: 6e20 7468 6520 7361 6d65 0a72 696e 672e  n the same.ring.
│ │ │ │ +0002b780: 2057 6520 6769 7665 2061 6e20 6578 616d   We give an exam
│ │ │ │ +0002b790: 706c 6520 6f66 2074 6865 2075 7365 206f  ple of the use o
│ │ │ │ +0002b7a0: 6620 7468 6973 206d 6574 686f 6420 746f  f this method to
│ │ │ │ +0002b7b0: 2077 6f72 6b20 7769 7468 2065 6c65 6d65   work with eleme
│ │ │ │ +0002b7c0: 6e74 7320 6f66 2074 6865 0a43 686f 7720  nts of the.Chow 
│ │ │ │ +0002b7d0: 7269 6e67 206f 6620 6120 746f 7269 6320  ring of a toric 
│ │ │ │ +0002b7e0: 7661 7269 6574 790a 0a2b 2d2d 2d2d 2d2d  variety..+------
│ │ │ │ +0002b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b840: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6e65  ------+.|i1 : ne
│ │ │ │ -0002b850: 6564 7350 6163 6b61 6765 2022 4e6f 726d  edsPackage "Norm
│ │ │ │ -0002b860: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │ -0002b870: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0002b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b890: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002b830: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e  -------+.|i1 : n
│ │ │ │ +0002b840: 6565 6473 5061 636b 6167 6520 224e 6f72  eedsPackage "Nor
│ │ │ │ +0002b850: 6d61 6c54 6f72 6963 5661 7269 6574 6965  malToricVarietie
│ │ │ │ +0002b860: 7322 2020 2020 2020 2020 2020 2020 2020  s"              
│ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b880: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8e0: 2020 2020 2020 7c0a 7c6f 3120 3d20 4e6f        |.|o1 = No
│ │ │ │ -0002b8f0: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ -0002b900: 6573 2020 2020 2020 2020 2020 2020 2020  es              
│ │ │ │ +0002b8d0: 2020 2020 2020 207c 0a7c 6f31 203d 204e         |.|o1 = N
│ │ │ │ +0002b8e0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002b8f0: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │ +0002b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b930: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002b920: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b980: 2020 2020 2020 7c0a 7c6f 3120 3a20 5061        |.|o1 : Pa
│ │ │ │ -0002b990: 636b 6167 6520 2020 2020 2020 2020 2020  ckage           
│ │ │ │ +0002b970: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +0002b980: 6163 6b61 6765 2020 2020 2020 2020 2020  ackage          
│ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002b9c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002b9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ba00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ba10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ba20: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5268  ------+.|i2 : Rh
│ │ │ │ -0002ba30: 6f20 3d20 7b7b 312c 302c 307d 2c7b 302c  o = {{1,0,0},{0,
│ │ │ │ -0002ba40: 312c 307d 2c7b 302c 302c 317d 2c7b 2d31  1,0},{0,0,1},{-1
│ │ │ │ -0002ba50: 2c2d 312c 307d 2c7b 302c 302c 2d31 7d7d  ,-1,0},{0,0,-1}}
│ │ │ │ -0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002ba10: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ +0002ba20: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ +0002ba30: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ +0002ba40: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ +0002ba50: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002ba60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bac0: 2020 2020 2020 7c0a 7c6f 3220 3d20 7b7b        |.|o2 = {{
│ │ │ │ -0002bad0: 312c 2030 2c20 307d 2c20 7b30 2c20 312c  1, 0, 0}, {0, 1,
│ │ │ │ -0002bae0: 2030 7d2c 207b 302c 2030 2c20 317d 2c20   0}, {0, 0, 1}, 
│ │ │ │ -0002baf0: 7b2d 312c 202d 312c 2030 7d2c 207b 302c  {-1, -1, 0}, {0,
│ │ │ │ -0002bb00: 2030 2c20 2d31 7d7d 2020 2020 2020 2020   0, -1}}        
│ │ │ │ -0002bb10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bab0: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ +0002bac0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031  {1, 0, 0}, {0, 1
│ │ │ │ +0002bad0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c  , 0}, {0, 0, 1},
│ │ │ │ +0002bae0: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30   {-1, -1, 0}, {0
│ │ │ │ +0002baf0: 2c20 302c 202d 317d 7d20 2020 2020 2020  , 0, -1}}       
│ │ │ │ +0002bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb60: 2020 2020 2020 7c0a 7c6f 3220 3a20 4c69        |.|o2 : Li
│ │ │ │ -0002bb70: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0002bb50: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ +0002bb60: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0002bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002bba0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bc00: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5369  ------+.|i3 : Si
│ │ │ │ -0002bc10: 676d 6120 3d20 7b7b 302c 312c 327d 2c7b  gma = {{0,1,2},{
│ │ │ │ -0002bc20: 312c 322c 337d 2c7b 302c 322c 337d 2c7b  1,2,3},{0,2,3},{
│ │ │ │ -0002bc30: 302c 312c 347d 2c7b 312c 332c 347d 2c7b  0,1,4},{1,3,4},{
│ │ │ │ -0002bc40: 302c 332c 347d 7d20 2020 2020 2020 2020  0,3,4}}         
│ │ │ │ -0002bc50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bbf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2053  -------+.|i3 : S
│ │ │ │ +0002bc00: 6967 6d61 203d 207b 7b30 2c31 2c32 7d2c  igma = {{0,1,2},
│ │ │ │ +0002bc10: 7b31 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c  {1,2,3},{0,2,3},
│ │ │ │ +0002bc20: 7b30 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c  {0,1,4},{1,3,4},
│ │ │ │ +0002bc30: 7b30 2c33 2c34 7d7d 2020 2020 2020 2020  {0,3,4}}        
│ │ │ │ +0002bc40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bca0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7b7b        |.|o3 = {{
│ │ │ │ -0002bcb0: 302c 2031 2c20 327d 2c20 7b31 2c20 322c  0, 1, 2}, {1, 2,
│ │ │ │ -0002bcc0: 2033 7d2c 207b 302c 2032 2c20 337d 2c20   3}, {0, 2, 3}, 
│ │ │ │ -0002bcd0: 7b30 2c20 312c 2034 7d2c 207b 312c 2033  {0, 1, 4}, {1, 3
│ │ │ │ -0002bce0: 2c20 347d 2c20 7b30 2c20 332c 2034 7d7d  , 4}, {0, 3, 4}}
│ │ │ │ -0002bcf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bc90: 2020 2020 2020 207c 0a7c 6f33 203d 207b         |.|o3 = {
│ │ │ │ +0002bca0: 7b30 2c20 312c 2032 7d2c 207b 312c 2032  {0, 1, 2}, {1, 2
│ │ │ │ +0002bcb0: 2c20 337d 2c20 7b30 2c20 322c 2033 7d2c  , 3}, {0, 2, 3},
│ │ │ │ +0002bcc0: 207b 302c 2031 2c20 347d 2c20 7b31 2c20   {0, 1, 4}, {1, 
│ │ │ │ +0002bcd0: 332c 2034 7d2c 207b 302c 2033 2c20 347d  3, 4}, {0, 3, 4}
│ │ │ │ +0002bce0: 7d20 2020 2020 207c 0a7c 2020 2020 2020  }      |.|      
│ │ │ │ +0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd40: 2020 2020 2020 7c0a 7c6f 3320 3a20 4c69        |.|o3 : Li
│ │ │ │ -0002bd50: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0002bd30: 2020 2020 2020 207c 0a7c 6f33 203a 204c         |.|o3 : L
│ │ │ │ +0002bd40: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0002bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002bd80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002bd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bde0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5820  ------+.|i4 : X 
│ │ │ │ -0002bdf0: 3d20 6e6f 726d 616c 546f 7269 6356 6172  = normalToricVar
│ │ │ │ -0002be00: 6965 7479 2852 686f 2c53 6967 6d61 2c43  iety(Rho,Sigma,C
│ │ │ │ -0002be10: 6f65 6666 6963 6965 6e74 5269 6e67 203d  oefficientRing =
│ │ │ │ -0002be20: 3e5a 5a2f 3332 3734 3929 2020 2020 2020  >ZZ/32749)      
│ │ │ │ -0002be30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bdd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2058  -------+.|i4 : X
│ │ │ │ +0002bde0: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ +0002bdf0: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ +0002be00: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ +0002be10: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ +0002be20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be80: 2020 2020 2020 7c0a 7c6f 3420 3d20 5820        |.|o4 = X 
│ │ │ │ +0002be70: 2020 2020 2020 207c 0a7c 6f34 203d 2058         |.|o4 = X
│ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bed0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf20: 2020 2020 2020 7c0a 7c6f 3420 3a20 4e6f        |.|o4 : No
│ │ │ │ -0002bf30: 726d 616c 546f 7269 6356 6172 6965 7479  rmalToricVariety
│ │ │ │ +0002bf10: 2020 2020 2020 207c 0a7c 6f34 203a 204e         |.|o4 : N
│ │ │ │ +0002bf20: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002bf30: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002bf60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bfc0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 523d  ------+.|i5 : R=
│ │ │ │ -0002bfd0: 7269 6e67 2058 2020 2020 2020 2020 2020  ring X          
│ │ │ │ +0002bfb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2052  -------+.|i5 : R
│ │ │ │ +0002bfc0: 3d72 696e 6720 5820 2020 2020 2020 2020  =ring X         
│ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c010: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c060: 2020 2020 2020 7c0a 7c6f 3520 3d20 5220        |.|o5 = R 
│ │ │ │ +0002c050: 2020 2020 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │ +0002c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c0a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c100: 2020 2020 2020 7c0a 7c6f 3520 3a20 506f        |.|o5 : Po
│ │ │ │ -0002c110: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +0002c0f0: 2020 2020 2020 207c 0a7c 6f35 203a 2050         |.|o5 : P
│ │ │ │ +0002c100: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +0002c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002c140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002c150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c1a0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4368  ------+.|i6 : Ch
│ │ │ │ -0002c1b0: 3d54 6f72 6963 4368 6f77 5269 6e67 2858  =ToricChowRing(X
│ │ │ │ -0002c1c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002c190: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2043  -------+.|i6 : C
│ │ │ │ +0002c1a0: 683d 546f 7269 6343 686f 7752 696e 6728  h=ToricChowRing(
│ │ │ │ +0002c1b0: 5829 2020 2020 2020 2020 2020 2020 2020  X)              
│ │ │ │ +0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c1e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c240: 2020 2020 2020 7c0a 7c6f 3620 3d20 4368        |.|o6 = Ch
│ │ │ │ +0002c230: 2020 2020 2020 207c 0a7c 6f36 203d 2043         |.|o6 = C
│ │ │ │ +0002c240: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │  0002c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c290: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2e0: 2020 2020 2020 7c0a 7c6f 3620 3a20 5175        |.|o6 : Qu
│ │ │ │ -0002c2f0: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +0002c2d0: 2020 2020 2020 207c 0a7c 6f36 203a 2051         |.|o6 : Q
│ │ │ │ +0002c2e0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c330: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002c320: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c380: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6465  ------+.|i7 : de
│ │ │ │ -0002c390: 7363 7269 6265 2043 6820 2020 2020 2020  scribe Ch       
│ │ │ │ +0002c370: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064  -------+.|i7 : d
│ │ │ │ +0002c380: 6573 6372 6962 6520 4368 2020 2020 2020  escribe Ch      
│ │ │ │ +0002c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c3d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c3c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c430: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ -0002c440: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │ +0002c410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c420: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +0002c430: 5a5b 7820 2e2e 7820 5d20 2020 2020 2020  Z[x ..x ]       
│ │ │ │ +0002c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c490: 2020 3020 2020 3420 2020 2020 2020 2020    0   4         
│ │ │ │ +0002c460: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c480: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +0002c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4c0: 2020 2020 2020 7c0a 7c6f 3720 3d20 2d2d        |.|o7 = --
│ │ │ │ +0002c4b0: 2020 2020 2020 207c 0a7c 6f37 203d 202d         |.|o7 = -
│ │ │ │ +0002c4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c4f0: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ -0002c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c510: 2020 2020 2020 7c0a 7c20 2020 2020 2878        |.|     (x
│ │ │ │ -0002c520: 2078 202c 2078 2078 2078 202c 2078 2020   x , x x x , x  
│ │ │ │ -0002c530: 2d20 7820 2c20 7820 202d 2078 202c 2078  - x , x  - x , x
│ │ │ │ -0002c540: 2020 2d20 7820 2920 2020 2020 2020 2020    - x )         
│ │ │ │ -0002c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c570: 3220 3420 2020 3020 3120 3320 2020 3020  2 4   0 1 3   0 
│ │ │ │ -0002c580: 2020 2033 2020 2031 2020 2020 3320 2020     3   1    3   
│ │ │ │ -0002c590: 3220 2020 2034 2020 2020 2020 2020 2020  2    4          
│ │ │ │ -0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002c4e0: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ +0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c500: 2020 2020 2020 207c 0a7c 2020 2020 2028         |.|     (
│ │ │ │ +0002c510: 7820 7820 2c20 7820 7820 7820 2c20 7820  x x , x x x , x 
│ │ │ │ +0002c520: 202d 2078 202c 2078 2020 2d20 7820 2c20   - x , x  - x , 
│ │ │ │ +0002c530: 7820 202d 2078 2029 2020 2020 2020 2020  x  - x )        
│ │ │ │ +0002c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c560: 2032 2034 2020 2030 2031 2033 2020 2030   2 4   0 1 3   0
│ │ │ │ +0002c570: 2020 2020 3320 2020 3120 2020 2033 2020      3   1    3  
│ │ │ │ +0002c580: 2032 2020 2020 3420 2020 2020 2020 2020   2    4         
│ │ │ │ +0002c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c600: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 723d  ------+.|i8 : r=
│ │ │ │ -0002c610: 6765 6e73 2052 2020 2020 2020 2020 2020  gens R          
│ │ │ │ +0002c5f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2072  -------+.|i8 : r
│ │ │ │ +0002c600: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +0002c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c640: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6a0: 2020 2020 2020 7c0a 7c6f 3820 3d20 7b78        |.|o8 = {x
│ │ │ │ -0002c6b0: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
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│ │ │ │ +0002c690: 2020 2020 2020 207c 0a7c 6f38 203d 207b         |.|o8 = {
│ │ │ │ +0002c6a0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +0002c6b0: 7820 7d20 2020 2020 2020 2020 2020 2020  x }             
│ │ │ │ +0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c700: 3020 2020 3120 2020 3220 2020 3320 2020  0   1   2   3   
│ │ │ │ -0002c710: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0002c6e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c6f0: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ +0002c700: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c740: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c790: 2020 2020 2020 7c0a 7c6f 3820 3a20 4c69        |.|o8 : Li
│ │ │ │ -0002c7a0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0002c780: 2020 2020 2020 207c 0a7c 6f38 203a 204c         |.|o8 : L
│ │ │ │ +0002c790: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002c7d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c830: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 493d  ------+.|i9 : I=
│ │ │ │ -0002c840: 6964 6561 6c28 7261 6e64 6f6d 287b 312c  ideal(random({1,
│ │ │ │ -0002c850: 307d 2c52 2929 2020 2020 2020 2020 2020  0},R))          
│ │ │ │ +0002c820: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2049  -------+.|i9 : I
│ │ │ │ +0002c830: 3d69 6465 616c 2872 616e 646f 6d28 7b31  =ideal(random({1
│ │ │ │ +0002c840: 2c30 7d2c 5229 2920 2020 2020 2020 2020  ,0},R))         
│ │ │ │ +0002c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c870: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8d0: 2020 2020 2020 7c0a 7c6f 3920 3d20 6964        |.|o9 = id
│ │ │ │ -0002c8e0: 6561 6c28 3130 3778 2020 2b20 3433 3736  eal(107x  + 4376
│ │ │ │ -0002c8f0: 7820 202d 2036 3331 3678 2029 2020 2020  x  - 6316x )    
│ │ │ │ +0002c8c0: 2020 2020 2020 207c 0a7c 6f39 203d 2069         |.|o9 = i
│ │ │ │ +0002c8d0: 6465 616c 2831 3037 7820 202b 2034 3337  deal(107x  + 437
│ │ │ │ +0002c8e0: 3678 2020 2d20 3633 3136 7820 2920 2020  6x  - 6316x )   
│ │ │ │ +0002c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c920: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c930: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0002c940: 2031 2020 2020 2020 2020 3320 2020 2020   1        3     
│ │ │ │ +0002c910: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c920: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0002c930: 2020 3120 2020 2020 2020 2033 2020 2020    1        3    
│ │ │ │ +0002c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c960: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9c0: 2020 2020 2020 7c0a 7c6f 3920 3a20 4964        |.|o9 : Id
│ │ │ │ -0002c9d0: 6561 6c20 6f66 2052 2020 2020 2020 2020  eal of R        
│ │ │ │ +0002c9b0: 2020 2020 2020 207c 0a7c 6f39 203a 2049         |.|o9 : I
│ │ │ │ +0002c9c0: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +0002c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002ca00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ca60: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 204b  ------+.|i10 : K
│ │ │ │ -0002ca70: 3d69 6465 616c 2872 616e 646f 6d28 7b31  =ideal(random({1
│ │ │ │ -0002ca80: 2c31 7d2c 5229 2920 2020 2020 2020 2020  ,1},R))         
│ │ │ │ +0002ca50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ +0002ca60: 4b3d 6964 6561 6c28 7261 6e64 6f6d 287b  K=ideal(random({
│ │ │ │ +0002ca70: 312c 317d 2c52 2929 2020 2020 2020 2020  1,1},R))        
│ │ │ │ +0002ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002caa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb00: 2020 2020 2020 7c0a 7c6f 3130 203d 2069        |.|o10 = i
│ │ │ │ -0002cb10: 6465 616c 2833 3138 3778 2078 2020 2d20  deal(3187x x  - 
│ │ │ │ -0002cb20: 3630 3533 7820 7820 202d 2031 3630 3930  6053x x  - 16090
│ │ │ │ -0002cb30: 7820 7820 202b 2033 3738 3378 2078 2020  x x  + 3783x x  
│ │ │ │ -0002cb40: 2b20 3835 3730 7820 7820 202b 2038 3434  + 8570x x  + 844
│ │ │ │ -0002cb50: 3478 2078 2029 7c0a 7c20 2020 2020 2020  4x x )|.|       
│ │ │ │ -0002cb60: 2020 2020 2020 2020 2020 3020 3220 2020            0 2   
│ │ │ │ -0002cb70: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ -0002cb80: 2032 2033 2020 2020 2020 2020 3020 3420   2 3        0 4 
│ │ │ │ -0002cb90: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ -0002cba0: 2020 3320 3420 7c0a 7c20 2020 2020 2020    3 4 |.|       
│ │ │ │ +0002caf0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +0002cb00: 6964 6561 6c28 3331 3837 7820 7820 202d  ideal(3187x x  -
│ │ │ │ +0002cb10: 2036 3035 3378 2078 2020 2d20 3136 3039   6053x x  - 1609
│ │ │ │ +0002cb20: 3078 2078 2020 2b20 3337 3833 7820 7820  0x x  + 3783x x 
│ │ │ │ +0002cb30: 202b 2038 3537 3078 2078 2020 2b20 3834   + 8570x x  + 84
│ │ │ │ +0002cb40: 3434 7820 7820 297c 0a7c 2020 2020 2020  44x x )|.|      
│ │ │ │ +0002cb50: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ +0002cb60: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ +0002cb70: 2020 3220 3320 2020 2020 2020 2030 2034    2 3        0 4
│ │ │ │ +0002cb80: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
│ │ │ │ +0002cb90: 2020 2033 2034 207c 0a7c 2020 2020 2020     3 4 |.|      
│ │ │ │ +0002cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbf0: 2020 2020 2020 7c0a 7c6f 3130 203a 2049        |.|o10 : I
│ │ │ │ -0002cc00: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +0002cbe0: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ +0002cbf0: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │ +0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002cc30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002cc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cc90: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2063  ------+.|i11 : c
│ │ │ │ -0002cca0: 3d43 6865 726e 2843 682c 582c 4929 2020  =Chern(Ch,X,I)  
│ │ │ │ +0002cc80: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ +0002cc90: 633d 4368 6572 6e28 4368 2c58 2c49 2920  c=Chern(Ch,X,I) 
│ │ │ │ +0002cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cce0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002ccd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002cd40: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ +0002cd20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cd30: 2020 3220 2020 2020 2020 3220 2020 2020    2       2     
│ │ │ │ +0002cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd80: 2020 2020 2020 7c0a 7c6f 3131 203d 2034        |.|o11 = 4
│ │ │ │ -0002cd90: 7820 7820 202b 2032 7820 202b 2032 7820  x x  + 2x  + 2x 
│ │ │ │ -0002cda0: 7820 202b 2078 2020 2020 2020 2020 2020  x  + x          
│ │ │ │ +0002cd70: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +0002cd80: 3478 2078 2020 2b20 3278 2020 2b20 3278  4x x  + 2x  + 2x
│ │ │ │ +0002cd90: 2078 2020 2b20 7820 2020 2020 2020 2020   x  + x         
│ │ │ │ +0002cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cdd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002cde0: 2033 2034 2020 2020 2033 2020 2020 2033   3 4     3     3
│ │ │ │ -0002cdf0: 2034 2020 2020 3320 2020 2020 2020 2020   4    3         
│ │ │ │ +0002cdc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cdd0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0002cde0: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0002cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002ce10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce70: 2020 2020 2020 7c0a 7c6f 3131 203a 2043        |.|o11 : C
│ │ │ │ -0002ce80: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002ce60: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ +0002ce70: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cec0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002ceb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002cec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ced0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cf10: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2073  ------+.|i12 : s
│ │ │ │ -0002cf20: 3d53 6567 7265 2843 682c 582c 4b29 2020  =Segre(Ch,X,K)  
│ │ │ │ +0002cf00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ +0002cf10: 733d 5365 6772 6528 4368 2c58 2c4b 2920  s=Segre(Ch,X,K) 
│ │ │ │ +0002cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002cf50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cfb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002cfc0: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +0002cfa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002cfb0: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │ +0002cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d000: 2020 2020 2020 7c0a 7c6f 3132 203d 2033        |.|o12 = 3
│ │ │ │ -0002d010: 7820 7820 202d 2078 2020 2d20 3278 2078  x x  - x  - 2x x
│ │ │ │ -0002d020: 2020 2b20 7820 202b 2078 2020 2020 2020    + x  + x      
│ │ │ │ +0002cff0: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +0002d000: 3378 2078 2020 2d20 7820 202d 2032 7820  3x x  - x  - 2x 
│ │ │ │ +0002d010: 7820 202b 2078 2020 2b20 7820 2020 2020  x  + x  + x     
│ │ │ │ +0002d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d050: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002d060: 2033 2034 2020 2020 3320 2020 2020 3320   3 4    3     3 
│ │ │ │ -0002d070: 3420 2020 2033 2020 2020 3420 2020 2020  4    3    4     
│ │ │ │ +0002d040: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d050: 2020 3320 3420 2020 2033 2020 2020 2033    3 4    3     3
│ │ │ │ +0002d060: 2034 2020 2020 3320 2020 2034 2020 2020   4    3    4    
│ │ │ │ +0002d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002d090: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0f0: 2020 2020 2020 7c0a 7c6f 3132 203a 2043        |.|o12 : C
│ │ │ │ -0002d100: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002d0e0: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ +0002d0f0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
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│ │ │ │  0002d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d190: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2073  ------+.|i13 : s
│ │ │ │ -0002d1a0: 2d63 2020 2020 2020 2020 2020 2020 2020  -c              
│ │ │ │ +0002d180: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ +0002d190: 732d 6320 2020 2020 2020 2020 2020 2020  s-c             
│ │ │ │ +0002d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002d1d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0002d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d230: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002d240: 2020 3220 2020 2020 2020 3220 2020 2020    2       2     
│ │ │ │ +0002d220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d230: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +0002d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d280: 2020 2020 2020 7c0a 7c6f 3133 203d 202d        |.|o13 = -
│ │ │ │ -0002d290: 2078 2078 2020 2d20 3378 2020 2d20 3478   x x  - 3x  - 4x
│ │ │ │ -0002d2a0: 2078 2020 2b20 7820 2020 2020 2020 2020   x  + x         
│ │ │ │ +0002d270: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ +0002d280: 2d20 7820 7820 202d 2033 7820 202d 2034  - x x  - 3x  - 4
│ │ │ │ +0002d290: 7820 7820 202b 2078 2020 2020 2020 2020  x x  + x        
│ │ │ │ +0002d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002d2e0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ -0002d2f0: 3320 3420 2020 2034 2020 2020 2020 2020  3 4    4        
│ │ │ │ +0002d2c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d2d0: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ +0002d2e0: 2033 2034 2020 2020 3420 2020 2020 2020   3 4    4       
│ │ │ │ +0002d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d370: 2020 2020 2020 7c0a 7c6f 3133 203a 2043        |.|o13 : C
│ │ │ │ -0002d380: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002d360: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ +0002d370: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
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│ │ │ │  0002d3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d410: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2073  ------+.|i14 : s
│ │ │ │ -0002d420: 2a63 2020 2020 2020 2020 2020 2020 2020  *c              
│ │ │ │ +0002d400: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +0002d410: 732a 6320 2020 2020 2020 2020 2020 2020  s*c             
│ │ │ │ +0002d420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d460: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │ +0002d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002d4c0: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +0002d4a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d4b0: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │ +0002d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d500: 2020 2020 2020 7c0a 7c6f 3134 203d 2032        |.|o14 = 2
│ │ │ │ -0002d510: 7820 7820 202b 2078 2020 2b20 7820 7820  x x  + x  + x x 
│ │ │ │ +0002d4f0: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ +0002d500: 3278 2078 2020 2b20 7820 202b 2078 2078  2x x  + x  + x x
│ │ │ │ +0002d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002d560: 2033 2034 2020 2020 3320 2020 2033 2034   3 4    3    3 4
│ │ │ │ +0002d540: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d550: 2020 3320 3420 2020 2033 2020 2020 3320    3 4    3    3 
│ │ │ │ +0002d560: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0002d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002d590: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5f0: 2020 2020 2020 7c0a 7c6f 3134 203a 2043        |.|o14 : C
│ │ │ │ -0002d600: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002d5e0: 2020 2020 2020 207c 0a7c 6f31 3420 3a20         |.|o14 : 
│ │ │ │ +0002d5f0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ +0002d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002d6e0: 546f 7269 6343 686f 7752 696e 672c 2069  ToricChowRing, i
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│ │ │ │ -0002d7a0: 6865 726e 7f33 3039 3035 0a4e 6f64 653a  hern.30905.Node:
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│ │ │ │ -0002d850: 3937 3436 370a 4e6f 6465 3a20 496e 6473  97467.Node: Inds
│ │ │ │ -0002d860: 4f66 536d 6f6f 7468 7f31 3135 3134 360a  OfSmooth.115146.
│ │ │ │ -0002d870: 4e6f 6465 3a20 496e 7075 7449 7353 6d6f  Node: InputIsSmo
│ │ │ │ -0002d880: 6f74 687f 3131 3838 3737 0a4e 6f64 653a  oth.118877.Node:
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│ │ │ │ -0002d8a0: 6f75 737f 3132 3236 3432 0a4e 6f64 653a  ous.122642.Node:
│ │ │ │ -0002d8b0: 204d 6574 686f 647f 3132 3635 3231 0a4e   Method.126521.N
│ │ │ │ -0002d8c0: 6f64 653a 204d 756c 7469 5072 6f6a 436f  ode: MultiProjCo
│ │ │ │ -0002d8d0: 6f72 6452 696e 677f 3133 3032 3532 0a4e  ordRing.130252.N
│ │ │ │ -0002d8e0: 6f64 653a 204f 7574 7075 747f 3133 3637  ode: Output.1367
│ │ │ │ -0002d8f0: 3231 0a4e 6f64 653a 2070 726f 6261 6269  21.Node: probabi
│ │ │ │ -0002d900: 6c69 7374 6963 2061 6c67 6f72 6974 686d  listic algorithm
│ │ │ │ -0002d910: 7f31 3535 3032 310a 4e6f 6465 3a20 5365  .155021.Node: Se
│ │ │ │ -0002d920: 6772 657f 3135 3934 3438 0a4e 6f64 653a  gre.159448.Node:
│ │ │ │ -0002d930: 2054 6f72 6963 4368 6f77 5269 6e67 7f31   ToricChowRing.1
│ │ │ │ -0002d940: 3737 3130 380a 1f0a 456e 6420 5461 6720  77108...End Tag 
│ │ │ │ -0002d950: 5461 626c 650a                           Table.
│ │ │ │ +0002d680: 2d2d 2d2d 2d2d 2d2b 0a0a 466f 7220 7468  -------+..For th
│ │ │ │ +0002d690: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +0002d6a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002d6b0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0002d6c0: 6520 546f 7269 6343 686f 7752 696e 673a  e ToricChowRing:
│ │ │ │ +0002d6d0: 2054 6f72 6963 4368 6f77 5269 6e67 2c20   ToricChowRing, 
│ │ │ │ +0002d6e0: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +0002d6f0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ +0002d700: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +0002d710: 4675 6e63 7469 6f6e 2c2e 0a1f 0a54 6167  Function,....Tag
│ │ │ │ +0002d720: 2054 6162 6c65 3a0a 4e6f 6465 3a20 546f   Table:.Node: To
│ │ │ │ +0002d730: 707f 3333 360a 4e6f 6465 3a20 6265 7274  p.336.Node: bert
│ │ │ │ +0002d740: 696e 6943 6865 636b 7f31 3435 3137 0a4e  iniCheck.14517.N
│ │ │ │ +0002d750: 6f64 653a 2043 6865 636b 536d 6f6f 7468  ode: CheckSmooth
│ │ │ │ +0002d760: 7f31 3535 3335 0a4e 6f64 653a 2043 6865  .15535.Node: Che
│ │ │ │ +0002d770: 636b 546f 7269 6356 6172 6965 7479 5661  ckToricVarietyVa
│ │ │ │ +0002d780: 6c69 647f 3230 3131 330a 4e6f 6465 3a20  lid.20113.Node: 
│ │ │ │ +0002d790: 4368 6572 6e7f 3330 3930 350a 4e6f 6465  Chern.30905.Node
│ │ │ │ +0002d7a0: 3a20 4368 6f77 5269 6e67 7f34 3831 3635  : ChowRing.48165
│ │ │ │ +0002d7b0: 0a4e 6f64 653a 2043 6c61 7373 496e 4368  .Node: ClassInCh
│ │ │ │ +0002d7c0: 6f77 5269 6e67 7f35 3536 3937 0a4e 6f64  owRing.55697.Nod
│ │ │ │ +0002d7d0: 653a 2043 6c61 7373 496e 546f 7269 6343  e: ClassInToricC
│ │ │ │ +0002d7e0: 686f 7752 696e 677f 3537 3835 380a 4e6f  howRing.57858.No
│ │ │ │ +0002d7f0: 6465 3a20 436f 6d70 4d65 7468 6f64 7f36  de: CompMethod.6
│ │ │ │ +0002d800: 3239 3032 0a4e 6f64 653a 2063 6f6e 6669  2902.Node: confi
│ │ │ │ +0002d810: 6775 7269 6e67 2042 6572 7469 6e69 7f37  guring Bertini.7
│ │ │ │ +0002d820: 3236 3131 0a4e 6f64 653a 2043 534d 7f37  2611.Node: CSM.7
│ │ │ │ +0002d830: 3430 3238 0a4e 6f64 653a 2045 756c 6572  4028.Node: Euler
│ │ │ │ +0002d840: 7f39 3734 3637 0a4e 6f64 653a 2049 6e64  .97467.Node: Ind
│ │ │ │ +0002d850: 734f 6653 6d6f 6f74 687f 3131 3531 3436  sOfSmooth.115146
│ │ │ │ +0002d860: 0a4e 6f64 653a 2049 6e70 7574 4973 536d  .Node: InputIsSm
│ │ │ │ +0002d870: 6f6f 7468 7f31 3138 3837 370a 4e6f 6465  ooth.118877.Node
│ │ │ │ +0002d880: 3a20 6973 4d75 6c74 6948 6f6d 6f67 656e  : isMultiHomogen
│ │ │ │ +0002d890: 656f 7573 7f31 3232 3632 370a 4e6f 6465  eous.122627.Node
│ │ │ │ +0002d8a0: 3a20 4d65 7468 6f64 7f31 3236 3530 360a  : Method.126506.
│ │ │ │ +0002d8b0: 4e6f 6465 3a20 4d75 6c74 6950 726f 6a43  Node: MultiProjC
│ │ │ │ +0002d8c0: 6f6f 7264 5269 6e67 7f31 3330 3233 370a  oordRing.130237.
│ │ │ │ +0002d8d0: 4e6f 6465 3a20 4f75 7470 7574 7f31 3336  Node: Output.136
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│ │ │ │ +0002d910: 6567 7265 7f31 3539 3433 330a 4e6f 6465  egre.159433.Node
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│ │ │ │ +0002d940: 2054 6162 6c65 0a                         Table.
│ │ ├── ./usr/share/info/CodingTheory.info.gz
│ │ │ ├── CodingTheory.info
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│ │ │ │ -0000fba0: 656e 6572 6174 6f72 7320 3d3e 207b 7b31  enerators => {{1
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│ │ │ │ @@ -4064,17 +4064,17 @@
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│ │ │ │  0000fe20: 6172 6974 7943 6865 636b 526f 7773 203d  arityCheckRows =
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│ │ │ │  0000fe90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fea0: 2020 2020 2020 506f 6c79 6e6f 6d69 616c        Polynomial
│ │ │ │  0000feb0: 5365 7420 3d3e 207b 7820 2b20 7920 2b20  Set => {x + y + 
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│ │ │ │ +00010620: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000108d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ @@ -4377,38 +4377,38 @@
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│ │ │ │  000111b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000111c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00011220: 312c 2030 2c20 302c 2061 202b 2031 2c20  1, 0, 0, a + 1, 
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│ │ │ │  000113c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000113d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ @@ -4507,18 +4507,18 @@
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│ │ │ │  000119b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000119c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000119d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000119e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000119f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00011a10: 0a7c 302c 2030 2c20 312c 2031 2c20 302c  .|0, 0, 1, 1, 0,
│ │ │ │ -00011a20: 2030 7d2c 207b 302c 2030 2c20 302c 2030   0}, {0, 0, 0, 0
│ │ │ │ -00011a30: 2c20 302c 2030 2c20 302c 2031 2c20 317d  , 0, 0, 0, 1, 1}
│ │ │ │ -00011a40: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00011a10: 0a7c 312c 2030 2c20 302c 2030 7d2c 207b  .|1, 0, 0, 0}, {
│ │ │ │ +00011a20: 302c 2030 2c20 302c 2030 2c20 302c 2030  0, 0, 0, 0, 0, 0
│ │ │ │ +00011a30: 2c20 312c 2030 2c20 307d 7d20 2020 2020  , 1, 0, 0}}     
│ │ │ │ +00011a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00011a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00011ab0: 0a7c 6935 203a 2043 2e4c 696e 6561 7243  .|i5 : C.LinearC
│ │ │ │ @@ -4549,112 +4549,112 @@
│ │ │ │  00011c40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011c50: 2020 6361 6368 6520 3d3e 2043 6163 6865    cache => Cache
│ │ │ │  00011c60: 5461 626c 657b 7d20 2020 2020 2020 2020  Table{}         
│ │ │ │  00011c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011c90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011ca0: 2020 436f 6465 203d 3e20 696d 6167 6520    Code => image 
│ │ │ │ -00011cb0: 7c20 3120 2020 612b 3120 7c20 2020 2020  | 1   a+1 |     
│ │ │ │ +00011cb0: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │  00011cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011ce0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011d00: 7c20 3120 2020 3020 2020 7c20 2020 2020  | 1   0   |     
│ │ │ │ +00011d00: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │  00011d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011d30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011d50: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │ +00011d50: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │  00011d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011d80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011da0: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │ +00011da0: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │  00011db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011dd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011df0: 7c20 3120 2020 3120 2020 7c20 2020 2020  | 1   1   |     
│ │ │ │ +00011df0: 7c20 3120 2020 3020 2020 7c20 2020 2020  | 1   0   |     
│ │ │ │  00011e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011e20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e40: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │ +00011e40: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │  00011e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011e70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e90: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │ +00011e90: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │  00011ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011eb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011ee0: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │ +00011ee0: 7c20 3120 2020 612b 3120 7c20 2020 2020  | 1   a+1 |     
│ │ │ │  00011ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011f10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011f30: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │ +00011f30: 7c20 3120 2020 3120 2020 7c20 2020 2020  | 1   1   |     
│ │ │ │  00011f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011f70: 2020 4765 6e65 7261 746f 724d 6174 7269    GeneratorMatri
│ │ │ │ -00011f80: 7820 3d3e 207c 2031 2020 2031 2030 2030  x => | 1   1 0 0
│ │ │ │ -00011f90: 2031 2061 2b31 2061 2b31 2061 2061 207c   1 a+1 a+1 a a |
│ │ │ │ +00011f80: 7820 3d3e 207c 2061 2b31 2061 2b31 2061  x => | a+1 a+1 a
│ │ │ │ +00011f90: 2061 2031 2030 2030 2031 2020 2031 207c   a 1 0 0 1   1 |
│ │ │ │  00011fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011fd0: 2020 2020 207c 2061 2b31 2030 2030 2030       | a+1 0 0 0
│ │ │ │ -00011fe0: 2031 2030 2020 2030 2020 2061 2061 207c   1 0   0   a a |
│ │ │ │ +00011fd0: 2020 2020 207c 2030 2020 2030 2020 2061       | 0   0   a
│ │ │ │ +00011fe0: 2061 2030 2030 2030 2061 2b31 2031 207c   a 0 0 0 a+1 1 |
│ │ │ │  00011ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012000: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012010: 2020 4765 6e65 7261 746f 7273 203d 3e20    Generators => 
│ │ │ │ -00012020: 7b7b 312c 2031 2c20 302c 2030 2c20 312c  {{1, 1, 0, 0, 1,
│ │ │ │ -00012030: 2061 202b 2031 2c20 6120 2b20 312c 2061   a + 1, a + 1, a
│ │ │ │ -00012040: 2c20 617d 2c20 7b61 202b 2031 2c20 207c  , a}, {a + 1,  |
│ │ │ │ +00012020: 7b7b 6120 2b20 312c 2061 202b 2031 2c20  {{a + 1, a + 1, 
│ │ │ │ +00012030: 612c 2061 2c20 312c 2030 2c20 302c 2031  a, a, 1, 0, 0, 1
│ │ │ │ +00012040: 2c20 317d 2c20 7b30 2c20 302c 2061 2c7c  , 1}, {0, 0, a,|
│ │ │ │  00012050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012060: 2020 5061 7269 7479 4368 6563 6b4d 6174    ParityCheckMat
│ │ │ │  00012070: 7269 7820 3d3e 207c 2031 2030 2030 2030  rix => | 1 0 0 0
│ │ │ │ -00012080: 2030 2061 2b31 2030 2061 2020 2030 207c   0 a+1 0 a   0 |
│ │ │ │ +00012080: 2030 2030 2030 2061 2020 2031 207c 2020   0 0 0 a   1 |  
│ │ │ │  00012090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000120a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000120b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000120c0: 2020 2020 2020 207c 2030 2031 2030 2030         | 0 1 0 0
│ │ │ │ -000120d0: 2030 2061 2020 2030 2030 2020 2030 207c   0 a   0 0   0 |
│ │ │ │ +000120d0: 2030 2030 2030 2061 2020 2031 207c 2020   0 0 0 a   1 |  
│ │ │ │  000120e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000120f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012110: 2020 2020 2020 207c 2030 2030 2031 2030         | 0 0 1 0
│ │ │ │ -00012120: 2030 2030 2020 2030 2030 2020 2030 207c   0 0   0 0   0 |
│ │ │ │ +00012120: 2030 2030 2030 2030 2020 2061 207c 2020   0 0 0 0   a |  
│ │ │ │  00012130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012160: 2020 2020 2020 207c 2030 2030 2030 2031         | 0 0 0 1
│ │ │ │ -00012170: 2030 2030 2020 2030 2030 2020 2030 207c   0 0   0 0   0 |
│ │ │ │ +00012170: 2030 2030 2030 2030 2020 2061 207c 2020   0 0 0 0   a |  
│ │ │ │  00012180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000121a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000121b0: 2020 2020 2020 207c 2030 2030 2030 2030         | 0 0 0 0
│ │ │ │ -000121c0: 2031 2030 2020 2030 2061 2b31 2030 207c   1 0   0 a+1 0 |
│ │ │ │ +000121c0: 2031 2030 2030 2061 2b31 2061 207c 2020   1 0 0 a+1 a |  
│ │ │ │  000121d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000121e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000121f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012200: 2020 2020 2020 207c 2030 2030 2030 2030         | 0 0 0 0
│ │ │ │ -00012210: 2030 2031 2020 2031 2030 2020 2030 207c   0 1   1 0   0 |
│ │ │ │ +00012210: 2030 2031 2030 2030 2020 2030 207c 2020   0 1 0 0   0 |  
│ │ │ │  00012220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012230: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012250: 2020 2020 2020 207c 2030 2030 2030 2030         | 0 0 0 0
│ │ │ │ -00012260: 2030 2030 2020 2030 2031 2020 2031 207c   0 0   0 1   1 |
│ │ │ │ +00012260: 2030 2030 2031 2030 2020 2030 207c 2020   0 0 1 0   0 |  
│ │ │ │  00012270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012280: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012290: 2020 5061 7269 7479 4368 6563 6b52 6f77    ParityCheckRow
│ │ │ │  000122a0: 7320 3d3e 207b 7b31 2c20 302c 2030 2c20  s => {{1, 0, 0, 
│ │ │ │ -000122b0: 302c 2030 2c20 6120 2b20 312c 2030 2c20  0, 0, a + 1, 0, 
│ │ │ │ -000122c0: 612c 2030 7d2c 207b 302c 2031 2c20 207c  a, 0}, {0, 1,  |
│ │ │ │ +000122b0: 302c 2030 2c20 302c 2030 2c20 612c 2031  0, 0, 0, 0, a, 1
│ │ │ │ +000122c0: 7d2c 207b 302c 2031 2c20 302c 2030 2c7c  }, {0, 1, 0, 0,|
│ │ │ │  000122d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000122e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000122f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012320: 0a7c 6f35 203a 204c 696e 6561 7243 6f64  .|o5 : LinearCod
│ │ │ │  00012330: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ @@ -4732,16 +4732,16 @@
│ │ │ │  000127b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000127d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000127e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012820: 0a7c 302c 2030 2c20 302c 2031 2c20 302c  .|0, 0, 0, 1, 0,
│ │ │ │ -00012830: 2030 2c20 612c 2061 7d7d 2020 2020 2020   0, a, a}}      
│ │ │ │ +00012820: 0a7c 612c 2030 2c20 302c 2030 2c20 6120  .|a, 0, 0, 0, a 
│ │ │ │ +00012830: 2b20 312c 2031 7d7d 2020 2020 2020 2020  + 1, 1}}        
│ │ │ │  00012840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012870: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4772,19 +4772,19 @@
│ │ │ │  00012a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012a50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012aa0: 0a7c 302c 2030 2c20 302c 2061 2c20 302c  .|0, 0, 0, a, 0,
│ │ │ │ -00012ab0: 2030 2c20 307d 2c20 7b30 2c20 302c 2031   0, 0}, {0, 0, 1
│ │ │ │ -00012ac0: 2c20 302c 2030 2c20 302c 2030 2c20 302c  , 0, 0, 0, 0, 0,
│ │ │ │ -00012ad0: 2030 7d2c 207b 302c 2030 2c20 302c 2031   0}, {0, 0, 0, 1
│ │ │ │ -00012ae0: 2c20 302c 2030 2c20 302c 2030 2c20 207c  , 0, 0, 0, 0,  |
│ │ │ │ +00012aa0: 0a7c 302c 2030 2c20 302c 2061 2c20 317d  .|0, 0, 0, a, 1}
│ │ │ │ +00012ab0: 2c20 7b30 2c20 302c 2031 2c20 302c 2030  , {0, 0, 1, 0, 0
│ │ │ │ +00012ac0: 2c20 302c 2030 2c20 302c 2061 7d2c 207b  , 0, 0, 0, a}, {
│ │ │ │ +00012ad0: 302c 2030 2c20 302c 2031 2c20 302c 2030  0, 0, 0, 1, 0, 0
│ │ │ │ +00012ae0: 2c20 302c 2030 2c20 617d 2c20 7b30 2c7c  , 0, 0, a}, {0,|
│ │ │ │  00012af0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00012b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │  00012b40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4892,26 +4892,26 @@
│ │ │ │  000131b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000131c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000131d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000131e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013220: 0a7c 307d 2c20 7b30 2c20 302c 2030 2c20  .|0}, {0, 0, 0, 
│ │ │ │ -00013230: 302c 2031 2c20 302c 2030 2c20 6120 2b20  0, 1, 0, 0, a + 
│ │ │ │ -00013240: 312c 2030 7d2c 207b 302c 2030 2c20 302c  1, 0}, {0, 0, 0,
│ │ │ │ -00013250: 2030 2c20 302c 2031 2c20 312c 2030 2c20   0, 0, 1, 1, 0, 
│ │ │ │ -00013260: 307d 2c20 7b30 2c20 302c 2030 2c20 207c  0}, {0, 0, 0,  |
│ │ │ │ +00013220: 0a7c 302c 2030 2c20 302c 2031 2c20 302c  .|0, 0, 0, 1, 0,
│ │ │ │ +00013230: 2030 2c20 6120 2b20 312c 2061 7d2c 207b   0, a + 1, a}, {
│ │ │ │ +00013240: 302c 2030 2c20 302c 2030 2c20 302c 2031  0, 0, 0, 0, 0, 1
│ │ │ │ +00013250: 2c20 302c 2030 2c20 307d 2c20 7b30 2c20  , 0, 0, 0}, {0, 
│ │ │ │ +00013260: 302c 2030 2c20 302c 2030 2c20 302c 207c  0, 0, 0, 0, 0, |
│ │ │ │  00013270: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00013280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000132a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000132b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000132c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000132d0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +000132c0: 0a7c 2020 2020 2020 2020 207d 2020 2020  .|         }    
│ │ │ │ +000132d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000132e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000132f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00013310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00013320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5012,16 +5012,16 @@
│ │ │ │  00013930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00013950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00013960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000139a0: 0a7c 302c 2030 2c20 302c 2030 2c20 312c  .|0, 0, 0, 0, 1,
│ │ │ │ -000139b0: 2031 7d7d 2020 2020 2020 2020 2020 2020   1}}            
│ │ │ │ +000139a0: 0a7c 312c 2030 2c20 307d 7d20 2020 2020  .|1, 0, 0}}     
│ │ │ │ +000139b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000139f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00013a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5126,116 +5126,116 @@
│ │ │ │  00014050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014060: 2020 2020 2020 2020 2020 2020 6361 6368              cach
│ │ │ │  00014070: 6520 3d3e 2043 6163 6865 5461 626c 657b  e => CacheTable{
│ │ │ │  00014080: 7d20 2020 2020 2020 2020 2020 7c0a 7c20  }           |.| 
│ │ │ │  00014090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140b0: 2020 2020 2020 2020 2020 2020 436f 6465              Code
│ │ │ │ -000140c0: 203d 3e20 696d 6167 6520 7c20 3020 2020   => image | 0   
│ │ │ │ -000140d0: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │ +000140c0: 203d 3e20 696d 6167 6520 7c20 612b 3120   => image | a+1 
│ │ │ │ +000140d0: 3120 2020 7c20 2020 2020 2020 7c0a 7c20  1   |       |.| 
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│ │ │ │ -000151d0: 207b 302c 2031 2c20 302c 2030 2c20 302c   {0, 1, 0, 0, 0,
│ │ │ │ -000151e0: 2030 2c20 302c 2030 2c20 307d 2c20 7b30   0, 0, 0, 0}, {0
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│ │ │ │ -00015200: 2030 2c20 302c 2061 7d2c 2020 7c0a 7c7d   0, 0, a},  |.|}
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│ │ │ │ +00015200: 2c20 302c 2030 2c20 302c 2020 7c0a 7c7d  , 0, 0, 0,  |.|}
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│ │ │ │ +00015220: 2c20 7b61 2c20 307d 7d20 2020 2020 2020  , {a, 0}}       
│ │ │ │  00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00015260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5567,17 +5567,17 @@
│ │ │ │  00015be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00015c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c50: 2020 2020 2020 2020 2020 2020 7c0a 7c7b              |.|{
│ │ │ │ -00015c60: 302c 2030 2c20 302c 2031 2c20 6120 2b20  0, 0, 0, 1, a + 
│ │ │ │ -00015c70: 312c 2030 2c20 302c 2030 2c20 317d 2c20  1, 0, 0, 0, 1}, 
│ │ │ │ +00015c50: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00015c60: 7d2c 207b 302c 2030 2c20 302c 2030 2c20  }, {0, 0, 0, 0, 
│ │ │ │ +00015c70: 312c 2030 2c20 302c 2030 2c20 307d 2c20  1, 0, 0, 0, 0}, 
│ │ │ │  00015c80: 7b30 2c20 302c 2030 2c20 302c 2030 2c20  {0, 0, 0, 0, 0, 
│ │ │ │  00015c90: 312c 2030 2c20 302c 2030 7d2c 207b 302c  1, 0, 0, 0}, {0,
│ │ │ │  00015ca0: 2030 2c20 302c 2030 2c20 302c 7c0a 7c2d   0, 0, 0, 0,|.|-
│ │ │ │  00015cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00016410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016470: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ -00016480: 2c20 312c 2030 2c20 307d 2c20 7b30 2c20  , 1, 0, 0}, {0, 
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│ │ │ │ -000164a0: 2c20 312c 2030 7d7d 2020 2020 2020 2020  , 1, 0}}        
│ │ │ │ +000164a0: 2c20 302c 2031 7d7d 2020 2020 2020 2020  , 0, 1}}        
│ │ │ │  000164b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000164c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  000164d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000164e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000164f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43  ------------+..C
│ │ │ │ @@ -5844,71 +5844,71 @@
│ │ │ │  00016d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d40: 2020 6361 6368 6520 3d3e 2043 6163 6865    cache => Cache
│ │ │ │  00016d50: 5461 626c 657b 7d20 2020 2020 2020 2020  Table{}         
│ │ │ │  00016d60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d90: 2020 436f 6465 203d 3e20 696d 6167 6520    Code => image 
│ │ │ │ -00016da0: 7c20 3120 2020 612b 3120 7c20 2020 2020  | 1   a+1 |     
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│ │ │ │  00016e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e40: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
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│ │ │ │  00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00016ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00016f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00016f30: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
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│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017070: 7820 3d3e 207c 2031 2020 2061 2b31 2061  x => | 1   a+1 a
│ │ │ │ -00017080: 2b20 7c0a 7c20 2020 2020 2020 2020 2020  + |.|           
│ │ │ │ +00017070: 7820 3d3e 207c 2031 2030 2030 2061 2b31  x => | 1 0 0 a+1
│ │ │ │ +00017080: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000170c0: 2020 2020 207c 2030 2030 2030 2030 2020       | 0 0 0 0  
│ │ │ │  000170d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017120: 3120 7c0a 7c20 2020 2020 2020 2020 2020  1 |.|           
│ │ │ │ +00017110: 7b7b 312c 2030 2c20 302c 2061 202b 2031  {{1, 0, 0, a + 1
│ │ │ │ +00017120: 2c20 7c0a 7c20 2020 2020 2020 2020 2020  , |.|           
│ │ │ │  00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017170: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017350: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017370: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000173a0: 3020 7c0a 7c20 2020 2020 2020 2020 2020  0 |.|           
│ │ │ │  000173b0: 2020 2020 2020 2020 2050 6f69 6e74 7320           Points 
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│ │ │ │ +000173f0: 3120 7c0a 7c20 2020 2020 2020 2020 2020  1 |.|           
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│ │ │ │ @@ -6050,71 +6050,71 @@
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│ │ │ │  00017a90: 2031 207c 2020 2020 2020 2020 2020 2020   1 |            
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│ │ │ │ +00017b40: 2030 2c20 302c 2030 2c20 612c 2061 202b   0, 0, 0, a, a +
│ │ │ │ +00017b50: 2031 2c20 612c 2031 7d7d 2020 2020 2020   1, a, 1}}      
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bc0: 2020 7c0a 7c61 2b31 2030 2030 2030 2030    |.|a+1 0 0 0 0
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│ │ │ │  00017be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c10: 2020 7c0a 7c61 2b31 2030 2030 2030 2030    |.|a+1 0 0 0 0
│ │ │ │ -00017c20: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00017c10: 2020 7c0a 7c30 2030 2030 2020 2030 2030    |.|0 0 0   0 0
│ │ │ │ +00017c20: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00017c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c60: 2020 7c0a 7c30 2020 2030 2030 2030 2061    |.|0   0 0 0 a
│ │ │ │ -00017c70: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00017c60: 2020 7c0a 7c30 2030 2061 2020 2030 2031    |.|0 0 a   0 1
│ │ │ │ +00017c70: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00017c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cb0: 2020 7c0a 7c30 2020 2031 2030 2030 2061    |.|0   1 0 0 a
│ │ │ │ -00017cc0: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00017cb0: 2020 7c0a 7c31 2030 2061 2020 2030 2031    |.|1 0 a   0 1
│ │ │ │ +00017cc0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00017cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d00: 2020 7c0a 7c30 2020 2030 2031 2030 2030    |.|0   0 1 0 0
│ │ │ │ -00017d10: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00017d00: 2020 7c0a 7c30 2031 2030 2020 2030 2061    |.|0 1 0   0 a
│ │ │ │ +00017d10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00017d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d50: 2020 7c0a 7c30 2020 2030 2030 2031 2030    |.|0   0 0 1 0
│ │ │ │ -00017d60: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00017d50: 2020 7c0a 7c30 2030 2030 2020 2031 2061    |.|0 0 0   1 a
│ │ │ │ +00017d60: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  00017d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017da0: 2020 7c0a 7c2c 2061 2c20 302c 2030 2c20    |.|, a, 0, 0, 
│ │ │ │ -00017db0: 302c 2061 202b 2031 7d2c 207b 302c 2031  0, a + 1}, {0, 1
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│ │ │ │ -00017dd0: 2c20 302c 2030 2c20 307d 2c20 7b30 2c20  , 0, 0, 0}, {0, 
│ │ │ │ -00017de0: 302c 2031 2c20 302c 2061 202b 2031 2c20  0, 1, 0, a + 1, 
│ │ │ │ -00017df0: 302c 7c0a 7c7d 2c20 7b31 2c20 307d 2c20  0,|.|}, {1, 0}, 
│ │ │ │ -00017e00: 7b30 2c20 317d 2c20 7b31 2c20 317d 7d20  {0, 1}, {1, 1}} 
│ │ │ │ +00017da0: 2020 7c0a 7c2c 2030 2c20 302c 2061 202b    |.|, 0, 0, a +
│ │ │ │ +00017db0: 2031 2c20 302c 2061 7d2c 207b 302c 2031   1, 0, a}, {0, 1
│ │ │ │ +00017dc0: 2c20 302c 2030 2c20 302c 2030 2c20 302c  , 0, 0, 0, 0, 0,
│ │ │ │ +00017dd0: 2030 2c20 307d 2c20 7b30 2c20 302c 2031   0, 0}, {0, 0, 1
│ │ │ │ +00017de0: 2c20 302c 2030 2c20 302c 2030 2c20 302c  , 0, 0, 0, 0, 0,
│ │ │ │ +00017df0: 2020 7c0a 7c7d 2c20 7b61 2c20 617d 2c20    |.|}, {a, a}, 
│ │ │ │ +00017e00: 7b31 2c20 617d 2c20 7b31 2c20 317d 7d20  {1, a}, {1, 1}} 
│ │ │ │  00017e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6255,38 +6255,38 @@
│ │ │ │  000186e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000186f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018750: 2020 7c0a 7c30 2c20 302c 2030 7d2c 207b    |.|0, 0, 0}, {
│ │ │ │ -00018760: 302c 2030 2c20 302c 2031 2c20 302c 2030  0, 0, 0, 1, 0, 0
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│ │ │ │  000187a0: 302c 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  0,|.|-----------
│ │ │ │  000187b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00018800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018820: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │ +00018810: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ +00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018840: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00018850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000188a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188c0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +000188c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000188d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000188e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000188f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018930: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ @@ -6385,18 +6385,18 @@
│ │ │ │  00018f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00018f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f70: 2020 7c0a 7c30 2c20 302c 2030 2c20 312c    |.|0, 0, 0, 1,
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│ │ │ │ +00018f80: 2c20 7b30 2c20 302c 2030 2c20 302c 2030  , {0, 0, 0, 0, 0
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│ │ │ │  00018fc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00018fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019010: 2d2d 2b0a 7c69 3420 3a20 432e 5365 7473  --+.|i4 : C.Sets
│ │ │ │ @@ -6618,96 +6618,96 @@
│ │ │ │  00019d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019da0: 2020 2020 2020 6361 6368 6520 3d3e 2043        cache => C
│ │ │ │  00019db0: 6163 6865 5461 626c 657b 7d20 2020 2020  acheTable{}     
│ │ │ │  00019dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019df0: 2020 2020 2020 436f 6465 203d 3e20 696d        Code => im
│ │ │ │ -00019e00: 6167 6520 7c20 3120 3020 2020 3020 2020  age | 1 0   0   
│ │ │ │ +00019e00: 6167 6520 7c20 3020 2020 3120 3020 2020  age | 0   1 0   
│ │ │ │  00019e10: 3020 2020 3020 7c0a 7c20 2020 2020 2020  0   0 |.|       
│ │ │ │  00019e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e50: 2020 2020 7c20 3120 612b 3120 3120 2020      | 1 a+1 1   
│ │ │ │ -00019e60: 6120 2020 3120 7c0a 7c20 2020 2020 2020  a   1 |.|       
│ │ │ │ +00019e50: 2020 2020 7c20 612b 3120 3120 612b 3120      | a+1 1 a+1 
│ │ │ │ +00019e60: 3120 2020 6120 7c0a 7c20 2020 2020 2020  1   a |.|       
│ │ │ │  00019e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ea0: 2020 2020 7c20 3120 6120 2020 612b 3120      | 1 a   a+1 
│ │ │ │ -00019eb0: 6120 2020 3120 7c0a 7c20 2020 2020 2020  a   1 |.|       
│ │ │ │ +00019ea0: 2020 2020 7c20 3120 2020 3120 6120 2020      | 1   1 a   
│ │ │ │ +00019eb0: 612b 3120 6120 7c0a 7c20 2020 2020 2020  a+1 a |.|       
│ │ │ │  00019ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ef0: 2020 2020 7c20 3120 3120 2020 6120 2020      | 1 1   a   
│ │ │ │ -00019f00: 6120 2020 3120 7c0a 7c20 2020 2020 2020  a   1 |.|       
│ │ │ │ +00019ef0: 2020 2020 7c20 6120 2020 3120 3120 2020      | a   1 1   
│ │ │ │ +00019f00: 6120 2020 6120 7c0a 7c20 2020 2020 2020  a   a |.|       
│ │ │ │  00019f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f40: 2020 2020 7c20 3120 612b 3120 6120 2020      | 1 a+1 a   
│ │ │ │ -00019f50: 612b 3120 3120 7c0a 7c20 2020 2020 2020  a+1 1 |.|       
│ │ │ │ +00019f40: 2020 2020 7c20 3120 2020 3120 612b 3120      | 1   1 a+1 
│ │ │ │ +00019f50: 6120 2020 6120 7c0a 7c20 2020 2020 2020  a   a |.|       
│ │ │ │  00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f90: 2020 2020 7c20 3120 6120 2020 3120 2020      | 1 a   1   
│ │ │ │ -00019fa0: 612b 3120 3120 7c0a 7c20 2020 2020 2020  a+1 1 |.|       
│ │ │ │ +00019f90: 2020 2020 7c20 6120 2020 3120 6120 2020      | a   1 a   
│ │ │ │ +00019fa0: 3120 2020 6120 7c0a 7c20 2020 2020 2020  1   a |.|       
│ │ │ │  00019fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019fe0: 2020 2020 7c20 3120 3120 2020 612b 3120      | 1 1   a+1 
│ │ │ │ -00019ff0: 612b 3120 3120 7c0a 7c20 2020 2020 2020  a+1 1 |.|       
│ │ │ │ +00019fe0: 2020 2020 7c20 612b 3120 3120 3120 2020      | a+1 1 1   
│ │ │ │ +00019ff0: 612b 3120 6120 7c0a 7c20 2020 2020 2020  a+1 a |.|       
│ │ │ │  0001a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a030: 2020 2020 7c20 3120 3020 2020 3020 2020      | 1 0   0   
│ │ │ │ -0001a040: 3120 2020 3020 7c0a 7c20 2020 2020 2020  1   0 |.|       
│ │ │ │ +0001a030: 2020 2020 7c20 3020 2020 3120 3020 2020      | 0   1 0   
│ │ │ │ +0001a040: 3020 2020 3120 7c0a 7c20 2020 2020 2020  0   1 |.|       
│ │ │ │  0001a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a070: 2020 2020 2020 4765 6e65 7261 746f 724d        GeneratorM
│ │ │ │ -0001a080: 6174 7269 7820 3d3e 207c 2031 2031 2020  atrix => | 1 1  
│ │ │ │ +0001a080: 6174 7269 7820 3d3e 207c 2030 2061 2b31  atrix => | 0 a+1
│ │ │ │  0001a090: 2031 2020 2020 7c0a 7c20 2020 2020 2020   1    |.|       
│ │ │ │  0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0d0: 2020 2020 2020 2020 207c 2030 2061 2b31           | 0 a+1
│ │ │ │ -0001a0e0: 2061 2020 2020 7c0a 7c20 2020 2020 2020   a    |.|       
│ │ │ │ +0001a0d0: 2020 2020 2020 2020 207c 2031 2031 2020           | 1 1  
│ │ │ │ +0001a0e0: 2031 2020 2020 7c0a 7c20 2020 2020 2020   1    |.|       
│ │ │ │  0001a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a120: 2020 2020 2020 2020 207c 2030 2031 2020           | 0 1  
│ │ │ │ -0001a130: 2061 2b31 2020 7c0a 7c20 2020 2020 2020   a+1  |.|       
│ │ │ │ +0001a120: 2020 2020 2020 2020 207c 2030 2061 2b31           | 0 a+1
│ │ │ │ +0001a130: 2061 2020 2020 7c0a 7c20 2020 2020 2020   a    |.|       
│ │ │ │  0001a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a170: 2020 2020 2020 2020 207c 2030 2061 2020           | 0 a  
│ │ │ │ -0001a180: 2061 2020 2020 7c0a 7c20 2020 2020 2020   a    |.|       
│ │ │ │ +0001a170: 2020 2020 2020 2020 207c 2030 2031 2020           | 0 1  
│ │ │ │ +0001a180: 2061 2b31 2020 7c0a 7c20 2020 2020 2020   a+1  |.|       
│ │ │ │  0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1c0: 2020 2020 2020 2020 207c 2030 2031 2020           | 0 1  
│ │ │ │ -0001a1d0: 2031 2020 2020 7c0a 7c20 2020 2020 2020   1    |.|       
│ │ │ │ +0001a1c0: 2020 2020 2020 2020 207c 2030 2061 2020           | 0 a  
│ │ │ │ +0001a1d0: 2061 2020 2020 7c0a 7c20 2020 2020 2020   a    |.|       
│ │ │ │  0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a210: 2020 2020 2020 2020 207c 2030 2061 2020           | 0 a  
│ │ │ │ -0001a220: 2061 2b31 2020 7c0a 7c20 2020 2020 2020   a+1  |.|       
│ │ │ │ +0001a210: 2020 2020 2020 2020 207c 2030 2031 2020           | 0 1  
│ │ │ │ +0001a220: 2031 2020 2020 7c0a 7c20 2020 2020 2020   1    |.|       
│ │ │ │  0001a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a260: 2020 2020 2020 2020 207c 2030 2061 2b31           | 0 a+1
│ │ │ │ +0001a260: 2020 2020 2020 2020 207c 2030 2061 2020           | 0 a  
│ │ │ │  0001a270: 2061 2b31 2020 7c0a 7c20 2020 2020 2020   a+1  |.|       
│ │ │ │  0001a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2b0: 2020 2020 2020 2020 207c 2030 2061 2b31           | 0 a+1
│ │ │ │ -0001a2c0: 2031 2020 2020 7c0a 7c20 2020 2020 2020   1    |.|       
│ │ │ │ +0001a2c0: 2061 2b31 2020 7c0a 7c20 2020 2020 2020   a+1  |.|       
│ │ │ │  0001a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2f0: 2020 2020 2020 4765 6e65 7261 746f 7273        Generators
│ │ │ │ -0001a300: 203d 3e20 7b7b 312c 2031 2c20 312c 2031   => {{1, 1, 1, 1
│ │ │ │ -0001a310: 2c20 312c 2020 7c0a 7c20 2020 2020 2020  , 1,  |.|       
│ │ │ │ +0001a300: 203d 3e20 7b7b 302c 2061 202b 2031 2c20   => {{0, a + 1, 
│ │ │ │ +0001a310: 312c 2061 2c20 7c0a 7c20 2020 2020 2020  1, a, |.|       
│ │ │ │  0001a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a340: 2020 2020 2020 5061 7269 7479 4368 6563        ParityChec
│ │ │ │  0001a350: 6b4d 6174 7269 7820 3d3e 207c 2031 2031  kMatrix => | 1 1
│ │ │ │  0001a360: 2031 2031 2020 7c0a 7c20 2020 2020 2020   1 1  |.|       
│ │ │ │  0001a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6727,26 +6727,26 @@
│ │ │ │  0001a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a480: 2020 2020 2020 2030 2031 2020 2020 3020         0 1    0 
│ │ │ │  0001a490: 3120 2020 2030 2020 2030 2020 2020 3120  1    0   0    1 
│ │ │ │  0001a4a0: 2020 2031 2020 7c0a 7c20 2020 2020 2020     1  |.|       
│ │ │ │  0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4d0: 2020 2032 2020 2032 2020 2020 2020 2020     2   2        
│ │ │ │ -0001a4e0: 2033 2020 2020 2020 2032 2020 2020 2020   3       2      
│ │ │ │ +0001a4d0: 2020 2020 2020 2020 2032 2020 2032 2020           2   2  
│ │ │ │ +0001a4e0: 2020 2020 2020 2033 2020 2020 2020 2032         3       2
│ │ │ │  0001a4f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001a500: 2020 2020 2020 2020 2020 2020 2050 6f6c               Pol
│ │ │ │ -0001a510: 796e 6f6d 6961 6c53 6574 203d 3e20 7b31  ynomialSet => {1
│ │ │ │ -0001a520: 2c20 7420 2c20 7420 7420 2c20 7420 2c20  , t , t t , t , 
│ │ │ │ -0001a530: 7420 2c20 7420 2c20 7420 2c20 7420 7420  t , t , t , t t 
│ │ │ │ +0001a510: 796e 6f6d 6961 6c53 6574 203d 3e20 7b74  ynomialSet => {t
│ │ │ │ +0001a520: 2074 202c 2031 2c20 7420 2c20 7420 7420   t , 1, t , t t 
│ │ │ │ +0001a530: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │  0001a540: 7d20 2020 2020 7c0a 7c20 2020 2020 2020  }     |.|       
│ │ │ │  0001a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a570: 2020 2030 2020 2030 2031 2020 2031 2020     0   0 1   1  
│ │ │ │ -0001a580: 2030 2020 2030 2020 2031 2020 2030 2031   0   0   1   0 1
│ │ │ │ +0001a570: 3020 3120 2020 2020 2030 2020 2030 2031  0 1      0   0 1
│ │ │ │ +0001a580: 2020 2031 2020 2030 2020 2030 2020 2031     1   0   0   1
│ │ │ │  0001a590: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001a5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5e0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0001a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6774,99 +6774,99 @@
│ │ │ │  0001a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7c0: 2020 2020 2020 7c0a 7c20 3020 2020 3020        |.| 0   0 
│ │ │ │ +0001a7c0: 2020 2020 2020 7c0a 7c20 2020 3020 3020        |.|   0 0 
│ │ │ │  0001a7d0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │  0001a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a810: 2020 2020 2020 7c0a 7c20 6120 2020 612b        |.| a   a+
│ │ │ │ -0001a820: 3120 612b 3120 7c20 2020 2020 2020 2020  1 a+1 |         
│ │ │ │ +0001a810: 2020 2020 2020 7c0a 7c20 2020 3120 6120        |.|   1 a 
│ │ │ │ +0001a820: 2020 612b 3120 7c20 2020 2020 2020 2020    a+1 |         
│ │ │ │  0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a860: 2020 2020 2020 7c0a 7c20 612b 3120 612b        |.| a+1 a+
│ │ │ │ -0001a870: 3120 3120 2020 7c20 2020 2020 2020 2020  1 1   |         
│ │ │ │ +0001a860: 2020 2020 2020 7c0a 7c20 2020 3120 612b        |.|   1 a+
│ │ │ │ +0001a870: 3120 612b 3120 7c20 2020 2020 2020 2020  1 a+1 |         
│ │ │ │  0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8b0: 2020 2020 2020 7c0a 7c20 3120 2020 612b        |.| 1   a+
│ │ │ │ -0001a8c0: 3120 6120 2020 7c20 2020 2020 2020 2020  1 a   |         
│ │ │ │ +0001a8b0: 2020 2020 2020 7c0a 7c20 2020 3120 3120        |.|   1 1 
│ │ │ │ +0001a8c0: 2020 612b 3120 7c20 2020 2020 2020 2020    a+1 |         
│ │ │ │  0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a900: 2020 2020 2020 7c0a 7c20 6120 2020 6120        |.| a   a 
│ │ │ │ -0001a910: 2020 3120 2020 7c20 2020 2020 2020 2020    1   |         
│ │ │ │ +0001a900: 2020 2020 2020 7c0a 7c2b 3120 3120 6120        |.|+1 1 a 
│ │ │ │ +0001a910: 2020 6120 2020 7c20 2020 2020 2020 2020    a   |         
│ │ │ │  0001a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a950: 2020 2020 2020 7c0a 7c20 612b 3120 6120        |.| a+1 a 
│ │ │ │ -0001a960: 2020 6120 2020 7c20 2020 2020 2020 2020    a   |         
│ │ │ │ +0001a950: 2020 2020 2020 7c0a 7c2b 3120 3120 612b        |.|+1 1 a+
│ │ │ │ +0001a960: 3120 6120 2020 7c20 2020 2020 2020 2020  1 a   |         
│ │ │ │  0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9a0: 2020 2020 2020 7c0a 7c20 3120 2020 6120        |.| 1   a 
│ │ │ │ -0001a9b0: 2020 612b 3120 7c20 2020 2020 2020 2020    a+1 |         
│ │ │ │ +0001a9a0: 2020 2020 2020 7c0a 7c2b 3120 3120 3120        |.|+1 1 1 
│ │ │ │ +0001a9b0: 2020 6120 2020 7c20 2020 2020 2020 2020    a   |         
│ │ │ │  0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9f0: 2020 2020 2020 7c0a 7c20 3020 2020 3120        |.| 0   1 
│ │ │ │ -0001aa00: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +0001a9f0: 2020 2020 2020 7c0a 7c20 2020 3020 3020        |.|   0 0 
│ │ │ │ +0001aa00: 2020 3120 2020 7c20 2020 2020 2020 2020    1   |         
│ │ │ │  0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa40: 2020 2020 2020 7c0a 7c31 2020 2031 2020        |.|1   1  
│ │ │ │ -0001aa50: 2031 2020 2031 2020 2031 207c 2020 2020   1   1   1 |    
│ │ │ │ +0001aa40: 2020 2020 2020 7c0a 7c61 2020 2031 2020        |.|a   1  
│ │ │ │ +0001aa50: 2061 2020 2061 2b31 2030 207c 2020 2020   a   a+1 0 |    
│ │ │ │  0001aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa90: 2020 2020 2020 7c0a 7c31 2020 2061 2b31        |.|1   a+1
│ │ │ │ -0001aaa0: 2061 2020 2031 2020 2030 207c 2020 2020   a   1   0 |    
│ │ │ │ +0001aa90: 2020 2020 2020 7c0a 7c31 2020 2031 2020        |.|1   1  
│ │ │ │ +0001aaa0: 2031 2020 2031 2020 2031 207c 2020 2020   1   1   1 |    
│ │ │ │  0001aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aae0: 2020 2020 2020 7c0a 7c61 2020 2061 2020        |.|a   a  
│ │ │ │ -0001aaf0: 2031 2020 2061 2b31 2030 207c 2020 2020   1   a+1 0 |    
│ │ │ │ +0001aae0: 2020 2020 2020 7c0a 7c31 2020 2061 2b31        |.|1   a+1
│ │ │ │ +0001aaf0: 2061 2020 2031 2020 2030 207c 2020 2020   a   1   0 |    
│ │ │ │  0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab30: 2020 2020 2020 7c0a 7c61 2020 2061 2b31        |.|a   a+1
│ │ │ │ -0001ab40: 2061 2b31 2061 2b31 2031 207c 2020 2020   a+1 a+1 1 |    
│ │ │ │ +0001ab30: 2020 2020 2020 7c0a 7c61 2020 2061 2020        |.|a   a  
│ │ │ │ +0001ab40: 2031 2020 2061 2b31 2030 207c 2020 2020   1   a+1 0 |    
│ │ │ │  0001ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab80: 2020 2020 2020 7c0a 7c31 2020 2031 2020        |.|1   1  
│ │ │ │ -0001ab90: 2031 2020 2031 2020 2030 207c 2020 2020   1   1   0 |    
│ │ │ │ +0001ab80: 2020 2020 2020 7c0a 7c61 2020 2061 2b31        |.|a   a+1
│ │ │ │ +0001ab90: 2061 2b31 2061 2b31 2031 207c 2020 2020   a+1 a+1 1 |    
│ │ │ │  0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abd0: 2020 2020 2020 7c0a 7c31 2020 2061 2020        |.|1   a  
│ │ │ │ -0001abe0: 2061 2b31 2031 2020 2030 207c 2020 2020   a+1 1   0 |    
│ │ │ │ +0001abd0: 2020 2020 2020 7c0a 7c31 2020 2031 2020        |.|1   1  
│ │ │ │ +0001abe0: 2031 2020 2031 2020 2030 207c 2020 2020   1   1   0 |    
│ │ │ │  0001abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac20: 2020 2020 2020 7c0a 7c61 2b31 2061 2020        |.|a+1 a  
│ │ │ │ -0001ac30: 2061 2020 2061 2020 2031 207c 2020 2020   a   a   1 |    
│ │ │ │ +0001ac20: 2020 2020 2020 7c0a 7c31 2020 2061 2020        |.|1   a  
│ │ │ │ +0001ac30: 2061 2b31 2031 2020 2030 207c 2020 2020   a+1 1   0 |    
│ │ │ │  0001ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac70: 2020 2020 2020 7c0a 7c61 2020 2031 2020        |.|a   1  
│ │ │ │ -0001ac80: 2061 2020 2061 2b31 2030 207c 2020 2020   a   a+1 0 |    
│ │ │ │ +0001ac70: 2020 2020 2020 7c0a 7c61 2b31 2061 2020        |.|a+1 a  
│ │ │ │ +0001ac80: 2061 2020 2061 2020 2031 207c 2020 2020   a   a   1 |    
│ │ │ │  0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001acc0: 2020 2020 2020 7c0a 7c31 2c20 312c 2031        |.|1, 1, 1
│ │ │ │ -0001acd0: 7d2c 207b 302c 2061 202b 2031 2c20 612c  }, {0, a + 1, a,
│ │ │ │ -0001ace0: 2031 2c20 6120 2b20 312c 2061 2c20 312c   1, a + 1, a, 1,
│ │ │ │ -0001acf0: 2030 7d2c 207b 302c 2031 2c20 6120 2b20   0}, {0, 1, a + 
│ │ │ │ -0001ad00: 312c 2061 2c20 612c 2031 2c20 6120 2b20  1, a, a, 1, a + 
│ │ │ │ +0001acc0: 2020 2020 2020 7c0a 7c20 312c 2061 2c20        |.| 1, a, 
│ │ │ │ +0001acd0: 6120 2b20 312c 2030 7d2c 207b 312c 2031  a + 1, 0}, {1, 1
│ │ │ │ +0001ace0: 2c20 312c 2031 2c20 312c 2031 2c20 312c  , 1, 1, 1, 1, 1,
│ │ │ │ +0001acf0: 2031 7d2c 207b 302c 2061 202b 2031 2c20   1}, {0, a + 1, 
│ │ │ │ +0001ad00: 612c 2031 2c20 6120 2b20 312c 2061 2c20  a, 1, a + 1, a, 
│ │ │ │  0001ad10: 312c 2030 7d2c 7c0a 7c31 2031 2031 2031  1, 0},|.|1 1 1 1
│ │ │ │  0001ad20: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │  0001ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad60: 2020 2020 2020 7c0a 7c2c 2031 2c20 312c        |.|, 1, 1,
│ │ │ │  0001ad70: 2031 2c20 317d 7d20 2020 2020 2020 2020   1, 1}}         
│ │ │ │ @@ -6999,20 +6999,20 @@
│ │ │ │  0001b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b580: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5d0: 2020 2020 2020 7c0a 7c7b 302c 2061 2c20        |.|{0, a, 
│ │ │ │ -0001b5e0: 612c 2061 2c20 6120 2b20 312c 2061 202b  a, a, a + 1, a +
│ │ │ │ -0001b5f0: 2031 2c20 6120 2b20 312c 2031 7d2c 207b   1, a + 1, 1}, {
│ │ │ │ -0001b600: 302c 2031 2c20 312c 2031 2c20 312c 2031  0, 1, 1, 1, 1, 1
│ │ │ │ -0001b610: 2c20 312c 2030 7d2c 207b 302c 2061 2c20  , 1, 0}, {0, a, 
│ │ │ │ -0001b620: 6120 2b20 312c 7c0a 7c2d 2d2d 2d2d 2d2d  a + 1,|.|-------
│ │ │ │ +0001b5d0: 2020 2020 2020 7c0a 7c7b 302c 2031 2c20        |.|{0, 1, 
│ │ │ │ +0001b5e0: 6120 2b20 312c 2061 2c20 612c 2031 2c20  a + 1, a, a, 1, 
│ │ │ │ +0001b5f0: 6120 2b20 312c 2030 7d2c 207b 302c 2061  a + 1, 0}, {0, a
│ │ │ │ +0001b600: 2c20 612c 2061 2c20 6120 2b20 312c 2061  , a, a, a + 1, a
│ │ │ │ +0001b610: 202b 2031 2c20 6120 2b20 312c 2031 7d2c   + 1, a + 1, 1},
│ │ │ │ +0001b620: 207b 302c 2020 7c0a 7c2d 2d2d 2d2d 2d2d   {0,  |.|-------
│ │ │ │  0001b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b670: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0001b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7119,41 +7119,41 @@
│ │ │ │  0001bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd50: 2020 2020 2020 7c0a 7c31 2c20 612c 2061        |.|1, a, a
│ │ │ │ -0001bd60: 202b 2031 2c20 312c 2030 7d2c 207b 302c   + 1, 1, 0}, {0,
│ │ │ │ -0001bd70: 2061 202b 2031 2c20 6120 2b20 312c 2061   a + 1, a + 1, a
│ │ │ │ -0001bd80: 202b 2031 2c20 612c 2061 2c20 612c 2031   + 1, a, a, a, 1
│ │ │ │ -0001bd90: 7d2c 207b 302c 2061 202b 2031 2c20 312c  }, {0, a + 1, 1,
│ │ │ │ -0001bda0: 2061 2c20 312c 7c0a 7c2d 2d2d 2d2d 2d2d   a, 1,|.|-------
│ │ │ │ +0001bd50: 2020 2020 2020 7c0a 7c31 2c20 312c 2031        |.|1, 1, 1
│ │ │ │ +0001bd60: 2c20 312c 2031 2c20 312c 2030 7d2c 207b  , 1, 1, 1, 0}, {
│ │ │ │ +0001bd70: 302c 2061 2c20 6120 2b20 312c 2031 2c20  0, a, a + 1, 1, 
│ │ │ │ +0001bd80: 612c 2061 202b 2031 2c20 312c 2030 7d2c  a, a + 1, 1, 0},
│ │ │ │ +0001bd90: 207b 302c 2061 202b 2031 2c20 6120 2b20   {0, a + 1, a + 
│ │ │ │ +0001bda0: 312c 2061 202b 7c0a 7c2d 2d2d 2d2d 2d2d  1, a +|.|-------
│ │ │ │  0001bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bdf0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001be00: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +0001be00: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │  0001be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bee0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001bef0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0001bef0: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │  0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7239,16 +7239,16 @@
│ │ │ │  0001c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c480: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4d0: 2020 2020 2020 7c0a 7c61 2c20 6120 2b20        |.|a, a + 
│ │ │ │ -0001c4e0: 312c 2030 7d7d 2020 2020 2020 2020 2020  1, 0}}          
│ │ │ │ +0001c4d0: 2020 2020 2020 7c0a 7c31 2c20 612c 2061        |.|1, a, a
│ │ │ │ +0001c4e0: 2c20 612c 2031 7d7d 2020 2020 2020 2020  , a, 1}}        
│ │ │ │  0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c520: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0001c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -12687,18 +12687,18 @@
│ │ │ │  000318e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000318f0: 2d2d 2b0a 7c69 3220 3a20 7665 6374 6f72  --+.|i2 : vector
│ │ │ │  00031900: 5370 6163 6520 4820 2020 2020 2020 2020  Space H         
│ │ │ │  00031910: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00031920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031940: 2020 2020 7c0a 7c6f 3220 3d20 696d 6167      |.|o2 = imag
│ │ │ │ -00031950: 6520 7c20 3120 3120 3020 3120 7c20 2020  e | 1 1 0 1 |   
│ │ │ │ +00031950: 6520 7c20 3120 3020 3120 3120 7c20 2020  e | 1 0 1 1 |   
│ │ │ │  00031960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00031970: 2020 2020 2020 2020 2020 207c 2031 2030             | 1 0
│ │ │ │ -00031980: 2031 2031 207c 2020 2020 2020 2020 2020   1 1 |          
│ │ │ │ +00031970: 2020 2020 2020 2020 2020 207c 2031 2031             | 1 1
│ │ │ │ +00031980: 2030 2031 207c 2020 2020 2020 2020 2020   0 1 |          
│ │ │ │  00031990: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000319a0: 2020 2020 7c20 3120 3120 3120 3020 7c20      | 1 1 1 0 | 
│ │ │ │  000319b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000319c0: 0a7c 2020 2020 2020 2020 2020 207c 2031  .|           | 1
│ │ │ │  000319d0: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │  000319e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000319f0: 2020 2020 2020 7c20 3020 3120 3020 3020        | 0 1 0 0 
│ │ │ │ @@ -13181,42 +13181,42 @@
│ │ │ │  000337c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000337d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000337e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000337f0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7b7b        |.|o3 = {{
│ │ │ │  00033800: 612c 2061 2c20 617d 2c20 7b61 202b 2031  a, a, a}, {a + 1
│ │ │ │  00033810: 2c20 612c 2031 7d2c 207b 312c 2031 2c20  , a, 1}, {1, 1, 
│ │ │ │  00033820: 317d 2c20 7b30 2c20 312c 2061 7d2c 207b  1}, {0, 1, a}, {
│ │ │ │ -00033830: 302c 2061 2c20 6120 2b20 317d 2c20 7b31  0, a, a + 1}, {1
│ │ │ │ -00033840: 2c20 6120 2b20 7c0a 7c20 2020 2020 2d2d  , a + |.|     --
│ │ │ │ +00033830: 612c 2031 2c20 6120 2b20 317d 2c20 7b30  a, 1, a + 1}, {0
│ │ │ │ +00033840: 2c20 612c 2061 7c0a 7c20 2020 2020 2d2d  , a, a|.|     --
│ │ │ │  00033850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033890: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 312c  ------|.|     1,
│ │ │ │ -000338a0: 2061 7d2c 207b 302c 2061 202b 2031 2c20   a}, {0, a + 1, 
│ │ │ │ -000338b0: 317d 2c20 7b61 2c20 312c 2061 202b 2031  1}, {a, 1, a + 1
│ │ │ │ -000338c0: 7d2c 207b 312c 2061 2c20 307d 2c20 7b61  }, {1, a, 0}, {a
│ │ │ │ -000338d0: 202b 2031 2c20 6120 2b20 312c 2061 202b   + 1, a + 1, a +
│ │ │ │ -000338e0: 2031 7d2c 2020 7c0a 7c20 2020 2020 2d2d   1},  |.|     --
│ │ │ │ +00033890: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2b20  ------|.|     + 
│ │ │ │ +000338a0: 317d 2c20 7b30 2c20 6120 2b20 312c 2031  1}, {0, a + 1, 1
│ │ │ │ +000338b0: 7d2c 207b 312c 2061 202b 2031 2c20 617d  }, {1, a + 1, a}
│ │ │ │ +000338c0: 2c20 7b61 202b 2031 2c20 302c 2061 7d2c  , {a + 1, 0, a},
│ │ │ │ +000338d0: 207b 612c 2030 2c20 317d 2c20 7b61 202b   {a, 0, 1}, {a +
│ │ │ │ +000338e0: 2031 2c20 312c 7c0a 7c20 2020 2020 2d2d   1, 1,|.|     --
│ │ │ │  000338f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033930: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7b61  ------|.|     {a
│ │ │ │ -00033940: 2c20 302c 2031 7d2c 207b 6120 2b20 312c  , 0, 1}, {a + 1,
│ │ │ │ -00033950: 2030 2c20 617d 2c20 7b61 202b 2031 2c20   0, a}, {a + 1, 
│ │ │ │ -00033960: 312c 2030 7d2c 207b 312c 2030 2c20 6120  1, 0}, {1, 0, a 
│ │ │ │ -00033970: 2b20 317d 2c20 7b61 2c20 6120 2b20 312c  + 1}, {a, a + 1,
│ │ │ │ -00033980: 2030 7d2c 2020 7c0a 7c20 2020 2020 2d2d   0},  |.|     --
│ │ │ │ +00033930: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 307d  ------|.|     0}
│ │ │ │ +00033940: 2c20 7b31 2c20 612c 2030 7d2c 207b 6120  , {1, a, 0}, {a 
│ │ │ │ +00033950: 2b20 312c 2061 202b 2031 2c20 6120 2b20  + 1, a + 1, a + 
│ │ │ │ +00033960: 317d 2c20 7b31 2c20 302c 2061 202b 2031  1}, {1, 0, a + 1
│ │ │ │ +00033970: 7d2c 207b 612c 2061 202b 2031 2c20 307d  }, {a, a + 1, 0}
│ │ │ │ +00033980: 2c20 7b30 2c20 7c0a 7c20 2020 2020 2d2d  , {0, |.|     --
│ │ │ │  00033990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000339a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000339b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000339c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000339d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7b30  ------|.|     {0
│ │ │ │ -000339e0: 2c20 302c 2030 7d7d 2020 2020 2020 2020  , 0, 0}}        
│ │ │ │ +000339d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 302c  ------|.|     0,
│ │ │ │ +000339e0: 2030 7d7d 2020 2020 2020 2020 2020 2020   0}}            
│ │ │ │  000339f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00033a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -19254,17 +19254,17 @@
│ │ │ │  0004b350: 2d2d 2d2b 0a7c 6932 203a 2043 312e 5061  ---+.|i2 : C1.Pa
│ │ │ │  0004b360: 7269 7479 4368 6563 6b4d 6174 7269 7820  rityCheckMatrix 
│ │ │ │  0004b370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0004b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b390: 2020 2020 2020 207c 0a7c 6f32 203d 207c         |.|o2 = |
│ │ │ │  0004b3a0: 2031 2031 2031 2031 2030 2030 2030 207c   1 1 1 1 0 0 0 |
│ │ │ │  0004b3b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0004b3c0: 207c 2030 2031 2030 2031 2031 2031 2030   | 0 1 0 1 1 1 0
│ │ │ │ +0004b3c0: 207c 2030 2030 2031 2031 2031 2031 2030   | 0 0 1 1 1 1 0
│ │ │ │  0004b3d0: 207c 2020 2020 2020 2020 207c 0a7c 2020   |         |.|  
│ │ │ │ -0004b3e0: 2020 207c 2031 2030 2030 2031 2031 2030     | 1 0 0 1 1 0
│ │ │ │ +0004b3e0: 2020 207c 2030 2031 2030 2031 2030 2031     | 0 1 0 1 0 1
│ │ │ │  0004b3f0: 2031 207c 2020 2020 2020 2020 207c 0a7c   1 |         |.|
│ │ │ │  0004b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0004b420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0004b430: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  0004b440: 377c 0a7c 6f32 203a 204d 6174 7269 7820  7|.|o2 : Matrix 
│ │ │ │  0004b450: 2847 4620 3229 2020 3c2d 2d20 2847 4620  (GF 2)  <-- (GF 
│ │ │ │ @@ -24045,17 +24045,17 @@
│ │ │ │  0005dec0: 462c 7b7b 312c 312c 317d 7d29 3b7c 0a2b  F,{{1,1,1}});|.+
│ │ │ │  0005ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0005def0: 0a7c 6933 203a 206d 6573 7361 6765 7320  .|i3 : messages 
│ │ │ │  0005df00: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0005df10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0005df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df30: 2020 207c 0a7c 6f33 203d 207b 7b31 7d2c     |.|o3 = {{1},
│ │ │ │ -0005df40: 207b 617d 2c20 7b61 202b 2031 7d2c 207b   {a}, {a + 1}, {
│ │ │ │ -0005df50: 307d 7d20 207c 0a7c 2020 2020 2020 2020  0}}  |.|        
│ │ │ │ +0005df30: 2020 207c 0a7c 6f33 203d 207b 7b30 7d2c     |.|o3 = {{0},
│ │ │ │ +0005df40: 207b 317d 2c20 7b61 7d2c 207b 6120 2b20   {1}, {a}, {a + 
│ │ │ │ +0005df50: 317d 7d20 207c 0a7c 2020 2020 2020 2020  1}}  |.|        
│ │ │ │  0005df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df70: 2020 2020 2020 207c 0a7c 6f33 203a 204c         |.|o3 : L
│ │ │ │  0005df80: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0005df90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0005dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d  -----------+.+--
│ │ │ │  0005dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -24953,19 +24953,19 @@
│ │ │ │  00061780: 2020 207c 2030 207c 2020 2020 2020 2020     | 0 |        
│ │ │ │  00061790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000617a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000617b0: 2020 2020 2020 7c20 3120 7c20 3d3e 207c        | 1 | => |
│ │ │ │  000617c0: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
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│ │ │ │  000675d0: 2020 2020 2020 2043 6f64 6520 3d3e 2069         Code => i
│ │ │ │ -000675e0: 6d61 6765 207c 2031 2031 2030 2031 207c  mage | 1 1 0 1 |
│ │ │ │ +000675e0: 6d61 6765 207c 2031 2031 2031 2030 207c  mage | 1 1 1 0 |
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│ │ │ │  00067690: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000676d0: 2020 2020 207c 2031 2030 2030 2030 207c       | 1 0 0 0 |
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│ │ │ │  000678a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000678f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00067940: 2020 2020 2020 2047 656e 6572 6174 6f72         Generator
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│ │ │ │  00067970: 2030 2c20 312c 2030 2c20 312c 2030 2c20   0, 1, 0, 1, 0, 
│ │ │ │  00067980: 307d 2c20 7c0a 7c20 2020 2020 2020 2020  0}, |.|         
│ │ │ │  00067990: 2020 2020 2020 2050 6172 6974 7943 6865         ParityChe
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│ │ │ │  00067a70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00067a80: 2020 2020 2020 2050 6172 6974 7943 6865         ParityChe
│ │ │ │  00067a90: 636b 526f 7773 203d 3e20 7b7b 312c 2031  ckRows => {{1, 1
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│ │ │ │  00067ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067b10: 2020 2020 7c0a 7c6f 3320 3a20 4c69 6e65      |.|o3 : Line
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│ │ │ │  00067fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00067fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068000: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00068040: 2031 7d7d 2020 2020 2020 2020 2020 2020   1}}            
│ │ │ │  00068050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00068070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000680a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -26863,20 +26863,20 @@
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│ │ │ │  00068f00: 2031 2030 2031 2031 207c 2020 2020 2020   1 0 1 1 |      
│ │ │ │  00068f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00068fa0: 2031 2031 2030 2031 207c 2020 2020 2020   1 1 0 1 |      
│ │ │ │  00068fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00068fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00068ff0: 2031 2030 2030 2030 207c 2020 2020 2020   1 0 0 0 |      
│ │ │ │  00069000: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000691e0: 3120 3020 7c20 2020 2020 2020 2020 2020  1 0 |           
│ │ │ │  000691f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  00069260: 2047 656e 6572 6174 6f72 7320 3d3e 207b   Generators => {
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│ │ │ │  00069290: 2030 2c20 312c 2030 2c20 307d 2c20 7c0a   0, 1, 0, 0}, |.
│ │ │ │  000692a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000692b0: 2050 6172 6974 7943 6865 636b 4d61 7472   ParityCheckMatr
│ │ │ │  000692c0: 6978 203d 3e20 7c20 3120 3120 3120 3120  ix => | 1 1 1 1 
│ │ │ │  000692d0: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │  000692e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000692f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00069300: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00069360: 2020 2020 2020 7c20 3020 3120 3020 3120        | 0 1 0 1 
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│ │ │ │ +00069370: 3020 3120 3120 7c20 2020 2020 2020 2020  0 1 1 |         
│ │ │ │  00069380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00069390: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000693a0: 2050 6172 6974 7943 6865 636b 526f 7773   ParityCheckRows
│ │ │ │  000693b0: 203d 3e20 7b7b 312c 2031 2c20 312c 2031   => {{1, 1, 1, 1
│ │ │ │  000693c0: 2c20 302c 2030 2c20 307d 2c20 7b30 2c20  , 0, 0, 0}, {0, 
│ │ │ │ -000693d0: 302c 2031 2c20 312c 2031 2c20 312c 7c0a  0, 1, 1, 1, 1,|.
│ │ │ │ +000693d0: 312c 2030 2c20 312c 2031 2c20 312c 7c0a  1, 0, 1, 1, 1,|.
│ │ │ │  000693e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000693f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00069410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00069430: 7c6f 3120 3a20 4c69 6e65 6172 436f 6465  |o1 : LinearCode
│ │ │ │  00069440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -27021,16 +27021,16 @@
│ │ │ │  000698c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000698d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000698e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000698f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00069930: 7c7b 312c 2030 2c20 312c 2030 2c20 302c  |{1, 0, 1, 0, 0,
│ │ │ │ -00069940: 2031 2c20 307d 2c20 7b31 2c20 312c 2030   1, 0}, {1, 1, 0
│ │ │ │ +00069930: 7c7b 312c 2031 2c20 302c 2030 2c20 302c  |{1, 1, 0, 0, 0,
│ │ │ │ +00069940: 2031 2c20 307d 2c20 7b31 2c20 302c 2031   1, 0}, {1, 0, 1
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│ │ │ │  00069960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  00069990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000699a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000699b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -27041,16 +27041,16 @@
│ │ │ │  00069a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069a10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  00069a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00069a70: 7c20 307d 2c20 7b30 2c20 312c 2030 2c20  | 0}, {0, 1, 0, 
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│ │ │ │ +00069a70: 7c20 307d 2c20 7b30 2c20 312c 2031 2c20  | 0}, {0, 1, 1, 
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│ │ │ │  00069a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00069ac0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00069ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/CohomCalg.info.gz
│ │ │ ├── CohomCalg.info
│ │ │ │ @@ -1042,15 +1042,15 @@
│ │ │ │  00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469  |i20 : elapsedTi
│ │ │ │  00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d  me hvecs = cohom
│ │ │ │  00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020  Calg(X, D2)     
│ │ │ │  00004160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00004180: 7c20 2d2d 2033 2e30 3833 3037 7320 656c  | -- 3.08307s el
│ │ │ │ +00004180: 7c20 2d2d 2034 2e31 3432 3936 7320 656c  | -- 4.14296s el
│ │ │ │  00004190: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000041a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000041d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000041e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1677,15 +1677,15 @@
│ │ │ │  000068c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000068d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000068e0: 7c69 3233 203a 2065 6c61 7073 6564 5469  |i23 : elapsedTi
│ │ │ │  000068f0: 6d65 2063 6f68 6f6d 7665 6331 203d 2063  me cohomvec1 = c
│ │ │ │  00006900: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058  ohomCalg(X_3 + X
│ │ │ │  00006910: 5f37 202b 2058 5f38 2920 2020 2020 2020  _7 + X_8)       
│ │ │ │  00006920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00006930: 7c20 2d2d 202e 3338 3336 3633 7320 656c  | -- .383663s el
│ │ │ │ +00006930: 7c20 2d2d 202e 3937 3837 3834 7320 656c  | -- .978784s el
│ │ │ │  00006940: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00006950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00006980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00006990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000069a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1712,15 +1712,15 @@
│ │ │ │  00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469  |i24 : elapsedTi
│ │ │ │  00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066  me cohomvec2 = f
│ │ │ │  00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064  or j from 0 to d
│ │ │ │  00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048  im X list rank H
│ │ │ │  00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a  H^j(X,        |.
│ │ │ │ -00006b60: 7c20 2d2d 2031 332e 3535 3836 7320 656c  | -- 13.5586s el
│ │ │ │ +00006b60: 7c20 2d2d 2031 312e 3532 3634 7320 656c  | -- 11.5264s el
│ │ │ │  00006b70: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00006b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00006bb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00006bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1797,15 +1797,15 @@
│ │ │ │  00007040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00007060: 7c69 3237 203a 2065 6c61 7073 6564 5469  |i27 : elapsedTi
│ │ │ │  00007070: 6d65 2063 6f68 6f6d 7665 6331 203d 2063  me cohomvec1 = c
│ │ │ │  00007080: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058  ohomCalg(X_3 + X
│ │ │ │  00007090: 5f37 202d 2058 5f38 2920 2020 2020 2020  _7 - X_8)       
│ │ │ │  000070a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000070b0: 7c20 2d2d 202e 3430 3436 3437 7320 656c  | -- .404647s el
│ │ │ │ +000070b0: 7c20 2d2d 2031 2e30 3338 3938 7320 656c  | -- 1.03898s el
│ │ │ │  000070c0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000070d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000070e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000070f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00007100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00007110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1832,20 +1832,20 @@
│ │ │ │  00007270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00007290: 7c69 3238 203a 2065 6c61 7073 6564 5469  |i28 : elapsedTi
│ │ │ │  000072a0: 6d65 2063 6f68 6f6d 7665 6332 203d 2065  me cohomvec2 = e
│ │ │ │  000072b0: 6c61 7073 6564 5469 6d65 2066 6f72 206a  lapsedTime for j
│ │ │ │  000072c0: 2066 726f 6d20 3020 746f 2064 696d 2058   from 0 to dim X
│ │ │ │  000072d0: 206c 6973 7420 7261 6e6b 2020 2020 7c0a   list rank    |.
│ │ │ │ -000072e0: 7c20 2d2d 202e 3437 3434 3835 7320 656c  | -- .474485s el
│ │ │ │ +000072e0: 7c20 2d2d 202e 3530 3331 3337 7320 656c  | -- .503137s el
│ │ │ │  000072f0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00007300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00007330: 7c20 2d2d 202e 3437 3435 3331 7320 656c  | -- .474531s el
│ │ │ │ +00007330: 7c20 2d2d 202e 3530 3331 3835 7320 656c  | -- .503185s el
│ │ │ │  00007340: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00007350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00007380: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00007390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000073a0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/CompleteIntersectionResolutions.info.gz
│ │ │ ├── CompleteIntersectionResolutions.info
│ │ │ │ @@ -4102,17 +4102,17 @@
│ │ │ │  00010050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010060: 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 696d  -----+.|i7 : tim
│ │ │ │  00010070: 6520 4720 3d20 4569 7365 6e62 7564 5368  e G = EisenbudSh
│ │ │ │  00010080: 616d 6173 6828 6666 2c46 2c6c 656e 2920  amash(ff,F,len) 
│ │ │ │  00010090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000100a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000100b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000100c0: 2039 2e36 3235 3635 7320 2863 7075 293b   9.62565s (cpu);
│ │ │ │ -000100d0: 2035 2e30 3834 3973 2028 7468 7265 6164   5.0849s (thread
│ │ │ │ -000100e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000100c0: 2031 332e 3632 3732 7320 2863 7075 293b   13.6272s (cpu);
│ │ │ │ +000100d0: 2037 2e38 3033 3537 7320 2874 6872 6561   7.80357s (threa
│ │ │ │ +000100e0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  000100f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010150: 2020 2020 207c 0a7c 2020 2020 202f 2020       |.|     /  
│ │ │ │ @@ -4642,17 +4642,17 @@
│ │ │ │  00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012230: 2d2d 2d2b 0a7c 6932 3020 3a20 4646 203d  ---+.|i20 : FF =
│ │ │ │  00012240: 2074 696d 6520 5368 616d 6173 6828 5231   time Shamash(R1
│ │ │ │  00012250: 2c46 2c34 2920 2020 2020 2020 2020 2020  ,F,4)           
│ │ │ │  00012260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012270: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00012280: 2030 2e31 3539 3736 3873 2028 6370 7529   0.159768s (cpu)
│ │ │ │ -00012290: 3b20 302e 3038 3830 3735 3673 2028 7468  ; 0.0880756s (th
│ │ │ │ -000122a0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00012280: 2030 2e31 3633 3833 3673 2028 6370 7529   0.163836s (cpu)
│ │ │ │ +00012290: 3b20 302e 3131 3339 3338 7320 2874 6872  ; 0.113938s (thr
│ │ │ │ +000122a0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  000122b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000122c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000122d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000122e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000122f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00012300: 2020 2020 3120 2020 2020 2020 3620 2020      1       6   
│ │ │ │  00012310: 2020 2020 3138 2020 2020 2020 2033 3820      18       38 
│ │ │ │ @@ -4683,17 +4683,17 @@
│ │ │ │  000124a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000124b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000124c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20  -------+.|i21 : 
│ │ │ │  000124d0: 4747 203d 2074 696d 6520 4569 7365 6e62  GG = time Eisenb
│ │ │ │  000124e0: 7564 5368 616d 6173 6828 6666 2c46 2c34  udShamash(ff,F,4
│ │ │ │  000124f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00012500: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00012510: 7573 6564 2031 2e35 3936 3834 7320 2863  used 1.59684s (c
│ │ │ │ -00012520: 7075 293b 2030 2e38 3333 3939 3773 2028  pu); 0.833997s (
│ │ │ │ -00012530: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +00012510: 7573 6564 2032 2e31 3031 3239 7320 2863  used 2.10129s (c
│ │ │ │ +00012520: 7075 293b 2031 2e32 3838 3873 2028 7468  pu); 1.2888s (th
│ │ │ │ +00012530: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00012540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00012550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00012590: 2020 2020 2020 2f20 525c 3120 2020 2020        / R\1     
│ │ │ │  000125a0: 2f20 525c 3620 2020 2020 2f20 525c 3138  / R\6     / R\18
│ │ │ │ @@ -4736,23612 +4736,23611 @@
│ │ │ │  000127f0: 6c73 6f20 6465 616c 7320 636f 7272 6563  lso deals correc
│ │ │ │  00012800: 746c 7920 7769 7468 2063 6f6d 706c 6578  tly with complex
│ │ │ │  00012810: 6573 2046 2077 6865 7265 206d 696e 2046  es F where min F
│ │ │ │  00012820: 2069 7320 6e6f 7420 303a 0a0a 2b2d 2d2d   is not 0:..+---
│ │ │ │  00012830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012860: 2d2b 0a7c 6932 3220 3a20 4747 203d 2074  -+.|i22 : GG = t
│ │ │ │ -00012870: 696d 6520 4569 7365 6e62 7564 5368 616d  ime EisenbudSham
│ │ │ │ -00012880: 6173 6828 5231 2c46 5b32 5d2c 3429 2020  ash(R1,F[2],4)  
│ │ │ │ -00012890: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000128a0: 7365 6420 312e 3536 3239 3473 2028 6370  sed 1.56294s (cp
│ │ │ │ -000128b0: 7529 3b20 302e 3737 3939 3233 7320 2874  u); 0.779923s (t
│ │ │ │ -000128c0: 6872 6561 6429 3b20 3073 2028 6763 297c  hread); 0s (gc)|
│ │ │ │ -000128d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012860: 2b0a 7c69 3232 203a 2047 4720 3d20 7469  +.|i22 : GG = ti
│ │ │ │ +00012870: 6d65 2045 6973 656e 6275 6453 6861 6d61  me EisenbudShama
│ │ │ │ +00012880: 7368 2852 312c 465b 325d 2c34 2920 2020  sh(R1,F[2],4)   
│ │ │ │ +00012890: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +000128a0: 6420 312e 3936 3433 3173 2028 6370 7529  d 1.96431s (cpu)
│ │ │ │ +000128b0: 3b20 312e 3231 3035 3473 2028 7468 7265  ; 1.21054s (thre
│ │ │ │ +000128c0: 6164 293b 2030 7320 2867 6329 7c0a 7c20  ad); 0s (gc)|.| 
│ │ │ │ +000128d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012900: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00012910: 2031 2020 2020 2020 2036 2020 2020 2020   1       6      
│ │ │ │ -00012920: 2031 3820 2020 2020 2020 3338 2020 2020   18       38    
│ │ │ │ -00012930: 2020 2036 3620 2020 2020 2020 207c 0a7c     66        |.|
│ │ │ │ -00012940: 6f32 3220 3d20 5231 2020 3c2d 2d20 5231  o22 = R1  <-- R1
│ │ │ │ -00012950: 2020 3c2d 2d20 5231 2020 203c 2d2d 2052    <-- R1   <-- R
│ │ │ │ -00012960: 3120 2020 3c2d 2d20 5231 2020 2020 2020  1   <-- R1      
│ │ │ │ -00012970: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00012900: 2020 7c0a 7c20 2020 2020 2020 2031 2020    |.|        1  
│ │ │ │ +00012910: 2020 2020 2036 2020 2020 2020 2031 3820       6       18 
│ │ │ │ +00012920: 2020 2020 2020 3338 2020 2020 2020 2036        38       6
│ │ │ │ +00012930: 3620 2020 2020 2020 7c0a 7c6f 3232 203d  6       |.|o22 =
│ │ │ │ +00012940: 2052 3120 203c 2d2d 2052 3120 203c 2d2d   R1  <-- R1  <--
│ │ │ │ +00012950: 2052 3120 2020 3c2d 2d20 5231 2020 203c   R1   <-- R1   <
│ │ │ │ +00012960: 2d2d 2052 3120 2020 2020 2020 2020 7c0a  -- R1         |.
│ │ │ │ +00012970: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00012980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000129a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000129b0: 2020 2020 2d32 2020 2020 2020 2d31 2020      -2      -1  
│ │ │ │ -000129c0: 2020 2020 3020 2020 2020 2020 2031 2020      0        1  
│ │ │ │ -000129d0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000129e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000129a0: 2020 2020 7c0a 7c20 2020 2020 202d 3220      |.|      -2 
│ │ │ │ +000129b0: 2020 2020 202d 3120 2020 2020 2030 2020       -1      0  
│ │ │ │ +000129c0: 2020 2020 2020 3120 2020 2020 2020 2032        1        2
│ │ │ │ +000129d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000129e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a10: 2020 2020 2020 2020 207c 0a7c 6f32 3220           |.|o22 
│ │ │ │ -00012a20: 3a20 4368 6169 6e43 6f6d 706c 6578 2020  : ChainComplex  
│ │ │ │ +00012a10: 7c0a 7c6f 3232 203a 2043 6861 696e 436f  |.|o22 : ChainCo
│ │ │ │ +00012a20: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │  00012a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00012a40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00012a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012a80: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -00012a90: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -00012aa0: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74   *note makeHomot
│ │ │ │ -00012ab0: 6f70 6965 733a 206d 616b 6548 6f6d 6f74  opies: makeHomot
│ │ │ │ -00012ac0: 6f70 6965 732c 202d 2d20 7265 7475 726e  opies, -- return
│ │ │ │ -00012ad0: 7320 6120 7379 7374 656d 206f 6620 6869  s a system of hi
│ │ │ │ -00012ae0: 6768 6572 0a20 2020 2068 6f6d 6f74 6f70  gher.    homotop
│ │ │ │ -00012af0: 6965 730a 2020 2a20 2a6e 6f74 6520 5368  ies.  * *note Sh
│ │ │ │ -00012b00: 616d 6173 683a 2053 6861 6d61 7368 2c20  amash: Shamash, 
│ │ │ │ -00012b10: 2d2d 2043 6f6d 7075 7465 7320 7468 6520  -- Computes the 
│ │ │ │ -00012b20: 5368 616d 6173 6820 436f 6d70 6c65 780a  Shamash Complex.
│ │ │ │ -00012b30: 2020 2a20 2a6e 6f74 6520 6578 706f 3a20    * *note expo: 
│ │ │ │ -00012b40: 6578 706f 2c20 2d2d 2072 6574 7572 6e73  expo, -- returns
│ │ │ │ -00012b50: 2061 2073 6574 2063 6f72 7265 7370 6f6e   a set correspon
│ │ │ │ -00012b60: 6469 6e67 2074 6f20 7468 6520 6261 7369  ding to the basi
│ │ │ │ -00012b70: 7320 6f66 2061 2064 6976 6964 6564 0a20  s of a divided. 
│ │ │ │ -00012b80: 2020 2070 6f77 6572 0a0a 5761 7973 2074     power..Ways t
│ │ │ │ -00012b90: 6f20 7573 6520 4569 7365 6e62 7564 5368  o use EisenbudSh
│ │ │ │ -00012ba0: 616d 6173 683a 0a3d 3d3d 3d3d 3d3d 3d3d  amash:.=========
│ │ │ │ -00012bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00012bc0: 3d3d 3d0a 0a20 202a 2022 4569 7365 6e62  ===..  * "Eisenb
│ │ │ │ -00012bd0: 7564 5368 616d 6173 6828 4d61 7472 6978  udShamash(Matrix
│ │ │ │ -00012be0: 2c43 6861 696e 436f 6d70 6c65 782c 5a5a  ,ChainComplex,ZZ
│ │ │ │ -00012bf0: 2922 0a20 202a 2022 4569 7365 6e62 7564  )".  * "Eisenbud
│ │ │ │ -00012c00: 5368 616d 6173 6828 5269 6e67 2c43 6861  Shamash(Ring,Cha
│ │ │ │ -00012c10: 696e 436f 6d70 6c65 782c 5a5a 2922 0a0a  inComplex,ZZ)"..
│ │ │ │ -00012c20: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00012c30: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00012c40: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00012c50: 7420 2a6e 6f74 6520 4569 7365 6e62 7564  t *note Eisenbud
│ │ │ │ -00012c60: 5368 616d 6173 683a 2045 6973 656e 6275  Shamash: Eisenbu
│ │ │ │ -00012c70: 6453 6861 6d61 7368 2c20 6973 2061 202a  dShamash, is a *
│ │ │ │ -00012c80: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -00012c90: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ -00012ca0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -00012cb0: 6f6e 2c2e 0a1f 0a46 696c 653a 2043 6f6d  on,....File: Com
│ │ │ │ -00012cc0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00012cd0: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ -00012ce0: 6f2c 204e 6f64 653a 2045 6973 656e 6275  o, Node: Eisenbu
│ │ │ │ -00012cf0: 6453 6861 6d61 7368 546f 7461 6c2c 204e  dShamashTotal, N
│ │ │ │ -00012d00: 6578 743a 2065 7665 6e45 7874 4d6f 6475  ext: evenExtModu
│ │ │ │ -00012d10: 6c65 2c20 5072 6576 3a20 4569 7365 6e62  le, Prev: Eisenb
│ │ │ │ -00012d20: 7564 5368 616d 6173 682c 2055 703a 2054  udShamash, Up: T
│ │ │ │ -00012d30: 6f70 0a0a 4569 7365 6e62 7564 5368 616d  op..EisenbudSham
│ │ │ │ -00012d40: 6173 6854 6f74 616c 202d 2d20 5072 6563  ashTotal -- Prec
│ │ │ │ -00012d50: 7572 736f 7220 636f 6d70 6c65 7820 6f66  ursor complex of
│ │ │ │ -00012d60: 2074 6f74 616c 2045 7874 0a2a 2a2a 2a2a   total Ext.*****
│ │ │ │ +00012a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00012a80: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00012a90: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 6b65  ..  * *note make
│ │ │ │ +00012aa0: 486f 6d6f 746f 7069 6573 3a20 6d61 6b65  Homotopies: make
│ │ │ │ +00012ab0: 486f 6d6f 746f 7069 6573 2c20 2d2d 2072  Homotopies, -- r
│ │ │ │ +00012ac0: 6574 7572 6e73 2061 2073 7973 7465 6d20  eturns a system 
│ │ │ │ +00012ad0: 6f66 2068 6967 6865 720a 2020 2020 686f  of higher.    ho
│ │ │ │ +00012ae0: 6d6f 746f 7069 6573 0a20 202a 202a 6e6f  motopies.  * *no
│ │ │ │ +00012af0: 7465 2053 6861 6d61 7368 3a20 5368 616d  te Shamash: Sham
│ │ │ │ +00012b00: 6173 682c 202d 2d20 436f 6d70 7574 6573  ash, -- Computes
│ │ │ │ +00012b10: 2074 6865 2053 6861 6d61 7368 2043 6f6d   the Shamash Com
│ │ │ │ +00012b20: 706c 6578 0a20 202a 202a 6e6f 7465 2065  plex.  * *note e
│ │ │ │ +00012b30: 7870 6f3a 2065 7870 6f2c 202d 2d20 7265  xpo: expo, -- re
│ │ │ │ +00012b40: 7475 726e 7320 6120 7365 7420 636f 7272  turns a set corr
│ │ │ │ +00012b50: 6573 706f 6e64 696e 6720 746f 2074 6865  esponding to the
│ │ │ │ +00012b60: 2062 6173 6973 206f 6620 6120 6469 7669   basis of a divi
│ │ │ │ +00012b70: 6465 640a 2020 2020 706f 7765 720a 0a57  ded.    power..W
│ │ │ │ +00012b80: 6179 7320 746f 2075 7365 2045 6973 656e  ays to use Eisen
│ │ │ │ +00012b90: 6275 6453 6861 6d61 7368 3a0a 3d3d 3d3d  budShamash:.====
│ │ │ │ +00012ba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00012bb0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245  ========..  * "E
│ │ │ │ +00012bc0: 6973 656e 6275 6453 6861 6d61 7368 284d  isenbudShamash(M
│ │ │ │ +00012bd0: 6174 7269 782c 4368 6169 6e43 6f6d 706c  atrix,ChainCompl
│ │ │ │ +00012be0: 6578 2c5a 5a29 220a 2020 2a20 2245 6973  ex,ZZ)".  * "Eis
│ │ │ │ +00012bf0: 656e 6275 6453 6861 6d61 7368 2852 696e  enbudShamash(Rin
│ │ │ │ +00012c00: 672c 4368 6169 6e43 6f6d 706c 6578 2c5a  g,ChainComplex,Z
│ │ │ │ +00012c10: 5a29 220a 0a46 6f72 2074 6865 2070 726f  Z)"..For the pro
│ │ │ │ +00012c20: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +00012c30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00012c40: 6f62 6a65 6374 202a 6e6f 7465 2045 6973  object *note Eis
│ │ │ │ +00012c50: 656e 6275 6453 6861 6d61 7368 3a20 4569  enbudShamash: Ei
│ │ │ │ +00012c60: 7365 6e62 7564 5368 616d 6173 682c 2069  senbudShamash, i
│ │ │ │ +00012c70: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +00012c80: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +00012c90: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +00012ca0: 756e 6374 696f 6e2c 2e0a 1f0a 4669 6c65  unction,....File
│ │ │ │ +00012cb0: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ +00012cc0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00012cd0: 732e 696e 666f 2c20 4e6f 6465 3a20 4569  s.info, Node: Ei
│ │ │ │ +00012ce0: 7365 6e62 7564 5368 616d 6173 6854 6f74  senbudShamashTot
│ │ │ │ +00012cf0: 616c 2c20 4e65 7874 3a20 6576 656e 4578  al, Next: evenEx
│ │ │ │ +00012d00: 744d 6f64 756c 652c 2050 7265 763a 2045  tModule, Prev: E
│ │ │ │ +00012d10: 6973 656e 6275 6453 6861 6d61 7368 2c20  isenbudShamash, 
│ │ │ │ +00012d20: 5570 3a20 546f 700a 0a45 6973 656e 6275  Up: Top..Eisenbu
│ │ │ │ +00012d30: 6453 6861 6d61 7368 546f 7461 6c20 2d2d  dShamashTotal --
│ │ │ │ +00012d40: 2050 7265 6375 7273 6f72 2063 6f6d 706c   Precursor compl
│ │ │ │ +00012d50: 6578 206f 6620 746f 7461 6c20 4578 740a  ex of total Ext.
│ │ │ │ +00012d60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00012d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00012d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00012da0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -00012db0: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -00012dc0: 200a 2020 2020 2020 2020 2864 302c 6431   .        (d0,d1
│ │ │ │ -00012dd0: 2920 3d20 2045 6973 656e 6275 6453 6861  ) =  EisenbudSha
│ │ │ │ -00012de0: 6d61 7368 546f 7461 6c20 4d0a 2020 2a20  mashTotal M.  * 
│ │ │ │ -00012df0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00012e00: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -00012e10: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00012e20: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ -00012e30: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00012e40: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465  ection.  * *note
│ │ │ │ -00012e50: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ -00012e60: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00012e70: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ -00012e80: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ -00012e90: 7075 7473 2c3a 0a20 2020 2020 202a 2043  puts,:.      * C
│ │ │ │ -00012ea0: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566  heck => ..., def
│ │ │ │ -00012eb0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00012ec0: 0a20 2020 2020 202a 2047 7261 6469 6e67  .      * Grading
│ │ │ │ -00012ed0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00012ee0: 2076 616c 7565 2032 0a20 2020 2020 202a   value 2.      *
│ │ │ │ -00012ef0: 2056 6172 6961 626c 6573 203d 3e20 2e2e   Variables => ..
│ │ │ │ -00012f00: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00012f10: 2073 0a20 202a 204f 7574 7075 7473 3a0a   s.  * Outputs:.
│ │ │ │ -00012f20: 2020 2020 2020 2a20 6430 2c20 6120 2a6e        * d0, a *n
│ │ │ │ -00012f30: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -00012f40: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -00012f50: 2c2c 206d 6170 206f 6620 6672 6565 206d  ,, map of free m
│ │ │ │ -00012f60: 6f64 756c 6573 206f 7665 7220 616e 0a20  odules over an. 
│ │ │ │ -00012f70: 2020 2020 2020 2065 6e6c 6172 6765 6420         enlarged 
│ │ │ │ -00012f80: 7269 6e67 0a20 2020 2020 202a 2064 312c  ring.      * d1,
│ │ │ │ -00012f90: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ -00012fa0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -00012fb0: 6174 7269 782c 2c20 6d61 7020 6f66 2066  atrix,, map of f
│ │ │ │ -00012fc0: 7265 6520 6d6f 6475 6c65 7320 6f76 6572  ree modules over
│ │ │ │ -00012fd0: 2061 6e0a 2020 2020 2020 2020 656e 6c61   an.        enla
│ │ │ │ -00012fe0: 7267 6564 2072 696e 670a 0a44 6573 6372  rged ring..Descr
│ │ │ │ -00012ff0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00013000: 3d3d 0a0a 4173 7375 6d65 2074 6861 7420  ==..Assume that 
│ │ │ │ -00013010: 4d20 6973 2064 6566 696e 6564 206f 7665  M is defined ove
│ │ │ │ -00013020: 7220 6120 7269 6e67 206f 6620 7468 6520  r a ring of the 
│ │ │ │ -00013030: 666f 726d 2052 6261 7220 3d20 522f 2866  form Rbar = R/(f
│ │ │ │ -00013040: 5f30 2e2e 665f 7b63 2d31 7d29 2c20 610a  _0..f_{c-1}), a.
│ │ │ │ -00013050: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ -00013060: 6374 696f 6e2c 2061 6e64 2074 6861 7420  ction, and that 
│ │ │ │ -00013070: 4d20 6861 7320 6120 6669 6e69 7465 2066  M has a finite f
│ │ │ │ -00013080: 7265 6520 7265 736f 6c75 7469 6f6e 2047  ree resolution G
│ │ │ │ -00013090: 206f 7665 7220 522e 2049 6e0a 7468 6973   over R. In.this
│ │ │ │ -000130a0: 2063 6173 6520 4d20 6861 7320 6120 6672   case M has a fr
│ │ │ │ -000130b0: 6565 2072 6573 6f6c 7574 696f 6e20 4620  ee resolution F 
│ │ │ │ -000130c0: 6f76 6572 2052 6261 7220 7768 6f73 6520  over Rbar whose 
│ │ │ │ -000130d0: 6475 616c 2c20 465e 2a20 6973 2061 2066  dual, F^* is a f
│ │ │ │ -000130e0: 696e 6974 656c 790a 6765 6e65 7261 7465  initely.generate
│ │ │ │ -000130f0: 642c 205a 2d67 7261 6465 6420 6672 6565  d, Z-graded free
│ │ │ │ -00013100: 206d 6f64 756c 6520 6f76 6572 2061 2072   module over a r
│ │ │ │ -00013110: 696e 6720 5362 6172 5c63 6f6e 6720 6b6b  ing Sbar\cong kk
│ │ │ │ -00013120: 5b73 5f30 2e2e 735f 7b63 2d31 7d2c 6765  [s_0..s_{c-1},ge
│ │ │ │ -00013130: 6e73 0a52 6261 725d 2c20 7768 6572 6520  ns.Rbar], where 
│ │ │ │ -00013140: 7468 6520 6465 6772 6565 7320 6f66 2074  the degrees of t
│ │ │ │ -00013150: 6865 2073 5f69 2061 7265 207b 2d32 2c20  he s_i are {-2, 
│ │ │ │ -00013160: 2d64 6567 7265 6520 665f 697d 2e20 5468  -degree f_i}. Th
│ │ │ │ -00013170: 6973 2072 6573 6f6c 7574 696f 6e20 6973  is resolution is
│ │ │ │ -00013180: 0a69 7320 636f 6e73 7472 7563 7465 6420  .is constructed 
│ │ │ │ -00013190: 6672 6f6d 2074 6865 2064 7561 6c20 6f66  from the dual of
│ │ │ │ -000131a0: 2047 2c20 746f 6765 7468 6572 2077 6974   G, together wit
│ │ │ │ -000131b0: 6820 7468 6520 6475 616c 7320 6f66 2074  h the duals of t
│ │ │ │ -000131c0: 6865 2068 6967 6865 720a 686f 6d6f 746f  he higher.homoto
│ │ │ │ -000131d0: 7069 6573 206f 6e20 4720 6465 6669 6e65  pies on G define
│ │ │ │ -000131e0: 6420 6279 2045 6973 656e 6275 642e 0a0a  d by Eisenbud...
│ │ │ │ -000131f0: 5468 6520 6675 6e63 7469 6f6e 2072 6574  The function ret
│ │ │ │ -00013200: 7572 6e73 2074 6865 2064 6966 6665 7265  urns the differe
│ │ │ │ -00013210: 6e74 6961 6c73 2064 303a 465e 2a5f 7b65  ntials d0:F^*_{e
│ │ │ │ -00013220: 7665 6e7d 205c 746f 2046 5e2a 5f7b 6f64  ven} \to F^*_{od
│ │ │ │ -00013230: 647d 2061 6e64 0a64 313a 465e 2a5f 7b6f  d} and.d1:F^*_{o
│ │ │ │ -00013240: 6464 7d5c 746f 2046 5e2a 5f7b 6576 656e  dd}\to F^*_{even
│ │ │ │ -00013250: 7d2e 0a0a 5468 6520 6d61 7073 2064 302c  }...The maps d0,
│ │ │ │ -00013260: 6431 2066 6f72 6d20 6120 6d61 7472 6978  d1 form a matrix
│ │ │ │ -00013270: 2066 6163 746f 7269 7a61 7469 6f6e 206f   factorization o
│ │ │ │ -00013280: 6620 7375 6d28 632c 2069 2d3e 735f 692a  f sum(c, i->s_i*
│ │ │ │ -00013290: 665f 6929 2e20 5468 6520 6861 7665 2074  f_i). The have t
│ │ │ │ -000132a0: 6865 0a70 726f 7065 7274 7920 7468 6174  he.property that
│ │ │ │ -000132b0: 2066 6f72 2061 6e79 2052 6261 7220 6d6f   for any Rbar mo
│ │ │ │ -000132c0: 6475 6c65 204e 2c0a 0a48 485f 3120 6368  dule N,..HH_1 ch
│ │ │ │ -000132d0: 6169 6e43 6f6d 706c 6578 205c 7b64 302a  ainComplex \{d0*
│ │ │ │ -000132e0: 2a4e 2c20 6431 2a2a 4e5c 7d20 3d20 4578  *N, d1**N\} = Ex
│ │ │ │ -000132f0: 745e 7b65 7665 6e7d 5f7b 5262 6172 7d28  t^{even}_{Rbar}(
│ │ │ │ -00013300: 4d2c 4e29 0a0a 535e 7b7b 312c 307d 7d2a  M,N)..S^{{1,0}}*
│ │ │ │ -00013310: 2a48 485f 3120 6368 6169 6e43 6f6d 706c  *HH_1 chainCompl
│ │ │ │ -00013320: 6578 205c 7b53 5e7b 7b2d 322c 307d 7d2a  ex \{S^{{-2,0}}*
│ │ │ │ -00013330: 2a64 312a 2a4e 2c20 6430 2a2a 4e5c 7d20  *d1**N, d0**N\} 
│ │ │ │ -00013340: 3d0a 4578 745e 7b6f 6464 7d5f 7b52 6261  =.Ext^{odd}_{Rba
│ │ │ │ -00013350: 727d 284d 2c4e 290a 0a54 6869 7320 6973  r}(M,N)..This is
│ │ │ │ -00013360: 2065 6e63 6f64 6564 2069 6e20 7468 6520   encoded in the 
│ │ │ │ -00013370: 7363 7269 7074 206e 6577 4578 740a 0a4f  script newExt..O
│ │ │ │ -00013380: 7074 696f 6e20 6465 6661 756c 7473 3a20  ption defaults: 
│ │ │ │ -00013390: 4368 6563 6b3d 3e66 616c 7365 2056 6172  Check=>false Var
│ │ │ │ -000133a0: 6961 626c 6573 3d3e 6765 7453 796d 626f  iables=>getSymbo
│ │ │ │ -000133b0: 6c20 2273 222c 2047 7261 6469 6e67 203d  l "s", Grading =
│ │ │ │ -000133c0: 3e32 7d0a 0a49 6620 4772 6164 696e 6720  >2}..If Grading 
│ │ │ │ -000133d0: 3d3e 312c 2074 6865 6e20 6120 7369 6e67  =>1, then a sing
│ │ │ │ -000133e0: 6c79 2067 7261 6465 6420 7265 7375 6c74  ly graded result
│ │ │ │ -000133f0: 2069 7320 7265 7475 726e 6564 2028 6a75   is returned (ju
│ │ │ │ -00013400: 7374 2066 6f72 6765 7474 696e 6720 7468  st forgetting th
│ │ │ │ -00013410: 650a 686f 6d6f 6c6f 6769 6361 6c20 6772  e.homological gr
│ │ │ │ -00013420: 6164 696e 672e 290a 0a0a 0a2b 2d2d 2d2d  ading.)....+----
│ │ │ │ +00012d90: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00012da0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +00012db0: 7361 6765 3a20 0a20 2020 2020 2020 2028  sage: .        (
│ │ │ │ +00012dc0: 6430 2c64 3129 203d 2020 4569 7365 6e62  d0,d1) =  Eisenb
│ │ │ │ +00012dd0: 7564 5368 616d 6173 6854 6f74 616c 204d  udShamashTotal M
│ │ │ │ +00012de0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00012df0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +00012e00: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +00012e10: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f  y2Doc)Module,, o
│ │ │ │ +00012e20: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ +00012e30: 6e74 6572 7365 6374 696f 6e0a 2020 2a20  ntersection.  * 
│ │ │ │ +00012e40: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ +00012e50: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ +00012e60: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ +00012e70: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00012e80: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ +00012e90: 2020 2a20 4368 6563 6b20 3d3e 202e 2e2e    * Check => ...
│ │ │ │ +00012ea0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00012eb0: 6661 6c73 650a 2020 2020 2020 2a20 4772  false.      * Gr
│ │ │ │ +00012ec0: 6164 696e 6720 3d3e 202e 2e2e 2c20 6465  ading => ..., de
│ │ │ │ +00012ed0: 6661 756c 7420 7661 6c75 6520 320a 2020  fault value 2.  
│ │ │ │ +00012ee0: 2020 2020 2a20 5661 7269 6162 6c65 7320      * Variables 
│ │ │ │ +00012ef0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00012f00: 7661 6c75 6520 730a 2020 2a20 4f75 7470  value s.  * Outp
│ │ │ │ +00012f10: 7574 733a 0a20 2020 2020 202a 2064 302c  uts:.      * d0,
│ │ │ │ +00012f20: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +00012f30: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00012f40: 6174 7269 782c 2c20 6d61 7020 6f66 2066  atrix,, map of f
│ │ │ │ +00012f50: 7265 6520 6d6f 6475 6c65 7320 6f76 6572  ree modules over
│ │ │ │ +00012f60: 2061 6e0a 2020 2020 2020 2020 656e 6c61   an.        enla
│ │ │ │ +00012f70: 7267 6564 2072 696e 670a 2020 2020 2020  rged ring.      
│ │ │ │ +00012f80: 2a20 6431 2c20 6120 2a6e 6f74 6520 6d61  * d1, a *note ma
│ │ │ │ +00012f90: 7472 6978 3a20 284d 6163 6175 6c61 7932  trix: (Macaulay2
│ │ │ │ +00012fa0: 446f 6329 4d61 7472 6978 2c2c 206d 6170  Doc)Matrix,, map
│ │ │ │ +00012fb0: 206f 6620 6672 6565 206d 6f64 756c 6573   of free modules
│ │ │ │ +00012fc0: 206f 7665 7220 616e 0a20 2020 2020 2020   over an.       
│ │ │ │ +00012fd0: 2065 6e6c 6172 6765 6420 7269 6e67 0a0a   enlarged ring..
│ │ │ │ +00012fe0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00012ff0: 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d 6520  =======..Assume 
│ │ │ │ +00013000: 7468 6174 204d 2069 7320 6465 6669 6e65  that M is define
│ │ │ │ +00013010: 6420 6f76 6572 2061 2072 696e 6720 6f66  d over a ring of
│ │ │ │ +00013020: 2074 6865 2066 6f72 6d20 5262 6172 203d   the form Rbar =
│ │ │ │ +00013030: 2052 2f28 665f 302e 2e66 5f7b 632d 317d   R/(f_0..f_{c-1}
│ │ │ │ +00013040: 292c 2061 0a63 6f6d 706c 6574 6520 696e  ), a.complete in
│ │ │ │ +00013050: 7465 7273 6563 7469 6f6e 2c20 616e 6420  tersection, and 
│ │ │ │ +00013060: 7468 6174 204d 2068 6173 2061 2066 696e  that M has a fin
│ │ │ │ +00013070: 6974 6520 6672 6565 2072 6573 6f6c 7574  ite free resolut
│ │ │ │ +00013080: 696f 6e20 4720 6f76 6572 2052 2e20 496e  ion G over R. In
│ │ │ │ +00013090: 0a74 6869 7320 6361 7365 204d 2068 6173  .this case M has
│ │ │ │ +000130a0: 2061 2066 7265 6520 7265 736f 6c75 7469   a free resoluti
│ │ │ │ +000130b0: 6f6e 2046 206f 7665 7220 5262 6172 2077  on F over Rbar w
│ │ │ │ +000130c0: 686f 7365 2064 7561 6c2c 2046 5e2a 2069  hose dual, F^* i
│ │ │ │ +000130d0: 7320 6120 6669 6e69 7465 6c79 0a67 656e  s a finitely.gen
│ │ │ │ +000130e0: 6572 6174 6564 2c20 5a2d 6772 6164 6564  erated, Z-graded
│ │ │ │ +000130f0: 2066 7265 6520 6d6f 6475 6c65 206f 7665   free module ove
│ │ │ │ +00013100: 7220 6120 7269 6e67 2053 6261 725c 636f  r a ring Sbar\co
│ │ │ │ +00013110: 6e67 206b 6b5b 735f 302e 2e73 5f7b 632d  ng kk[s_0..s_{c-
│ │ │ │ +00013120: 317d 2c67 656e 730a 5262 6172 5d2c 2077  1},gens.Rbar], w
│ │ │ │ +00013130: 6865 7265 2074 6865 2064 6567 7265 6573  here the degrees
│ │ │ │ +00013140: 206f 6620 7468 6520 735f 6920 6172 6520   of the s_i are 
│ │ │ │ +00013150: 7b2d 322c 202d 6465 6772 6565 2066 5f69  {-2, -degree f_i
│ │ │ │ +00013160: 7d2e 2054 6869 7320 7265 736f 6c75 7469  }. This resoluti
│ │ │ │ +00013170: 6f6e 2069 730a 6973 2063 6f6e 7374 7275  on is.is constru
│ │ │ │ +00013180: 6374 6564 2066 726f 6d20 7468 6520 6475  cted from the du
│ │ │ │ +00013190: 616c 206f 6620 472c 2074 6f67 6574 6865  al of G, togethe
│ │ │ │ +000131a0: 7220 7769 7468 2074 6865 2064 7561 6c73  r with the duals
│ │ │ │ +000131b0: 206f 6620 7468 6520 6869 6768 6572 0a68   of the higher.h
│ │ │ │ +000131c0: 6f6d 6f74 6f70 6965 7320 6f6e 2047 2064  omotopies on G d
│ │ │ │ +000131d0: 6566 696e 6564 2062 7920 4569 7365 6e62  efined by Eisenb
│ │ │ │ +000131e0: 7564 2e0a 0a54 6865 2066 756e 6374 696f  ud...The functio
│ │ │ │ +000131f0: 6e20 7265 7475 726e 7320 7468 6520 6469  n returns the di
│ │ │ │ +00013200: 6666 6572 656e 7469 616c 7320 6430 3a46  fferentials d0:F
│ │ │ │ +00013210: 5e2a 5f7b 6576 656e 7d20 5c74 6f20 465e  ^*_{even} \to F^
│ │ │ │ +00013220: 2a5f 7b6f 6464 7d20 616e 640a 6431 3a46  *_{odd} and.d1:F
│ │ │ │ +00013230: 5e2a 5f7b 6f64 647d 5c74 6f20 465e 2a5f  ^*_{odd}\to F^*_
│ │ │ │ +00013240: 7b65 7665 6e7d 2e0a 0a54 6865 206d 6170  {even}...The map
│ │ │ │ +00013250: 7320 6430 2c64 3120 666f 726d 2061 206d  s d0,d1 form a m
│ │ │ │ +00013260: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ +00013270: 696f 6e20 6f66 2073 756d 2863 2c20 692d  ion of sum(c, i-
│ │ │ │ +00013280: 3e73 5f69 2a66 5f69 292e 2054 6865 2068  >s_i*f_i). The h
│ │ │ │ +00013290: 6176 6520 7468 650a 7072 6f70 6572 7479  ave the.property
│ │ │ │ +000132a0: 2074 6861 7420 666f 7220 616e 7920 5262   that for any Rb
│ │ │ │ +000132b0: 6172 206d 6f64 756c 6520 4e2c 0a0a 4848  ar module N,..HH
│ │ │ │ +000132c0: 5f31 2063 6861 696e 436f 6d70 6c65 7820  _1 chainComplex 
│ │ │ │ +000132d0: 5c7b 6430 2a2a 4e2c 2064 312a 2a4e 5c7d  \{d0**N, d1**N\}
│ │ │ │ +000132e0: 203d 2045 7874 5e7b 6576 656e 7d5f 7b52   = Ext^{even}_{R
│ │ │ │ +000132f0: 6261 727d 284d 2c4e 290a 0a53 5e7b 7b31  bar}(M,N)..S^{{1
│ │ │ │ +00013300: 2c30 7d7d 2a2a 4848 5f31 2063 6861 696e  ,0}}**HH_1 chain
│ │ │ │ +00013310: 436f 6d70 6c65 7820 5c7b 535e 7b7b 2d32  Complex \{S^{{-2
│ │ │ │ +00013320: 2c30 7d7d 2a2a 6431 2a2a 4e2c 2064 302a  ,0}}**d1**N, d0*
│ │ │ │ +00013330: 2a4e 5c7d 203d 0a45 7874 5e7b 6f64 647d  *N\} =.Ext^{odd}
│ │ │ │ +00013340: 5f7b 5262 6172 7d28 4d2c 4e29 0a0a 5468  _{Rbar}(M,N)..Th
│ │ │ │ +00013350: 6973 2069 7320 656e 636f 6465 6420 696e  is is encoded in
│ │ │ │ +00013360: 2074 6865 2073 6372 6970 7420 6e65 7745   the script newE
│ │ │ │ +00013370: 7874 0a0a 4f70 7469 6f6e 2064 6566 6175  xt..Option defau
│ │ │ │ +00013380: 6c74 733a 2043 6865 636b 3d3e 6661 6c73  lts: Check=>fals
│ │ │ │ +00013390: 6520 5661 7269 6162 6c65 733d 3e67 6574  e Variables=>get
│ │ │ │ +000133a0: 5379 6d62 6f6c 2022 7322 2c20 4772 6164  Symbol "s", Grad
│ │ │ │ +000133b0: 696e 6720 3d3e 327d 0a0a 4966 2047 7261  ing =>2}..If Gra
│ │ │ │ +000133c0: 6469 6e67 203d 3e31 2c20 7468 656e 2061  ding =>1, then a
│ │ │ │ +000133d0: 2073 696e 676c 7920 6772 6164 6564 2072   singly graded r
│ │ │ │ +000133e0: 6573 756c 7420 6973 2072 6574 7572 6e65  esult is returne
│ │ │ │ +000133f0: 6420 286a 7573 7420 666f 7267 6574 7469  d (just forgetti
│ │ │ │ +00013400: 6e67 2074 6865 0a68 6f6d 6f6c 6f67 6963  ng the.homologic
│ │ │ │ +00013410: 616c 2067 7261 6469 6e67 2e29 0a0a 0a0a  al grading.)....
│ │ │ │ +00013420: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013470: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -00013480: 206e 203d 2033 2020 2020 2020 2020 2020   n = 3          
│ │ │ │ +00013460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013470: 7c69 3120 3a20 6e20 3d20 3320 2020 2020  |i1 : n = 3     
│ │ │ │ +00013480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000134b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000134c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000134b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000134c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000134d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134e0: 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                  
│ │ │ │ -00013510: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -00013520: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00013500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013510: 7c6f 3120 3d20 3320 2020 2020 2020 2020  |o1 = 3         
│ │ │ │ +00013520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013560: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00013550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013560: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000135a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000135b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -000135c0: 2063 203d 2032 2020 2020 2020 2020 2020   c = 2          
│ │ │ │ +000135a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000135b0: 7c69 3220 3a20 6320 3d20 3220 2020 2020  |i2 : c = 2     
│ │ │ │ +000135c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000135d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000135e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000135f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013600: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000135f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013600: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013650: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -00013660: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00013640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013650: 7c6f 3220 3d20 3220 2020 2020 2020 2020  |o2 = 2         
│ │ │ │ +00013660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000136a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00013690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000136a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000136b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000136c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000136d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000136e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000136f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -00013700: 206b 6b20 3d20 5a5a 2f31 3031 2020 2020   kk = ZZ/101    
│ │ │ │ +000136e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000136f0: 7c69 3320 3a20 6b6b 203d 205a 5a2f 3130  |i3 : kk = ZZ/10
│ │ │ │ +00013700: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00013710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013740: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013790: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ -000137a0: 206b 6b20 2020 2020 2020 2020 2020 2020   kk             
│ │ │ │ +00013780: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013790: 7c6f 3320 3d20 6b6b 2020 2020 2020 2020  |o3 = kk        
│ │ │ │ +000137a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000137b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000137c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000137d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000137e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000137d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000137e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000137f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013830: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -00013840: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00013820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013830: 7c6f 3320 3a20 5175 6f74 6965 6e74 5269  |o3 : QuotientRi
│ │ │ │ +00013840: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00013850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013880: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00013870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013880: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000138a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000138b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000138c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000138d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -000138e0: 2052 203d 206b 6b5b 785f 302e 2e78 5f28   R = kk[x_0..x_(
│ │ │ │ -000138f0: 6e2d 3129 5d20 2020 2020 2020 2020 2020  n-1)]           
│ │ │ │ +000138c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000138d0: 7c69 3420 3a20 5220 3d20 6b6b 5b78 5f30  |i4 : R = kk[x_0
│ │ │ │ +000138e0: 2e2e 785f 286e 2d31 295d 2020 2020 2020  ..x_(n-1)]      
│ │ │ │ +000138f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013920: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013920: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013970: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -00013980: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00013960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013970: 7c6f 3420 3d20 5220 2020 2020 2020 2020  |o4 = R         
│ │ │ │ +00013980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000139b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000139c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000139b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000139c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000139d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000139f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a10: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -00013a20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ +00013a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013a10: 7c6f 3420 3a20 506f 6c79 6e6f 6d69 616c  |o4 : Polynomial
│ │ │ │ +00013a20: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  00013a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013a60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00013a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013a60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013ab0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -00013ac0: 2049 203d 2069 6465 616c 2878 5f30 5e32   I = ideal(x_0^2
│ │ │ │ -00013ad0: 2c20 785f 325e 3329 2020 2020 2020 2020  , x_2^3)        
│ │ │ │ +00013aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013ab0: 7c69 3520 3a20 4920 3d20 6964 6561 6c28  |i5 : I = ideal(
│ │ │ │ +00013ac0: 785f 305e 322c 2078 5f32 5e33 2920 2020  x_0^2, x_2^3)   
│ │ │ │ +00013ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013af0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013b00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00013b60: 2020 2020 2020 2020 2032 2020 2033 2020           2   3  
│ │ │ │ +00013b40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013b50: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +00013b60: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  00013b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ba0: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ -00013bb0: 2069 6465 616c 2028 7820 2c20 7820 2920   ideal (x , x ) 
│ │ │ │ +00013b90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013ba0: 7c6f 3520 3d20 6964 6561 6c20 2878 202c  |o5 = ideal (x ,
│ │ │ │ +00013bb0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │  00013bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013bf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00013c00: 2020 2020 2020 2020 2030 2020 2032 2020           0   2  
│ │ │ │ +00013be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013bf0: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +00013c00: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00013c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013c30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013c40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013c90: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -00013ca0: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +00013c80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013c90: 7c6f 3520 3a20 4964 6561 6c20 6f66 2052  |o5 : Ideal of R
│ │ │ │ +00013ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ce0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00013cd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013ce0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -00013d40: 2066 6620 3d20 6765 6e73 2049 2020 2020   ff = gens I    
│ │ │ │ +00013d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013d30: 7c69 3620 3a20 6666 203d 2067 656e 7320  |i6 : ff = gens 
│ │ │ │ +00013d40: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │  00013d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013d70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013d80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013dd0: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ -00013de0: 207c 2078 5f30 5e32 2078 5f32 5e33 207c   | x_0^2 x_2^3 |
│ │ │ │ +00013dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013dd0: 7c6f 3620 3d20 7c20 785f 305e 3220 785f  |o6 = | x_0^2 x_
│ │ │ │ +00013de0: 325e 3320 7c20 2020 2020 2020 2020 2020  2^3 |           
│ │ │ │  00013df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013e10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013e20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00013e80: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00013e90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00013e60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013e70: 7c20 2020 2020 2020 2020 2020 2020 3120  |             1 
│ │ │ │ +00013e80: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00013e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ec0: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ -00013ed0: 204d 6174 7269 7820 5220 203c 2d2d 2052   Matrix R  <-- R
│ │ │ │ +00013eb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013ec0: 7c6f 3620 3a20 4d61 7472 6978 2052 2020  |o6 : Matrix R  
│ │ │ │ +00013ed0: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │  00013ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00013f00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013f10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013f60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ -00013f70: 2052 6261 7220 3d20 522f 4920 2020 2020   Rbar = R/I     
│ │ │ │ +00013f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013f60: 7c69 3720 3a20 5262 6172 203d 2052 2f49  |i7 : Rbar = R/I
│ │ │ │ +00013f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013fb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00013fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014000: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ -00014010: 2052 6261 7220 2020 2020 2020 2020 2020   Rbar           
│ │ │ │ +00013ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014000: 7c6f 3720 3d20 5262 6172 2020 2020 2020  |o7 = Rbar      
│ │ │ │ +00014010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00014060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000140a0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -000140b0: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00014090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000140a0: 7c6f 3720 3a20 5175 6f74 6965 6e74 5269  |o7 : QuotientRi
│ │ │ │ +000140b0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  000140c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000140e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000140f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000140e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000140f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00014100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -00014150: 2062 6172 203d 206d 6170 2852 6261 722c   bar = map(Rbar,
│ │ │ │ -00014160: 2052 2920 2020 2020 2020 2020 2020 2020   R)             
│ │ │ │ +00014130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014140: 7c69 3820 3a20 6261 7220 3d20 6d61 7028  |i8 : bar = map(
│ │ │ │ +00014150: 5262 6172 2c20 5229 2020 2020 2020 2020  Rbar, R)        
│ │ │ │ +00014160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014190: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014190: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000141a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000141d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000141e0: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ -000141f0: 206d 6170 2028 5262 6172 2c20 522c 207b   map (Rbar, R, {
│ │ │ │ -00014200: 7820 2c20 7820 2c20 7820 7d29 2020 2020  x , x , x })    
│ │ │ │ +000141d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000141e0: 7c6f 3820 3d20 6d61 7020 2852 6261 722c  |o8 = map (Rbar,
│ │ │ │ +000141f0: 2052 2c20 7b78 202c 2078 202c 2078 207d   R, {x , x , x }
│ │ │ │ +00014200: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00014210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014230: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014250: 2030 2020 2031 2020 2032 2020 2020 2020   0   1   2      
│ │ │ │ +00014220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014230: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014240: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ +00014250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014280: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014280: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00014290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000142c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000142d0: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ -000142e0: 2052 696e 674d 6170 2052 6261 7220 3c2d   RingMap Rbar <-
│ │ │ │ -000142f0: 2d20 5220 2020 2020 2020 2020 2020 2020  - R             
│ │ │ │ +000142c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000142d0: 7c6f 3820 3a20 5269 6e67 4d61 7020 5262  |o8 : RingMap Rb
│ │ │ │ +000142e0: 6172 203c 2d2d 2052 2020 2020 2020 2020  ar <-- R        
│ │ │ │ +000142f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014320: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00014310: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014320: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00014330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014370: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ -00014380: 204d 6261 7220 3d20 7072 756e 6520 636f   Mbar = prune co
│ │ │ │ -00014390: 6b65 7220 7261 6e64 6f6d 2852 6261 725e  ker random(Rbar^
│ │ │ │ -000143a0: 312c 2052 6261 725e 7b2d 327d 2920 2020  1, Rbar^{-2})   
│ │ │ │ -000143b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000143c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014370: 7c69 3920 3a20 4d62 6172 203d 2070 7275  |i9 : Mbar = pru
│ │ │ │ +00014380: 6e65 2063 6f6b 6572 2072 616e 646f 6d28  ne coker random(
│ │ │ │ +00014390: 5262 6172 5e31 2c20 5262 6172 5e7b 2d32  Rbar^1, Rbar^{-2
│ │ │ │ +000143a0: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │ +000143b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000143c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000143d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000143e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000143f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014410: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ -00014420: 2063 6f6b 6572 6e65 6c20 7c20 785f 3078   cokernel | x_0x
│ │ │ │ -00014430: 5f31 2b32 3478 5f31 5e32 2b34 3978 5f30  _1+24x_1^2+49x_0
│ │ │ │ -00014440: 785f 322b 3378 5f31 785f 322b 3578 5f32  x_2+3x_1x_2+5x_2
│ │ │ │ -00014450: 5e32 207c 2020 2020 2020 2020 2020 2020  ^2 |            
│ │ │ │ -00014460: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014410: 7c6f 3920 3d20 636f 6b65 726e 656c 207c  |o9 = cokernel |
│ │ │ │ +00014420: 2078 5f30 785f 312b 3234 785f 315e 322b   x_0x_1+24x_1^2+
│ │ │ │ +00014430: 3439 785f 3078 5f32 2b33 785f 3178 5f32  49x_0x_2+3x_1x_2
│ │ │ │ +00014440: 2b35 785f 325e 3220 7c20 2020 2020 2020  +5x_2^2 |       
│ │ │ │ +00014450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014460: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00014470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000144a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000144b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000144c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144d0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000144d0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  000144e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014500: 2020 2020 2020 2020 207c 0a7c 6f39 203a           |.|o9 :
│ │ │ │ -00014510: 2052 6261 722d 6d6f 6475 6c65 2c20 7175   Rbar-module, qu
│ │ │ │ -00014520: 6f74 6965 6e74 206f 6620 5262 6172 2020  otient of Rbar  
│ │ │ │ +000144f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014500: 7c6f 3920 3a20 5262 6172 2d6d 6f64 756c  |o9 : Rbar-modul
│ │ │ │ +00014510: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +00014520: 6261 7220 2020 2020 2020 2020 2020 2020  bar             
│ │ │ │  00014530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014550: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00014540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014550: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000145a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ -000145b0: 3a20 2864 302c 6431 2920 3d20 4569 7365  : (d0,d1) = Eise
│ │ │ │ -000145c0: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ -000145d0: 284d 6261 722c 4772 6164 696e 6720 3d3e  (Mbar,Grading =>
│ │ │ │ -000145e0: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ -000145f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000145a0: 7c69 3130 203a 2028 6430 2c64 3129 203d  |i10 : (d0,d1) =
│ │ │ │ +000145b0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ +000145c0: 546f 7461 6c28 4d62 6172 2c47 7261 6469  Total(Mbar,Gradi
│ │ │ │ +000145d0: 6e67 203d 3e31 2920 2020 2020 2020 2020  ng =>1)         
│ │ │ │ +000145e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000145f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00014600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014640: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -00014650: 3d20 287b 2d32 7d20 7c20 785f 305e 3220  = ({-2} | x_0^2 
│ │ │ │ +00014630: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014640: 7c6f 3130 203d 2028 7b2d 327d 207c 2078  |o10 = ({-2} | x
│ │ │ │ +00014650: 5f30 5e32 2020 2020 2020 2020 2020 2020  _0^2            
│ │ │ │  00014660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014680: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00014690: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000146a0: 2020 207b 2d32 7d20 7c20 785f 3078 5f31     {-2} | x_0x_1
│ │ │ │ -000146b0: 2b32 3478 5f31 5e32 2b34 3978 5f30 785f  +24x_1^2+49x_0x_
│ │ │ │ -000146c0: 322b 3378 5f31 785f 322b 3578 5f32 5e32  2+3x_1x_2+5x_2^2
│ │ │ │ -000146d0: 2033 3073 5f30 2020 2020 2020 2020 2020   30s_0          
│ │ │ │ -000146e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000146f0: 2020 207b 2d33 7d20 7c20 785f 325e 3320     {-3} | x_2^3 
│ │ │ │ +00014670: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00014680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014690: 7c20 2020 2020 2020 7b2d 327d 207c 2078  |       {-2} | x
│ │ │ │ +000146a0: 5f30 785f 312b 3234 785f 315e 322b 3439  _0x_1+24x_1^2+49
│ │ │ │ +000146b0: 785f 3078 5f32 2b33 785f 3178 5f32 2b35  x_0x_2+3x_1x_2+5
│ │ │ │ +000146c0: 785f 325e 3220 3330 735f 3020 2020 2020  x_2^2 30s_0     
│ │ │ │ +000146d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000146e0: 7c20 2020 2020 2020 7b2d 337d 207c 2078  |       {-3} | x
│ │ │ │ +000146f0: 5f32 5e33 2020 2020 2020 2020 2020 2020  _2^3            
│ │ │ │  00014700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014720: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00014730: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014740: 2020 207b 2d37 7d20 7c20 3020 2020 2020     {-7} | 0     
│ │ │ │ +00014710: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00014720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014730: 7c20 2020 2020 2020 7b2d 377d 207c 2030  |       {-7} | 0
│ │ │ │ +00014740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014770: 2078 5f32 5e33 2020 2020 2020 2020 2020   x_2^3          
│ │ │ │ -00014780: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014790: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +00014760: 2020 2020 2020 785f 325e 3320 2020 2020        x_2^3     
│ │ │ │ +00014770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014780: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00014790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000147a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000147b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000147c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000147d0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -000147e0: 2020 2d73 5f31 2020 2020 2020 2020 2020    -s_1          
│ │ │ │ +000147c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000147d0: 7c20 2020 2020 202d 735f 3120 2020 2020  |      -s_1     
│ │ │ │ +000147e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014800: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00014810: 2020 2020 207c 2c20 7b30 7d20 207c 2020       |, {0}  |  
│ │ │ │ -00014820: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014830: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00014800: 2030 2020 2020 2020 2020 7c2c 207b 307d   0        |, {0}
│ │ │ │ +00014810: 2020 7c20 2020 2020 2020 2020 2020 7c0a    |           |.
│ │ │ │ +00014820: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00014830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014850: 2020 2020 2020 2020 2020 2020 2d73 5f31              -s_1
│ │ │ │ -00014860: 2020 2020 207c 2020 7b2d 347d 207c 2020       |  {-4} |  
│ │ │ │ -00014870: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014880: 2020 735f 3020 2020 2020 2020 2020 2020    s_0           
│ │ │ │ +00014850: 202d 735f 3120 2020 2020 7c20 207b 2d34   -s_1     |  {-4
│ │ │ │ +00014860: 7d20 7c20 2020 2020 2020 2020 2020 7c0a  } |           |.
│ │ │ │ +00014870: 7c20 2020 2020 2073 5f30 2020 2020 2020  |      s_0      
│ │ │ │ +00014880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000148a0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -000148b0: 2020 2020 207c 2020 7b2d 357d 207c 2020       |  {-5} |  
│ │ │ │ -000148c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000148d0: 2020 3337 785f 3078 5f31 2d32 3178 5f31    37x_0x_1-21x_1
│ │ │ │ -000148e0: 5e32 2d35 785f 3078 5f32 2b31 3078 5f31  ^2-5x_0x_2+10x_1
│ │ │ │ -000148f0: 785f 322d 3137 785f 325e 3220 2d33 3778  x_2-17x_2^2 -37x
│ │ │ │ -00014900: 5f30 5e32 207c 2020 7b2d 357d 207c 2020  _0^2 |  {-5} |  
│ │ │ │ -00014910: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014920: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +000148a0: 2030 2020 2020 2020 2020 7c20 207b 2d35   0        |  {-5
│ │ │ │ +000148b0: 7d20 7c20 2020 2020 2020 2020 2020 7c0a  } |           |.
│ │ │ │ +000148c0: 7c20 2020 2020 2033 3778 5f30 785f 312d  |      37x_0x_1-
│ │ │ │ +000148d0: 3231 785f 315e 322d 3578 5f30 785f 322b  21x_1^2-5x_0x_2+
│ │ │ │ +000148e0: 3130 785f 3178 5f32 2d31 3778 5f32 5e32  10x_1x_2-17x_2^2
│ │ │ │ +000148f0: 202d 3337 785f 305e 3220 7c20 207b 2d35   -37x_0^2 |  {-5
│ │ │ │ +00014900: 7d20 7c20 2020 2020 2020 2020 2020 7c0a  } |           |.
│ │ │ │ +00014910: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014960: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -00014970: 2020 735f 3020 2020 2020 2020 2020 2020    s_0           
│ │ │ │ +00014950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00014960: 7c20 2020 2020 2073 5f30 2020 2020 2020  |      s_0      
│ │ │ │ +00014970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014990: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000149c0: 2020 3337 785f 3078 5f31 2d32 3178 5f31    37x_0x_1-21x_1
│ │ │ │ -000149d0: 5e32 2d35 785f 3078 5f32 2b31 3078 5f31  ^2-5x_0x_2+10x_1
│ │ │ │ -000149e0: 785f 322d 3137 785f 325e 3220 2d33 3778  x_2-17x_2^2 -37x
│ │ │ │ -000149f0: 5f30 5e32 2020 2020 2020 2020 2020 2020  _0^2            
│ │ │ │ -00014a00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014a10: 2020 2d78 5f32 5e33 2020 2020 2020 2020    -x_2^3        
│ │ │ │ +00014990: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +000149a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000149b0: 7c20 2020 2020 2033 3778 5f30 785f 312d  |      37x_0x_1-
│ │ │ │ +000149c0: 3231 785f 315e 322d 3578 5f30 785f 322b  21x_1^2-5x_0x_2+
│ │ │ │ +000149d0: 3130 785f 3178 5f32 2d31 3778 5f32 5e32  10x_1x_2-17x_2^2
│ │ │ │ +000149e0: 202d 3337 785f 305e 3220 2020 2020 2020   -37x_0^2       
│ │ │ │ +000149f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014a00: 7c20 2020 2020 202d 785f 325e 3320 2020  |      -x_2^3   
│ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a30: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014a60: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00014a30: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00014a40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014a50: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00014a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a80: 2020 2020 2020 2020 2020 2020 2d78 5f32              -x_2
│ │ │ │ -00014a90: 5e33 2020 2020 2020 2020 2020 2020 2020  ^3              
│ │ │ │ -00014aa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014ab0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +00014a80: 202d 785f 325e 3320 2020 2020 2020 2020   -x_2^3         
│ │ │ │ +00014a90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014aa0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014af0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -00014b00: 2020 735f 3120 2020 2020 2020 2020 2020    s_1           
│ │ │ │ -00014b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b20: 2020 2020 2020 2020 2030 2020 2020 207c           0     |
│ │ │ │ -00014b30: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00014b40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014b50: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00014b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b70: 2020 2020 2020 2020 2073 5f31 2020 207c           s_1   |
│ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014ba0: 2020 785f 305e 3220 2020 2020 2020 2020    x_0^2         
│ │ │ │ -00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bc0: 2020 2020 2020 2020 2030 2020 2020 207c           0     |
│ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014be0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014bf0: 2020 785f 3078 5f31 2b32 3478 5f31 5e32    x_0x_1+24x_1^2
│ │ │ │ -00014c00: 2b34 3978 5f30 785f 322b 3378 5f31 785f  +49x_0x_2+3x_1x_
│ │ │ │ -00014c10: 322b 3578 5f32 5e32 2033 3073 5f30 207c  2+5x_2^2 30s_0 |
│ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00014af0: 7c20 2020 2020 2073 5f31 2020 2020 2020  |      s_1      
│ │ │ │ +00014b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b10: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00014b20: 2020 2020 7c29 2020 2020 2020 2020 2020      |)          
│ │ │ │ +00014b30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014b40: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00014b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b60: 2020 2020 2020 2020 2020 2020 2020 735f                s_
│ │ │ │ +00014b70: 3120 2020 7c20 2020 2020 2020 2020 2020  1   |           
│ │ │ │ +00014b80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014b90: 7c20 2020 2020 2078 5f30 5e32 2020 2020  |      x_0^2    
│ │ │ │ +00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014bb0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00014bc0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00014bd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014be0: 7c20 2020 2020 2078 5f30 785f 312b 3234  |      x_0x_1+24
│ │ │ │ +00014bf0: 785f 315e 322b 3439 785f 3078 5f32 2b33  x_1^2+49x_0x_2+3
│ │ │ │ +00014c00: 785f 3178 5f32 2b35 785f 325e 3220 3330  x_1x_2+5x_2^2 30
│ │ │ │ +00014c10: 735f 3020 7c20 2020 2020 2020 2020 2020  s_0 |           
│ │ │ │ +00014c20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014c30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00014c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c80: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -00014c90: 3a20 5365 7175 656e 6365 2020 2020 2020  : Sequence      
│ │ │ │ +00014c70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014c80: 7c6f 3130 203a 2053 6571 7565 6e63 6520  |o10 : Sequence 
│ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cd0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00014cc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014cd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00014ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ -00014d30: 3a20 6430 2a64 3120 2020 2020 2020 2020  : d0*d1         
│ │ │ │ +00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014d20: 7c69 3131 203a 2064 302a 6431 2020 2020  |i11 : d0*d1    
│ │ │ │ +00014d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014d60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014d70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00014d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014dc0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -00014dd0: 3d20 7b2d 327d 207c 2073 5f31 785f 325e  = {-2} | s_1x_2^
│ │ │ │ -00014de0: 332b 735f 3078 5f30 5e32 2030 2020 2020  3+s_0x_0^2 0    
│ │ │ │ -00014df0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014e20: 2020 7b2d 327d 207c 2030 2020 2020 2020    {-2} | 0      
│ │ │ │ -00014e30: 2020 2020 2020 2020 2020 2073 5f31 785f             s_1x_
│ │ │ │ -00014e40: 325e 332b 735f 3078 5f30 5e32 2030 2020  2^3+s_0x_0^2 0  
│ │ │ │ -00014e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014e70: 2020 7b2d 337d 207c 2030 2020 2020 2020    {-3} | 0      
│ │ │ │ -00014e80: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -00014e90: 2020 2020 2020 2020 2020 2020 2073 5f31               s_1
│ │ │ │ -00014ea0: 785f 325e 332b 735f 3078 5f30 5e32 2020  x_2^3+s_0x_0^2  
│ │ │ │ -00014eb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014ec0: 2020 7b2d 377d 207c 2030 2020 2020 2020    {-7} | 0      
│ │ │ │ -00014ed0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -00014ee0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014f10: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +00014db0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014dc0: 7c6f 3131 203d 207b 2d32 7d20 7c20 735f  |o11 = {-2} | s_
│ │ │ │ +00014dd0: 3178 5f32 5e33 2b73 5f30 785f 305e 3220  1x_2^3+s_0x_0^2 
│ │ │ │ +00014de0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00014df0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00014e00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014e10: 7c20 2020 2020 207b 2d32 7d20 7c20 3020  |      {-2} | 0 
│ │ │ │ +00014e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e30: 735f 3178 5f32 5e33 2b73 5f30 785f 305e  s_1x_2^3+s_0x_0^
│ │ │ │ +00014e40: 3220 3020 2020 2020 2020 2020 2020 2020  2 0             
│ │ │ │ +00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014e60: 7c20 2020 2020 207b 2d33 7d20 7c20 3020  |      {-3} | 0 
│ │ │ │ +00014e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e80: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00014e90: 2020 735f 3178 5f32 5e33 2b73 5f30 785f    s_1x_2^3+s_0x_
│ │ │ │ +00014ea0: 305e 3220 2020 2020 2020 2020 2020 7c0a  0^2           |.
│ │ │ │ +00014eb0: 7c20 2020 2020 207b 2d37 7d20 7c20 3020  |      {-7} | 0 
│ │ │ │ +00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ed0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00014ee0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00014ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014f00: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f50: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -00014f60: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00014f70: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00014f50: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00014f60: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00014f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014fa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014fb0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00014fc0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00014f90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014fa0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00014fb0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00014fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ff0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015000: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015010: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00014fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014ff0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015000: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015040: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015050: 2020 735f 3178 5f32 5e33 2b73 5f30 785f    s_1x_2^3+s_0x_
│ │ │ │ -00015060: 305e 3220 7c20 2020 2020 2020 2020 2020  0^2 |           
│ │ │ │ +00015030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015040: 7c20 2020 2020 2073 5f31 785f 325e 332b  |      s_1x_2^3+
│ │ │ │ +00015050: 735f 3078 5f30 5e32 207c 2020 2020 2020  s_0x_0^2 |      
│ │ │ │ +00015060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015090: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015090: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000150a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000150d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000150e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000150f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015100: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00015110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015120: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ -00015130: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -00015140: 3a20 4d61 7472 6978 2028 6b6b 5b73 202e  : Matrix (kk[s .
│ │ │ │ -00015150: 2e73 202c 2078 202e 2e78 205d 2920 203c  .s , x ..x ])  <
│ │ │ │ -00015160: 2d2d 2028 6b6b 5b73 202e 2e73 202c 2078  -- (kk[s ..s , x
│ │ │ │ -00015170: 202e 2e78 205d 2920 2020 2020 2020 2020   ..x ])         
│ │ │ │ -00015180: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015190: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000151a0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ -000151b0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ -000151c0: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │ -000151d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00015100: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00015110: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00015120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015130: 7c6f 3131 203a 204d 6174 7269 7820 286b  |o11 : Matrix (k
│ │ │ │ +00015140: 6b5b 7320 2e2e 7320 2c20 7820 2e2e 7820  k[s ..s , x ..x 
│ │ │ │ +00015150: 5d29 2020 3c2d 2d20 286b 6b5b 7320 2e2e  ])  <-- (kk[s ..
│ │ │ │ +00015160: 7320 2c20 7820 2e2e 7820 5d29 2020 2020  s , x ..x ])    
│ │ │ │ +00015170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015180: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015190: 2020 2030 2020 2031 2020 2030 2020 2032     0   1   0   2
│ │ │ │ +000151a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +000151b0: 2031 2020 2030 2020 2032 2020 2020 2020   1   0   2      
│ │ │ │ +000151c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000151d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000151f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015220: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220  ---------+.|i12 
│ │ │ │ -00015230: 3a20 6431 2a64 3020 2020 2020 2020 2020  : d1*d0         
│ │ │ │ +00015210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015220: 7c69 3132 203a 2064 312a 6430 2020 2020  |i12 : d1*d0    
│ │ │ │ +00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015270: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015270: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000152a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152c0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -000152d0: 3d20 7b30 7d20 207c 2073 5f31 785f 325e  = {0}  | s_1x_2^
│ │ │ │ -000152e0: 332b 735f 3078 5f30 5e32 2030 2020 2020  3+s_0x_0^2 0    
│ │ │ │ -000152f0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00015300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015320: 2020 7b2d 347d 207c 2030 2020 2020 2020    {-4} | 0      
│ │ │ │ -00015330: 2020 2020 2020 2020 2020 2073 5f31 785f             s_1x_
│ │ │ │ -00015340: 325e 332b 735f 3078 5f30 5e32 2030 2020  2^3+s_0x_0^2 0  
│ │ │ │ -00015350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015360: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015370: 2020 7b2d 357d 207c 2030 2020 2020 2020    {-5} | 0      
│ │ │ │ -00015380: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -00015390: 2020 2020 2020 2020 2020 2020 2073 5f31               s_1
│ │ │ │ -000153a0: 785f 325e 332b 735f 3078 5f30 5e32 2020  x_2^3+s_0x_0^2  
│ │ │ │ -000153b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000153c0: 2020 7b2d 357d 207c 2030 2020 2020 2020    {-5} | 0      
│ │ │ │ -000153d0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -000153e0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015410: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +000152b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000152c0: 7c6f 3132 203d 207b 307d 2020 7c20 735f  |o12 = {0}  | s_
│ │ │ │ +000152d0: 3178 5f32 5e33 2b73 5f30 785f 305e 3220  1x_2^3+s_0x_0^2 
│ │ │ │ +000152e0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000152f0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00015300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015310: 7c20 2020 2020 207b 2d34 7d20 7c20 3020  |      {-4} | 0 
│ │ │ │ +00015320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015330: 735f 3178 5f32 5e33 2b73 5f30 785f 305e  s_1x_2^3+s_0x_0^
│ │ │ │ +00015340: 3220 3020 2020 2020 2020 2020 2020 2020  2 0             
│ │ │ │ +00015350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015360: 7c20 2020 2020 207b 2d35 7d20 7c20 3020  |      {-5} | 0 
│ │ │ │ +00015370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015380: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00015390: 2020 735f 3178 5f32 5e33 2b73 5f30 785f    s_1x_2^3+s_0x_
│ │ │ │ +000153a0: 305e 3220 2020 2020 2020 2020 2020 7c0a  0^2           |.
│ │ │ │ +000153b0: 7c20 2020 2020 207b 2d35 7d20 7c20 3020  |      {-5} | 0 
│ │ │ │ +000153c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000153d0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000153e0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000153f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015400: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00015410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015450: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -00015460: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015470: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00015440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00015450: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015460: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00015470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000154b0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -000154c0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00015490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000154a0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +000154b0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000154d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015500: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015510: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +000154e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000154f0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015500: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015550: 2020 735f 3178 5f32 5e33 2b73 5f30 785f    s_1x_2^3+s_0x_
│ │ │ │ -00015560: 305e 3220 7c20 2020 2020 2020 2020 2020  0^2 |           
│ │ │ │ +00015530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015540: 7c20 2020 2020 2073 5f31 785f 325e 332b  |      s_1x_2^3+
│ │ │ │ +00015550: 735f 3078 5f30 5e32 207c 2020 2020 2020  s_0x_0^2 |      
│ │ │ │ +00015560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015590: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000155a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000155b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000155c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000155d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000155e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000155f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015600: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00015610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015620: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ -00015630: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -00015640: 3a20 4d61 7472 6978 2028 6b6b 5b73 202e  : Matrix (kk[s .
│ │ │ │ -00015650: 2e73 202c 2078 202e 2e78 205d 2920 203c  .s , x ..x ])  <
│ │ │ │ -00015660: 2d2d 2028 6b6b 5b73 202e 2e73 202c 2078  -- (kk[s ..s , x
│ │ │ │ -00015670: 202e 2e78 205d 2920 2020 2020 2020 2020   ..x ])         
│ │ │ │ -00015680: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015690: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000156a0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ -000156b0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ -000156c0: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │ -000156d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00015600: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00015610: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00015620: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015630: 7c6f 3132 203a 204d 6174 7269 7820 286b  |o12 : Matrix (k
│ │ │ │ +00015640: 6b5b 7320 2e2e 7320 2c20 7820 2e2e 7820  k[s ..s , x ..x 
│ │ │ │ +00015650: 5d29 2020 3c2d 2d20 286b 6b5b 7320 2e2e  ])  <-- (kk[s ..
│ │ │ │ +00015660: 7320 2c20 7820 2e2e 7820 5d29 2020 2020  s , x ..x ])    
│ │ │ │ +00015670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015680: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015690: 2020 2030 2020 2031 2020 2030 2020 2032     0   1   0   2
│ │ │ │ +000156a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +000156b0: 2031 2020 2030 2020 2032 2020 2020 2020   1   0   2      
│ │ │ │ +000156c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000156d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000156e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000156f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015720: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ -00015730: 3a20 5320 3d20 7269 6e67 2064 3020 2020  : S = ring d0   
│ │ │ │ +00015710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015720: 7c69 3133 203a 2053 203d 2072 696e 6720  |i13 : S = ring 
│ │ │ │ +00015730: 6430 2020 2020 2020 2020 2020 2020 2020  d0              
│ │ │ │  00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015770: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000157a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157c0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ -000157d0: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │ +000157b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000157c0: 7c6f 3133 203d 2053 2020 2020 2020 2020  |o13 = S        
│ │ │ │ +000157d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000157e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000157f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015810: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015810: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015860: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ -00015870: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +00015850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015860: 7c6f 3133 203a 2050 6f6c 796e 6f6d 6961  |o13 : Polynomia
│ │ │ │ +00015870: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │  00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000158a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000158b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000158e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015900: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420  ---------+.|i14 
│ │ │ │ -00015910: 3a20 7068 6920 3d20 6d61 7028 532c 5229  : phi = map(S,R)
│ │ │ │ +000158f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015900: 7c69 3134 203a 2070 6869 203d 206d 6170  |i14 : phi = map
│ │ │ │ +00015910: 2853 2c52 2920 2020 2020 2020 2020 2020  (S,R)           
│ │ │ │  00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015950: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159a0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -000159b0: 3d20 6d61 7020 2853 2c20 522c 207b 7820  = map (S, R, {x 
│ │ │ │ -000159c0: 2c20 7820 2c20 7820 7d29 2020 2020 2020  , x , x })      
│ │ │ │ +00015990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000159a0: 7c6f 3134 203d 206d 6170 2028 532c 2052  |o14 = map (S, R
│ │ │ │ +000159b0: 2c20 7b78 202c 2078 202c 2078 207d 2920  , {x , x , x }) 
│ │ │ │ +000159c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000159d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015a00: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00015a10: 2020 2031 2020 2032 2020 2020 2020 2020     1   2        
│ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000159f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015a00: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a90: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -00015aa0: 3a20 5269 6e67 4d61 7020 5320 3c2d 2d20  : RingMap S <-- 
│ │ │ │ -00015ab0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00015a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015a90: 7c6f 3134 203a 2052 696e 674d 6170 2053  |o14 : RingMap S
│ │ │ │ +00015aa0: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +00015ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ae0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00015ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015ae0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00015af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520  ---------+.|i15 
│ │ │ │ -00015b40: 3a20 4953 203d 2070 6869 2049 2020 2020  : IS = phi I    
│ │ │ │ +00015b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015b30: 7c69 3135 203a 2049 5320 3d20 7068 6920  |i15 : IS = phi 
│ │ │ │ +00015b40: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │  00015b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015b80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015be0: 2020 2020 2020 2020 2020 3220 2020 3320            2   3 
│ │ │ │ +00015bc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015bd0: 7c20 2020 2020 2020 2020 2020 2020 2032  |              2
│ │ │ │ +00015be0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c20: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ -00015c30: 3d20 6964 6561 6c20 2878 202c 2078 2029  = ideal (x , x )
│ │ │ │ +00015c10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c20: 7c6f 3135 203d 2069 6465 616c 2028 7820  |o15 = ideal (x 
│ │ │ │ +00015c30: 2c20 7820 2920 2020 2020 2020 2020 2020  , x )           
│ │ │ │  00015c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015c80: 2020 2020 2020 2020 2020 3020 2020 3220            0   2 
│ │ │ │ +00015c60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c70: 7c20 2020 2020 2020 2020 2020 2020 2030  |              0
│ │ │ │ +00015c80: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00015c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015cc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d10: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ -00015d20: 3a20 4964 6561 6c20 6f66 2053 2020 2020  : Ideal of S    
│ │ │ │ +00015d00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015d10: 7c6f 3135 203a 2049 6465 616c 206f 6620  |o15 : Ideal of 
│ │ │ │ +00015d20: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00015d50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015d60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00015d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015db0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ -00015dc0: 3a20 5362 6172 203d 2053 2f49 5320 2020  : Sbar = S/IS   
│ │ │ │ +00015da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015db0: 7c69 3136 203a 2053 6261 7220 3d20 532f  |i16 : Sbar = S/
│ │ │ │ +00015dc0: 4953 2020 2020 2020 2020 2020 2020 2020  IS              
│ │ │ │  00015dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015e00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e50: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ -00015e60: 3d20 5362 6172 2020 2020 2020 2020 2020  = Sbar          
│ │ │ │ +00015e40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015e50: 7c6f 3136 203d 2053 6261 7220 2020 2020  |o16 = Sbar     
│ │ │ │ +00015e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015e90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015ea0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ef0: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ -00015f00: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +00015ee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015ef0: 7c6f 3136 203a 2051 756f 7469 656e 7452  |o16 : QuotientR
│ │ │ │ +00015f00: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  00015f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00015f30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015f40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00015f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015f90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -00015fa0: 3a20 534d 6261 7220 3d20 5362 6172 2a2a  : SMbar = Sbar**
│ │ │ │ -00015fb0: 4d62 6172 2020 2020 2020 2020 2020 2020  Mbar            
│ │ │ │ +00015f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015f90: 7c69 3137 203a 2053 4d62 6172 203d 2053  |i17 : SMbar = S
│ │ │ │ +00015fa0: 6261 722a 2a4d 6261 7220 2020 2020 2020  bar**Mbar       
│ │ │ │ +00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015fe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015fd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015fe0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016030: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ -00016040: 3d20 636f 6b65 726e 656c 207c 2078 5f30  = cokernel | x_0
│ │ │ │ -00016050: 785f 312b 3234 785f 315e 322b 3439 785f  x_1+24x_1^2+49x_
│ │ │ │ -00016060: 3078 5f32 2b33 785f 3178 5f32 2b35 785f  0x_2+3x_1x_2+5x_
│ │ │ │ -00016070: 325e 3220 7c20 2020 2020 2020 2020 2020  2^2 |           
│ │ │ │ -00016080: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00016020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016030: 7c6f 3137 203d 2063 6f6b 6572 6e65 6c20  |o17 = cokernel 
│ │ │ │ +00016040: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32  | x_0x_1+24x_1^2
│ │ │ │ +00016050: 2b34 3978 5f30 785f 322b 3378 5f31 785f  +49x_0x_2+3x_1x_
│ │ │ │ +00016060: 322b 3578 5f32 5e32 207c 2020 2020 2020  2+5x_2^2 |      
│ │ │ │ +00016070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016080: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00016090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000160a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000160b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000160c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000160d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000160e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160f0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +000160f0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00016100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016120: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ -00016130: 3a20 5362 6172 2d6d 6f64 756c 652c 2071  : Sbar-module, q
│ │ │ │ -00016140: 756f 7469 656e 7420 6f66 2053 6261 7220  uotient of Sbar 
│ │ │ │ +00016110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016120: 7c6f 3137 203a 2053 6261 722d 6d6f 6475  |o17 : Sbar-modu
│ │ │ │ +00016130: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ +00016140: 5362 6172 2020 2020 2020 2020 2020 2020  Sbar            
│ │ │ │  00016150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016170: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00016160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016170: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00016180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000161a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 486f 6d28  ---------+..Hom(
│ │ │ │ -000161d0: 6430 2c53 6261 7229 2061 6e64 2048 6f6d  d0,Sbar) and Hom
│ │ │ │ -000161e0: 2864 312c 5362 6172 2920 746f 6765 7468  (d1,Sbar) togeth
│ │ │ │ -000161f0: 6572 2066 6f72 6d20 7468 6520 7265 736f  er form the reso
│ │ │ │ -00016200: 6c75 7469 6f6e 206f 6620 4d62 6172 3b20  lution of Mbar; 
│ │ │ │ -00016210: 7468 7573 2074 6865 0a68 6f6d 6f6c 6f67  thus the.homolog
│ │ │ │ -00016220: 7920 6f66 206f 6e65 2063 6f6d 706f 7369  y of one composi
│ │ │ │ -00016230: 7469 6f6e 2069 7320 302c 2077 6869 6c65  tion is 0, while
│ │ │ │ -00016240: 2074 6865 206f 7468 6572 2069 7320 4d62   the other is Mb
│ │ │ │ -00016250: 6172 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ar..+-----------
│ │ │ │ +000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000161c0: 0a48 6f6d 2864 302c 5362 6172 2920 616e  .Hom(d0,Sbar) an
│ │ │ │ +000161d0: 6420 486f 6d28 6431 2c53 6261 7229 2074  d Hom(d1,Sbar) t
│ │ │ │ +000161e0: 6f67 6574 6865 7220 666f 726d 2074 6865  ogether form the
│ │ │ │ +000161f0: 2072 6573 6f6c 7574 696f 6e20 6f66 204d   resolution of M
│ │ │ │ +00016200: 6261 723b 2074 6875 7320 7468 650a 686f  bar; thus the.ho
│ │ │ │ +00016210: 6d6f 6c6f 6779 206f 6620 6f6e 6520 636f  mology of one co
│ │ │ │ +00016220: 6d70 6f73 6974 696f 6e20 6973 2030 2c20  mposition is 0, 
│ │ │ │ +00016230: 7768 696c 6520 7468 6520 6f74 6865 7220  while the other 
│ │ │ │ +00016240: 6973 204d 6261 720a 0a2b 2d2d 2d2d 2d2d  is Mbar..+------
│ │ │ │ +00016250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000162a0: 2d2d 2b0a 7c69 3138 203a 2070 7275 6e65  --+.|i18 : prune
│ │ │ │ -000162b0: 2048 485f 3120 6368 6169 6e43 6f6d 706c   HH_1 chainCompl
│ │ │ │ -000162c0: 6578 7b64 7561 6c20 2853 6261 722a 2a64  ex{dual (Sbar**d
│ │ │ │ -000162d0: 3029 2c20 6475 616c 2853 6261 722a 2a64  0), dual(Sbar**d
│ │ │ │ -000162e0: 3129 7d20 3d3d 2030 2020 2020 2020 2020  1)} == 0        
│ │ │ │ -000162f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016290: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20  -------+.|i18 : 
│ │ │ │ +000162a0: 7072 756e 6520 4848 5f31 2063 6861 696e  prune HH_1 chain
│ │ │ │ +000162b0: 436f 6d70 6c65 787b 6475 616c 2028 5362  Complex{dual (Sb
│ │ │ │ +000162c0: 6172 2a2a 6430 292c 2064 7561 6c28 5362  ar**d0), dual(Sb
│ │ │ │ +000162d0: 6172 2a2a 6431 297d 203d 3d20 3020 2020  ar**d1)} == 0   
│ │ │ │ +000162e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000162f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016340: 2020 7c0a 7c6f 3138 203d 2074 7275 6520    |.|o18 = true 
│ │ │ │ +00016330: 2020 2020 2020 207c 0a7c 6f31 3820 3d20         |.|o18 = 
│ │ │ │ +00016340: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │  00016350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016390: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00016380: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00016390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000163a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000163b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000163c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163e0: 2d2d 2b0a 7c69 3139 203a 204d 6261 7227  --+.|i19 : Mbar'
│ │ │ │ -000163f0: 203d 2053 6261 725e 312f 2853 6261 725f   = Sbar^1/(Sbar_
│ │ │ │ -00016400: 302c 2053 6261 725f 3129 2a2a 534d 6261  0, Sbar_1)**SMba
│ │ │ │ -00016410: 7220 2020 2020 2020 2020 2020 2020 2020  r               
│ │ │ │ -00016420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016430: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000163d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20  -------+.|i19 : 
│ │ │ │ +000163e0: 4d62 6172 2720 3d20 5362 6172 5e31 2f28  Mbar' = Sbar^1/(
│ │ │ │ +000163f0: 5362 6172 5f30 2c20 5362 6172 5f31 292a  Sbar_0, Sbar_1)*
│ │ │ │ +00016400: 2a53 4d62 6172 2020 2020 2020 2020 2020  *SMbar          
│ │ │ │ +00016410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016420: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00016430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016480: 2020 7c0a 7c6f 3139 203d 2063 6f6b 6572    |.|o19 = coker
│ │ │ │ -00016490: 6e65 6c20 7c20 785f 3078 5f31 2b32 3478  nel | x_0x_1+24x
│ │ │ │ -000164a0: 5f31 5e32 2b34 3978 5f30 785f 322b 3378  _1^2+49x_0x_2+3x
│ │ │ │ -000164b0: 5f31 785f 322b 3578 5f32 5e32 2073 5f30  _1x_2+5x_2^2 s_0
│ │ │ │ -000164c0: 2073 5f31 207c 2020 2020 2020 2020 2020   s_1 |          
│ │ │ │ -000164d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016470: 2020 2020 2020 207c 0a7c 6f31 3920 3d20         |.|o19 = 
│ │ │ │ +00016480: 636f 6b65 726e 656c 207c 2078 5f30 785f  cokernel | x_0x_
│ │ │ │ +00016490: 312b 3234 785f 315e 322b 3439 785f 3078  1+24x_1^2+49x_0x
│ │ │ │ +000164a0: 5f32 2b33 785f 3178 5f32 2b35 785f 325e  _2+3x_1x_2+5x_2^
│ │ │ │ +000164b0: 3220 735f 3020 735f 3120 7c20 2020 2020  2 s_0 s_1 |     
│ │ │ │ +000164c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000164d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000164e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000164f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016520: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00016530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016540: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00016510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00016520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016530: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00016540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016570: 2020 7c0a 7c6f 3139 203a 2053 6261 722d    |.|o19 : Sbar-
│ │ │ │ -00016580: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -00016590: 206f 6620 5362 6172 2020 2020 2020 2020   of Sbar        
│ │ │ │ +00016560: 2020 2020 2020 207c 0a7c 6f31 3920 3a20         |.|o19 : 
│ │ │ │ +00016570: 5362 6172 2d6d 6f64 756c 652c 2071 756f  Sbar-module, quo
│ │ │ │ +00016580: 7469 656e 7420 6f66 2053 6261 7220 2020  tient of Sbar   
│ │ │ │ +00016590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000165a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000165b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000165c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000165d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000165e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000165f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016610: 2d2d 2b0a 7c69 3230 203a 2069 6465 616c  --+.|i20 : ideal
│ │ │ │ -00016620: 2070 7265 7365 6e74 6174 696f 6e20 7072   presentation pr
│ │ │ │ -00016630: 756e 6520 4848 5f31 2063 6861 696e 436f  une HH_1 chainCo
│ │ │ │ -00016640: 6d70 6c65 787b 6475 616c 2028 5362 6172  mplex{dual (Sbar
│ │ │ │ -00016650: 2a2a 6431 292c 2020 2020 2020 2020 2020  **d1),          
│ │ │ │ -00016660: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016600: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20  -------+.|i20 : 
│ │ │ │ +00016610: 6964 6561 6c20 7072 6573 656e 7461 7469  ideal presentati
│ │ │ │ +00016620: 6f6e 2070 7275 6e65 2048 485f 3120 6368  on prune HH_1 ch
│ │ │ │ +00016630: 6169 6e43 6f6d 706c 6578 7b64 7561 6c20  ainComplex{dual 
│ │ │ │ +00016640: 2853 6261 722a 2a64 3129 2c20 2020 2020  (Sbar**d1),     
│ │ │ │ +00016650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00016660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000166a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000166b0: 2020 7c0a 7c6f 3230 203d 2074 7275 6520    |.|o20 = true 
│ │ │ │ +000166a0: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +000166b0: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │  000166c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000166d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000166e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000166f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016700: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +000166f0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +00016700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016750: 2d2d 7c0a 7c64 7561 6c28 5362 6172 2a2a  --|.|dual(Sbar**
│ │ │ │ -00016760: 6430 297d 203d 3d20 6964 6561 6c20 7072  d0)} == ideal pr
│ │ │ │ -00016770: 6573 656e 7461 7469 6f6e 204d 6261 7227  esentation Mbar'
│ │ │ │ +00016740: 2d2d 2d2d 2d2d 2d7c 0a7c 6475 616c 2853  -------|.|dual(S
│ │ │ │ +00016750: 6261 722a 2a64 3029 7d20 3d3d 2069 6465  bar**d0)} == ide
│ │ │ │ +00016760: 616c 2070 7265 7365 6e74 6174 696f 6e20  al presentation 
│ │ │ │ +00016770: 4d62 6172 2720 2020 2020 2020 2020 2020  Mbar'           
│ │ │ │  00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00016790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000167a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000167b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000167c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167f0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
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│ │ │ │ -00016850: 4578 743a 206e 6577 4578 742c 202d 2d20  Ext: newExt, -- 
│ │ │ │ -00016860: 476c 6f62 616c 2045 7874 2066 6f72 206d  Global Ext for m
│ │ │ │ -00016870: 6f64 756c 6573 206f 7665 7220 6120 636f  odules over a co
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│ │ │ │ -000168a0: 6520 6d61 6b65 486f 6d6f 746f 7069 6573  e makeHomotopies
│ │ │ │ -000168b0: 3a20 6d61 6b65 486f 6d6f 746f 7069 6573  : makeHomotopies
│ │ │ │ -000168c0: 2c20 2d2d 2072 6574 7572 6e73 2061 2073  , -- returns a s
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│ │ │ │ -000168e0: 2020 2020 686f 6d6f 746f 7069 6573 0a0a      homotopies..
│ │ │ │ -000168f0: 5761 7973 2074 6f20 7573 6520 4569 7365  Ways to use Eise
│ │ │ │ -00016900: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ -00016910: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00016920: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00016960: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00016970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00016980: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00016990: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ -000169a0: 546f 7461 6c3a 2045 6973 656e 6275 6453  Total: EisenbudS
│ │ │ │ -000169b0: 6861 6d61 7368 546f 7461 6c2c 2069 7320  hamashTotal, is 
│ │ │ │ -000169c0: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66  a *note method.f
│ │ │ │ -000169d0: 756e 6374 696f 6e20 7769 7468 206f 7074  unction with opt
│ │ │ │ -000169e0: 696f 6e73 3a20 284d 6163 6175 6c61 7932  ions: (Macaulay2
│ │ │ │ -000169f0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -00016a00: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a  onWithOptions,..
│ │ │ │ -00016a10: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ -00016a20: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -00016a30: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ -00016a40: 6465 3a20 6576 656e 4578 744d 6f64 756c  de: evenExtModul
│ │ │ │ -00016a50: 652c 204e 6578 743a 2065 7870 6f2c 2050  e, Next: expo, P
│ │ │ │ -00016a60: 7265 763a 2045 6973 656e 6275 6453 6861  rev: EisenbudSha
│ │ │ │ -00016a70: 6d61 7368 546f 7461 6c2c 2055 703a 2054  mashTotal, Up: T
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│ │ │ │ -00016a90: 6520 2d2d 2065 7665 6e20 7061 7274 206f  e -- even part o
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│ │ │ │ +000167e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +000167f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +00016800: 202a 6e6f 7465 2045 7874 3a20 284d 6163   *note Ext: (Mac
│ │ │ │ +00016810: 6175 6c61 7932 446f 6329 4578 742c 202d  aulay2Doc)Ext, -
│ │ │ │ +00016820: 2d20 636f 6d70 7574 6520 616e 2045 7874  - compute an Ext
│ │ │ │ +00016830: 206d 6f64 756c 650a 2020 2a20 2a6e 6f74   module.  * *not
│ │ │ │ +00016840: 6520 6e65 7745 7874 3a20 6e65 7745 7874  e newExt: newExt
│ │ │ │ +00016850: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420  , -- Global Ext 
│ │ │ │ +00016860: 666f 7220 6d6f 6475 6c65 7320 6f76 6572  for modules over
│ │ │ │ +00016870: 2061 2063 6f6d 706c 6574 650a 2020 2020   a complete.    
│ │ │ │ +00016880: 496e 7465 7273 6563 7469 6f6e 0a20 202a  Intersection.  *
│ │ │ │ +00016890: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74   *note makeHomot
│ │ │ │ +000168a0: 6f70 6965 733a 206d 616b 6548 6f6d 6f74  opies: makeHomot
│ │ │ │ +000168b0: 6f70 6965 732c 202d 2d20 7265 7475 726e  opies, -- return
│ │ │ │ +000168c0: 7320 6120 7379 7374 656d 206f 6620 6869  s a system of hi
│ │ │ │ +000168d0: 6768 6572 0a20 2020 2068 6f6d 6f74 6f70  gher.    homotop
│ │ │ │ +000168e0: 6965 730a 0a57 6179 7320 746f 2075 7365  ies..Ways to use
│ │ │ │ +000168f0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ +00016900: 546f 7461 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d  Total:.=========
│ │ │ │ +00016910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00016ad0: 206f 7065 7261 746f 7220 7269 6e67 0a2a   operator ring.*
│ │ │ │ +00016ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016b00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016b10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
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│ │ │ │ -00016b70: 2020 2020 2020 2020 4520 3d20 6576 656e          E = even
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│ │ │ │ -00016bb0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
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│ │ │ │ -00016bd0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
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│ │ │ │ -00016c10: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
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│ │ │ │ -00016c70: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
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│ │ │ │ -00016cb0: 6120 706f 6c79 6e6f 6d69 616c 2072 696e  a polynomial rin
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│ │ │ │ -00016d20: 4d6f 6475 6c65 204d 2049 6620 7468 6520  Module M If the 
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│ │ │ │ -00016d40: 7420 4f75 7452 696e 670a 3d3e 2054 2069  t OutRing.=> T i
│ │ │ │ -00016d50: 7320 6769 7665 6e2c 2061 6e64 2063 6c61  s given, and cla
│ │ │ │ -00016d60: 7373 2054 203d 3d3d 2050 6f6c 796e 6f6d  ss T === Polynom
│ │ │ │ -00016d70: 6961 6c52 696e 672c 2074 6865 6e20 7468  ialRing, then th
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│ │ │ │ -00016d90: 2061 206d 6f64 756c 650a 6f76 6572 2054   a module.over T
│ │ │ │ -00016da0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
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│ │ │ │ +00016ba0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
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│ │ │ │ +00016be0: 2020 2020 7269 6e67 0a20 202a 202a 6e6f      ring.  * *no
│ │ │ │ +00016bf0: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ +00016c00: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ +00016c10: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ +00016c20: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +00016c30: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ +00016c40: 204f 7574 5269 6e67 203d 3e20 2e2e 2e2c   OutRing => ...,
│ │ │ │ +00016c50: 2064 6566 6175 6c74 2076 616c 7565 2030   default value 0
│ │ │ │ +00016c60: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +00016c70: 2020 2020 2a20 452c 2061 202a 6e6f 7465      * E, a *note
│ │ │ │ +00016c80: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00016c90: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00016ca0: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961  over a polynomia
│ │ │ │ +00016cb0: 6c20 7269 6e67 2077 6974 680a 2020 2020  l ring with.    
│ │ │ │ +00016cc0: 2020 2020 6765 6e73 2069 6e20 6465 6772      gens in degr
│ │ │ │ +00016cd0: 6565 2031 0a0a 4465 7363 7269 7074 696f  ee 1..Descriptio
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│ │ │ │ +00016cf0: 7874 7261 6374 7320 7468 6520 6576 656e  xtracts the even
│ │ │ │ +00016d00: 2064 6567 7265 6520 7061 7274 2066 726f   degree part fro
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│ │ │ │ +00016d20: 2074 6865 206f 7074 696f 6e61 6c20 6172   the optional ar
│ │ │ │ +00016d30: 6775 6d65 6e74 204f 7574 5269 6e67 0a3d  gument OutRing.=
│ │ │ │ +00016d40: 3e20 5420 6973 2067 6976 656e 2c20 616e  > T is given, an
│ │ │ │ +00016d50: 6420 636c 6173 7320 5420 3d3d 3d20 506f  d class T === Po
│ │ │ │ +00016d60: 6c79 6e6f 6d69 616c 5269 6e67 2c20 7468  lynomialRing, th
│ │ │ │ +00016d70: 656e 2074 6865 206f 7574 7075 7420 7769  en the output wi
│ │ │ │ +00016d80: 6c6c 2062 6520 6120 6d6f 6475 6c65 0a6f  ll be a module.o
│ │ │ │ +00016d90: 7665 7220 542e 0a0a 2b2d 2d2d 2d2d 2d2d  ver T...+-------
│ │ │ │ +00016da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00016de0: 6931 203a 206b 6b3d 205a 5a2f 3130 3120  i1 : kk= ZZ/101 
│ │ │ │ +00016dd0: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a  --+.|i1 : kk= ZZ
│ │ │ │ +00016de0: 2f31 3031 2020 2020 2020 2020 2020 2020  /101            
│ │ │ │  00016df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00016e00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016e10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00016e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e50: 2020 2020 207c 0a7c 6f31 203d 206b 6b20       |.|o1 = kk 
│ │ │ │ +00016e40: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00016e50: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │  00016e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00016e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016ed0: 6f31 203a 2051 756f 7469 656e 7452 696e  o1 : QuotientRin
│ │ │ │ -00016ee0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f00: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00016ec0: 2020 7c0a 7c6f 3120 3a20 5175 6f74 6965    |.|o1 : Quotie
│ │ │ │ +00016ed0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +00016ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016f00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00016f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016f40: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d  -----+.|i2 : S =
│ │ │ │ -00016f50: 206b 6b5b 782c 792c 7a5d 2020 2020 2020   kk[x,y,z]      
│ │ │ │ +00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00016f40: 3a20 5320 3d20 6b6b 5b78 2c79 2c7a 5d20  : S = kk[x,y,z] 
│ │ │ │ +00016f50: 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                  
│ │ │ │ -00016f80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00016f70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016fc0: 6f32 203d 2053 2020 2020 2020 2020 2020  o2 = S          
│ │ │ │ +00016fb0: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020    |.|o2 = S     
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ff0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00016fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016ff0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017030: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -00017040: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +00017020: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00017030: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017070: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00017060: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000170b0: 6933 203a 2049 3220 3d20 6964 6561 6c22  i3 : I2 = ideal"
│ │ │ │ -000170c0: 7833 2c79 7a22 2020 2020 2020 2020 2020  x3,yz"          
│ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000170a0: 2d2d 2b0a 7c69 3320 3a20 4932 203d 2069  --+.|i3 : I2 = i
│ │ │ │ +000170b0: 6465 616c 2278 332c 797a 2220 2020 2020  deal"x3,yz"     
│ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000170e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017130: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +00017110: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00017120: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017160: 207c 0a7c 6f33 203d 2069 6465 616c 2028   |.|o3 = ideal (
│ │ │ │ -00017170: 7820 2c20 792a 7a29 2020 2020 2020 2020  x , y*z)        
│ │ │ │ +00017150: 2020 2020 2020 7c0a 7c6f 3320 3d20 6964        |.|o3 = id
│ │ │ │ +00017160: 6561 6c20 2878 202c 2079 2a7a 2920 2020  eal (x , y*z)   
│ │ │ │ +00017170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017190: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000171a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000171b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171d0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -000171e0: 2049 6465 616c 206f 6620 5320 2020 2020   Ideal of S     
│ │ │ │ +000171c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000171d0: 7c6f 3320 3a20 4964 6561 6c20 6f66 2053  |o3 : Ideal of S
│ │ │ │ +000171e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000171f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017210: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00017200: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00017210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017250: 2d2b 0a7c 6934 203a 2052 3220 3d20 532f  -+.|i4 : R2 = S/
│ │ │ │ -00017260: 4932 2020 2020 2020 2020 2020 2020 2020  I2              
│ │ │ │ +00017240: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5232  ------+.|i4 : R2
│ │ │ │ +00017250: 203d 2053 2f49 3220 2020 2020 2020 2020   = S/I2         
│ │ │ │ +00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017280: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172c0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -000172d0: 2052 3220 2020 2020 2020 2020 2020 2020   R2             
│ │ │ │ +000172b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000172c0: 7c6f 3420 3d20 5232 2020 2020 2020 2020  |o4 = R2        
│ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000172f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00017300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017340: 207c 0a7c 6f34 203a 2051 756f 7469 656e   |.|o4 : Quotien
│ │ │ │ -00017350: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +00017330: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175        |.|o4 : Qu
│ │ │ │ +00017340: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00017350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017370: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00017370: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -000173c0: 204d 3220 3d20 5232 5e31 2f69 6465 616c   M2 = R2^1/ideal
│ │ │ │ -000173d0: 2278 322c 792c 7a22 2020 2020 2020 2020  "x2,y,z"        
│ │ │ │ -000173e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000173f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000173b0: 7c69 3520 3a20 4d32 203d 2052 325e 312f  |i5 : M2 = R2^1/
│ │ │ │ +000173c0: 6964 6561 6c22 7832 2c79 2c7a 2220 2020  ideal"x2,y,z"   
│ │ │ │ +000173d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000173f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017430: 207c 0a7c 6f35 203d 2063 6f6b 6572 6e65   |.|o5 = cokerne
│ │ │ │ -00017440: 6c20 7c20 7832 2079 207a 207c 2020 2020  l | x2 y z |    
│ │ │ │ +00017420: 2020 2020 2020 7c0a 7c6f 3520 3d20 636f        |.|o5 = co
│ │ │ │ +00017430: 6b65 726e 656c 207c 2078 3220 7920 7a20  kernel | x2 y z 
│ │ │ │ +00017440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000174b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174c0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -000174d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174e0: 2020 2020 207c 0a7c 6f35 203a 2052 322d       |.|o5 : R2-
│ │ │ │ -000174f0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -00017500: 206f 6620 5232 2020 2020 2020 2020 2020   of R2          
│ │ │ │ -00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017520: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00017490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000174a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +000174c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +000174e0: 3a20 5232 2d6d 6f64 756c 652c 2071 756f  : R2-module, quo
│ │ │ │ +000174f0: 7469 656e 7420 6f66 2052 3220 2020 2020  tient of R2     
│ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017510: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00017520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00017560: 6936 203a 2062 6574 7469 2072 6573 2028  i6 : betti res (
│ │ │ │ -00017570: 4d32 2c20 4c65 6e67 7468 4c69 6d69 7420  M2, LengthLimit 
│ │ │ │ -00017580: 3d3e 3130 2920 2020 2020 2020 2020 2020  =>10)           
│ │ │ │ -00017590: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017550: 2d2d 2b0a 7c69 3620 3a20 6265 7474 6920  --+.|i6 : betti 
│ │ │ │ +00017560: 7265 7320 284d 322c 204c 656e 6774 684c  res (M2, LengthL
│ │ │ │ +00017570: 696d 6974 203d 3e31 3029 2020 2020 2020  imit =>10)      
│ │ │ │ +00017580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000175a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000175b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000175e0: 2020 2020 3020 3120 3220 3320 3420 2035      0 1 2 3 4  5
│ │ │ │ -000175f0: 2020 3620 2037 2020 3820 2039 2031 3020    6  7  8  9 10 
│ │ │ │ -00017600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017610: 207c 0a7c 6f36 203d 2074 6f74 616c 3a20   |.|o6 = total: 
│ │ │ │ -00017620: 3120 3320 3520 3720 3920 3131 2031 3320  1 3 5 7 9 11 13 
│ │ │ │ -00017630: 3135 2031 3720 3139 2032 3120 2020 2020  15 17 19 21     
│ │ │ │ -00017640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017650: 2020 2020 2020 2020 2030 3a20 3120 3220           0: 1 2 
│ │ │ │ -00017660: 3220 3220 3220 2032 2020 3220 2032 2020  2 2 2  2  2  2  
│ │ │ │ -00017670: 3220 2032 2020 3220 2020 2020 2020 2020  2  2  2         
│ │ │ │ -00017680: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017690: 2020 2020 2031 3a20 2e20 3120 3320 3420       1: . 1 3 4 
│ │ │ │ -000176a0: 3420 2034 2020 3420 2034 2020 3420 2034  4  4  4  4  4  4
│ │ │ │ -000176b0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ -000176c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000176d0: 2032 3a20 2e20 2e20 2e20 3120 3320 2034   2: . . . 1 3  4
│ │ │ │ -000176e0: 2020 3420 2034 2020 3420 2034 2020 3420    4  4  4  4  4 
│ │ │ │ -000176f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017700: 207c 0a7c 2020 2020 2020 2020 2033 3a20   |.|         3: 
│ │ │ │ -00017710: 2e20 2e20 2e20 2e20 2e20 2031 2020 3320  . . . . .  1  3 
│ │ │ │ -00017720: 2034 2020 3420 2034 2020 3420 2020 2020   4  4  4  4     
│ │ │ │ -00017730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017740: 2020 2020 2020 2020 2034 3a20 2e20 2e20           4: . . 
│ │ │ │ -00017750: 2e20 2e20 2e20 202e 2020 2e20 2031 2020  . . .  .  .  1  
│ │ │ │ -00017760: 3320 2034 2020 3420 2020 2020 2020 2020  3  4  4         
│ │ │ │ -00017770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017780: 2020 2020 2035 3a20 2e20 2e20 2e20 2e20       5: . . . . 
│ │ │ │ -00017790: 2e20 202e 2020 2e20 202e 2020 2e20 2031  .  .  .  .  .  1
│ │ │ │ -000177a0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -000177b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000175c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000175d0: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ +000175e0: 2034 2020 3520 2036 2020 3720 2038 2020   4  5  6  7  8  
│ │ │ │ +000175f0: 3920 3130 2020 2020 2020 2020 2020 2020  9 10            
│ │ │ │ +00017600: 2020 2020 2020 7c0a 7c6f 3620 3d20 746f        |.|o6 = to
│ │ │ │ +00017610: 7461 6c3a 2031 2033 2035 2037 2039 2031  tal: 1 3 5 7 9 1
│ │ │ │ +00017620: 3120 3133 2031 3520 3137 2031 3920 3231  1 13 15 17 19 21
│ │ │ │ +00017630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017640: 2020 7c0a 7c20 2020 2020 2020 2020 303a    |.|         0:
│ │ │ │ +00017650: 2031 2032 2032 2032 2032 2020 3220 2032   1 2 2 2 2  2  2
│ │ │ │ +00017660: 2020 3220 2032 2020 3220 2032 2020 2020    2  2  2  2    
│ │ │ │ +00017670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017680: 7c20 2020 2020 2020 2020 313a 202e 2031  |         1: . 1
│ │ │ │ +00017690: 2033 2034 2034 2020 3420 2034 2020 3420   3 4 4  4  4  4 
│ │ │ │ +000176a0: 2034 2020 3420 2034 2020 2020 2020 2020   4  4  4        
│ │ │ │ +000176b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000176c0: 2020 2020 2020 323a 202e 202e 202e 2031        2: . . . 1
│ │ │ │ +000176d0: 2033 2020 3420 2034 2020 3420 2034 2020   3  4  4  4  4  
│ │ │ │ +000176e0: 3420 2034 2020 2020 2020 2020 2020 2020  4  4            
│ │ │ │ +000176f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017700: 2020 333a 202e 202e 202e 202e 202e 2020    3: . . . . .  
│ │ │ │ +00017710: 3120 2033 2020 3420 2034 2020 3420 2034  1  3  4  4  4  4
│ │ │ │ +00017720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017730: 2020 7c0a 7c20 2020 2020 2020 2020 343a    |.|         4:
│ │ │ │ +00017740: 202e 202e 202e 202e 202e 2020 2e20 202e   . . . . .  .  .
│ │ │ │ +00017750: 2020 3120 2033 2020 3420 2034 2020 2020    1  3  4  4    
│ │ │ │ +00017760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017770: 7c20 2020 2020 2020 2020 353a 202e 202e  |         5: . .
│ │ │ │ +00017780: 202e 202e 202e 2020 2e20 202e 2020 2e20   . . .  .  .  . 
│ │ │ │ +00017790: 202e 2020 3120 2033 2020 2020 2020 2020   .  1  3        
│ │ │ │ +000177a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000177b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000177e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000177f0: 207c 0a7c 6f36 203a 2042 6574 7469 5461   |.|o6 : BettiTa
│ │ │ │ -00017800: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ +000177e0: 2020 2020 2020 7c0a 7c6f 3620 3a20 4265        |.|o6 : Be
│ │ │ │ +000177f0: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +00017800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017820: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00017820: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00017830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017860: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ -00017870: 2045 203d 2045 7874 4d6f 6475 6c65 204d   E = ExtModule M
│ │ │ │ -00017880: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00017890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000178a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00017850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00017860: 7c69 3720 3a20 4520 3d20 4578 744d 6f64  |i7 : E = ExtMod
│ │ │ │ +00017870: 756c 6520 4d32 2020 2020 2020 2020 2020  ule M2          
│ │ │ │ +00017880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017890: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000178a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000178b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000178c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000178d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000178e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000178f0: 2020 2020 2038 2020 2020 2020 2020 2020       8          
│ │ │ │ +000178d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000178e0: 2020 2020 2020 2020 2020 3820 2020 2020            8     
│ │ │ │ +000178f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017920: 6f37 203d 2028 6b6b 5b58 202e 2e58 205d  o7 = (kk[X ..X ]
│ │ │ │ -00017930: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00017940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017950: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017960: 2020 2020 2020 3020 2020 3120 2020 2020        0   1     
│ │ │ │ +00017910: 2020 7c0a 7c6f 3720 3d20 286b 6b5b 5820    |.|o7 = (kk[X 
│ │ │ │ +00017920: 2e2e 5820 5d29 2020 2020 2020 2020 2020  ..X ])          
│ │ │ │ +00017930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017950: 7c20 2020 2020 2020 2020 2030 2020 2031  |          0   1
│ │ │ │ +00017960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00017980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00017990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179d0: 207c 0a7c 6f37 203a 206b 6b5b 5820 2e2e   |.|o7 : kk[X ..
│ │ │ │ -000179e0: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565  X ]-module, free
│ │ │ │ -000179f0: 2c20 6465 6772 6565 7320 7b30 2e2e 312c  , degrees {0..1,
│ │ │ │ -00017a00: 2032 3a31 2c20 333a 322c 2033 7d7c 0a7c   2:1, 3:2, 3}|.|
│ │ │ │ -00017a10: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +000179c0: 2020 2020 2020 7c0a 7c6f 3720 3a20 6b6b        |.|o7 : kk
│ │ │ │ +000179d0: 5b58 202e 2e58 205d 2d6d 6f64 756c 652c  [X ..X ]-module,
│ │ │ │ +000179e0: 2066 7265 652c 2064 6567 7265 6573 207b   free, degrees {
│ │ │ │ +000179f0: 302e 2e31 2c20 323a 312c 2033 3a32 2c20  0..1, 2:1, 3:2, 
│ │ │ │ +00017a00: 337d 7c0a 7c20 2020 2020 2020 2020 3020  3}|.|         0 
│ │ │ │ +00017a10: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  00017a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00017a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017a40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00017a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017a80: 2d2d 2d2d 2d2b 0a7c 6938 203a 2061 7070  -----+.|i8 : app
│ │ │ │ -00017a90: 6c79 2874 6f4c 6973 7428 302e 2e31 3029  ly(toList(0..10)
│ │ │ │ -00017aa0: 2c20 692d 3e68 696c 6265 7274 4675 6e63  , i->hilbertFunc
│ │ │ │ -00017ab0: 7469 6f6e 2869 2c20 4529 2920 2020 2020  tion(i, E))     
│ │ │ │ -00017ac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00017a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +00017a80: 3a20 6170 706c 7928 746f 4c69 7374 2830  : apply(toList(0
│ │ │ │ +00017a90: 2e2e 3130 292c 2069 2d3e 6869 6c62 6572  ..10), i->hilber
│ │ │ │ +00017aa0: 7446 756e 6374 696f 6e28 692c 2045 2929  tFunction(i, E))
│ │ │ │ +00017ab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017b00: 6f38 203d 207b 312c 2033 2c20 352c 2037  o8 = {1, 3, 5, 7
│ │ │ │ -00017b10: 2c20 392c 2031 312c 2031 332c 2031 352c  , 9, 11, 13, 15,
│ │ │ │ -00017b20: 2031 372c 2031 392c 2032 317d 2020 2020   17, 19, 21}    
│ │ │ │ -00017b30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017af0: 2020 7c0a 7c6f 3820 3d20 7b31 2c20 332c    |.|o8 = {1, 3,
│ │ │ │ +00017b00: 2035 2c20 372c 2039 2c20 3131 2c20 3133   5, 7, 9, 11, 13
│ │ │ │ +00017b10: 2c20 3135 2c20 3137 2c20 3139 2c20 3231  , 15, 17, 19, 21
│ │ │ │ +00017b20: 7d20 2020 2020 2020 2020 2020 2020 7c0a  }             |.
│ │ │ │ +00017b30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b70: 2020 2020 207c 0a7c 6f38 203a 204c 6973       |.|o8 : Lis
│ │ │ │ -00017b80: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00017b60: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00017b70: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bb0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
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│ │ │ │ +00017bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00017bf0: 6939 203a 2045 6576 656e 203d 2065 7665  i9 : Eeven = eve
│ │ │ │ -00017c00: 6e45 7874 4d6f 6475 6c65 204d 3220 2020  nExtModule M2   
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│ │ │ │ -00017c20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017be0: 2d2d 2b0a 7c69 3920 3a20 4565 7665 6e20  --+.|i9 : Eeven 
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│ │ │ │ +00017c00: 4d32 2020 2020 2020 2020 2020 2020 2020  M2              
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│ │ │ │  00017c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017c70: 2020 2020 2020 2020 2034 2020 2020 2020           4      
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│ │ │ │ +00017c60: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ca0: 207c 0a7c 6f39 203d 2028 6b6b 5b58 202e   |.|o9 = (kk[X .
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│ │ │ │ +00017c90: 2020 2020 2020 7c0a 7c6f 3920 3d20 286b        |.|o9 = (k
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│ │ │ │ +00017cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017ce0: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
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│ │ │ │ +00017ce0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017d00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017d10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d50: 2020 2020 207c 0a7c 6f39 203a 206b 6b5b       |.|o9 : kk[
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│ │ │ │ -00017d90: 207c 0a7c 2020 2020 2020 2020 2030 2020   |.|         0  
│ │ │ │ -00017da0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00017d40: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
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│ │ │ │ +00017d80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017d90: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ +00017da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017dc0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00017dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00017dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
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│ │ │ │ +00017df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00017e00: 7c69 3130 203a 2061 7070 6c79 2874 6f4c  |i10 : apply(toL
│ │ │ │ +00017e10: 6973 7428 302e 2e35 292c 2069 2d3e 6869  ist(0..5), i->hi
│ │ │ │ +00017e20: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
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│ │ │ │ +00017e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e80: 207c 0a7c 6f31 3020 3d20 7b31 2c20 352c   |.|o10 = {1, 5,
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│ │ │ │ +00017e70: 2020 2020 2020 7c0a 7c6f 3130 203d 207b        |.|o10 = {
│ │ │ │ +00017e80: 312c 2035 2c20 392c 2031 332c 2031 372c  1, 5, 9, 13, 17,
│ │ │ │ +00017e90: 2032 317d 2020 2020 2020 2020 2020 2020   21}            
│ │ │ │  00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017eb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ef0: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -00017f00: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00017ee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017ef0: 7c6f 3130 203a 204c 6973 7420 2020 2020  |o10 : List     
│ │ │ │ +00017f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00017f20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00017f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f70: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -00017f80: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00017f90: 2045 7874 4d6f 6475 6c65 3a20 4578 744d   ExtModule: ExtM
│ │ │ │ -00017fa0: 6f64 756c 652c 202d 2d20 4578 745e 2a28  odule, -- Ext^*(
│ │ │ │ -00017fb0: 4d2c 6b29 206f 7665 7220 6120 636f 6d70  M,k) over a comp
│ │ │ │ -00017fc0: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ -00017fd0: 6e20 6173 0a20 2020 206d 6f64 756c 6520  n as.    module 
│ │ │ │ -00017fe0: 6f76 6572 2043 4920 6f70 6572 6174 6f72  over CI operator
│ │ │ │ -00017ff0: 2072 696e 670a 2020 2a20 2a6e 6f74 6520   ring.  * *note 
│ │ │ │ -00018000: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64  oddExtModule: od
│ │ │ │ -00018010: 6445 7874 4d6f 6475 6c65 2c20 2d2d 206f  dExtModule, -- o
│ │ │ │ -00018020: 6464 2070 6172 7420 6f66 2045 7874 5e2a  dd part of Ext^*
│ │ │ │ -00018030: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d  (M,k) over a com
│ │ │ │ -00018040: 706c 6574 650a 2020 2020 696e 7465 7273  plete.    inters
│ │ │ │ -00018050: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65  ection as module
│ │ │ │ -00018060: 206f 7665 7220 4349 206f 7065 7261 746f   over CI operato
│ │ │ │ -00018070: 7220 7269 6e67 0a20 202a 202a 6e6f 7465  r ring.  * *note
│ │ │ │ -00018080: 204f 7574 5269 6e67 3a20 4f75 7452 696e   OutRing: OutRin
│ │ │ │ -00018090: 672c 202d 2d20 4f70 7469 6f6e 2061 6c6c  g, -- Option all
│ │ │ │ -000180a0: 6f77 696e 6720 7370 6563 6966 6963 6174  owing specificat
│ │ │ │ -000180b0: 696f 6e20 6f66 2074 6865 2072 696e 6720  ion of the ring 
│ │ │ │ -000180c0: 6f76 6572 0a20 2020 2077 6869 6368 2074  over.    which t
│ │ │ │ -000180d0: 6865 206f 7574 7075 7420 6973 2064 6566  he output is def
│ │ │ │ -000180e0: 696e 6564 0a0a 5761 7973 2074 6f20 7573  ined..Ways to us
│ │ │ │ -000180f0: 6520 6576 656e 4578 744d 6f64 756c 653a  e evenExtModule:
│ │ │ │ -00018100: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00018110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00018120: 2022 6576 656e 4578 744d 6f64 756c 6528   "evenExtModule(
│ │ │ │ -00018130: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ -00018140: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00018150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00018160: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00018170: 6520 6576 656e 4578 744d 6f64 756c 653a  e evenExtModule:
│ │ │ │ -00018180: 2065 7665 6e45 7874 4d6f 6475 6c65 2c20   evenExtModule, 
│ │ │ │ -00018190: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -000181a0: 6420 6675 6e63 7469 6f6e 2077 6974 680a  d function with.
│ │ │ │ -000181b0: 6f70 7469 6f6e 733a 2028 4d61 6361 756c  options: (Macaul
│ │ │ │ -000181c0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -000181d0: 6374 696f 6e57 6974 684f 7074 696f 6e73  ctionWithOptions
│ │ │ │ -000181e0: 2c2e 0a1f 0a46 696c 653a 2043 6f6d 706c  ,....File: Compl
│ │ │ │ -000181f0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -00018200: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -00018210: 204e 6f64 653a 2065 7870 6f2c 204e 6578   Node: expo, Nex
│ │ │ │ -00018220: 743a 2065 7874 6572 696f 7245 7874 4d6f  t: exteriorExtMo
│ │ │ │ -00018230: 6475 6c65 2c20 5072 6576 3a20 6576 656e  dule, Prev: even
│ │ │ │ -00018240: 4578 744d 6f64 756c 652c 2055 703a 2054  ExtModule, Up: T
│ │ │ │ -00018250: 6f70 0a0a 6578 706f 202d 2d20 7265 7475  op..expo -- retu
│ │ │ │ -00018260: 726e 7320 6120 7365 7420 636f 7272 6573  rns a set corres
│ │ │ │ -00018270: 706f 6e64 696e 6720 746f 2074 6865 2062  ponding to the b
│ │ │ │ -00018280: 6173 6973 206f 6620 6120 6469 7669 6465  asis of a divide
│ │ │ │ -00018290: 6420 706f 7765 720a 2a2a 2a2a 2a2a 2a2a  d power.********
│ │ │ │ +00017f60: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00017f70: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00017f80: 2a6e 6f74 6520 4578 744d 6f64 756c 653a  *note ExtModule:
│ │ │ │ +00017f90: 2045 7874 4d6f 6475 6c65 2c20 2d2d 2045   ExtModule, -- E
│ │ │ │ +00017fa0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ +00017fb0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +00017fc0: 6563 7469 6f6e 2061 730a 2020 2020 6d6f  ection as.    mo
│ │ │ │ +00017fd0: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ +00017fe0: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ +00017ff0: 6e6f 7465 206f 6464 4578 744d 6f64 756c  note oddExtModul
│ │ │ │ +00018000: 653a 206f 6464 4578 744d 6f64 756c 652c  e: oddExtModule,
│ │ │ │ +00018010: 202d 2d20 6f64 6420 7061 7274 206f 6620   -- odd part of 
│ │ │ │ +00018020: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ +00018030: 6120 636f 6d70 6c65 7465 0a20 2020 2069  a complete.    i
│ │ │ │ +00018040: 6e74 6572 7365 6374 696f 6e20 6173 206d  ntersection as m
│ │ │ │ +00018050: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ +00018060: 6572 6174 6f72 2072 696e 670a 2020 2a20  erator ring.  * 
│ │ │ │ +00018070: 2a6e 6f74 6520 4f75 7452 696e 673a 204f  *note OutRing: O
│ │ │ │ +00018080: 7574 5269 6e67 2c20 2d2d 204f 7074 696f  utRing, -- Optio
│ │ │ │ +00018090: 6e20 616c 6c6f 7769 6e67 2073 7065 6369  n allowing speci
│ │ │ │ +000180a0: 6669 6361 7469 6f6e 206f 6620 7468 6520  fication of the 
│ │ │ │ +000180b0: 7269 6e67 206f 7665 720a 2020 2020 7768  ring over.    wh
│ │ │ │ +000180c0: 6963 6820 7468 6520 6f75 7470 7574 2069  ich the output i
│ │ │ │ +000180d0: 7320 6465 6669 6e65 640a 0a57 6179 7320  s defined..Ways 
│ │ │ │ +000180e0: 746f 2075 7365 2065 7665 6e45 7874 4d6f  to use evenExtMo
│ │ │ │ +000180f0: 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  dule:.==========
│ │ │ │ +00018100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00018110: 0a0a 2020 2a20 2265 7665 6e45 7874 4d6f  ..  * "evenExtMo
│ │ │ │ +00018120: 6475 6c65 284d 6f64 756c 6529 220a 0a46  dule(Module)"..F
│ │ │ │ +00018130: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +00018140: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +00018150: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +00018160: 202a 6e6f 7465 2065 7665 6e45 7874 4d6f   *note evenExtMo
│ │ │ │ +00018170: 6475 6c65 3a20 6576 656e 4578 744d 6f64  dule: evenExtMod
│ │ │ │ +00018180: 756c 652c 2069 7320 6120 2a6e 6f74 6520  ule, is a *note 
│ │ │ │ +00018190: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ +000181a0: 7769 7468 0a6f 7074 696f 6e73 3a20 284d  with.options: (M
│ │ │ │ +000181b0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +000181c0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ +000181d0: 7469 6f6e 732c 2e0a 1f0a 4669 6c65 3a20  tions,....File: 
│ │ │ │ +000181e0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +000181f0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +00018200: 696e 666f 2c20 4e6f 6465 3a20 6578 706f  info, Node: expo
│ │ │ │ +00018210: 2c20 4e65 7874 3a20 6578 7465 7269 6f72  , Next: exterior
│ │ │ │ +00018220: 4578 744d 6f64 756c 652c 2050 7265 763a  ExtModule, Prev:
│ │ │ │ +00018230: 2065 7665 6e45 7874 4d6f 6475 6c65 2c20   evenExtModule, 
│ │ │ │ +00018240: 5570 3a20 546f 700a 0a65 7870 6f20 2d2d  Up: Top..expo --
│ │ │ │ +00018250: 2072 6574 7572 6e73 2061 2073 6574 2063   returns a set c
│ │ │ │ +00018260: 6f72 7265 7370 6f6e 6469 6e67 2074 6f20  orresponding to 
│ │ │ │ +00018270: 7468 6520 6261 7369 7320 6f66 2061 2064  the basis of a d
│ │ │ │ +00018280: 6976 6964 6564 2070 6f77 6572 0a2a 2a2a  ivided power.***
│ │ │ │ +00018290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000182a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000182b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000182c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000182d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ -000182e0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ -000182f0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00018300: 2020 2020 4220 3d20 6578 706f 2863 2c4e      B = expo(c,N
│ │ │ │ -00018310: 290a 2020 2020 2020 2020 4220 3d20 6578  ).        B = ex
│ │ │ │ -00018320: 706f 2863 2c4c 290a 2020 2a20 496e 7075  po(c,L).  * Inpu
│ │ │ │ -00018330: 7473 3a0a 2020 2020 2020 2a20 4e2c 2061  ts:.      * N, a
│ │ │ │ -00018340: 6e20 2a6e 6f74 6520 696e 7465 6765 723a  n *note integer:
│ │ │ │ -00018350: 2028 4d61 6361 756c 6179 3244 6f63 295a   (Macaulay2Doc)Z
│ │ │ │ -00018360: 5a2c 2c20 0a20 2020 2020 202a 2063 2c20  Z,, .      * c, 
│ │ │ │ -00018370: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ -00018380: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00018390: 5a5a 2c2c 200a 2020 2020 2020 2a20 4c2c  ZZ,, .      * L,
│ │ │ │ -000183a0: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ -000183b0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ -000183c0: 742c 2c20 6f66 2063 206e 6f6e 2d6e 6567  t,, of c non-neg
│ │ │ │ -000183d0: 6174 6976 6520 696e 7465 6765 7273 0a20  ative integers. 
│ │ │ │ -000183e0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -000183f0: 2020 2a20 422c 2061 202a 6e6f 7465 206c    * B, a *note l
│ │ │ │ -00018400: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ -00018410: 6f63 294c 6973 742c 2c20 7061 7274 6974  oc)List,, partit
│ │ │ │ -00018420: 696f 6e73 2077 6974 6820 6320 6e6f 6e2d  ions with c non-
│ │ │ │ -00018430: 6e65 6761 7469 7665 0a20 2020 2020 2020  negative.       
│ │ │ │ -00018440: 2070 6172 7473 0a0a 4465 7363 7269 7074   parts..Descript
│ │ │ │ -00018450: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00018460: 0a54 6865 2066 6f72 6d20 6578 706f 2863  .The form expo(c
│ │ │ │ -00018470: 2c4e 2920 7265 7475 726e 7320 7061 7274  ,N) returns part
│ │ │ │ -00018480: 6974 696f 6e73 206f 6620 4e20 7769 7468  itions of N with
│ │ │ │ -00018490: 2063 206e 6f6e 2d6e 6567 6174 6976 6520   c non-negative 
│ │ │ │ -000184a0: 7061 7274 732e 2054 6865 2066 6f72 6d0a  parts. The form.
│ │ │ │ -000184b0: 6578 706f 2863 2c20 4c29 2072 6574 7572  expo(c, L) retur
│ │ │ │ -000184c0: 6e73 2070 6172 7469 7469 6f6e 7320 7769  ns partitions wi
│ │ │ │ -000184d0: 7468 206e 6f6e 2d6e 6567 6174 6976 6520  th non-negative 
│ │ │ │ -000184e0: 7061 7274 7320 7468 6174 2061 7265 2063  parts that are c
│ │ │ │ -000184f0: 6f6d 706f 6e65 6e74 7769 7365 203c 3d0a  omponentwise <=.
│ │ │ │ -00018500: 4c20 2861 6e64 2061 6e79 2073 756d 203c  L (and any sum <
│ │ │ │ -00018510: 3d20 7375 6d20 4c29 2e0a 0a54 6865 206c  = sum L)...The l
│ │ │ │ -00018520: 6973 7420 6578 706f 2863 2c4e 2920 206d  ist expo(c,N)  m
│ │ │ │ -00018530: 6179 2062 6520 7468 6f75 6768 7420 6f66  ay be thought of
│ │ │ │ -00018540: 2061 7320 7468 6520 6c69 7374 206f 6620   as the list of 
│ │ │ │ -00018550: 6578 706f 6e65 6e74 2076 6563 746f 7273  exponent vectors
│ │ │ │ -00018560: 206f 6620 7468 650a 6d6f 6e6f 6d69 616c   of the.monomial
│ │ │ │ -00018570: 7320 6f66 2064 6567 7265 6520 4e20 696e  s of degree N in
│ │ │ │ -00018580: 2063 2076 6172 6961 626c 6573 2e20 5468   c variables. Th
│ │ │ │ -00018590: 6973 2069 7320 7573 6564 2069 6e20 7468  is is used in th
│ │ │ │ -000185a0: 6520 636f 6e73 7472 7563 7469 6f6e 206f  e construction o
│ │ │ │ -000185b0: 6620 7468 650a 4569 7365 6e62 7564 2d53  f the.Eisenbud-S
│ │ │ │ -000185c0: 6861 6d61 7368 2072 6573 6f6c 7574 696f  hamash resolutio
│ │ │ │ -000185d0: 6e2e 0a0a 5468 6520 6c69 7374 2065 7870  n...The list exp
│ │ │ │ -000185e0: 6f28 632c 204c 292c 206f 6e20 7468 6520  o(c, L), on the 
│ │ │ │ -000185f0: 6f74 6865 7220 6861 6e64 2c20 6d61 7920  other hand, may 
│ │ │ │ -00018600: 6265 2074 686f 7567 6874 206f 6620 6173  be thought of as
│ │ │ │ -00018610: 2074 6865 206c 6973 7420 6f66 0a64 6976   the list of.div
│ │ │ │ -00018620: 6973 6f72 7320 6f66 2065 5e4c 203d 2065  isors of e^L = e
│ │ │ │ -00018630: 5f30 5e7b 4c5f 307d 202e 2e2e 2065 5f63  _0^{L_0} ... e_c
│ │ │ │ -00018640: 5e7b 4c5f 637d 2e20 5468 6973 2069 7320  ^{L_c}. This is 
│ │ │ │ -00018650: 7573 6564 2069 6e20 7468 6520 636f 6e73  used in the cons
│ │ │ │ -00018660: 7472 7563 7469 6f6e 206f 660a 7468 6520  truction of.the 
│ │ │ │ -00018670: 6869 6768 6572 2068 6f6d 6f74 6f70 6965  higher homotopie
│ │ │ │ -00018680: 7320 6f6e 2061 2063 6f6d 706c 6578 2e0a  s on a complex..
│ │ │ │ -00018690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000182d0: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ +000182e0: 3d3d 3d0a 0a20 202a 2055 7361 6765 3a20  ===..  * Usage: 
│ │ │ │ +000182f0: 0a20 2020 2020 2020 2042 203d 2065 7870  .        B = exp
│ │ │ │ +00018300: 6f28 632c 4e29 0a20 2020 2020 2020 2042  o(c,N).        B
│ │ │ │ +00018310: 203d 2065 7870 6f28 632c 4c29 0a20 202a   = expo(c,L).  *
│ │ │ │ +00018320: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00018330: 204e 2c20 616e 202a 6e6f 7465 2069 6e74   N, an *note int
│ │ │ │ +00018340: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ +00018350: 446f 6329 5a5a 2c2c 200a 2020 2020 2020  Doc)ZZ,, .      
│ │ │ │ +00018360: 2a20 632c 2061 6e20 2a6e 6f74 6520 696e  * c, an *note in
│ │ │ │ +00018370: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ +00018380: 3244 6f63 295a 5a2c 2c20 0a20 2020 2020  2Doc)ZZ,, .     
│ │ │ │ +00018390: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69   * L, a *note li
│ │ │ │ +000183a0: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +000183b0: 6329 4c69 7374 2c2c 206f 6620 6320 6e6f  c)List,, of c no
│ │ │ │ +000183c0: 6e2d 6e65 6761 7469 7665 2069 6e74 6567  n-negative integ
│ │ │ │ +000183d0: 6572 730a 2020 2a20 4f75 7470 7574 733a  ers.  * Outputs:
│ │ │ │ +000183e0: 0a20 2020 2020 202a 2042 2c20 6120 2a6e  .      * B, a *n
│ │ │ │ +000183f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00018400: 6c61 7932 446f 6329 4c69 7374 2c2c 2070  lay2Doc)List,, p
│ │ │ │ +00018410: 6172 7469 7469 6f6e 7320 7769 7468 2063  artitions with c
│ │ │ │ +00018420: 206e 6f6e 2d6e 6567 6174 6976 650a 2020   non-negative.  
│ │ │ │ +00018430: 2020 2020 2020 7061 7274 730a 0a44 6573        parts..Des
│ │ │ │ +00018440: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00018450: 3d3d 3d3d 0a0a 5468 6520 666f 726d 2065  ====..The form e
│ │ │ │ +00018460: 7870 6f28 632c 4e29 2072 6574 7572 6e73  xpo(c,N) returns
│ │ │ │ +00018470: 2070 6172 7469 7469 6f6e 7320 6f66 204e   partitions of N
│ │ │ │ +00018480: 2077 6974 6820 6320 6e6f 6e2d 6e65 6761   with c non-nega
│ │ │ │ +00018490: 7469 7665 2070 6172 7473 2e20 5468 6520  tive parts. The 
│ │ │ │ +000184a0: 666f 726d 0a65 7870 6f28 632c 204c 2920  form.expo(c, L) 
│ │ │ │ +000184b0: 7265 7475 726e 7320 7061 7274 6974 696f  returns partitio
│ │ │ │ +000184c0: 6e73 2077 6974 6820 6e6f 6e2d 6e65 6761  ns with non-nega
│ │ │ │ +000184d0: 7469 7665 2070 6172 7473 2074 6861 7420  tive parts that 
│ │ │ │ +000184e0: 6172 6520 636f 6d70 6f6e 656e 7477 6973  are componentwis
│ │ │ │ +000184f0: 6520 3c3d 0a4c 2028 616e 6420 616e 7920  e <=.L (and any 
│ │ │ │ +00018500: 7375 6d20 3c3d 2073 756d 204c 292e 0a0a  sum <= sum L)...
│ │ │ │ +00018510: 5468 6520 6c69 7374 2065 7870 6f28 632c  The list expo(c,
│ │ │ │ +00018520: 4e29 2020 6d61 7920 6265 2074 686f 7567  N)  may be thoug
│ │ │ │ +00018530: 6874 206f 6620 6173 2074 6865 206c 6973  ht of as the lis
│ │ │ │ +00018540: 7420 6f66 2065 7870 6f6e 656e 7420 7665  t of exponent ve
│ │ │ │ +00018550: 6374 6f72 7320 6f66 2074 6865 0a6d 6f6e  ctors of the.mon
│ │ │ │ +00018560: 6f6d 6961 6c73 206f 6620 6465 6772 6565  omials of degree
│ │ │ │ +00018570: 204e 2069 6e20 6320 7661 7269 6162 6c65   N in c variable
│ │ │ │ +00018580: 732e 2054 6869 7320 6973 2075 7365 6420  s. This is used 
│ │ │ │ +00018590: 696e 2074 6865 2063 6f6e 7374 7275 6374  in the construct
│ │ │ │ +000185a0: 696f 6e20 6f66 2074 6865 0a45 6973 656e  ion of the.Eisen
│ │ │ │ +000185b0: 6275 642d 5368 616d 6173 6820 7265 736f  bud-Shamash reso
│ │ │ │ +000185c0: 6c75 7469 6f6e 2e0a 0a54 6865 206c 6973  lution...The lis
│ │ │ │ +000185d0: 7420 6578 706f 2863 2c20 4c29 2c20 6f6e  t expo(c, L), on
│ │ │ │ +000185e0: 2074 6865 206f 7468 6572 2068 616e 642c   the other hand,
│ │ │ │ +000185f0: 206d 6179 2062 6520 7468 6f75 6768 7420   may be thought 
│ │ │ │ +00018600: 6f66 2061 7320 7468 6520 6c69 7374 206f  of as the list o
│ │ │ │ +00018610: 660a 6469 7669 736f 7273 206f 6620 655e  f.divisors of e^
│ │ │ │ +00018620: 4c20 3d20 655f 305e 7b4c 5f30 7d20 2e2e  L = e_0^{L_0} ..
│ │ │ │ +00018630: 2e20 655f 635e 7b4c 5f63 7d2e 2054 6869  . e_c^{L_c}. Thi
│ │ │ │ +00018640: 7320 6973 2075 7365 6420 696e 2074 6865  s is used in the
│ │ │ │ +00018650: 2063 6f6e 7374 7275 6374 696f 6e20 6f66   construction of
│ │ │ │ +00018660: 0a74 6865 2068 6967 6865 7220 686f 6d6f  .the higher homo
│ │ │ │ +00018670: 746f 7069 6573 206f 6e20 6120 636f 6d70  topies on a comp
│ │ │ │ +00018680: 6c65 782e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  lex...+---------
│ │ │ │ +00018690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000186d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000186e0: 0a7c 6931 203a 2065 7870 6f28 332c 3529  .|i1 : expo(3,5)
│ │ │ │ +000186d0: 2d2d 2d2d 2b0a 7c69 3120 3a20 6578 706f  ----+.|i1 : expo
│ │ │ │ +000186e0: 2833 2c35 2920 2020 2020 2020 2020 2020  (3,5)           
│ │ │ │  000186f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018780: 0a7c 6f31 203d 207b 7b35 2c20 302c 2030  .|o1 = {{5, 0, 0
│ │ │ │ -00018790: 7d2c 207b 342c 2031 2c20 307d 2c20 7b34  }, {4, 1, 0}, {4
│ │ │ │ -000187a0: 2c20 302c 2031 7d2c 207b 332c 2032 2c20  , 0, 1}, {3, 2, 
│ │ │ │ -000187b0: 307d 2c20 7b33 2c20 312c 2031 7d2c 207b  0}, {3, 1, 1}, {
│ │ │ │ -000187c0: 332c 2030 2c20 327d 2c20 7b32 2c20 207c  3, 0, 2}, {2,  |
│ │ │ │ -000187d0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00018770: 2020 2020 7c0a 7c6f 3120 3d20 7b7b 352c      |.|o1 = {{5,
│ │ │ │ +00018780: 2030 2c20 307d 2c20 7b34 2c20 312c 2030   0, 0}, {4, 1, 0
│ │ │ │ +00018790: 7d2c 207b 342c 2030 2c20 317d 2c20 7b33  }, {4, 0, 1}, {3
│ │ │ │ +000187a0: 2c20 322c 2030 7d2c 207b 332c 2031 2c20  , 2, 0}, {3, 1, 
│ │ │ │ +000187b0: 317d 2c20 7b33 2c20 302c 2032 7d2c 207b  1}, {3, 0, 2}, {
│ │ │ │ +000187c0: 322c 2020 7c0a 7c20 2020 2020 2d2d 2d2d  2,  |.|     ----
│ │ │ │ +000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00018820: 0a7c 2020 2020 2033 2c20 307d 2c20 7b32  .|     3, 0}, {2
│ │ │ │ -00018830: 2c20 322c 2031 7d2c 207b 322c 2031 2c20  , 2, 1}, {2, 1, 
│ │ │ │ -00018840: 327d 2c20 7b32 2c20 302c 2033 7d2c 207b  2}, {2, 0, 3}, {
│ │ │ │ -00018850: 312c 2034 2c20 307d 2c20 7b31 2c20 332c  1, 4, 0}, {1, 3,
│ │ │ │ -00018860: 2031 7d2c 207b 312c 2032 2c20 327d 2c7c   1}, {1, 2, 2},|
│ │ │ │ -00018870: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00018810: 2d2d 2d2d 7c0a 7c20 2020 2020 332c 2030  ----|.|     3, 0
│ │ │ │ +00018820: 7d2c 207b 322c 2032 2c20 317d 2c20 7b32  }, {2, 2, 1}, {2
│ │ │ │ +00018830: 2c20 312c 2032 7d2c 207b 322c 2030 2c20  , 1, 2}, {2, 0, 
│ │ │ │ +00018840: 337d 2c20 7b31 2c20 342c 2030 7d2c 207b  3}, {1, 4, 0}, {
│ │ │ │ +00018850: 312c 2033 2c20 317d 2c20 7b31 2c20 322c  1, 3, 1}, {1, 2,
│ │ │ │ +00018860: 2032 7d2c 7c0a 7c20 2020 2020 2d2d 2d2d   2},|.|     ----
│ │ │ │ +00018870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000188a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000188b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000188c0: 0a7c 2020 2020 207b 312c 2031 2c20 337d  .|     {1, 1, 3}
│ │ │ │ -000188d0: 2c20 7b31 2c20 302c 2034 7d2c 207b 302c  , {1, 0, 4}, {0,
│ │ │ │ -000188e0: 2035 2c20 307d 2c20 7b30 2c20 342c 2031   5, 0}, {0, 4, 1
│ │ │ │ -000188f0: 7d2c 207b 302c 2033 2c20 327d 2c20 7b30  }, {0, 3, 2}, {0
│ │ │ │ -00018900: 2c20 322c 2033 7d2c 207b 302c 2031 2c7c  , 2, 3}, {0, 1,|
│ │ │ │ -00018910: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +000188b0: 2d2d 2d2d 7c0a 7c20 2020 2020 7b31 2c20  ----|.|     {1, 
│ │ │ │ +000188c0: 312c 2033 7d2c 207b 312c 2030 2c20 347d  1, 3}, {1, 0, 4}
│ │ │ │ +000188d0: 2c20 7b30 2c20 352c 2030 7d2c 207b 302c  , {0, 5, 0}, {0,
│ │ │ │ +000188e0: 2034 2c20 317d 2c20 7b30 2c20 332c 2032   4, 1}, {0, 3, 2
│ │ │ │ +000188f0: 7d2c 207b 302c 2032 2c20 337d 2c20 7b30  }, {0, 2, 3}, {0
│ │ │ │ +00018900: 2c20 312c 7c0a 7c20 2020 2020 2d2d 2d2d  , 1,|.|     ----
│ │ │ │ +00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00018960: 0a7c 2020 2020 2034 7d2c 207b 302c 2030  .|     4}, {0, 0
│ │ │ │ -00018970: 2c20 357d 7d20 2020 2020 2020 2020 2020  , 5}}           
│ │ │ │ +00018950: 2d2d 2d2d 7c0a 7c20 2020 2020 347d 2c20  ----|.|     4}, 
│ │ │ │ +00018960: 7b30 2c20 302c 2035 7d7d 2020 2020 2020  {0, 0, 5}}      
│ │ │ │ +00018970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000189a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000189b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000189a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000189b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000189c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000189d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000189e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000189f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018a00: 0a7c 6f31 203a 204c 6973 7420 2020 2020  .|o1 : List     
│ │ │ │ +000189f0: 2020 2020 7c0a 7c6f 3120 3a20 4c69 7374      |.|o1 : List
│ │ │ │ +00018a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00018a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00018a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00018aa0: 0a7c 6932 203a 2065 7870 6f28 332c 207b  .|i2 : expo(3, {
│ │ │ │ -00018ab0: 332c 322c 317d 2920 2020 2020 2020 2020  3,2,1})         
│ │ │ │ +00018a90: 2d2d 2d2d 2b0a 7c69 3220 3a20 6578 706f  ----+.|i2 : expo
│ │ │ │ +00018aa0: 2833 2c20 7b33 2c32 2c31 7d29 2020 2020  (3, {3,2,1})    
│ │ │ │ +00018ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018b40: 0a7c 6f32 203d 207b 7b30 2c20 302c 2030  .|o2 = {{0, 0, 0
│ │ │ │ -00018b50: 7d2c 207b 312c 2030 2c20 307d 2c20 7b30  }, {1, 0, 0}, {0
│ │ │ │ -00018b60: 2c20 312c 2030 7d2c 207b 302c 2030 2c20  , 1, 0}, {0, 0, 
│ │ │ │ -00018b70: 317d 2c20 7b32 2c20 302c 2030 7d2c 207b  1}, {2, 0, 0}, {
│ │ │ │ -00018b80: 312c 2031 2c20 307d 2c20 7b31 2c20 207c  1, 1, 0}, {1,  |
│ │ │ │ -00018b90: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00018b30: 2020 2020 7c0a 7c6f 3220 3d20 7b7b 302c      |.|o2 = {{0,
│ │ │ │ +00018b40: 2030 2c20 307d 2c20 7b31 2c20 302c 2030   0, 0}, {1, 0, 0
│ │ │ │ +00018b50: 7d2c 207b 302c 2031 2c20 307d 2c20 7b30  }, {0, 1, 0}, {0
│ │ │ │ +00018b60: 2c20 302c 2031 7d2c 207b 322c 2030 2c20  , 0, 1}, {2, 0, 
│ │ │ │ +00018b70: 307d 2c20 7b31 2c20 312c 2030 7d2c 207b  0}, {1, 1, 0}, {
│ │ │ │ +00018b80: 312c 2020 7c0a 7c20 2020 2020 2d2d 2d2d  1,  |.|     ----
│ │ │ │ +00018b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00018be0: 0a7c 2020 2020 2030 2c20 317d 2c20 7b30  .|     0, 1}, {0
│ │ │ │ -00018bf0: 2c20 322c 2030 7d2c 207b 302c 2031 2c20  , 2, 0}, {0, 1, 
│ │ │ │ -00018c00: 317d 2c20 7b33 2c20 302c 2030 7d2c 207b  1}, {3, 0, 0}, {
│ │ │ │ -00018c10: 322c 2031 2c20 307d 2c20 7b32 2c20 302c  2, 1, 0}, {2, 0,
│ │ │ │ -00018c20: 2031 7d2c 207b 312c 2032 2c20 307d 2c7c   1}, {1, 2, 0},|
│ │ │ │ -00018c30: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00018bd0: 2d2d 2d2d 7c0a 7c20 2020 2020 302c 2031  ----|.|     0, 1
│ │ │ │ +00018be0: 7d2c 207b 302c 2032 2c20 307d 2c20 7b30  }, {0, 2, 0}, {0
│ │ │ │ +00018bf0: 2c20 312c 2031 7d2c 207b 332c 2030 2c20  , 1, 1}, {3, 0, 
│ │ │ │ +00018c00: 307d 2c20 7b32 2c20 312c 2030 7d2c 207b  0}, {2, 1, 0}, {
│ │ │ │ +00018c10: 322c 2030 2c20 317d 2c20 7b31 2c20 322c  2, 0, 1}, {1, 2,
│ │ │ │ +00018c20: 2030 7d2c 7c0a 7c20 2020 2020 2d2d 2d2d   0},|.|     ----
│ │ │ │ +00018c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00018c80: 0a7c 2020 2020 207b 312c 2031 2c20 317d  .|     {1, 1, 1}
│ │ │ │ -00018c90: 2c20 7b30 2c20 322c 2031 7d2c 207b 332c  , {0, 2, 1}, {3,
│ │ │ │ -00018ca0: 2031 2c20 307d 2c20 7b33 2c20 302c 2031   1, 0}, {3, 0, 1
│ │ │ │ -00018cb0: 7d2c 207b 322c 2032 2c20 307d 2c20 7b32  }, {2, 2, 0}, {2
│ │ │ │ -00018cc0: 2c20 312c 2031 7d2c 207b 312c 2032 2c7c  , 1, 1}, {1, 2,|
│ │ │ │ -00018cd0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00018c70: 2d2d 2d2d 7c0a 7c20 2020 2020 7b31 2c20  ----|.|     {1, 
│ │ │ │ +00018c80: 312c 2031 7d2c 207b 302c 2032 2c20 317d  1, 1}, {0, 2, 1}
│ │ │ │ +00018c90: 2c20 7b33 2c20 312c 2030 7d2c 207b 332c  , {3, 1, 0}, {3,
│ │ │ │ +00018ca0: 2030 2c20 317d 2c20 7b32 2c20 322c 2030   0, 1}, {2, 2, 0
│ │ │ │ +00018cb0: 7d2c 207b 322c 2031 2c20 317d 2c20 7b31  }, {2, 1, 1}, {1
│ │ │ │ +00018cc0: 2c20 322c 7c0a 7c20 2020 2020 2d2d 2d2d  , 2,|.|     ----
│ │ │ │ +00018cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00018d20: 0a7c 2020 2020 2031 7d2c 207b 332c 2032  .|     1}, {3, 2
│ │ │ │ -00018d30: 2c20 307d 2c20 7b33 2c20 312c 2031 7d2c  , 0}, {3, 1, 1},
│ │ │ │ -00018d40: 207b 322c 2032 2c20 317d 7d20 2020 2020   {2, 2, 1}}     
│ │ │ │ +00018d10: 2d2d 2d2d 7c0a 7c20 2020 2020 317d 2c20  ----|.|     1}, 
│ │ │ │ +00018d20: 7b33 2c20 322c 2030 7d2c 207b 332c 2031  {3, 2, 0}, {3, 1
│ │ │ │ +00018d30: 2c20 317d 2c20 7b32 2c20 322c 2031 7d7d  , 1}, {2, 2, 1}}
│ │ │ │ +00018d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018dc0: 0a7c 6f32 203a 204c 6973 7420 2020 2020  .|o2 : List     
│ │ │ │ +00018db0: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │ +00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00018e10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00018e00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00018e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00018e60: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00018e70: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2045  ===..  * *note E
│ │ │ │ -00018e80: 6973 656e 6275 6453 6861 6d61 7368 3a20  isenbudShamash: 
│ │ │ │ -00018e90: 4569 7365 6e62 7564 5368 616d 6173 682c  EisenbudShamash,
│ │ │ │ -00018ea0: 202d 2d20 436f 6d70 7574 6573 2074 6865   -- Computes the
│ │ │ │ -00018eb0: 2045 6973 656e 6275 642d 5368 616d 6173   Eisenbud-Shamas
│ │ │ │ -00018ec0: 680a 2020 2020 436f 6d70 6c65 780a 2020  h.    Complex.  
│ │ │ │ -00018ed0: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f  * *note makeHomo
│ │ │ │ -00018ee0: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f  topies: makeHomo
│ │ │ │ -00018ef0: 746f 7069 6573 2c20 2d2d 2072 6574 7572  topies, -- retur
│ │ │ │ -00018f00: 6e73 2061 2073 7973 7465 6d20 6f66 2068  ns a system of h
│ │ │ │ -00018f10: 6967 6865 720a 2020 2020 686f 6d6f 746f  igher.    homoto
│ │ │ │ -00018f20: 7069 6573 0a0a 5761 7973 2074 6f20 7573  pies..Ways to us
│ │ │ │ -00018f30: 6520 6578 706f 3a0a 3d3d 3d3d 3d3d 3d3d  e expo:.========
│ │ │ │ -00018f40: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -00018f50: 6578 706f 285a 5a2c 4c69 7374 2922 0a20  expo(ZZ,List)". 
│ │ │ │ -00018f60: 202a 2022 6578 706f 285a 5a2c 5a5a 2922   * "expo(ZZ,ZZ)"
│ │ │ │ -00018f70: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -00018f80: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -00018f90: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -00018fa0: 6563 7420 2a6e 6f74 6520 6578 706f 3a20  ect *note expo: 
│ │ │ │ -00018fb0: 6578 706f 2c20 6973 2061 202a 6e6f 7465  expo, is a *note
│ │ │ │ -00018fc0: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ -00018fd0: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ -00018fe0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ -00018ff0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -00019000: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -00019010: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -00019020: 6f64 653a 2065 7874 6572 696f 7245 7874  ode: exteriorExt
│ │ │ │ -00019030: 4d6f 6475 6c65 2c20 4e65 7874 3a20 6578  Module, Next: ex
│ │ │ │ -00019040: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f  teriorHomologyMo
│ │ │ │ -00019050: 6475 6c65 2c20 5072 6576 3a20 6578 706f  dule, Prev: expo
│ │ │ │ -00019060: 2c20 5570 3a20 546f 700a 0a65 7874 6572  , Up: Top..exter
│ │ │ │ -00019070: 696f 7245 7874 4d6f 6475 6c65 202d 2d20  iorExtModule -- 
│ │ │ │ -00019080: 4578 7428 4d2c 6b29 206f 7220 4578 7428  Ext(M,k) or Ext(
│ │ │ │ -00019090: 4d2c 4e29 2061 7320 6120 6d6f 6475 6c65  M,N) as a module
│ │ │ │ -000190a0: 206f 7665 7220 616e 2065 7874 6572 696f   over an exterio
│ │ │ │ -000190b0: 7220 616c 6765 6272 610a 2a2a 2a2a 2a2a  r algebra.******
│ │ │ │ +00018e50: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ +00018e60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00018e70: 6f74 6520 4569 7365 6e62 7564 5368 616d  ote EisenbudSham
│ │ │ │ +00018e80: 6173 683a 2045 6973 656e 6275 6453 6861  ash: EisenbudSha
│ │ │ │ +00018e90: 6d61 7368 2c20 2d2d 2043 6f6d 7075 7465  mash, -- Compute
│ │ │ │ +00018ea0: 7320 7468 6520 4569 7365 6e62 7564 2d53  s the Eisenbud-S
│ │ │ │ +00018eb0: 6861 6d61 7368 0a20 2020 2043 6f6d 706c  hamash.    Compl
│ │ │ │ +00018ec0: 6578 0a20 202a 202a 6e6f 7465 206d 616b  ex.  * *note mak
│ │ │ │ +00018ed0: 6548 6f6d 6f74 6f70 6965 733a 206d 616b  eHomotopies: mak
│ │ │ │ +00018ee0: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20  eHomotopies, -- 
│ │ │ │ +00018ef0: 7265 7475 726e 7320 6120 7379 7374 656d  returns a system
│ │ │ │ +00018f00: 206f 6620 6869 6768 6572 0a20 2020 2068   of higher.    h
│ │ │ │ +00018f10: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320  omotopies..Ways 
│ │ │ │ +00018f20: 746f 2075 7365 2065 7870 6f3a 0a3d 3d3d  to use expo:.===
│ │ │ │ +00018f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00018f40: 2020 2a20 2265 7870 6f28 5a5a 2c4c 6973    * "expo(ZZ,Lis
│ │ │ │ +00018f50: 7429 220a 2020 2a20 2265 7870 6f28 5a5a  t)".  * "expo(ZZ
│ │ │ │ +00018f60: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070  ,ZZ)"..For the p
│ │ │ │ +00018f70: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00018f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00018f90: 6520 6f62 6a65 6374 202a 6e6f 7465 2065  e object *note e
│ │ │ │ +00018fa0: 7870 6f3a 2065 7870 6f2c 2069 7320 6120  xpo: expo, is a 
│ │ │ │ +00018fb0: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +00018fc0: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ +00018fd0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +00018fe0: 696f 6e2c 2e0a 1f0a 4669 6c65 3a20 436f  ion,....File: Co
│ │ │ │ +00018ff0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +00019000: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +00019010: 666f 2c20 4e6f 6465 3a20 6578 7465 7269  fo, Node: exteri
│ │ │ │ +00019020: 6f72 4578 744d 6f64 756c 652c 204e 6578  orExtModule, Nex
│ │ │ │ +00019030: 743a 2065 7874 6572 696f 7248 6f6d 6f6c  t: exteriorHomol
│ │ │ │ +00019040: 6f67 794d 6f64 756c 652c 2050 7265 763a  ogyModule, Prev:
│ │ │ │ +00019050: 2065 7870 6f2c 2055 703a 2054 6f70 0a0a   expo, Up: Top..
│ │ │ │ +00019060: 6578 7465 7269 6f72 4578 744d 6f64 756c  exteriorExtModul
│ │ │ │ +00019070: 6520 2d2d 2045 7874 284d 2c6b 2920 6f72  e -- Ext(M,k) or
│ │ │ │ +00019080: 2045 7874 284d 2c4e 2920 6173 2061 206d   Ext(M,N) as a m
│ │ │ │ +00019090: 6f64 756c 6520 6f76 6572 2061 6e20 6578  odule over an ex
│ │ │ │ +000190a0: 7465 7269 6f72 2061 6c67 6562 7261 0a2a  terior algebra.*
│ │ │ │ +000190b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000190c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000190d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000190e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000190f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019100: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ -00019110: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ -00019120: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00019130: 2045 203d 2065 7874 6572 696f 7245 7874   E = exteriorExt
│ │ │ │ -00019140: 4d6f 6475 6c65 2866 2c4d 290a 2020 2a20  Module(f,M).  * 
│ │ │ │ -00019150: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00019160: 662c 2061 202a 6e6f 7465 206d 6174 7269  f, a *note matri
│ │ │ │ -00019170: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -00019180: 294d 6174 7269 782c 2c20 3120 7820 632c  )Matrix,, 1 x c,
│ │ │ │ -00019190: 2065 6e74 7269 6573 206d 7573 7420 6265   entries must be
│ │ │ │ -000191a0: 0a20 2020 2020 2020 2068 6f6d 6f74 6f70  .        homotop
│ │ │ │ -000191b0: 6963 2074 6f20 3020 6f6e 2046 0a20 2020  ic to 0 on F.   
│ │ │ │ -000191c0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ -000191d0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ -000191e0: 7932 446f 6329 4d6f 6475 6c65 2c2c 2061  y2Doc)Module,, a
│ │ │ │ -000191f0: 6e6e 6968 696c 6174 6564 2062 7920 7468  nnihilated by th
│ │ │ │ -00019200: 6520 656c 656d 656e 7473 0a20 2020 2020  e elements.     
│ │ │ │ -00019210: 2020 206f 6620 6666 0a20 2020 2020 202a     of ff.      *
│ │ │ │ -00019220: 204e 2c20 6120 2a6e 6f74 6520 6d6f 6475   N, a *note modu
│ │ │ │ -00019230: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00019240: 6329 4d6f 6475 6c65 2c2c 2061 6e6e 6968  c)Module,, annih
│ │ │ │ -00019250: 696c 6174 6564 2062 7920 7468 6520 656c  ilated by the el
│ │ │ │ -00019260: 656d 656e 7473 0a20 2020 2020 2020 206f  ements.        o
│ │ │ │ -00019270: 6620 6666 0a20 202a 204f 7574 7075 7473  f ff.  * Outputs
│ │ │ │ -00019280: 3a0a 2020 2020 2020 2a20 452c 2061 202a  :.      * E, a *
│ │ │ │ -00019290: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -000192a0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -000192b0: 652c 2c20 4d6f 6475 6c65 206f 7665 7220  e,, Module over 
│ │ │ │ -000192c0: 616e 2065 7874 6572 696f 720a 2020 2020  an exterior.    
│ │ │ │ -000192d0: 2020 2020 616c 6765 6272 6120 7769 7468      algebra with
│ │ │ │ -000192e0: 2076 6172 6961 626c 6573 2063 6f72 7265   variables corre
│ │ │ │ -000192f0: 7370 6f6e 6469 6e67 2074 6f20 656c 656d  sponding to elem
│ │ │ │ -00019300: 656e 7473 206f 6620 660a 0a44 6573 6372  ents of f..Descr
│ │ │ │ -00019310: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00019320: 3d3d 0a0a 4966 204d 2c4e 2061 7265 2053  ==..If M,N are S
│ │ │ │ -00019330: 2d6d 6f64 756c 6573 2061 6e6e 6968 696c  -modules annihil
│ │ │ │ -00019340: 6174 6564 2062 7920 7468 6520 656c 656d  ated by the elem
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│ │ │ │ -000193d0: 6173 2061 206d 6f64 756c 6520 6f76 6572  as a module over
│ │ │ │ -000193e0: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
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│ │ │ │ -00019410: 6865 0a65 5f69 2068 6176 6520 6465 6772  he.e_i have degr
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│ │ │ │ -00019440: 616c 206f 6620 6578 7465 7269 6f72 546f  al of exteriorTo
│ │ │ │ -00019450: 724d 6f64 756c 652e 0a0a 5468 6520 7363  rModule...The sc
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│ │ │ │ -00019470: 4d6f 6475 6c65 2866 2c4d 2c4e 2920 7265  Module(f,M,N) re
│ │ │ │ -00019480: 7475 726e 7320 4578 745f 5328 4d2c 4e29  turns Ext_S(M,N)
│ │ │ │ -00019490: 2061 7320 6120 6d6f 6475 6c65 206f 7665   as a module ove
│ │ │ │ -000194a0: 7220 610a 6269 6772 6164 6564 2072 696e  r a.bigraded rin
│ │ │ │ -000194b0: 6720 5345 203d 2053 3c65 5f31 2c2e 2e2c  g SE = S, where the 
│ │ │ │ -000194d0: 655f 6920 6861 7665 2064 6567 7265 6573  e_i have degrees
│ │ │ │ -000194e0: 207b 645f 692c 317d 2c20 7768 6572 6520   {d_i,1}, where 
│ │ │ │ -000194f0: 645f 690a 6973 2074 6865 2064 6567 7265  d_i.is the degre
│ │ │ │ -00019500: 6520 6f66 2066 5f69 2e20 5468 6520 6d6f  e of f_i. The mo
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│ │ │ │ -00019520: 696e 2065 6974 6865 7220 6361 7365 2c20  in either case, 
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│ │ │ │ -00019550: 2074 6865 2066 5f69 206f 6e20 7468 6520   the f_i on the 
│ │ │ │ -00019560: 7265 736f 6c75 7469 6f6e 206f 6620 4d2c  resolution of M,
│ │ │ │ -00019570: 2063 6f6d 7075 7465 6420 6279 2074 6865   computed by the
│ │ │ │ -00019580: 2073 6372 6970 740a 6d61 6b65 486f 6d6f   script.makeHomo
│ │ │ │ -00019590: 746f 7069 6573 312e 5468 6520 7363 7269  topies1.The scri
│ │ │ │ -000195a0: 7074 2063 616c 6c73 206d 616b 654d 6f64  pt calls makeMod
│ │ │ │ -000195b0: 756c 6520 746f 2063 6f6d 7075 7465 2061  ule to compute a
│ │ │ │ -000195c0: 2028 6e6f 6e2d 6d69 6e69 6d61 6c29 0a70   (non-minimal).p
│ │ │ │ -000195d0: 7265 7365 6e74 6174 696f 6e20 6f66 2074  resentation of t
│ │ │ │ -000195e0: 6869 7320 6d6f 6475 6c65 2e0a 0a2b 2d2d  his module...+--
│ │ │ │ +000190f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ +00019100: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ +00019110: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +00019120: 2020 2020 2020 4520 3d20 6578 7465 7269        E = exteri
│ │ │ │ +00019130: 6f72 4578 744d 6f64 756c 6528 662c 4d29  orExtModule(f,M)
│ │ │ │ +00019140: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00019150: 2020 202a 2066 2c20 6120 2a6e 6f74 6520     * f, a *note 
│ │ │ │ +00019160: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ +00019170: 7932 446f 6329 4d61 7472 6978 2c2c 2031  y2Doc)Matrix,, 1
│ │ │ │ +00019180: 2078 2063 2c20 656e 7472 6965 7320 6d75   x c, entries mu
│ │ │ │ +00019190: 7374 2062 650a 2020 2020 2020 2020 686f  st be.        ho
│ │ │ │ +000191a0: 6d6f 746f 7069 6320 746f 2030 206f 6e20  motopic to 0 on 
│ │ │ │ +000191b0: 460a 2020 2020 2020 2a20 4d2c 2061 202a  F.      * M, a *
│ │ │ │ +000191c0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +000191d0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +000191e0: 652c 2c20 616e 6e69 6869 6c61 7465 6420  e,, annihilated 
│ │ │ │ +000191f0: 6279 2074 6865 2065 6c65 6d65 6e74 730a  by the elements.
│ │ │ │ +00019200: 2020 2020 2020 2020 6f66 2066 660a 2020          of ff.  
│ │ │ │ +00019210: 2020 2020 2a20 4e2c 2061 202a 6e6f 7465      * N, a *note
│ │ │ │ +00019220: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00019230: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00019240: 616e 6e69 6869 6c61 7465 6420 6279 2074  annihilated by t
│ │ │ │ +00019250: 6865 2065 6c65 6d65 6e74 730a 2020 2020  he elements.    
│ │ │ │ +00019260: 2020 2020 6f66 2066 660a 2020 2a20 4f75      of ff.  * Ou
│ │ │ │ +00019270: 7470 7574 733a 0a20 2020 2020 202a 2045  tputs:.      * E
│ │ │ │ +00019280: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +00019290: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000192a0: 4d6f 6475 6c65 2c2c 204d 6f64 756c 6520  Module,, Module 
│ │ │ │ +000192b0: 6f76 6572 2061 6e20 6578 7465 7269 6f72  over an exterior
│ │ │ │ +000192c0: 0a20 2020 2020 2020 2061 6c67 6562 7261  .        algebra
│ │ │ │ +000192d0: 2077 6974 6820 7661 7269 6162 6c65 7320   with variables 
│ │ │ │ +000192e0: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ +000192f0: 2065 6c65 6d65 6e74 7320 6f66 2066 0a0a   elements of f..
│ │ │ │ +00019300: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00019310: 3d3d 3d3d 3d3d 3d0a 0a49 6620 4d2c 4e20  =======..If M,N 
│ │ │ │ +00019320: 6172 6520 532d 6d6f 6475 6c65 7320 616e  are S-modules an
│ │ │ │ +00019330: 6e69 6869 6c61 7465 6420 6279 2074 6865  nihilated by the
│ │ │ │ +00019340: 2065 6c65 6d65 6e74 7320 6f66 2074 6865   elements of the
│ │ │ │ +00019350: 206d 6174 7269 7820 6666 203d 2028 665f   matrix ff = (f_
│ │ │ │ +00019360: 312e 2e66 5f63 292c 0a61 6e64 206b 2069  1..f_c),.and k i
│ │ │ │ +00019370: 7320 7468 6520 7265 7369 6475 6520 6669  s the residue fi
│ │ │ │ +00019380: 656c 6420 6f66 2053 2c20 7468 656e 2074  eld of S, then t
│ │ │ │ +00019390: 6865 2073 6372 6970 7420 6578 7465 7269  he script exteri
│ │ │ │ +000193a0: 6f72 4578 744d 6f64 756c 6528 662c 4d29  orExtModule(f,M)
│ │ │ │ +000193b0: 2072 6574 7572 6e73 0a45 7874 5f53 284d   returns.Ext_S(M
│ │ │ │ +000193c0: 2c20 6b29 2061 7320 6120 6d6f 6475 6c65  , k) as a module
│ │ │ │ +000193d0: 206f 7665 7220 616e 2065 7874 6572 696f   over an exterio
│ │ │ │ +000193e0: 7220 616c 6765 6272 6120 4520 3d20 6b3c  r algebra E = k<
│ │ │ │ +000193f0: 655f 312c 2e2e 2e2c 655f 633e 2c20 7768  e_1,...,e_c>, wh
│ │ │ │ +00019400: 6572 6520 7468 650a 655f 6920 6861 7665  ere the.e_i have
│ │ │ │ +00019410: 2064 6567 7265 6520 312e 2049 7420 6973   degree 1. It is
│ │ │ │ +00019420: 2063 6f6d 7075 7465 6420 6173 2074 6865   computed as the
│ │ │ │ +00019430: 2045 2d64 7561 6c20 6f66 2065 7874 6572   E-dual of exter
│ │ │ │ +00019440: 696f 7254 6f72 4d6f 6475 6c65 2e0a 0a54  iorTorModule...T
│ │ │ │ +00019450: 6865 2073 6372 6970 7420 6578 7465 7269  he script exteri
│ │ │ │ +00019460: 6f72 546f 724d 6f64 756c 6528 662c 4d2c  orTorModule(f,M,
│ │ │ │ +00019470: 4e29 2072 6574 7572 6e73 2045 7874 5f53  N) returns Ext_S
│ │ │ │ +00019480: 284d 2c4e 2920 6173 2061 206d 6f64 756c  (M,N) as a modul
│ │ │ │ +00019490: 6520 6f76 6572 2061 0a62 6967 7261 6465  e over a.bigrade
│ │ │ │ +000194a0: 6420 7269 6e67 2053 4520 3d20 533c 655f  d ring SE = S, where
│ │ │ │ +000194c0: 2074 6865 2065 5f69 2068 6176 6520 6465   the e_i have de
│ │ │ │ +000194d0: 6772 6565 7320 7b64 5f69 2c31 7d2c 2077  grees {d_i,1}, w
│ │ │ │ +000194e0: 6865 7265 2064 5f69 0a69 7320 7468 6520  here d_i.is the 
│ │ │ │ +000194f0: 6465 6772 6565 206f 6620 665f 692e 2054  degree of f_i. T
│ │ │ │ +00019500: 6865 206d 6f64 756c 6520 7374 7275 6374  he module struct
│ │ │ │ +00019510: 7572 652c 2069 6e20 6569 7468 6572 2063  ure, in either c
│ │ │ │ +00019520: 6173 652c 2069 7320 6465 6669 6e65 6420  ase, is defined 
│ │ │ │ +00019530: 6279 2074 6865 0a68 6f6d 6f74 6f70 6965  by the.homotopie
│ │ │ │ +00019540: 7320 666f 7220 7468 6520 665f 6920 6f6e  s for the f_i on
│ │ │ │ +00019550: 2074 6865 2072 6573 6f6c 7574 696f 6e20   the resolution 
│ │ │ │ +00019560: 6f66 204d 2c20 636f 6d70 7574 6564 2062  of M, computed b
│ │ │ │ +00019570: 7920 7468 6520 7363 7269 7074 0a6d 616b  y the script.mak
│ │ │ │ +00019580: 6548 6f6d 6f74 6f70 6965 7331 2e54 6865  eHomotopies1.The
│ │ │ │ +00019590: 2073 6372 6970 7420 6361 6c6c 7320 6d61   script calls ma
│ │ │ │ +000195a0: 6b65 4d6f 6475 6c65 2074 6f20 636f 6d70  keModule to comp
│ │ │ │ +000195b0: 7574 6520 6120 286e 6f6e 2d6d 696e 696d  ute a (non-minim
│ │ │ │ +000195c0: 616c 290a 7072 6573 656e 7461 7469 6f6e  al).presentation
│ │ │ │ +000195d0: 206f 6620 7468 6973 206d 6f64 756c 652e   of this module.
│ │ │ │ +000195e0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  000195f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019620: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 203d  ----+.|i1 : kk =
│ │ │ │ -00019630: 205a 5a2f 3130 3120 2020 2020 2020 2020   ZZ/101         
│ │ │ │ +00019610: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00019620: 206b 6b20 3d20 5a5a 2f31 3031 2020 2020   kk = ZZ/101    
│ │ │ │ +00019630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00019650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019690: 2020 2020 2020 7c0a 7c6f 3120 3d20 6b6b        |.|o1 = kk
│ │ │ │ +00019680: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00019690: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │  000196a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000196b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000196c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000196d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000196c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000196d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000196e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000196f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019700: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -00019710: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +000196f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00019700: 6f31 203a 2051 756f 7469 656e 7452 696e  o1 : QuotientRin
│ │ │ │ +00019710: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00019720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019740: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00019730: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00019740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019770: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00019780: 3a20 5320 3d20 6b6b 5b61 2c62 2c63 5d20  : S = kk[a,b,c] 
│ │ │ │ +00019760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00019770: 0a7c 6932 203a 2053 203d 206b 6b5b 612c  .|i2 : S = kk[a,
│ │ │ │ +00019780: 622c 635d 2020 2020 2020 2020 2020 2020  b,c]            
│ │ │ │  00019790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000197a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000197b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000197a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000197b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000197c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000197d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000197e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000197f0: 3220 3d20 5320 2020 2020 2020 2020 2020  2 = S           
│ │ │ │ +000197e0: 207c 0a7c 6f32 203d 2053 2020 2020 2020   |.|o2 = S      
│ │ │ │ +000197f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019820: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00019820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019860: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ -00019870: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ -00019880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019890: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00019850: 2020 207c 0a7c 6f32 203a 2050 6f6c 796e     |.|o2 : Polyn
│ │ │ │ +00019860: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +00019870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019880: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00019890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000198a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000198b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000198c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000198d0: 2b0a 7c69 3320 3a20 6620 3d20 6d61 7472  +.|i3 : f = matr
│ │ │ │ -000198e0: 6978 2261 342c 6234 2c63 3422 2020 2020  ix"a4,b4,c4"    
│ │ │ │ -000198f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000198c0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 203d  -----+.|i3 : f =
│ │ │ │ +000198d0: 206d 6174 7269 7822 6134 2c62 342c 6334   matrix"a4,b4,c4
│ │ │ │ +000198e0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +000198f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00019900: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00019910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019940: 2020 7c0a 7c6f 3320 3d20 7c20 6134 2062    |.|o3 = | a4 b
│ │ │ │ -00019950: 3420 6334 207c 2020 2020 2020 2020 2020  4 c4 |          
│ │ │ │ +00019930: 2020 2020 2020 207c 0a7c 6f33 203d 207c         |.|o3 = |
│ │ │ │ +00019940: 2061 3420 6234 2063 3420 7c20 2020 2020   a4 b4 c4 |     
│ │ │ │ +00019950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019970: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00019970: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00019980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000199a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000199b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000199c0: 2020 2020 3120 2020 2020 2033 2020 2020      1      3    
│ │ │ │ +000199a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000199b0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +000199c0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  000199d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000199e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000199f0: 6f33 203a 204d 6174 7269 7820 5320 203c  o3 : Matrix S  <
│ │ │ │ -00019a00: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ -00019a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019a20: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000199e0: 2020 7c0a 7c6f 3320 3a20 4d61 7472 6978    |.|o3 : Matrix
│ │ │ │ +000199f0: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ +00019a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019a10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00019a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00019a60: 0a7c 6934 203a 2052 203d 2053 2f69 6465  .|i4 : R = S/ide
│ │ │ │ -00019a70: 616c 2066 2020 2020 2020 2020 2020 2020  al f            
│ │ │ │ -00019a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019a90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00019a50: 2d2d 2d2d 2b0a 7c69 3420 3a20 5220 3d20  ----+.|i4 : R = 
│ │ │ │ +00019a60: 532f 6964 6561 6c20 6620 2020 2020 2020  S/ideal f       
│ │ │ │ +00019a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019a80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00019a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ad0: 207c 0a7c 6f34 203d 2052 2020 2020 2020   |.|o4 = R      
│ │ │ │ +00019ac0: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220        |.|o4 = R 
│ │ │ │ +00019ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00019af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00019b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00019b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b40: 2020 207c 0a7c 6f34 203a 2051 756f 7469     |.|o4 : Quoti
│ │ │ │ -00019b50: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00019b30: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +00019b40: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00019b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00019b70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00019b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019bb0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 203d  -----+.|i5 : p =
│ │ │ │ -00019bc0: 206d 6170 2852 2c53 2920 2020 2020 2020   map(R,S)       
│ │ │ │ +00019ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00019bb0: 3a20 7020 3d20 6d61 7028 522c 5329 2020  : p = map(R,S)  
│ │ │ │ +00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019bf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00019be0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00019bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c20: 2020 2020 2020 207c 0a7c 6f35 203d 206d         |.|o5 = m
│ │ │ │ -00019c30: 6170 2028 522c 2053 2c20 7b61 2c20 622c  ap (R, S, {a, b,
│ │ │ │ -00019c40: 2063 7d29 2020 2020 2020 2020 2020 2020   c})            
│ │ │ │ -00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00019c10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00019c20: 3520 3d20 6d61 7020 2852 2c20 532c 207b  5 = map (R, S, {
│ │ │ │ +00019c30: 612c 2062 2c20 637d 2920 2020 2020 2020  a, b, c})       
│ │ │ │ +00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c90: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -00019ca0: 2052 696e 674d 6170 2052 203c 2d2d 2053   RingMap R <-- S
│ │ │ │ +00019c80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00019c90: 7c6f 3520 3a20 5269 6e67 4d61 7020 5220  |o5 : RingMap R 
│ │ │ │ +00019ca0: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │  00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00019cc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00019cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00019d10: 203a 204d 203d 2063 6f6b 6572 206d 6170   : M = coker map
│ │ │ │ -00019d20: 2852 5e32 2c20 525e 7b33 3a2d 317d 2c20  (R^2, R^{3:-1}, 
│ │ │ │ -00019d30: 7b7b 612c 622c 637d 2c7b 622c 632c 617d  {{a,b,c},{b,c,a}
│ │ │ │ -00019d40: 7d29 2020 7c0a 7c20 2020 2020 2020 2020  })  |.|         
│ │ │ │ +00019d00: 2b0a 7c69 3620 3a20 4d20 3d20 636f 6b65  +.|i6 : M = coke
│ │ │ │ +00019d10: 7220 6d61 7028 525e 322c 2052 5e7b 333a  r map(R^2, R^{3:
│ │ │ │ +00019d20: 2d31 7d2c 207b 7b61 2c62 2c63 7d2c 7b62  -1}, {{a,b,c},{b
│ │ │ │ +00019d30: 2c63 2c61 7d7d 2920 207c 0a7c 2020 2020  ,c,a}})  |.|    
│ │ │ │ +00019d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00019d80: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20  o6 = cokernel | 
│ │ │ │ -00019d90: 6120 6220 6320 7c20 2020 2020 2020 2020  a b c |         
│ │ │ │ -00019da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019db0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00019dc0: 2020 2020 2020 207c 2062 2063 2061 207c         | b c a |
│ │ │ │ +00019d70: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ +00019d80: 656c 207c 2061 2062 2063 207c 2020 2020  el | a b c |    
│ │ │ │ +00019d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019da0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00019db0: 2020 2020 2020 2020 2020 2020 7c20 6220              | b 
│ │ │ │ +00019dc0: 6320 6120 7c20 2020 2020 2020 2020 2020  c a |           
│ │ │ │  00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00019df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00019de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00019df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00019e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e40: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00019e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e60: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -00019e70: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -00019e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00019e10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00019e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019e30: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00019e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019e50: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ +00019e60: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +00019e70: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +00019e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00019e90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00019ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ed0: 2d2d 2d2b 0a7c 6937 203a 2062 6574 7469  ---+.|i7 : betti
│ │ │ │ -00019ee0: 2028 4646 203d 7265 7328 204d 2c20 4c65   (FF =res( M, Le
│ │ │ │ -00019ef0: 6e67 7468 4c69 6d69 7420 3d3e 3629 2920  ngthLimit =>6)) 
│ │ │ │ -00019f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00019ec0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +00019ed0: 6265 7474 6920 2846 4620 3d72 6573 2820  betti (FF =res( 
│ │ │ │ +00019ee0: 4d2c 204c 656e 6774 684c 696d 6974 203d  M, LengthLimit =
│ │ │ │ +00019ef0: 3e36 2929 2020 2020 2020 2020 2020 2020  >6))            
│ │ │ │ +00019f00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00019f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00019f50: 2020 2020 3020 3120 3220 3320 3420 2035      0 1 2 3 4  5
│ │ │ │ -00019f60: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ -00019f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019f80: 7c6f 3720 3d20 746f 7461 6c3a 2032 2033  |o7 = total: 2 3
│ │ │ │ -00019f90: 2034 2036 2039 2031 3320 3138 2020 2020   4 6 9 13 18    
│ │ │ │ -00019fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019fb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00019fc0: 2020 2030 3a20 3220 3320 2e20 2e20 2e20     0: 2 3 . . . 
│ │ │ │ -00019fd0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ -00019fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ff0: 7c0a 7c20 2020 2020 2020 2020 313a 202e  |.|         1: .
│ │ │ │ -0001a000: 202e 2031 202e 202e 2020 2e20 202e 2020   . 1 . .  .  .  
│ │ │ │ -0001a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001a030: 2020 2020 2032 3a20 2e20 2e20 3320 3320       2: . . 3 3 
│ │ │ │ -0001a040: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -0001a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a060: 2020 7c0a 7c20 2020 2020 2020 2020 333a    |.|         3:
│ │ │ │ -0001a070: 202e 202e 202e 2033 2033 2020 2e20 202e   . . . 3 3  .  .
│ │ │ │ +00019f30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00019f40: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ +00019f50: 2034 2020 3520 2036 2020 2020 2020 2020   4  5  6        
│ │ │ │ +00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f70: 2020 207c 0a7c 6f37 203d 2074 6f74 616c     |.|o7 = total
│ │ │ │ +00019f80: 3a20 3220 3320 3420 3620 3920 3133 2031  : 2 3 4 6 9 13 1
│ │ │ │ +00019f90: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ +00019fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00019fb0: 2020 2020 2020 2020 303a 2032 2033 202e          0: 2 3 .
│ │ │ │ +00019fc0: 202e 202e 2020 2e20 202e 2020 2020 2020   . .  .  .      
│ │ │ │ +00019fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019ff0: 2031 3a20 2e20 2e20 3120 2e20 2e20 202e   1: . . 1 . .  .
│ │ │ │ +0001a000: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +0001a010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001a020: 7c20 2020 2020 2020 2020 323a 202e 202e  |         2: . .
│ │ │ │ +0001a030: 2033 2033 202e 2020 2e20 202e 2020 2020   3 3 .  .  .    
│ │ │ │ +0001a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a050: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001a060: 2020 2033 3a20 2e20 2e20 2e20 3320 3320     3: . . . 3 3 
│ │ │ │ +0001a070: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │  0001a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a0a0: 2020 2020 2020 2034 3a20 2e20 2e20 2e20         4: . . . 
│ │ │ │ -0001a0b0: 2e20 3320 2033 2020 2e20 2020 2020 2020  . 3  3  .       
│ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001a0e0: 353a 202e 202e 202e 202e 2033 2020 3920  5: . . . . 3  9 
│ │ │ │ -0001a0f0: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ -0001a100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001a110: 2020 2020 2020 2020 2036 3a20 2e20 2e20           6: . . 
│ │ │ │ -0001a120: 2e20 2e20 2e20 202e 2020 3320 2020 2020  . . .  .  3     
│ │ │ │ -0001a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a150: 2020 373a 202e 202e 202e 202e 202e 2020    7: . . . . .  
│ │ │ │ -0001a160: 3120 2039 2020 2020 2020 2020 2020 2020  1  9            
│ │ │ │ -0001a170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001a180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001a090: 7c0a 7c20 2020 2020 2020 2020 343a 202e  |.|         4: .
│ │ │ │ +0001a0a0: 202e 202e 202e 2033 2020 3320 202e 2020   . . . 3  3  .  
│ │ │ │ +0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a0c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001a0d0: 2020 2020 2035 3a20 2e20 2e20 2e20 2e20       5: . . . . 
│ │ │ │ +0001a0e0: 3320 2039 2020 3620 2020 2020 2020 2020  3  9  6         
│ │ │ │ +0001a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a100: 2020 7c0a 7c20 2020 2020 2020 2020 363a    |.|         6:
│ │ │ │ +0001a110: 202e 202e 202e 202e 202e 2020 2e20 2033   . . . . .  .  3
│ │ │ │ +0001a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001a140: 2020 2020 2020 2037 3a20 2e20 2e20 2e20         7: . . . 
│ │ │ │ +0001a150: 2e20 2e20 2031 2020 3920 2020 2020 2020  . .  1  9       
│ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a170: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1b0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -0001a1c0: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0001a1a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001a1b0: 6f37 203a 2042 6574 7469 5461 6c6c 7920  o7 : BettiTally 
│ │ │ │ +0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001a1e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -0001a230: 3a20 4d53 203d 2070 7275 6e65 2070 7573  : MS = prune pus
│ │ │ │ -0001a240: 6846 6f72 7761 7264 2870 2c20 636f 6b65  hForward(p, coke
│ │ │ │ -0001a250: 7220 4646 2e64 645f 3629 3b20 2020 2020  r FF.dd_6);     
│ │ │ │ -0001a260: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001a220: 0a7c 6938 203a 204d 5320 3d20 7072 756e  .|i8 : MS = prun
│ │ │ │ +0001a230: 6520 7075 7368 466f 7277 6172 6428 702c  e pushForward(p,
│ │ │ │ +0001a240: 2063 6f6b 6572 2046 462e 6464 5f36 293b   coker FF.dd_6);
│ │ │ │ +0001a250: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001a2a0: 3920 3a20 7265 7346 6c64 203a 3d20 7075  9 : resFld := pu
│ │ │ │ -0001a2b0: 7368 466f 7277 6172 6428 702c 2063 6f6b  shForward(p, cok
│ │ │ │ -0001a2c0: 6572 2076 6172 7320 5229 3b20 2020 2020  er vars R);     
│ │ │ │ -0001a2d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a290: 2d2b 0a7c 6939 203a 2072 6573 466c 6420  -+.|i9 : resFld 
│ │ │ │ +0001a2a0: 3a3d 2070 7573 6846 6f72 7761 7264 2870  := pushForward(p
│ │ │ │ +0001a2b0: 2c20 636f 6b65 7220 7661 7273 2052 293b  , coker vars R);
│ │ │ │ +0001a2c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001a310: 7c69 3130 203a 2054 203d 2065 7874 6572  |i10 : T = exter
│ │ │ │ -0001a320: 696f 7254 6f72 4d6f 6475 6c65 2866 2c4d  iorTorModule(f,M
│ │ │ │ -0001a330: 5329 3b20 2020 2020 2020 2020 2020 2020  S);             
│ │ │ │ -0001a340: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001a300: 2d2d 2d2b 0a7c 6931 3020 3a20 5420 3d20  ---+.|i10 : T = 
│ │ │ │ +0001a310: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +0001a320: 6528 662c 4d53 293b 2020 2020 2020 2020  e(f,MS);        
│ │ │ │ +0001a330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a380: 2b0a 7c69 3131 203a 2045 203d 2065 7874  +.|i11 : E = ext
│ │ │ │ -0001a390: 6572 696f 7245 7874 4d6f 6475 6c65 2866  eriorExtModule(f
│ │ │ │ -0001a3a0: 2c4d 5329 3b20 2020 2020 2020 2020 2020  ,MS);           
│ │ │ │ -0001a3b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001a370: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 4520  -----+.|i11 : E 
│ │ │ │ +0001a380: 3d20 6578 7465 7269 6f72 4578 744d 6f64  = exteriorExtMod
│ │ │ │ +0001a390: 756c 6528 662c 4d53 293b 2020 2020 2020  ule(f,MS);      
│ │ │ │ +0001a3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001a3b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3f0: 2d2d 2b0a 7c69 3132 203a 2068 6628 2d34  --+.|i12 : hf(-4
│ │ │ │ -0001a400: 2e2e 302c 4529 2020 2020 2020 2020 2020  ..0,E)          
│ │ │ │ +0001a3e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ +0001a3f0: 6866 282d 342e 2e30 2c45 2920 2020 2020  hf(-4..0,E)     
│ │ │ │ +0001a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001a420: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a460: 2020 2020 7c0a 7c6f 3132 203d 207b 302c      |.|o12 = {0,
│ │ │ │ -0001a470: 2039 2c20 3239 2c20 3333 2c20 3133 7d20   9, 29, 33, 13} 
│ │ │ │ +0001a450: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ +0001a460: 3d20 7b30 2c20 392c 2032 392c 2033 332c  = {0, 9, 29, 33,
│ │ │ │ +0001a470: 2031 337d 2020 2020 2020 2020 2020 2020   13}            
│ │ │ │  0001a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001a490: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4d0: 2020 2020 2020 7c0a 7c6f 3132 203a 204c        |.|o12 : L
│ │ │ │ -0001a4e0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +0001a4c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001a4d0: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ +0001a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001a510: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001a500: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a540: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ -0001a550: 2062 6574 7469 2072 6573 204d 5320 2020   betti res MS   
│ │ │ │ +0001a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001a540: 6931 3320 3a20 6265 7474 6920 7265 7320  i13 : betti res 
│ │ │ │ +0001a550: 4d53 2020 2020 2020 2020 2020 2020 2020  MS              
│ │ │ │  0001a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a580: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a570: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ -0001a5d0: 2032 2033 2020 2020 2020 2020 2020 2020   2 3            
│ │ │ │ -0001a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5f0: 2020 207c 0a7c 6f31 3320 3d20 746f 7461     |.|o13 = tota
│ │ │ │ -0001a600: 6c3a 2031 3320 3333 2032 3920 3920 2020  l: 13 33 29 9   
│ │ │ │ +0001a5a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001a5b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001a5c0: 3020 2031 2020 3220 3320 2020 2020 2020  0  1  2 3       
│ │ │ │ +0001a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a5e0: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +0001a5f0: 2074 6f74 616c 3a20 3133 2033 3320 3239   total: 13 33 29
│ │ │ │ +0001a600: 2039 2020 2020 2020 2020 2020 2020 2020   9              
│ │ │ │  0001a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a620: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001a630: 2020 2020 2020 2020 2039 3a20 2033 2020           9:  3  
│ │ │ │ -0001a640: 2e20 202e 202e 2020 2020 2020 2020 2020  .  . .          
│ │ │ │ -0001a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a660: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a670: 2031 303a 2020 3920 2036 2020 2e20 2e20   10:  9  6  . . 
│ │ │ │ +0001a620: 207c 0a7c 2020 2020 2020 2020 2020 393a   |.|          9:
│ │ │ │ +0001a630: 2020 3320 202e 2020 2e20 2e20 2020 2020    3  .  . .     
│ │ │ │ +0001a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a650: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001a660: 2020 2020 2020 3130 3a20 2039 2020 3620        10:  9  6 
│ │ │ │ +0001a670: 202e 202e 2020 2020 2020 2020 2020 2020   . .            
│ │ │ │  0001a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001a6a0: 7c20 2020 2020 2020 2020 3131 3a20 202e  |         11:  .
│ │ │ │ -0001a6b0: 2020 3320 202e 202e 2020 2020 2020 2020    3  . .        
│ │ │ │ -0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001a6e0: 2020 2031 323a 2020 3120 3135 2020 2e20     12:  1 15  . 
│ │ │ │ -0001a6f0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0001a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a710: 7c0a 7c20 2020 2020 2020 2020 3133 3a20  |.|         13: 
│ │ │ │ -0001a720: 202e 2020 3920 2038 202e 2020 2020 2020   .  9  8 .      
│ │ │ │ -0001a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001a750: 2020 2020 2031 343a 2020 2e20 202e 2020       14:  .  .  
│ │ │ │ -0001a760: 3620 2e20 2020 2020 2020 2020 2020 2020  6 .             
│ │ │ │ -0001a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a780: 2020 7c0a 7c20 2020 2020 2020 2020 3135    |.|         15
│ │ │ │ -0001a790: 3a20 202e 2020 2e20 3132 202e 2020 2020  :  .  . 12 .    
│ │ │ │ +0001a690: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ +0001a6a0: 313a 2020 2e20 2033 2020 2e20 2e20 2020  1:  .  3  . .   
│ │ │ │ +0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a6d0: 2020 2020 2020 2020 3132 3a20 2031 2031          12:  1 1
│ │ │ │ +0001a6e0: 3520 202e 202e 2020 2020 2020 2020 2020  5  . .          
│ │ │ │ +0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a710: 2031 333a 2020 2e20 2039 2020 3820 2e20   13:  .  9  8 . 
│ │ │ │ +0001a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001a740: 7c20 2020 2020 2020 2020 3134 3a20 202e  |         14:  .
│ │ │ │ +0001a750: 2020 2e20 2036 202e 2020 2020 2020 2020    .  6 .        
│ │ │ │ +0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001a780: 2020 2031 353a 2020 2e20 202e 2031 3220     15:  .  . 12 
│ │ │ │ +0001a790: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │  0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a7c0: 2020 2020 2020 2031 363a 2020 2e20 202e         16:  .  .
│ │ │ │ -0001a7d0: 2020 3320 3320 2020 2020 2020 2020 2020    3 3           
│ │ │ │ -0001a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001a800: 3137 3a20 202e 2020 2e20 202e 2033 2020  17:  .  .  . 3  
│ │ │ │ +0001a7b0: 7c0a 7c20 2020 2020 2020 2020 3136 3a20  |.|         16: 
│ │ │ │ +0001a7c0: 202e 2020 2e20 2033 2033 2020 2020 2020   .  .  3 3      
│ │ │ │ +0001a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001a7f0: 2020 2020 2031 373a 2020 2e20 202e 2020       17:  .  .  
│ │ │ │ +0001a800: 2e20 3320 2020 2020 2020 2020 2020 2020  . 3             
│ │ │ │  0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001a830: 2020 2020 2020 2020 2031 383a 2020 2e20           18:  . 
│ │ │ │ -0001a840: 202e 2020 2e20 3320 2020 2020 2020 2020   .  . 3         
│ │ │ │ -0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a820: 2020 7c0a 7c20 2020 2020 2020 2020 3138    |.|         18
│ │ │ │ +0001a830: 3a20 202e 2020 2e20 202e 2033 2020 2020  :  .  .  . 3    
│ │ │ │ +0001a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001a8a0: 0a7c 6f31 3320 3a20 4265 7474 6954 616c  .|o13 : BettiTal
│ │ │ │ -0001a8b0: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ -0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001a890: 2020 2020 7c0a 7c6f 3133 203a 2042 6574      |.|o13 : Bet
│ │ │ │ +0001a8a0: 7469 5461 6c6c 7920 2020 2020 2020 2020  tiTally         
│ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a910: 2d2b 0a7c 6931 3420 3a20 6265 7474 6920  -+.|i14 : betti 
│ │ │ │ -0001a920: 7265 7320 2850 4520 3d20 7072 756e 6520  res (PE = prune 
│ │ │ │ -0001a930: 4529 2020 2020 2020 2020 2020 2020 2020  E)              
│ │ │ │ -0001a940: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001a900: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2062  ------+.|i14 : b
│ │ │ │ +0001a910: 6574 7469 2072 6573 2028 5045 203d 2070  etti res (PE = p
│ │ │ │ +0001a920: 7275 6e65 2045 2920 2020 2020 2020 2020  rune E)         
│ │ │ │ +0001a930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001a940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001a990: 2020 2020 3020 2031 2020 3220 2033 2020      0  1  2  3  
│ │ │ │ -0001a9a0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0001a9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001a9c0: 3134 203d 2074 6f74 616c 3a20 3136 2031  14 = total: 16 1
│ │ │ │ -0001a9d0: 3320 3235 2034 3920 3831 2020 2020 2020  3 25 49 81      
│ │ │ │ -0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001aa00: 202d 333a 2020 3920 2034 2020 3320 2033   -3:  9  4  3  3
│ │ │ │ -0001aa10: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -0001aa20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001aa30: 7c20 2020 2020 2020 2020 2d32 3a20 2036  |         -2:  6
│ │ │ │ -0001aa40: 2020 3320 202e 2020 2e20 202e 2020 2020    3  .  .  .    
│ │ │ │ -0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001aa70: 2020 202d 313a 2020 2e20 202e 2020 3720     -1:  .  .  7 
│ │ │ │ -0001aa80: 3138 2033 3320 2020 2020 2020 2020 2020  18 33           
│ │ │ │ -0001aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aaa0: 7c0a 7c20 2020 2020 2020 2020 2030 3a20  |.|          0: 
│ │ │ │ -0001aab0: 2031 2020 3620 3135 2032 3820 3435 2020   1  6 15 28 45  
│ │ │ │ -0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aad0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001a970: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001a980: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ +0001a990: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9b0: 207c 0a7c 6f31 3420 3d20 746f 7461 6c3a   |.|o14 = total:
│ │ │ │ +0001a9c0: 2031 3620 3133 2032 3520 3439 2038 3120   16 13 25 49 81 
│ │ │ │ +0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001a9f0: 2020 2020 2020 2d33 3a20 2039 2020 3420        -3:  9  4 
│ │ │ │ +0001aa00: 2033 2020 3320 2033 2020 2020 2020 2020   3  3  3        
│ │ │ │ +0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa20: 2020 207c 0a7c 2020 2020 2020 2020 202d     |.|         -
│ │ │ │ +0001aa30: 323a 2020 3620 2033 2020 2e20 202e 2020  2:  6  3  .  .  
│ │ │ │ +0001aa40: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0001aa50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001aa60: 2020 2020 2020 2020 2d31 3a20 202e 2020          -1:  .  
│ │ │ │ +0001aa70: 2e20 2037 2031 3820 3333 2020 2020 2020  .  7 18 33      
│ │ │ │ +0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001aaa0: 2020 303a 2020 3120 2036 2031 3520 3238    0:  1  6 15 28
│ │ │ │ +0001aab0: 2034 3520 2020 2020 2020 2020 2020 2020   45             
│ │ │ │ +0001aac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001aad0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab10: 2020 7c0a 7c6f 3134 203a 2042 6574 7469    |.|o14 : Betti
│ │ │ │ -0001ab20: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0001ab00: 2020 2020 2020 207c 0a7c 6f31 3420 3a20         |.|o14 : 
│ │ │ │ +0001ab10: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0001ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001ab40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab80: 2d2d 2d2d 2b0a 7c69 3135 203a 2062 6574  ----+.|i15 : bet
│ │ │ │ -0001ab90: 7469 2072 6573 2028 5054 203d 2070 7275  ti res (PT = pru
│ │ │ │ -0001aba0: 6e65 2054 2920 2020 2020 2020 2020 2020  ne T)           
│ │ │ │ -0001abb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ab70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520  ---------+.|i15 
│ │ │ │ +0001ab80: 3a20 6265 7474 6920 7265 7320 2850 5420  : betti res (PT 
│ │ │ │ +0001ab90: 3d20 7072 756e 6520 5429 2020 2020 2020  = prune T)      
│ │ │ │ +0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001abb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ac00: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -0001ac10: 2033 2020 2034 2020 2020 2020 2020 2020   3   4          
│ │ │ │ -0001ac20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001ac30: 0a7c 6f31 3520 3d20 746f 7461 6c3a 2033  .|o15 = total: 3
│ │ │ │ -0001ac40: 3120 3535 2038 3720 3132 3720 3137 3520  1 55 87 127 175 
│ │ │ │ -0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001ac70: 2020 2020 2030 3a20 3133 2032 3420 3339       0: 13 24 39
│ │ │ │ -0001ac80: 2020 3538 2020 3831 2020 2020 2020 2020    58  81        
│ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 207c 0a7c 2020 2020 2020 2020 2020 313a   |.|          1:
│ │ │ │ -0001acb0: 2031 3820 3331 2034 3820 2036 3920 2039   18 31 48  69  9
│ │ │ │ -0001acc0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0001acd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001abe0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001abf0: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ +0001ac00: 2020 3220 2020 3320 2020 3420 2020 2020    2   3   4     
│ │ │ │ +0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac20: 2020 2020 7c0a 7c6f 3135 203d 2074 6f74      |.|o15 = tot
│ │ │ │ +0001ac30: 616c 3a20 3331 2035 3520 3837 2031 3237  al: 31 55 87 127
│ │ │ │ +0001ac40: 2031 3735 2020 2020 2020 2020 2020 2020   175            
│ │ │ │ +0001ac50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ac60: 2020 2020 2020 2020 2020 303a 2031 3320            0: 13 
│ │ │ │ +0001ac70: 3234 2033 3920 2035 3820 2038 3120 2020  24 39  58  81   
│ │ │ │ +0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001aca0: 2020 2031 3a20 3138 2033 3120 3438 2020     1: 18 31 48  
│ │ │ │ +0001acb0: 3639 2020 3934 2020 2020 2020 2020 2020  69  94          
│ │ │ │ +0001acc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001acd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad10: 2020 207c 0a7c 6f31 3520 3a20 4265 7474     |.|o15 : Bett
│ │ │ │ -0001ad20: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +0001ad00: 2020 2020 2020 2020 7c0a 7c6f 3135 203a          |.|o15 :
│ │ │ │ +0001ad10: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0001ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001ad40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad80: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 4531  -----+.|i16 : E1
│ │ │ │ -0001ad90: 203d 2070 7275 6e65 2065 7874 6572 696f   = prune exterio
│ │ │ │ -0001ada0: 7245 7874 4d6f 6475 6c65 2866 2c20 4d53  rExtModule(f, MS
│ │ │ │ -0001adb0: 2c20 7265 7346 6c64 293b 2020 2020 7c0a  , resFld);    |.
│ │ │ │ -0001adc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │ +0001ad80: 203a 2045 3120 3d20 7072 756e 6520 6578   : E1 = prune ex
│ │ │ │ +0001ad90: 7465 7269 6f72 4578 744d 6f64 756c 6528  teriorExtModule(
│ │ │ │ +0001ada0: 662c 204d 532c 2072 6573 466c 6429 3b20  f, MS, resFld); 
│ │ │ │ +0001adb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001adf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20  -------+.|i17 : 
│ │ │ │ -0001ae00: 7269 6e67 2045 3120 2020 2020 2020 2020  ring E1         
│ │ │ │ +0001ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001adf0: 3137 203a 2072 696e 6720 4531 2020 2020  17 : ring E1    
│ │ │ │ +0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001ae20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae60: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ -0001ae70: 3d20 6b6b 5b58 202e 2e58 202c 2065 202e  = kk[X ..X , e .
│ │ │ │ -0001ae80: 2e65 205d 2020 2020 2020 2020 2020 2020  .e ]            
│ │ │ │ -0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aea0: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ -0001aeb0: 2020 2032 2020 2030 2020 2032 2020 2020     2   0   2    
│ │ │ │ +0001ae50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001ae60: 7c6f 3137 203d 206b 6b5b 5820 2e2e 5820  |o17 = kk[X ..X 
│ │ │ │ +0001ae70: 2c20 6520 2e2e 6520 5d20 2020 2020 2020  , e ..e ]       
│ │ │ │ +0001ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001aea0: 2020 2020 3020 2020 3220 2020 3020 2020      0   2   0   
│ │ │ │ +0001aeb0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001aed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af10: 2020 2020 7c0a 7c6f 3137 203a 2050 6f6c      |.|o17 : Pol
│ │ │ │ -0001af20: 796e 6f6d 6961 6c52 696e 672c 2033 2073  ynomialRing, 3 s
│ │ │ │ -0001af30: 6b65 7720 636f 6d6d 7574 6174 6976 6520  kew commutative 
│ │ │ │ -0001af40: 7661 7269 6162 6c65 2873 2920 207c 0a2b  variable(s)  |.+
│ │ │ │ +0001af00: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ +0001af10: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +0001af20: 2c20 3320 736b 6577 2063 6f6d 6d75 7461  , 3 skew commuta
│ │ │ │ +0001af30: 7469 7665 2076 6172 6961 626c 6528 7329  tive variable(s)
│ │ │ │ +0001af40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0001af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af80: 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a 2065  ------+.|i18 : e
│ │ │ │ -0001af90: 7852 696e 6720 3d20 6b6b 5b65 5f30 2c65  xRing = kk[e_0,e
│ │ │ │ -0001afa0: 5f31 2c65 5f32 2c20 536b 6577 436f 6d6d  _1,e_2, SkewComm
│ │ │ │ -0001afb0: 7574 6174 6976 6520 3d3e 7472 7565 5d7c  utative =>true]|
│ │ │ │ -0001afc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001af80: 3820 3a20 6578 5269 6e67 203d 206b 6b5b  8 : exRing = kk[
│ │ │ │ +0001af90: 655f 302c 655f 312c 655f 322c 2053 6b65  e_0,e_1,e_2, Ske
│ │ │ │ +0001afa0: 7743 6f6d 6d75 7461 7469 7665 203d 3e74  wCommutative =>t
│ │ │ │ +0001afb0: 7275 655d 7c0a 7c20 2020 2020 2020 2020  rue]|.|         
│ │ │ │ +0001afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aff0: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ -0001b000: 2065 7852 696e 6720 2020 2020 2020 2020   exRing         
│ │ │ │ +0001afe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001aff0: 6f31 3820 3d20 6578 5269 6e67 2020 2020  o18 = exRing    
│ │ │ │ +0001b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b030: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b020: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b060: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -0001b070: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -0001b080: 672c 2033 2073 6b65 7720 636f 6d6d 7574  g, 3 skew commut
│ │ │ │ -0001b090: 6174 6976 6520 7661 7269 6162 6c65 2873  ative variable(s
│ │ │ │ -0001b0a0: 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  )  |.+----------
│ │ │ │ +0001b050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b060: 0a7c 6f31 3820 3a20 506f 6c79 6e6f 6d69  .|o18 : Polynomi
│ │ │ │ +0001b070: 616c 5269 6e67 2c20 3320 736b 6577 2063  alRing, 3 skew c
│ │ │ │ +0001b080: 6f6d 6d75 7461 7469 7665 2076 6172 6961  ommutative varia
│ │ │ │ +0001b090: 626c 6528 7329 2020 7c0a 2b2d 2d2d 2d2d  ble(s)  |.+-----
│ │ │ │ +0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ -0001b0e0: 6520 6361 6e20 616c 736f 2063 6f6e 7374  e can also const
│ │ │ │ -0001b0f0: 7275 6374 2074 6865 2065 7874 6572 696f  ruct the exterio
│ │ │ │ -0001b100: 7245 7874 4d6f 6475 6c65 2061 7320 6120  rExtModule as a 
│ │ │ │ -0001b110: 6269 6772 6164 6564 206d 6f64 756c 652c  bigraded module,
│ │ │ │ -0001b120: 206f 7665 7220 6120 7269 6e67 0a53 4520   over a ring.SE 
│ │ │ │ -0001b130: 7468 6174 2068 6173 2062 6f74 6820 706f  that has both po
│ │ │ │ -0001b140: 6c79 6e6f 6d69 616c 2076 6172 6961 626c  lynomial variabl
│ │ │ │ -0001b150: 6573 206c 696b 6520 5320 616e 6420 6578  es like S and ex
│ │ │ │ -0001b160: 7465 7269 6f72 2076 6172 6961 626c 6573  terior variables
│ │ │ │ -0001b170: 206c 696b 6520 452e 2054 6865 0a70 6f6c   like E. The.pol
│ │ │ │ -0001b180: 796e 6f6d 6961 6c20 7661 7269 6162 6c65  ynomial variable
│ │ │ │ -0001b190: 7320 6861 7665 2064 6567 7265 6573 207b  s have degrees {
│ │ │ │ -0001b1a0: 312c 307d 2e20 5468 6520 6578 7465 7269  1,0}. The exteri
│ │ │ │ -0001b1b0: 6f72 2076 6172 6961 626c 6573 2068 6176  or variables hav
│ │ │ │ -0001b1c0: 6520 6465 6772 6565 730a 7b64 6567 2066  e degrees.{deg f
│ │ │ │ -0001b1d0: 665f 692c 2031 7d2e 0a0a 2b2d 2d2d 2d2d  f_i, 1}...+-----
│ │ │ │ +0001b0d0: 2d2b 0a0a 5765 2063 616e 2061 6c73 6f20  -+..We can also 
│ │ │ │ +0001b0e0: 636f 6e73 7472 7563 7420 7468 6520 6578  construct the ex
│ │ │ │ +0001b0f0: 7465 7269 6f72 4578 744d 6f64 756c 6520  teriorExtModule 
│ │ │ │ +0001b100: 6173 2061 2062 6967 7261 6465 6420 6d6f  as a bigraded mo
│ │ │ │ +0001b110: 6475 6c65 2c20 6f76 6572 2061 2072 696e  dule, over a rin
│ │ │ │ +0001b120: 670a 5345 2074 6861 7420 6861 7320 626f  g.SE that has bo
│ │ │ │ +0001b130: 7468 2070 6f6c 796e 6f6d 6961 6c20 7661  th polynomial va
│ │ │ │ +0001b140: 7269 6162 6c65 7320 6c69 6b65 2053 2061  riables like S a
│ │ │ │ +0001b150: 6e64 2065 7874 6572 696f 7220 7661 7269  nd exterior vari
│ │ │ │ +0001b160: 6162 6c65 7320 6c69 6b65 2045 2e20 5468  ables like E. Th
│ │ │ │ +0001b170: 650a 706f 6c79 6e6f 6d69 616c 2076 6172  e.polynomial var
│ │ │ │ +0001b180: 6961 626c 6573 2068 6176 6520 6465 6772  iables have degr
│ │ │ │ +0001b190: 6565 7320 7b31 2c30 7d2e 2054 6865 2065  ees {1,0}. The e
│ │ │ │ +0001b1a0: 7874 6572 696f 7220 7661 7269 6162 6c65  xterior variable
│ │ │ │ +0001b1b0: 7320 6861 7665 2064 6567 7265 6573 0a7b  s have degrees.{
│ │ │ │ +0001b1c0: 6465 6720 6666 5f69 2c20 317d 2e0a 0a2b  deg ff_i, 1}...+
│ │ │ │ +0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b210: 2d2b 0a7c 6931 3920 3a20 4531 203d 2070  -+.|i19 : E1 = p
│ │ │ │ -0001b220: 7275 6e65 2065 7874 6572 696f 7245 7874  rune exteriorExt
│ │ │ │ -0001b230: 4d6f 6475 6c65 2866 2c20 4d53 2c20 7265  Module(f, MS, re
│ │ │ │ -0001b240: 7346 6c64 293b 2020 2020 7c0a 2b2d 2d2d  sFld);    |.+---
│ │ │ │ +0001b200: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2045  ------+.|i19 : E
│ │ │ │ +0001b210: 3120 3d20 7072 756e 6520 6578 7465 7269  1 = prune exteri
│ │ │ │ +0001b220: 6f72 4578 744d 6f64 756c 6528 662c 204d  orExtModule(f, M
│ │ │ │ +0001b230: 532c 2072 6573 466c 6429 3b20 2020 207c  S, resFld);    |
│ │ │ │ +0001b240: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b280: 2d2d 2d2b 0a7c 6932 3020 3a20 7269 6e67  ---+.|i20 : ring
│ │ │ │ -0001b290: 2045 3120 2020 2020 2020 2020 2020 2020   E1             
│ │ │ │ +0001b270: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +0001b280: 2072 696e 6720 4531 2020 2020 2020 2020   ring E1        
│ │ │ │ +0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b2b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2f0: 2020 2020 207c 0a7c 6f32 3020 3d20 6b6b       |.|o20 = kk
│ │ │ │ -0001b300: 5b58 202e 2e58 202c 2065 202e 2e65 205d  [X ..X , e ..e ]
│ │ │ │ +0001b2e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ +0001b2f0: 203d 206b 6b5b 5820 2e2e 5820 2c20 6520   = kk[X ..X , e 
│ │ │ │ +0001b300: 2e2e 6520 5d20 2020 2020 2020 2020 2020  ..e ]           
│ │ │ │  0001b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b330: 7c20 2020 2020 2020 2020 2030 2020 2032  |          0   2
│ │ │ │ -0001b340: 2020 2030 2020 2032 2020 2020 2020 2020     0   2        
│ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b320: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001b330: 3020 2020 3220 2020 3020 2020 3220 2020  0   2   0   2   
│ │ │ │ +0001b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b350: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b3a0: 7c0a 7c6f 3230 203a 2050 6f6c 796e 6f6d  |.|o20 : Polynom
│ │ │ │ -0001b3b0: 6961 6c52 696e 672c 2033 2073 6b65 7720  ialRing, 3 skew 
│ │ │ │ -0001b3c0: 636f 6d6d 7574 6174 6976 6520 7661 7269  commutative vari
│ │ │ │ -0001b3d0: 6162 6c65 2873 2920 207c 0a2b 2d2d 2d2d  able(s)  |.+----
│ │ │ │ +0001b390: 2020 2020 207c 0a7c 6f32 3020 3a20 506f       |.|o20 : Po
│ │ │ │ +0001b3a0: 6c79 6e6f 6d69 616c 5269 6e67 2c20 3320  lynomialRing, 3 
│ │ │ │ +0001b3b0: 736b 6577 2063 6f6d 6d75 7461 7469 7665  skew commutative
│ │ │ │ +0001b3c0: 2076 6172 6961 626c 6528 7329 2020 7c0a   variable(s)  |.
│ │ │ │ +0001b3d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b410: 2d2d 2b0a 7c69 3231 203a 2065 7852 696e  --+.|i21 : exRin
│ │ │ │ -0001b420: 6720 3d20 6b6b 5b65 5f30 2c65 5f31 2c65  g = kk[e_0,e_1,e
│ │ │ │ -0001b430: 5f32 2c20 536b 6577 436f 6d6d 7574 6174  _2, SkewCommutat
│ │ │ │ -0001b440: 6976 6520 3d3e 7472 7565 5d7c 0a7c 2020  ive =>true]|.|  
│ │ │ │ +0001b400: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20  -------+.|i21 : 
│ │ │ │ +0001b410: 6578 5269 6e67 203d 206b 6b5b 655f 302c  exRing = kk[e_0,
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│ │ │ │ -0001bd30: 640a 6675 6e63 7469 6f6e 3a20 284d 6163  d.function: (Mac
│ │ │ │ -0001bd40: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -0001bd50: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ -0001bd60: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -0001bd70: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -0001bd80: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2065  ns.info, Node: e
│ │ │ │ -0001bd90: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d  xteriorHomologyM
│ │ │ │ -0001bda0: 6f64 756c 652c 204e 6578 743a 2065 7874  odule, Next: ext
│ │ │ │ -0001bdb0: 6572 696f 7254 6f72 4d6f 6475 6c65 2c20  eriorTorModule, 
│ │ │ │ -0001bdc0: 5072 6576 3a20 6578 7465 7269 6f72 4578  Prev: exteriorEx
│ │ │ │ -0001bdd0: 744d 6f64 756c 652c 2055 703a 2054 6f70  tModule, Up: Top
│ │ │ │ -0001bde0: 0a0a 6578 7465 7269 6f72 486f 6d6f 6c6f  ..exteriorHomolo
│ │ │ │ -0001bdf0: 6779 4d6f 6475 6c65 202d 2d20 4d61 6b65  gyModule -- Make
│ │ │ │ -0001be00: 2074 6865 2068 6f6d 6f6c 6f67 7920 6f66   the homology of
│ │ │ │ -0001be10: 2061 2063 6f6d 706c 6578 2069 6e74 6f20   a complex into 
│ │ │ │ -0001be20: 6120 6d6f 6475 6c65 206f 7665 7220 616e  a module over an
│ │ │ │ -0001be30: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ -0001be40: 610a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  a.**************
│ │ │ │ +0001bb30: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +0001bb40: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +0001bb50: 202a 6e6f 7465 2065 7874 6572 696f 7254   *note exteriorT
│ │ │ │ +0001bb60: 6f72 4d6f 6475 6c65 3a20 6578 7465 7269  orModule: exteri
│ │ │ │ +0001bb70: 6f72 546f 724d 6f64 756c 652c 202d 2d20  orTorModule, -- 
│ │ │ │ +0001bb80: 546f 7220 6173 2061 206d 6f64 756c 6520  Tor as a module 
│ │ │ │ +0001bb90: 6f76 6572 2061 6e0a 2020 2020 6578 7465  over an.    exte
│ │ │ │ +0001bba0: 7269 6f72 2061 6c67 6562 7261 206f 7220  rior algebra or 
│ │ │ │ +0001bbb0: 6269 6772 6164 6564 2061 6c67 6562 7261  bigraded algebra
│ │ │ │ +0001bbc0: 0a20 202a 202a 6e6f 7465 206d 616b 654d  .  * *note makeM
│ │ │ │ +0001bbd0: 6f64 756c 653a 206d 616b 654d 6f64 756c  odule: makeModul
│ │ │ │ +0001bbe0: 652c 202d 2d20 6d61 6b65 7320 6120 4d6f  e, -- makes a Mo
│ │ │ │ +0001bbf0: 6475 6c65 206f 7574 206f 6620 6120 636f  dule out of a co
│ │ │ │ +0001bc00: 6c6c 6563 7469 6f6e 206f 660a 2020 2020  llection of.    
│ │ │ │ +0001bc10: 6d6f 6475 6c65 7320 616e 6420 6d61 7073  modules and maps
│ │ │ │ +0001bc20: 0a0a 5761 7973 2074 6f20 7573 6520 6578  ..Ways to use ex
│ │ │ │ +0001bc30: 7465 7269 6f72 4578 744d 6f64 756c 653a  teriorExtModule:
│ │ │ │ +0001bc40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001bc50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0001bc60: 0a20 202a 2022 6578 7465 7269 6f72 4578  .  * "exteriorEx
│ │ │ │ +0001bc70: 744d 6f64 756c 6528 4d61 7472 6978 2c4d  tModule(Matrix,M
│ │ │ │ +0001bc80: 6f64 756c 6529 220a 2020 2a20 2265 7874  odule)".  * "ext
│ │ │ │ +0001bc90: 6572 696f 7245 7874 4d6f 6475 6c65 284d  eriorExtModule(M
│ │ │ │ +0001bca0: 6174 7269 782c 4d6f 6475 6c65 2c4d 6f64  atrix,Module,Mod
│ │ │ │ +0001bcb0: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ +0001bcc0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0001bcd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0001bce0: 6520 6f62 6a65 6374 202a 6e6f 7465 2065  e object *note e
│ │ │ │ +0001bcf0: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ +0001bd00: 3a20 6578 7465 7269 6f72 4578 744d 6f64  : exteriorExtMod
│ │ │ │ +0001bd10: 756c 652c 2069 7320 6120 2a6e 6f74 6520  ule, is a *note 
│ │ │ │ +0001bd20: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a  method.function:
│ │ │ │ +0001bd30: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0001bd40: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +0001bd50: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +0001bd60: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +0001bd70: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +0001bd80: 6465 3a20 6578 7465 7269 6f72 486f 6d6f  de: exteriorHomo
│ │ │ │ +0001bd90: 6c6f 6779 4d6f 6475 6c65 2c20 4e65 7874  logyModule, Next
│ │ │ │ +0001bda0: 3a20 6578 7465 7269 6f72 546f 724d 6f64  : exteriorTorMod
│ │ │ │ +0001bdb0: 756c 652c 2050 7265 763a 2065 7874 6572  ule, Prev: exter
│ │ │ │ +0001bdc0: 696f 7245 7874 4d6f 6475 6c65 2c20 5570  iorExtModule, Up
│ │ │ │ +0001bdd0: 3a20 546f 700a 0a65 7874 6572 696f 7248  : Top..exteriorH
│ │ │ │ +0001bde0: 6f6d 6f6c 6f67 794d 6f64 756c 6520 2d2d  omologyModule --
│ │ │ │ +0001bdf0: 204d 616b 6520 7468 6520 686f 6d6f 6c6f   Make the homolo
│ │ │ │ +0001be00: 6779 206f 6620 6120 636f 6d70 6c65 7820  gy of a complex 
│ │ │ │ +0001be10: 696e 746f 2061 206d 6f64 756c 6520 6f76  into a module ov
│ │ │ │ +0001be20: 6572 2061 6e20 6578 7465 7269 6f72 2061  er an exterior a
│ │ │ │ +0001be30: 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a 2a2a  lgebra.*********
│ │ │ │ +0001be40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001be50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001be60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001be70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001be80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001be90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001bea0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -0001beb0: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -0001bec0: 200a 2020 2020 2020 2020 4d20 3d20 6578   .        M = ex
│ │ │ │ -0001bed0: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f  teriorHomologyMo
│ │ │ │ -0001bee0: 6475 6c65 2866 662c 2043 290a 2020 2a20  dule(ff, C).  * 
│ │ │ │ -0001bef0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0001bf00: 6666 2c20 6120 2a6e 6f74 6520 6d61 7472  ff, a *note matr
│ │ │ │ -0001bf10: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -0001bf20: 6329 4d61 7472 6978 2c2c 204d 6174 7269  c)Matrix,, Matri
│ │ │ │ -0001bf30: 7820 6f66 2065 6c65 6d65 6e74 7320 7468  x of elements th
│ │ │ │ -0001bf40: 6174 2061 7265 0a20 2020 2020 2020 2068  at are.        h
│ │ │ │ -0001bf50: 6f6d 6f74 6f70 6963 2074 6f20 3020 6f6e  omotopic to 0 on
│ │ │ │ -0001bf60: 2043 0a20 2020 2020 202a 2043 2c20 6120   C.      * C, a 
│ │ │ │ -0001bf70: 2a6e 6f74 6520 6368 6169 6e20 636f 6d70  *note chain comp
│ │ │ │ -0001bf80: 6c65 783a 2028 4d61 6361 756c 6179 3244  lex: (Macaulay2D
│ │ │ │ -0001bf90: 6f63 2943 6861 696e 436f 6d70 6c65 782c  oc)ChainComplex,
│ │ │ │ -0001bfa0: 2c20 0a20 202a 204f 7574 7075 7473 3a0a  , .  * Outputs:.
│ │ │ │ -0001bfb0: 2020 2020 2020 2a20 4d2c 2061 202a 6e6f        * M, a *no
│ │ │ │ -0001bfc0: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -0001bfd0: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -0001bfe0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a  , ..Description.
│ │ │ │ -0001bff0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 7373  ===========..Ass
│ │ │ │ -0001c000: 756d 696e 6720 7468 6174 2074 6865 2065  uming that the e
│ │ │ │ -0001c010: 6c65 6d65 6e74 7320 6f66 2074 6865 2031  lements of the 1
│ │ │ │ -0001c020: 7863 206d 6174 7269 7820 6666 2061 7265  xc matrix ff are
│ │ │ │ -0001c030: 206e 756c 6c2d 686f 6d6f 746f 7069 6320   null-homotopic 
│ │ │ │ -0001c040: 6f6e 2043 2c20 7468 650a 7363 7269 7074  on C, the.script
│ │ │ │ -0001c050: 2072 6574 7572 6e73 2074 6865 2064 6972   returns the dir
│ │ │ │ -0001c060: 6563 7420 7375 6d20 6f66 2074 6865 2068  ect sum of the h
│ │ │ │ -0001c070: 6f6d 6f6c 6f67 7920 6f66 2043 2061 7320  omology of C as 
│ │ │ │ -0001c080: 6120 6d6f 6475 6c65 206f 7665 7220 6120  a module over a 
│ │ │ │ -0001c090: 6e65 7720 7269 6e67 2c0a 636f 6e73 6973  new ring,.consis
│ │ │ │ -0001c0a0: 7469 6e67 206f 6620 7269 6e67 2043 2077  ting of ring C w
│ │ │ │ -0001c0b0: 6974 6820 6320 6578 7465 7269 6f72 2076  ith c exterior v
│ │ │ │ -0001c0c0: 6172 6961 626c 6573 2061 646a 6f69 6e65  ariables adjoine
│ │ │ │ -0001c0d0: 642e 2054 6865 2073 6372 6970 7420 6973  d. The script is
│ │ │ │ -0001c0e0: 2074 6865 206d 6169 6e0a 636f 6d70 6f6e   the main.compon
│ │ │ │ -0001c0f0: 656e 7420 6f66 2065 7874 6572 696f 7254  ent of exteriorT
│ │ │ │ -0001c100: 6f72 4d6f 6475 6c65 0a0a 5365 6520 616c  orModule..See al
│ │ │ │ -0001c110: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -0001c120: 202a 6e6f 7465 2065 7874 6572 696f 7254   *note exteriorT
│ │ │ │ -0001c130: 6f72 4d6f 6475 6c65 3a20 6578 7465 7269  orModule: exteri
│ │ │ │ -0001c140: 6f72 546f 724d 6f64 756c 652c 202d 2d20  orTorModule, -- 
│ │ │ │ -0001c150: 546f 7220 6173 2061 206d 6f64 756c 6520  Tor as a module 
│ │ │ │ -0001c160: 6f76 6572 2061 6e0a 2020 2020 6578 7465  over an.    exte
│ │ │ │ -0001c170: 7269 6f72 2061 6c67 6562 7261 206f 7220  rior algebra or 
│ │ │ │ -0001c180: 6269 6772 6164 6564 2061 6c67 6562 7261  bigraded algebra
│ │ │ │ -0001c190: 0a20 202a 202a 6e6f 7465 206d 616b 6548  .  * *note makeH
│ │ │ │ -0001c1a0: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c  omotopiesOnHomol
│ │ │ │ -0001c1b0: 6f67 793a 206d 616b 6548 6f6d 6f74 6f70  ogy: makeHomotop
│ │ │ │ -0001c1c0: 6965 734f 6e48 6f6d 6f6c 6f67 792c 202d  iesOnHomology, -
│ │ │ │ -0001c1d0: 2d20 486f 6d6f 6c6f 6779 206f 6620 610a  - Homology of a.
│ │ │ │ -0001c1e0: 2020 2020 636f 6d70 6c65 7820 6173 2065      complex as e
│ │ │ │ -0001c1f0: 7874 6572 696f 7220 6d6f 6475 6c65 0a0a  xterior module..
│ │ │ │ -0001c200: 5761 7973 2074 6f20 7573 6520 6578 7465  Ways to use exte
│ │ │ │ -0001c210: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475  riorHomologyModu
│ │ │ │ -0001c220: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  le:.============
│ │ │ │ -0001c230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001c240: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6578  =======..  * "ex
│ │ │ │ -0001c250: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f  teriorHomologyMo
│ │ │ │ -0001c260: 6475 6c65 284d 6174 7269 782c 4368 6169  dule(Matrix,Chai
│ │ │ │ -0001c270: 6e43 6f6d 706c 6578 2922 0a0a 466f 7220  nComplex)"..For 
│ │ │ │ -0001c280: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0001c290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001c2a0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0001c2b0: 6f74 6520 6578 7465 7269 6f72 486f 6d6f  ote exteriorHomo
│ │ │ │ -0001c2c0: 6c6f 6779 4d6f 6475 6c65 3a20 6578 7465  logyModule: exte
│ │ │ │ -0001c2d0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475  riorHomologyModu
│ │ │ │ -0001c2e0: 6c65 2c20 6973 2061 202a 6e6f 7465 0a6d  le, is a *note.m
│ │ │ │ -0001c2f0: 6574 686f 6420 6675 6e63 7469 6f6e 3a20  ethod function: 
│ │ │ │ -0001c300: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -0001c310: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a1f  thodFunction,...
│ │ │ │ -0001c320: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ -0001c330: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -0001c340: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -0001c350: 653a 2065 7874 6572 696f 7254 6f72 4d6f  e: exteriorTorMo
│ │ │ │ -0001c360: 6475 6c65 2c20 4e65 7874 3a20 6578 7449  dule, Next: extI
│ │ │ │ -0001c370: 734f 6e65 506f 6c79 6e6f 6d69 616c 2c20  sOnePolynomial, 
│ │ │ │ -0001c380: 5072 6576 3a20 6578 7465 7269 6f72 486f  Prev: exteriorHo
│ │ │ │ -0001c390: 6d6f 6c6f 6779 4d6f 6475 6c65 2c20 5570  mologyModule, Up
│ │ │ │ -0001c3a0: 3a20 546f 700a 0a65 7874 6572 696f 7254  : Top..exteriorT
│ │ │ │ -0001c3b0: 6f72 4d6f 6475 6c65 202d 2d20 546f 7220  orModule -- Tor 
│ │ │ │ -0001c3c0: 6173 2061 206d 6f64 756c 6520 6f76 6572  as a module over
│ │ │ │ -0001c3d0: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ -0001c3e0: 6562 7261 206f 7220 6269 6772 6164 6564  ebra or bigraded
│ │ │ │ -0001c3f0: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a   algebra.*******
│ │ │ │ +0001be90: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +0001bea0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +0001beb0: 7361 6765 3a20 0a20 2020 2020 2020 204d  sage: .        M
│ │ │ │ +0001bec0: 203d 2065 7874 6572 696f 7248 6f6d 6f6c   = exteriorHomol
│ │ │ │ +0001bed0: 6f67 794d 6f64 756c 6528 6666 2c20 4329  ogyModule(ff, C)
│ │ │ │ +0001bee0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0001bef0: 2020 202a 2066 662c 2061 202a 6e6f 7465     * ff, a *note
│ │ │ │ +0001bf00: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +0001bf10: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +0001bf20: 4d61 7472 6978 206f 6620 656c 656d 656e  Matrix of elemen
│ │ │ │ +0001bf30: 7473 2074 6861 7420 6172 650a 2020 2020  ts that are.    
│ │ │ │ +0001bf40: 2020 2020 686f 6d6f 746f 7069 6320 746f      homotopic to
│ │ │ │ +0001bf50: 2030 206f 6e20 430a 2020 2020 2020 2a20   0 on C.      * 
│ │ │ │ +0001bf60: 432c 2061 202a 6e6f 7465 2063 6861 696e  C, a *note chain
│ │ │ │ +0001bf70: 2063 6f6d 706c 6578 3a20 284d 6163 6175   complex: (Macau
│ │ │ │ +0001bf80: 6c61 7932 446f 6329 4368 6169 6e43 6f6d  lay2Doc)ChainCom
│ │ │ │ +0001bf90: 706c 6578 2c2c 200a 2020 2a20 4f75 7470  plex,, .  * Outp
│ │ │ │ +0001bfa0: 7574 733a 0a20 2020 2020 202a 204d 2c20  uts:.      * M, 
│ │ │ │ +0001bfb0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ +0001bfc0: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ +0001bfd0: 6475 6c65 2c2c 200a 0a44 6573 6372 6970  dule,, ..Descrip
│ │ │ │ +0001bfe0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +0001bff0: 0a0a 4173 7375 6d69 6e67 2074 6861 7420  ..Assuming that 
│ │ │ │ +0001c000: 7468 6520 656c 656d 656e 7473 206f 6620  the elements of 
│ │ │ │ +0001c010: 7468 6520 3178 6320 6d61 7472 6978 2066  the 1xc matrix f
│ │ │ │ +0001c020: 6620 6172 6520 6e75 6c6c 2d68 6f6d 6f74  f are null-homot
│ │ │ │ +0001c030: 6f70 6963 206f 6e20 432c 2074 6865 0a73  opic on C, the.s
│ │ │ │ +0001c040: 6372 6970 7420 7265 7475 726e 7320 7468  cript returns th
│ │ │ │ +0001c050: 6520 6469 7265 6374 2073 756d 206f 6620  e direct sum of 
│ │ │ │ +0001c060: 7468 6520 686f 6d6f 6c6f 6779 206f 6620  the homology of 
│ │ │ │ +0001c070: 4320 6173 2061 206d 6f64 756c 6520 6f76  C as a module ov
│ │ │ │ +0001c080: 6572 2061 206e 6577 2072 696e 672c 0a63  er a new ring,.c
│ │ │ │ +0001c090: 6f6e 7369 7374 696e 6720 6f66 2072 696e  onsisting of rin
│ │ │ │ +0001c0a0: 6720 4320 7769 7468 2063 2065 7874 6572  g C with c exter
│ │ │ │ +0001c0b0: 696f 7220 7661 7269 6162 6c65 7320 6164  ior variables ad
│ │ │ │ +0001c0c0: 6a6f 696e 6564 2e20 5468 6520 7363 7269  joined. The scri
│ │ │ │ +0001c0d0: 7074 2069 7320 7468 6520 6d61 696e 0a63  pt is the main.c
│ │ │ │ +0001c0e0: 6f6d 706f 6e65 6e74 206f 6620 6578 7465  omponent of exte
│ │ │ │ +0001c0f0: 7269 6f72 546f 724d 6f64 756c 650a 0a53  riorTorModule..S
│ │ │ │ +0001c100: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +0001c110: 0a0a 2020 2a20 2a6e 6f74 6520 6578 7465  ..  * *note exte
│ │ │ │ +0001c120: 7269 6f72 546f 724d 6f64 756c 653a 2065  riorTorModule: e
│ │ │ │ +0001c130: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ +0001c140: 2c20 2d2d 2054 6f72 2061 7320 6120 6d6f  , -- Tor as a mo
│ │ │ │ +0001c150: 6475 6c65 206f 7665 7220 616e 0a20 2020  dule over an.   
│ │ │ │ +0001c160: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ +0001c170: 6120 6f72 2062 6967 7261 6465 6420 616c  a or bigraded al
│ │ │ │ +0001c180: 6765 6272 610a 2020 2a20 2a6e 6f74 6520  gebra.  * *note 
│ │ │ │ +0001c190: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e  makeHomotopiesOn
│ │ │ │ +0001c1a0: 486f 6d6f 6c6f 6779 3a20 6d61 6b65 486f  Homology: makeHo
│ │ │ │ +0001c1b0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f  motopiesOnHomolo
│ │ │ │ +0001c1c0: 6779 2c20 2d2d 2048 6f6d 6f6c 6f67 7920  gy, -- Homology 
│ │ │ │ +0001c1d0: 6f66 2061 0a20 2020 2063 6f6d 706c 6578  of a.    complex
│ │ │ │ +0001c1e0: 2061 7320 6578 7465 7269 6f72 206d 6f64   as exterior mod
│ │ │ │ +0001c1f0: 756c 650a 0a57 6179 7320 746f 2075 7365  ule..Ways to use
│ │ │ │ +0001c200: 2065 7874 6572 696f 7248 6f6d 6f6c 6f67   exteriorHomolog
│ │ │ │ +0001c210: 794d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d  yModule:.=======
│ │ │ │ +0001c220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001c230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +0001c240: 2a20 2265 7874 6572 696f 7248 6f6d 6f6c  * "exteriorHomol
│ │ │ │ +0001c250: 6f67 794d 6f64 756c 6528 4d61 7472 6978  ogyModule(Matrix
│ │ │ │ +0001c260: 2c43 6861 696e 436f 6d70 6c65 7829 220a  ,ChainComplex)".
│ │ │ │ +0001c270: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0001c280: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0001c290: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +0001c2a0: 6374 202a 6e6f 7465 2065 7874 6572 696f  ct *note exterio
│ │ │ │ +0001c2b0: 7248 6f6d 6f6c 6f67 794d 6f64 756c 653a  rHomologyModule:
│ │ │ │ +0001c2c0: 2065 7874 6572 696f 7248 6f6d 6f6c 6f67   exteriorHomolog
│ │ │ │ +0001c2d0: 794d 6f64 756c 652c 2069 7320 6120 2a6e  yModule, is a *n
│ │ │ │ +0001c2e0: 6f74 650a 6d65 7468 6f64 2066 756e 6374  ote.method funct
│ │ │ │ +0001c2f0: 696f 6e3a 2028 4d61 6361 756c 6179 3244  ion: (Macaulay2D
│ │ │ │ +0001c300: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ +0001c310: 6e2c 2e0a 1f0a 4669 6c65 3a20 436f 6d70  n,....File: Comp
│ │ │ │ +0001c320: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +0001c330: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ +0001c340: 2c20 4e6f 6465 3a20 6578 7465 7269 6f72  , Node: exterior
│ │ │ │ +0001c350: 546f 724d 6f64 756c 652c 204e 6578 743a  TorModule, Next:
│ │ │ │ +0001c360: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ +0001c370: 6961 6c2c 2050 7265 763a 2065 7874 6572  ial, Prev: exter
│ │ │ │ +0001c380: 696f 7248 6f6d 6f6c 6f67 794d 6f64 756c  iorHomologyModul
│ │ │ │ +0001c390: 652c 2055 703a 2054 6f70 0a0a 6578 7465  e, Up: Top..exte
│ │ │ │ +0001c3a0: 7269 6f72 546f 724d 6f64 756c 6520 2d2d  riorTorModule --
│ │ │ │ +0001c3b0: 2054 6f72 2061 7320 6120 6d6f 6475 6c65   Tor as a module
│ │ │ │ +0001c3c0: 206f 7665 7220 616e 2065 7874 6572 696f   over an exterio
│ │ │ │ +0001c3d0: 7220 616c 6765 6272 6120 6f72 2062 6967  r algebra or big
│ │ │ │ +0001c3e0: 7261 6465 6420 616c 6765 6272 610a 2a2a  raded algebra.**
│ │ │ │ +0001c3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001c400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001c410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001c420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001c430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001c440: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -0001c450: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -0001c460: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -0001c470: 2020 2054 203d 2065 7874 6572 696f 7254     T = exteriorT
│ │ │ │ -0001c480: 6f72 4d6f 6475 6c65 2866 2c46 290a 2020  orModule(f,F).  
│ │ │ │ -0001c490: 2020 2020 2020 5420 3d20 6578 7465 7269        T = exteri
│ │ │ │ -0001c4a0: 6f72 546f 724d 6f64 756c 6528 662c 4d2c  orTorModule(f,M,
│ │ │ │ -0001c4b0: 4e29 0a20 202a 2049 6e70 7574 733a 0a20  N).  * Inputs:. 
│ │ │ │ -0001c4c0: 2020 2020 202a 2066 2c20 6120 2a6e 6f74       * f, a *not
│ │ │ │ -0001c4d0: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ -0001c4e0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ -0001c4f0: 2031 2078 2063 2c20 656e 7472 6965 7320   1 x c, entries 
│ │ │ │ -0001c500: 6d75 7374 2062 650a 2020 2020 2020 2020  must be.        
│ │ │ │ -0001c510: 686f 6d6f 746f 7069 6320 746f 2030 206f  homotopic to 0 o
│ │ │ │ -0001c520: 6e20 460a 2020 2020 2020 2a20 4d2c 2061  n F.      * M, a
│ │ │ │ -0001c530: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ -0001c540: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ -0001c550: 756c 652c 2c20 532d 6d6f 6475 6c65 2061  ule,, S-module a
│ │ │ │ -0001c560: 6e6e 6968 696c 6174 6564 2062 7920 6964  nnihilated by id
│ │ │ │ -0001c570: 6561 6c0a 2020 2020 2020 2020 660a 2020  eal.        f.  
│ │ │ │ -0001c580: 2020 2020 2a20 4e2c 2061 202a 6e6f 7465      * N, a *note
│ │ │ │ -0001c590: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ -0001c5a0: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ -0001c5b0: 532d 6d6f 6475 6c65 2061 6e6e 6968 696c  S-module annihil
│ │ │ │ -0001c5c0: 6174 6564 2062 7920 6964 6561 6c0a 2020  ated by ideal.  
│ │ │ │ -0001c5d0: 2020 2020 2020 660a 2020 2a20 4f75 7470        f.  * Outp
│ │ │ │ -0001c5e0: 7574 733a 0a20 2020 2020 202a 2054 2c20  uts:.      * T, 
│ │ │ │ -0001c5f0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ -0001c600: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ -0001c610: 6475 6c65 2c2c 2054 6f72 5e53 284d 2c4e  dule,, Tor^S(M,N
│ │ │ │ -0001c620: 2920 6173 2061 204d 6f64 756c 6520 6f76  ) as a Module ov
│ │ │ │ -0001c630: 6572 0a20 2020 2020 2020 2061 6e20 6578  er.        an ex
│ │ │ │ -0001c640: 7465 7269 6f72 2061 6c67 6562 7261 0a0a  terior algebra..
│ │ │ │ -0001c650: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -0001c660: 3d3d 3d3d 3d3d 3d0a 0a49 6620 4d2c 4e20  =======..If M,N 
│ │ │ │ -0001c670: 6172 6520 532d 6d6f 6475 6c65 7320 616e  are S-modules an
│ │ │ │ -0001c680: 6e69 6869 6c61 7465 6420 6279 2074 6865  nihilated by the
│ │ │ │ -0001c690: 2065 6c65 6d65 6e74 7320 6f66 2074 6865   elements of the
│ │ │ │ -0001c6a0: 206d 6174 7269 7820 6666 203d 2028 665f   matrix ff = (f_
│ │ │ │ -0001c6b0: 312e 2e66 5f63 292c 0a61 6e64 206b 2069  1..f_c),.and k i
│ │ │ │ -0001c6c0: 7320 7468 6520 7265 7369 6475 6520 6669  s the residue fi
│ │ │ │ -0001c6d0: 656c 6420 6f66 2053 2c20 7468 656e 2074  eld of S, then t
│ │ │ │ -0001c6e0: 6865 2073 6372 6970 7420 6578 7465 7269  he script exteri
│ │ │ │ -0001c6f0: 6f72 546f 724d 6f64 756c 6528 662c 4d29  orTorModule(f,M)
│ │ │ │ -0001c700: 2072 6574 7572 6e73 0a54 6f72 5e53 284d   returns.Tor^S(M
│ │ │ │ -0001c710: 2c20 6b29 2061 7320 6120 6d6f 6475 6c65  , k) as a module
│ │ │ │ -0001c720: 206f 7665 7220 616e 2065 7874 6572 696f   over an exterio
│ │ │ │ -0001c730: 7220 616c 6765 6272 6120 6b3c 655f 312c  r algebra k, where 
│ │ │ │ -0001c750: 7468 6520 655f 690a 6861 7665 2064 6567  the e_i.have deg
│ │ │ │ -0001c760: 7265 6520 312c 2077 6869 6c65 2065 7874  ree 1, while ext
│ │ │ │ -0001c770: 6572 696f 7254 6f72 4d6f 6475 6c65 2866  eriorTorModule(f
│ │ │ │ -0001c780: 2c4d 2c4e 2920 7265 7475 726e 7320 546f  ,M,N) returns To
│ │ │ │ -0001c790: 725e 5328 4d2c 4e29 2061 7320 6120 6d6f  r^S(M,N) as a mo
│ │ │ │ -0001c7a0: 6475 6c65 0a6f 7665 7220 6120 6269 6772  dule.over a bigr
│ │ │ │ -0001c7b0: 6164 6564 2072 696e 6720 5345 203d 2053  aded ring SE = S
│ │ │ │ -0001c7c0: 3c65 5f31 2c2e 2e2c 655f 633e 2c20 7768  , wh
│ │ │ │ -0001c7d0: 6572 6520 7468 6520 655f 6920 6861 7665  ere the e_i have
│ │ │ │ -0001c7e0: 2064 6567 7265 6573 207b 645f 692c 317d   degrees {d_i,1}
│ │ │ │ -0001c7f0: 2c0a 7768 6572 6520 645f 6920 6973 2074  ,.where d_i is t
│ │ │ │ -0001c800: 6865 2064 6567 7265 6520 6f66 2066 5f69  he degree of f_i
│ │ │ │ -0001c810: 2e20 5468 6520 6d6f 6475 6c65 2073 7472  . The module str
│ │ │ │ -0001c820: 7563 7475 7265 2c20 696e 2065 6974 6865  ucture, in eithe
│ │ │ │ -0001c830: 7220 6361 7365 2c20 6973 0a64 6566 696e  r case, is.defin
│ │ │ │ -0001c840: 6564 2062 7920 7468 6520 686f 6d6f 746f  ed by the homoto
│ │ │ │ -0001c850: 7069 6573 2066 6f72 2074 6865 2066 5f69  pies for the f_i
│ │ │ │ -0001c860: 206f 6e20 7468 6520 7265 736f 6c75 7469   on the resoluti
│ │ │ │ -0001c870: 6f6e 206f 6620 4d2c 2063 6f6d 7075 7465  on of M, compute
│ │ │ │ -0001c880: 6420 6279 2074 6865 0a73 6372 6970 7420  d by the.script 
│ │ │ │ -0001c890: 6d61 6b65 486f 6d6f 746f 7069 6573 312e  makeHomotopies1.
│ │ │ │ -0001c8a0: 0a0a 5468 6520 7363 7269 7074 7320 6361  ..The scripts ca
│ │ │ │ -0001c8b0: 6c6c 206d 616b 654d 6f64 756c 6520 746f  ll makeModule to
│ │ │ │ -0001c8c0: 2063 6f6d 7075 7465 2061 2028 6e6f 6e2d   compute a (non-
│ │ │ │ -0001c8d0: 6d69 6e69 6d61 6c29 2070 7265 7365 6e74  minimal) present
│ │ │ │ -0001c8e0: 6174 696f 6e20 6f66 2074 6869 730a 6d6f  ation of this.mo
│ │ │ │ -0001c8f0: 6475 6c65 2e0a 0a46 726f 6d20 7468 6520  dule...From the 
│ │ │ │ -0001c900: 6465 7363 7269 7074 696f 6e20 6279 206d  description by m
│ │ │ │ -0001c910: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ -0001c920: 696f 6e73 2061 6e64 2074 6865 2070 6170  ions and the pap
│ │ │ │ -0001c930: 6572 2022 546f 7220 6173 2061 206d 6f64  er "Tor as a mod
│ │ │ │ -0001c940: 756c 650a 6f76 6572 2061 6e20 6578 7465  ule.over an exte
│ │ │ │ -0001c950: 7269 6f72 2061 6c67 6562 7261 2220 6f66  rior algebra" of
│ │ │ │ -0001c960: 2045 6973 656e 6275 642c 2050 6565 7661   Eisenbud, Peeva
│ │ │ │ -0001c970: 2061 6e64 2053 6368 7265 7965 7220 6974   and Schreyer it
│ │ │ │ -0001c980: 2066 6f6c 6c6f 7773 2074 6861 7420 7768   follows that wh
│ │ │ │ -0001c990: 656e 0a4d 2069 7320 6120 6869 6768 2073  en.M is a high s
│ │ │ │ -0001c9a0: 797a 7967 7920 616e 6420 4620 6973 2069  yzygy and F is i
│ │ │ │ -0001c9b0: 7473 2072 6573 6f6c 7574 696f 6e2c 2074  ts resolution, t
│ │ │ │ -0001c9c0: 6865 6e20 7468 6520 7072 6573 656e 7461  hen the presenta
│ │ │ │ -0001c9d0: 7469 6f6e 206f 660a 546f 7228 4d2c 535e  tion of.Tor(M,S^
│ │ │ │ -0001c9e0: 312f 6d6d 2920 616c 7761 7973 2068 6173  1/mm) always has
│ │ │ │ -0001c9f0: 2067 656e 6572 6174 6f72 7320 696e 2064   generators in d
│ │ │ │ -0001ca00: 6567 7265 6573 2030 2c31 2c20 636f 7272  egrees 0,1, corr
│ │ │ │ -0001ca10: 6573 706f 6e64 696e 6720 746f 2074 6865  esponding to the
│ │ │ │ -0001ca20: 0a74 6172 6765 7473 2061 6e64 2073 6f75  .targets and sou
│ │ │ │ -0001ca30: 7263 6573 206f 6620 7468 6520 7374 6163  rces of the stac
│ │ │ │ -0001ca40: 6b20 6f66 206d 6170 7320 4228 6929 2c20  k of maps B(i), 
│ │ │ │ -0001ca50: 616e 6420 7468 6174 2074 6865 2072 6573  and that the res
│ │ │ │ -0001ca60: 6f6c 7574 696f 6e20 6973 0a63 6f6d 706f  olution is.compo
│ │ │ │ -0001ca70: 6e65 6e74 7769 7365 206c 696e 6561 7220  nentwise linear 
│ │ │ │ -0001ca80: 696e 2061 2073 7569 7461 626c 6520 7365  in a suitable se
│ │ │ │ -0001ca90: 6e73 652e 2049 6e20 7468 6520 666f 6c6c  nse. In the foll
│ │ │ │ -0001caa0: 6f77 696e 6720 6578 616d 706c 652c 2074  owing example, t
│ │ │ │ -0001cab0: 6865 7365 2066 6163 7473 0a61 7265 2076  hese facts.are v
│ │ │ │ -0001cac0: 6572 6966 6965 642e 2054 6865 2054 6f72  erified. The Tor
│ │ │ │ -0001cad0: 206d 6f64 756c 6520 646f 6573 204e 4f54   module does NOT
│ │ │ │ -0001cae0: 2073 706c 6974 2069 6e74 6f20 7468 6520   split into the 
│ │ │ │ -0001caf0: 6469 7265 6374 2073 756d 206f 6620 7468  direct sum of th
│ │ │ │ -0001cb00: 650a 7375 626d 6f64 756c 6573 2067 656e  e.submodules gen
│ │ │ │ -0001cb10: 6572 6174 6564 2069 6e20 6465 6772 6565  erated in degree
│ │ │ │ -0001cb20: 7320 3020 616e 6420 312c 2068 6f77 6576  s 0 and 1, howev
│ │ │ │ -0001cb30: 6572 2e0a 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d  er.....+--------
│ │ │ │ +0001c430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +0001c440: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +0001c450: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +0001c460: 2020 2020 2020 2020 5420 3d20 6578 7465          T = exte
│ │ │ │ +0001c470: 7269 6f72 546f 724d 6f64 756c 6528 662c  riorTorModule(f,
│ │ │ │ +0001c480: 4629 0a20 2020 2020 2020 2054 203d 2065  F).        T = e
│ │ │ │ +0001c490: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ +0001c4a0: 2866 2c4d 2c4e 290a 2020 2a20 496e 7075  (f,M,N).  * Inpu
│ │ │ │ +0001c4b0: 7473 3a0a 2020 2020 2020 2a20 662c 2061  ts:.      * f, a
│ │ │ │ +0001c4c0: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ +0001c4d0: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ +0001c4e0: 7269 782c 2c20 3120 7820 632c 2065 6e74  rix,, 1 x c, ent
│ │ │ │ +0001c4f0: 7269 6573 206d 7573 7420 6265 0a20 2020  ries must be.   
│ │ │ │ +0001c500: 2020 2020 2068 6f6d 6f74 6f70 6963 2074       homotopic t
│ │ │ │ +0001c510: 6f20 3020 6f6e 2046 0a20 2020 2020 202a  o 0 on F.      *
│ │ │ │ +0001c520: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ +0001c530: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +0001c540: 6329 4d6f 6475 6c65 2c2c 2053 2d6d 6f64  c)Module,, S-mod
│ │ │ │ +0001c550: 756c 6520 616e 6e69 6869 6c61 7465 6420  ule annihilated 
│ │ │ │ +0001c560: 6279 2069 6465 616c 0a20 2020 2020 2020  by ideal.       
│ │ │ │ +0001c570: 2066 0a20 2020 2020 202a 204e 2c20 6120   f.      * N, a 
│ │ │ │ +0001c580: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +0001c590: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +0001c5a0: 6c65 2c2c 2053 2d6d 6f64 756c 6520 616e  le,, S-module an
│ │ │ │ +0001c5b0: 6e69 6869 6c61 7465 6420 6279 2069 6465  nihilated by ide
│ │ │ │ +0001c5c0: 616c 0a20 2020 2020 2020 2066 0a20 202a  al.        f.  *
│ │ │ │ +0001c5d0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0001c5e0: 2a20 542c 2061 202a 6e6f 7465 206d 6f64  * T, a *note mod
│ │ │ │ +0001c5f0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ +0001c600: 6f63 294d 6f64 756c 652c 2c20 546f 725e  oc)Module,, Tor^
│ │ │ │ +0001c610: 5328 4d2c 4e29 2061 7320 6120 4d6f 6475  S(M,N) as a Modu
│ │ │ │ +0001c620: 6c65 206f 7665 720a 2020 2020 2020 2020  le over.        
│ │ │ │ +0001c630: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ +0001c640: 6272 610a 0a44 6573 6372 6970 7469 6f6e  bra..Description
│ │ │ │ +0001c650: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966  .===========..If
│ │ │ │ +0001c660: 204d 2c4e 2061 7265 2053 2d6d 6f64 756c   M,N are S-modul
│ │ │ │ +0001c670: 6573 2061 6e6e 6968 696c 6174 6564 2062  es annihilated b
│ │ │ │ +0001c680: 7920 7468 6520 656c 656d 656e 7473 206f  y the elements o
│ │ │ │ +0001c690: 6620 7468 6520 6d61 7472 6978 2066 6620  f the matrix ff 
│ │ │ │ +0001c6a0: 3d20 2866 5f31 2e2e 665f 6329 2c0a 616e  = (f_1..f_c),.an
│ │ │ │ +0001c6b0: 6420 6b20 6973 2074 6865 2072 6573 6964  d k is the resid
│ │ │ │ +0001c6c0: 7565 2066 6965 6c64 206f 6620 532c 2074  ue field of S, t
│ │ │ │ +0001c6d0: 6865 6e20 7468 6520 7363 7269 7074 2065  hen the script e
│ │ │ │ +0001c6e0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ +0001c6f0: 2866 2c4d 2920 7265 7475 726e 730a 546f  (f,M) returns.To
│ │ │ │ +0001c700: 725e 5328 4d2c 206b 2920 6173 2061 206d  r^S(M, k) as a m
│ │ │ │ +0001c710: 6f64 756c 6520 6f76 6572 2061 6e20 6578  odule over an ex
│ │ │ │ +0001c720: 7465 7269 6f72 2061 6c67 6562 7261 206b  terior algebra k
│ │ │ │ +0001c730: 3c65 5f31 2c2e 2e2e 2c65 5f63 3e2c 2077  , w
│ │ │ │ +0001c740: 6865 7265 2074 6865 2065 5f69 0a68 6176  here the e_i.hav
│ │ │ │ +0001c750: 6520 6465 6772 6565 2031 2c20 7768 696c  e degree 1, whil
│ │ │ │ +0001c760: 6520 6578 7465 7269 6f72 546f 724d 6f64  e exteriorTorMod
│ │ │ │ +0001c770: 756c 6528 662c 4d2c 4e29 2072 6574 7572  ule(f,M,N) retur
│ │ │ │ +0001c780: 6e73 2054 6f72 5e53 284d 2c4e 2920 6173  ns Tor^S(M,N) as
│ │ │ │ +0001c790: 2061 206d 6f64 756c 650a 6f76 6572 2061   a module.over a
│ │ │ │ +0001c7a0: 2062 6967 7261 6465 6420 7269 6e67 2053   bigraded ring S
│ │ │ │ +0001c7b0: 4520 3d20 533c 655f 312c 2e2e 2c65 5f63  E = S, where the e_i
│ │ │ │ +0001c7d0: 2068 6176 6520 6465 6772 6565 7320 7b64   have degrees {d
│ │ │ │ +0001c7e0: 5f69 2c31 7d2c 0a77 6865 7265 2064 5f69  _i,1},.where d_i
│ │ │ │ +0001c7f0: 2069 7320 7468 6520 6465 6772 6565 206f   is the degree o
│ │ │ │ +0001c800: 6620 665f 692e 2054 6865 206d 6f64 756c  f f_i. The modul
│ │ │ │ +0001c810: 6520 7374 7275 6374 7572 652c 2069 6e20  e structure, in 
│ │ │ │ +0001c820: 6569 7468 6572 2063 6173 652c 2069 730a  either case, is.
│ │ │ │ +0001c830: 6465 6669 6e65 6420 6279 2074 6865 2068  defined by the h
│ │ │ │ +0001c840: 6f6d 6f74 6f70 6965 7320 666f 7220 7468  omotopies for th
│ │ │ │ +0001c850: 6520 665f 6920 6f6e 2074 6865 2072 6573  e f_i on the res
│ │ │ │ +0001c860: 6f6c 7574 696f 6e20 6f66 204d 2c20 636f  olution of M, co
│ │ │ │ +0001c870: 6d70 7574 6564 2062 7920 7468 650a 7363  mputed by the.sc
│ │ │ │ +0001c880: 7269 7074 206d 616b 6548 6f6d 6f74 6f70  ript makeHomotop
│ │ │ │ +0001c890: 6965 7331 2e0a 0a54 6865 2073 6372 6970  ies1...The scrip
│ │ │ │ +0001c8a0: 7473 2063 616c 6c20 6d61 6b65 4d6f 6475  ts call makeModu
│ │ │ │ +0001c8b0: 6c65 2074 6f20 636f 6d70 7574 6520 6120  le to compute a 
│ │ │ │ +0001c8c0: 286e 6f6e 2d6d 696e 696d 616c 2920 7072  (non-minimal) pr
│ │ │ │ +0001c8d0: 6573 656e 7461 7469 6f6e 206f 6620 7468  esentation of th
│ │ │ │ +0001c8e0: 6973 0a6d 6f64 756c 652e 0a0a 4672 6f6d  is.module...From
│ │ │ │ +0001c8f0: 2074 6865 2064 6573 6372 6970 7469 6f6e   the description
│ │ │ │ +0001c900: 2062 7920 6d61 7472 6978 2066 6163 746f   by matrix facto
│ │ │ │ +0001c910: 7269 7a61 7469 6f6e 7320 616e 6420 7468  rizations and th
│ │ │ │ +0001c920: 6520 7061 7065 7220 2254 6f72 2061 7320  e paper "Tor as 
│ │ │ │ +0001c930: 6120 6d6f 6475 6c65 0a6f 7665 7220 616e  a module.over an
│ │ │ │ +0001c940: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ +0001c950: 6122 206f 6620 4569 7365 6e62 7564 2c20  a" of Eisenbud, 
│ │ │ │ +0001c960: 5065 6576 6120 616e 6420 5363 6872 6579  Peeva and Schrey
│ │ │ │ +0001c970: 6572 2069 7420 666f 6c6c 6f77 7320 7468  er it follows th
│ │ │ │ +0001c980: 6174 2077 6865 6e0a 4d20 6973 2061 2068  at when.M is a h
│ │ │ │ +0001c990: 6967 6820 7379 7a79 6779 2061 6e64 2046  igh syzygy and F
│ │ │ │ +0001c9a0: 2069 7320 6974 7320 7265 736f 6c75 7469   is its resoluti
│ │ │ │ +0001c9b0: 6f6e 2c20 7468 656e 2074 6865 2070 7265  on, then the pre
│ │ │ │ +0001c9c0: 7365 6e74 6174 696f 6e20 6f66 0a54 6f72  sentation of.Tor
│ │ │ │ +0001c9d0: 284d 2c53 5e31 2f6d 6d29 2061 6c77 6179  (M,S^1/mm) alway
│ │ │ │ +0001c9e0: 7320 6861 7320 6765 6e65 7261 746f 7273  s has generators
│ │ │ │ +0001c9f0: 2069 6e20 6465 6772 6565 7320 302c 312c   in degrees 0,1,
│ │ │ │ +0001ca00: 2063 6f72 7265 7370 6f6e 6469 6e67 2074   corresponding t
│ │ │ │ +0001ca10: 6f20 7468 650a 7461 7267 6574 7320 616e  o the.targets an
│ │ │ │ +0001ca20: 6420 736f 7572 6365 7320 6f66 2074 6865  d sources of the
│ │ │ │ +0001ca30: 2073 7461 636b 206f 6620 6d61 7073 2042   stack of maps B
│ │ │ │ +0001ca40: 2869 292c 2061 6e64 2074 6861 7420 7468  (i), and that th
│ │ │ │ +0001ca50: 6520 7265 736f 6c75 7469 6f6e 2069 730a  e resolution is.
│ │ │ │ +0001ca60: 636f 6d70 6f6e 656e 7477 6973 6520 6c69  componentwise li
│ │ │ │ +0001ca70: 6e65 6172 2069 6e20 6120 7375 6974 6162  near in a suitab
│ │ │ │ +0001ca80: 6c65 2073 656e 7365 2e20 496e 2074 6865  le sense. In the
│ │ │ │ +0001ca90: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70   following examp
│ │ │ │ +0001caa0: 6c65 2c20 7468 6573 6520 6661 6374 730a  le, these facts.
│ │ │ │ +0001cab0: 6172 6520 7665 7269 6669 6564 2e20 5468  are verified. Th
│ │ │ │ +0001cac0: 6520 546f 7220 6d6f 6475 6c65 2064 6f65  e Tor module doe
│ │ │ │ +0001cad0: 7320 4e4f 5420 7370 6c69 7420 696e 746f  s NOT split into
│ │ │ │ +0001cae0: 2074 6865 2064 6972 6563 7420 7375 6d20   the direct sum 
│ │ │ │ +0001caf0: 6f66 2074 6865 0a73 7562 6d6f 6475 6c65  of the.submodule
│ │ │ │ +0001cb00: 7320 6765 6e65 7261 7465 6420 696e 2064  s generated in d
│ │ │ │ +0001cb10: 6567 7265 6573 2030 2061 6e64 2031 2c20  egrees 0 and 1, 
│ │ │ │ +0001cb20: 686f 7765 7665 722e 0a0a 0a0a 2b2d 2d2d  however.....+---
│ │ │ │ +0001cb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0001cb80: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ +0001cb70: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f  +.|i1 : kk = ZZ/
│ │ │ │ +0001cb80: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │  0001cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001cbb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc00: 2020 2020 2020 207c 0a7c 6f31 203d 206b         |.|o1 = k
│ │ │ │ -0001cc10: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ +0001cbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001cc00: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001cc40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc90: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ -0001cca0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0001cc80: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +0001cc90: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0001cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccd0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001ccc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001ccd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001cd20: 0a7c 6932 203a 2053 203d 206b 6b5b 612c  .|i2 : S = kk[a,
│ │ │ │ -0001cd30: 622c 635d 2020 2020 2020 2020 2020 2020  b,c]            
│ │ │ │ +0001cd10: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20  ----+.|i2 : S = 
│ │ │ │ +0001cd20: 6b6b 5b61 2c62 2c63 5d20 2020 2020 2020  kk[a,b,c]       
│ │ │ │ +0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0001cdb0: 203d 2053 2020 2020 2020 2020 2020 2020   = S            
│ │ │ │ +0001cda0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020  |.|o2 = S       
│ │ │ │ +0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001cde0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce30: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ -0001ce40: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +0001ce20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001ce30: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ +0001ce40: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce70: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001ce70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0001ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cec0: 2d2d 2d2b 0a7c 6933 203a 2066 203d 206d  ---+.|i3 : f = m
│ │ │ │ -0001ced0: 6174 7269 7822 6134 2c62 342c 6334 2220  atrix"a4,b4,c4" 
│ │ │ │ +0001ceb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0001cec0: 6620 3d20 6d61 7472 6978 2261 342c 6234  f = matrix"a4,b4
│ │ │ │ +0001ced0: 2c63 3422 2020 2020 2020 2020 2020 2020  ,c4"            
│ │ │ │  0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001cef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001cf00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001cf50: 0a7c 6f33 203d 207c 2061 3420 6234 2063  .|o3 = | a4 b4 c
│ │ │ │ -0001cf60: 3420 7c20 2020 2020 2020 2020 2020 2020  4 |             
│ │ │ │ +0001cf40: 2020 2020 7c0a 7c6f 3320 3d20 7c20 6134      |.|o3 = | a4
│ │ │ │ +0001cf50: 2062 3420 6334 207c 2020 2020 2020 2020   b4 c4 |        
│ │ │ │ +0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001cf80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cfd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001cfe0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0001cff0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0001cfd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001cfe0: 3120 2020 2020 2033 2020 2020 2020 2020  1      3        
│ │ │ │ +0001cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d020: 207c 0a7c 6f33 203a 204d 6174 7269 7820   |.|o3 : Matrix 
│ │ │ │ -0001d030: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ +0001d010: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61        |.|o3 : Ma
│ │ │ │ +0001d020: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ +0001d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d060: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001d050: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001d060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0001d0b0: 6934 203a 2052 203d 2053 2f69 6465 616c  i4 : R = S/ideal
│ │ │ │ -0001d0c0: 2066 2020 2020 2020 2020 2020 2020 2020   f              
│ │ │ │ +0001d0a0: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f  --+.|i4 : R = S/
│ │ │ │ +0001d0b0: 6964 6561 6c20 6620 2020 2020 2020 2020  ideal f         
│ │ │ │ +0001d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d0f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001d0e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d130: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -0001d140: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0001d120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001d130: 7c6f 3420 3d20 5220 2020 2020 2020 2020  |o4 = R         
│ │ │ │ +0001d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001d180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001d170: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d1c0: 2020 2020 207c 0a7c 6f34 203a 2051 756f       |.|o4 : Quo
│ │ │ │ -0001d1d0: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +0001d1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +0001d1c0: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +0001d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d200: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001d200: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0001d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d250: 2d2b 0a7c 6935 203a 2070 203d 206d 6170  -+.|i5 : p = map
│ │ │ │ -0001d260: 2852 2c53 2920 2020 2020 2020 2020 2020  (R,S)           
│ │ │ │ +0001d240: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7020  ------+.|i5 : p 
│ │ │ │ +0001d250: 3d20 6d61 7028 522c 5329 2020 2020 2020  = map(R,S)      
│ │ │ │ +0001d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d290: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001d280: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d2d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001d2e0: 6f35 203d 206d 6170 2028 522c 2053 2c20  o5 = map (R, S, 
│ │ │ │ -0001d2f0: 7b61 2c20 622c 2063 7d29 2020 2020 2020  {a, b, c})      
│ │ │ │ +0001d2d0: 2020 7c0a 7c6f 3520 3d20 6d61 7020 2852    |.|o5 = map (R
│ │ │ │ +0001d2e0: 2c20 532c 207b 612c 2062 2c20 637d 2920  , S, {a, b, c}) 
│ │ │ │ +0001d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d320: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001d310: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d360: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -0001d370: 2052 696e 674d 6170 2052 203c 2d2d 2053   RingMap R <-- S
│ │ │ │ +0001d350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001d360: 7c6f 3520 3a20 5269 6e67 4d61 7020 5220  |o5 : RingMap R 
│ │ │ │ +0001d370: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │  0001d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001d3b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001d3a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d3f0: 2d2d 2d2d 2d2b 0a7c 6936 203a 204d 203d  -----+.|i6 : M =
│ │ │ │ -0001d400: 2063 6f6b 6572 206d 6170 2852 5e32 2c20   coker map(R^2, 
│ │ │ │ -0001d410: 525e 7b33 3a2d 317d 2c20 7b7b 612c 622c  R^{3:-1}, {{a,b,
│ │ │ │ -0001d420: 637d 2c7b 622c 632c 617d 7d29 2020 2020  c},{b,c,a}})    
│ │ │ │ -0001d430: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +0001d3f0: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028  : M = coker map(
│ │ │ │ +0001d400: 525e 322c 2052 5e7b 333a 2d31 7d2c 207b  R^2, R^{3:-1}, {
│ │ │ │ +0001d410: 7b61 2c62 2c63 7d2c 7b62 2c63 2c61 7d7d  {a,b,c},{b,c,a}}
│ │ │ │ +0001d420: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001d430: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d480: 207c 0a7c 6f36 203d 2063 6f6b 6572 6e65   |.|o6 = cokerne
│ │ │ │ -0001d490: 6c20 7c20 6120 6220 6320 7c20 2020 2020  l | a b c |     
│ │ │ │ +0001d470: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ +0001d480: 6b65 726e 656c 207c 2061 2062 2063 207c  kernel | a b c |
│ │ │ │ +0001d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d4c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001d4d0: 2020 2020 2020 2020 7c20 6220 6320 6120          | b c a 
│ │ │ │ -0001d4e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001d4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d4c0: 2020 2020 2020 2020 2020 2020 207c 2062               | b
│ │ │ │ +0001d4d0: 2063 2061 207c 2020 2020 2020 2020 2020   c a |          
│ │ │ │ +0001d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d570: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0001d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d590: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ -0001d5a0: 2052 2d6d 6f64 756c 652c 2071 756f 7469   R-module, quoti
│ │ │ │ -0001d5b0: 656e 7420 6f66 2052 2020 2020 2020 2020  ent of R        
│ │ │ │ +0001d540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d560: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001d590: 7c6f 3620 3a20 522d 6d6f 6475 6c65 2c20  |o6 : R-module, 
│ │ │ │ +0001d5a0: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ +0001d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d5d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001d5e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001d5d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2062 6574  -----+.|i7 : bet
│ │ │ │ -0001d630: 7469 2028 4646 203d 7265 7328 204d 2c20  ti (FF =res( M, 
│ │ │ │ -0001d640: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3629  LengthLimit =>6)
│ │ │ │ -0001d650: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001d660: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0001d620: 3a20 6265 7474 6920 2846 4620 3d72 6573  : betti (FF =res
│ │ │ │ +0001d630: 2820 4d2c 204c 656e 6774 684c 696d 6974  ( M, LengthLimit
│ │ │ │ +0001d640: 203d 3e36 2929 2020 2020 2020 2020 2020   =>6))          
│ │ │ │ +0001d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d660: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001d6c0: 3020 3120 3220 3320 3420 2035 2020 3620  0 1 2 3 4  5  6 
│ │ │ │ +0001d6a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d6b0: 2020 2020 2030 2031 2032 2033 2034 2020       0 1 2 3 4  
│ │ │ │ +0001d6c0: 3520 2036 2020 2020 2020 2020 2020 2020  5  6            
│ │ │ │  0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6f0: 2020 2020 2020 207c 0a7c 6f37 203d 2074         |.|o7 = t
│ │ │ │ -0001d700: 6f74 616c 3a20 3220 3320 3420 3620 3920  otal: 2 3 4 6 9 
│ │ │ │ -0001d710: 3133 2031 3820 2020 2020 2020 2020 2020  13 18           
│ │ │ │ +0001d6e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001d6f0: 3720 3d20 746f 7461 6c3a 2032 2033 2034  7 = total: 2 3 4
│ │ │ │ +0001d700: 2036 2039 2031 3320 3138 2020 2020 2020   6 9 13 18      
│ │ │ │ +0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001d740: 2020 2020 2020 2020 2030 3a20 3220 3320           0: 2 3 
│ │ │ │ -0001d750: 2e20 2e20 2e20 202e 2020 2e20 2020 2020  . . .  .  .     
│ │ │ │ +0001d730: 2020 7c0a 7c20 2020 2020 2020 2020 303a    |.|         0:
│ │ │ │ +0001d740: 2032 2033 202e 202e 202e 2020 2e20 202e   2 3 . . .  .  .
│ │ │ │ +0001d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d780: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ -0001d790: 3a20 2e20 2e20 3120 2e20 2e20 202e 2020  : . . 1 . .  .  
│ │ │ │ -0001d7a0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0001d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001d7d0: 2020 2020 2032 3a20 2e20 2e20 3320 3320       2: . . 3 3 
│ │ │ │ -0001d7e0: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +0001d770: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d780: 2020 2020 313a 202e 202e 2031 202e 202e      1: . . 1 . .
│ │ │ │ +0001d790: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +0001d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d7b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001d7c0: 7c20 2020 2020 2020 2020 323a 202e 202e  |         2: . .
│ │ │ │ +0001d7d0: 2033 2033 202e 2020 2e20 202e 2020 2020   3 3 .  .  .    
│ │ │ │ +0001d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001d810: 0a7c 2020 2020 2020 2020 2033 3a20 2e20  .|         3: . 
│ │ │ │ -0001d820: 2e20 2e20 3320 3320 202e 2020 2e20 2020  . . 3 3  .  .   
│ │ │ │ +0001d800: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001d810: 333a 202e 202e 202e 2033 2033 2020 2e20  3: . . . 3 3  . 
│ │ │ │ +0001d820: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001d860: 2034 3a20 2e20 2e20 2e20 2e20 3320 2033   4: . . . . 3  3
│ │ │ │ -0001d870: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +0001d840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001d850: 2020 2020 2020 343a 202e 202e 202e 202e        4: . . . .
│ │ │ │ +0001d860: 2033 2020 3320 202e 2020 2020 2020 2020   3  3  .        
│ │ │ │ +0001d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d890: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001d8a0: 2020 2020 2020 2035 3a20 2e20 2e20 2e20         5: . . . 
│ │ │ │ -0001d8b0: 2e20 3320 2039 2020 3620 2020 2020 2020  . 3  9  6       
│ │ │ │ +0001d890: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ +0001d8a0: 202e 202e 202e 2033 2020 3920 2036 2020   . . . 3  9  6  
│ │ │ │ +0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8e0: 207c 0a7c 2020 2020 2020 2020 2036 3a20   |.|         6: 
│ │ │ │ -0001d8f0: 2e20 2e20 2e20 2e20 2e20 202e 2020 3320  . . . . .  .  3 
│ │ │ │ +0001d8d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d8e0: 2020 363a 202e 202e 202e 202e 202e 2020    6: . . . . .  
│ │ │ │ +0001d8f0: 2e20 2033 2020 2020 2020 2020 2020 2020  .  3            
│ │ │ │  0001d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d920: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001d930: 2020 2037 3a20 2e20 2e20 2e20 2e20 2e20     7: . . . . . 
│ │ │ │ -0001d940: 2031 2020 3920 2020 2020 2020 2020 2020   1  9           
│ │ │ │ +0001d910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d920: 2020 2020 2020 2020 373a 202e 202e 202e          7: . . .
│ │ │ │ +0001d930: 202e 202e 2020 3120 2039 2020 2020 2020   . .  1  9      
│ │ │ │ +0001d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9b0: 2020 207c 0a7c 6f37 203a 2042 6574 7469     |.|o7 : Betti
│ │ │ │ -0001d9c0: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0001d9a0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ +0001d9b0: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0001d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001d9e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001d9f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001da40: 0a7c 6938 203a 204d 5320 3d20 7072 756e  .|i8 : MS = prun
│ │ │ │ -0001da50: 6520 7075 7368 466f 7277 6172 6428 702c  e pushForward(p,
│ │ │ │ -0001da60: 2063 6f6b 6572 2046 462e 6464 5f36 293b   coker FF.dd_6);
│ │ │ │ -0001da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001da30: 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 203d  ----+.|i8 : MS =
│ │ │ │ +0001da40: 2070 7275 6e65 2070 7573 6846 6f72 7761   prune pushForwa
│ │ │ │ +0001da50: 7264 2870 2c20 636f 6b65 7220 4646 2e64  rd(p, coker FF.d
│ │ │ │ +0001da60: 645f 3629 3b20 2020 2020 2020 2020 2020  d_6);           
│ │ │ │ +0001da70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ -0001dad0: 203a 2054 203d 2065 7874 6572 696f 7254   : T = exteriorT
│ │ │ │ -0001dae0: 6f72 4d6f 6475 6c65 2866 2c4d 5329 3b20  orModule(f,MS); 
│ │ │ │ +0001dac0: 2b0a 7c69 3920 3a20 5420 3d20 6578 7465  +.|i9 : T = exte
│ │ │ │ +0001dad0: 7269 6f72 546f 724d 6f64 756c 6528 662c  riorTorModule(f,
│ │ │ │ +0001dae0: 4d53 293b 2020 2020 2020 2020 2020 2020  MS);            
│ │ │ │  0001daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001db00: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001db50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -0001db60: 6265 7474 6920 5420 2020 2020 2020 2020  betti T         
│ │ │ │ +0001db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001db50: 3130 203a 2062 6574 7469 2054 2020 2020  10 : betti T    
│ │ │ │ +0001db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001db90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dbe0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001dbf0: 2020 2020 3020 2020 3120 2020 2020 2020      0   1       
│ │ │ │ +0001dbd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001dbe0: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +0001dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc20: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -0001dc30: 3d20 746f 7461 6c3a 2038 3420 3235 3220  = total: 84 252 
│ │ │ │ +0001dc10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001dc20: 7c6f 3130 203d 2074 6f74 616c 3a20 3834  |o10 = total: 84
│ │ │ │ +0001dc30: 2032 3532 2020 2020 2020 2020 2020 2020   252            
│ │ │ │  0001dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001dc70: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ -0001dc80: 3320 2033 3920 2020 2020 2020 2020 2020  3  39           
│ │ │ │ +0001dc60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001dc70: 2030 3a20 3133 2020 3339 2020 2020 2020   0: 13  39      
│ │ │ │ +0001dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001dcc0: 2020 313a 2033 3320 2039 3920 2020 2020    1: 33  99     
│ │ │ │ +0001dca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001dcb0: 2020 2020 2020 2031 3a20 3333 2020 3939         1: 33  99
│ │ │ │ +0001dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001dd00: 2020 2020 2020 2020 323a 2032 3920 2038          2: 29  8
│ │ │ │ -0001dd10: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +0001dcf0: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ +0001dd00: 3239 2020 3837 2020 2020 2020 2020 2020  29  87          
│ │ │ │ +0001dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd40: 207c 0a7c 2020 2020 2020 2020 2020 333a   |.|          3:
│ │ │ │ -0001dd50: 2020 3920 2032 3720 2020 2020 2020 2020    9  27         
│ │ │ │ +0001dd30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dd40: 2020 2033 3a20 2039 2020 3237 2020 2020     3:  9  27    
│ │ │ │ +0001dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001dd70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ddc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001ddd0: 6f31 3020 3a20 4265 7474 6954 616c 6c79  o10 : BettiTally
│ │ │ │ +0001ddc0: 2020 7c0a 7c6f 3130 203a 2042 6574 7469    |.|o10 : Betti
│ │ │ │ +0001ddd0: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │  0001dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001de00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ -0001de60: 3a20 6265 7474 6920 7265 7320 2850 5420  : betti res (PT 
│ │ │ │ -0001de70: 3d20 7072 756e 6520 5429 2020 2020 2020  = prune T)      
│ │ │ │ +0001de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0001de50: 7c69 3131 203a 2062 6574 7469 2072 6573  |i11 : betti res
│ │ │ │ +0001de60: 2028 5054 203d 2070 7275 6e65 2054 2920   (PT = prune T) 
│ │ │ │ +0001de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001dea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001de90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001def0: 2020 2020 2020 3020 2031 2020 3220 2020        0  1  2   
│ │ │ │ -0001df00: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │ +0001ded0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001dee0: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ +0001def0: 2032 2020 2033 2020 2034 2020 2020 2020   2   3   4      
│ │ │ │ +0001df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df20: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001df30: 3120 3d20 746f 7461 6c3a 2033 3120 3535  1 = total: 31 55
│ │ │ │ -0001df40: 2038 3720 3132 3720 3137 3520 2020 2020   87 127 175     
│ │ │ │ +0001df20: 7c0a 7c6f 3131 203d 2074 6f74 616c 3a20  |.|o11 = total: 
│ │ │ │ +0001df30: 3331 2035 3520 3837 2031 3237 2031 3735  31 55 87 127 175
│ │ │ │ +0001df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df70: 207c 0a7c 2020 2020 2020 2020 2020 303a   |.|          0:
│ │ │ │ -0001df80: 2031 3320 3234 2033 3920 2035 3820 2038   13 24 39  58  8
│ │ │ │ -0001df90: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -0001dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dfb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001dfc0: 2020 2020 313a 2031 3820 3331 2034 3820      1: 18 31 48 
│ │ │ │ -0001dfd0: 2036 3920 2039 3420 2020 2020 2020 2020   69  94         
│ │ │ │ +0001df60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001df70: 2020 2030 3a20 3133 2032 3420 3339 2020     0: 13 24 39  
│ │ │ │ +0001df80: 3538 2020 3831 2020 2020 2020 2020 2020  58  81          
│ │ │ │ +0001df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dfb0: 2020 2020 2020 2020 2031 3a20 3138 2033           1: 18 3
│ │ │ │ +0001dfc0: 3120 3438 2020 3639 2020 3934 2020 2020  1 48  69  94    
│ │ │ │ +0001dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001dff0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e040: 2020 207c 0a7c 6f31 3120 3a20 4265 7474     |.|o11 : Bett
│ │ │ │ -0001e050: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +0001e030: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ +0001e040: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0001e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e080: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001e070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e080: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001e090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001e0d0: 0a7c 6931 3220 3a20 616e 6e20 5054 2020  .|i12 : ann PT  
│ │ │ │ +0001e0c0: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 6e6e  ----+.|i12 : ann
│ │ │ │ +0001e0d0: 2050 5420 2020 2020 2020 2020 2020 2020   PT             
│ │ │ │  0001e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001e100: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e150: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001e160: 3220 3d20 6964 6561 6c28 6520 6520 6520  2 = ideal(e e e 
│ │ │ │ -0001e170: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001e150: 7c0a 7c6f 3132 203d 2069 6465 616c 2865  |.|o12 = ideal(e
│ │ │ │ +0001e160: 2065 2065 2029 2020 2020 2020 2020 2020   e e )          
│ │ │ │ +0001e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001e1b0: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ +0001e190: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e1a0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e230: 6f31 3220 3a20 4964 6561 6c20 6f66 206b  o12 : Ideal of k
│ │ │ │ -0001e240: 6b5b 6520 2e2e 6520 5d20 2020 2020 2020  k[e ..e ]       
│ │ │ │ +0001e220: 2020 7c0a 7c6f 3132 203a 2049 6465 616c    |.|o12 : Ideal
│ │ │ │ +0001e230: 206f 6620 6b6b 5b65 202e 2e65 205d 2020   of kk[e ..e ]  
│ │ │ │ +0001e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e270: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001e280: 2020 2020 2020 2020 2030 2020 2032 2020           0   2  
│ │ │ │ +0001e260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e270: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0001e280: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001e2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e2b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001e300: 0a7c 6931 3320 3a20 5054 3020 3d20 696d  .|i13 : PT0 = im
│ │ │ │ -0001e310: 6167 6520 2869 6e64 7563 6564 4d61 7028  age (inducedMap(
│ │ │ │ -0001e320: 5054 2c63 6f76 6572 2050 5429 2a20 2828  PT,cover PT)* ((
│ │ │ │ -0001e330: 636f 7665 7220 5054 295f 7b30 2e2e 3132  cover PT)_{0..12
│ │ │ │ -0001e340: 7d29 293b 207c 0a2b 2d2d 2d2d 2d2d 2d2d  })); |.+--------
│ │ │ │ +0001e2f0: 2d2d 2d2d 2b0a 7c69 3133 203a 2050 5430  ----+.|i13 : PT0
│ │ │ │ +0001e300: 203d 2069 6d61 6765 2028 696e 6475 6365   = image (induce
│ │ │ │ +0001e310: 644d 6170 2850 542c 636f 7665 7220 5054  dMap(PT,cover PT
│ │ │ │ +0001e320: 292a 2028 2863 6f76 6572 2050 5429 5f7b  )* ((cover PT)_{
│ │ │ │ +0001e330: 302e 2e31 327d 2929 3b20 7c0a 2b2d 2d2d  0..12})); |.+---
│ │ │ │ +0001e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0001e390: 3420 3a20 5054 3120 3d20 696d 6167 6520  4 : PT1 = image 
│ │ │ │ -0001e3a0: 2869 6e64 7563 6564 4d61 7028 5054 2c63  (inducedMap(PT,c
│ │ │ │ -0001e3b0: 6f76 6572 2050 5429 2a20 2828 636f 7665  over PT)* ((cove
│ │ │ │ -0001e3c0: 7220 5054 295f 7b31 332e 2e33 307d 2929  r PT)_{13..30}))
│ │ │ │ -0001e3d0: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ;|.+------------
│ │ │ │ +0001e380: 2b0a 7c69 3134 203a 2050 5431 203d 2069  +.|i14 : PT1 = i
│ │ │ │ +0001e390: 6d61 6765 2028 696e 6475 6365 644d 6170  mage (inducedMap
│ │ │ │ +0001e3a0: 2850 542c 636f 7665 7220 5054 292a 2028  (PT,cover PT)* (
│ │ │ │ +0001e3b0: 2863 6f76 6572 2050 5429 5f7b 3133 2e2e  (cover PT)_{13..
│ │ │ │ +0001e3c0: 3330 7d29 293b 7c0a 2b2d 2d2d 2d2d 2d2d  30}));|.+-------
│ │ │ │ +0001e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e410: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20  -------+.|i15 : 
│ │ │ │ -0001e420: 6265 7474 6920 7265 7320 7072 756e 6520  betti res prune 
│ │ │ │ -0001e430: 5054 3020 2020 2020 2020 2020 2020 2020  PT0             
│ │ │ │ +0001e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001e410: 3135 203a 2062 6574 7469 2072 6573 2070  15 : betti res p
│ │ │ │ +0001e420: 7275 6e65 2050 5430 2020 2020 2020 2020  rune PT0        
│ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e450: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001e4b0: 2020 2020 3020 2031 2020 3220 2033 2020      0  1  2  3  
│ │ │ │ -0001e4c0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0001e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4e0: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ -0001e4f0: 3d20 746f 7461 6c3a 2031 3320 3234 2033  = total: 13 24 3
│ │ │ │ -0001e500: 3920 3538 2038 3120 2020 2020 2020 2020  9 58 81         
│ │ │ │ +0001e490: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e4a0: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ +0001e4b0: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +0001e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e4d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e4e0: 7c6f 3135 203d 2074 6f74 616c 3a20 3133  |o15 = total: 13
│ │ │ │ +0001e4f0: 2032 3420 3339 2035 3820 3831 2020 2020   24 39 58 81    
│ │ │ │ +0001e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e530: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ -0001e540: 3320 3234 2033 3920 3538 2038 3120 2020  3 24 39 58 81   
│ │ │ │ +0001e520: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001e530: 2030 3a20 3133 2032 3420 3339 2035 3820   0: 13 24 39 58 
│ │ │ │ +0001e540: 3831 2020 2020 2020 2020 2020 2020 2020  81              
│ │ │ │  0001e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001e560: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001e5c0: 3520 3a20 4265 7474 6954 616c 6c79 2020  5 : BettiTally  
│ │ │ │ +0001e5b0: 7c0a 7c6f 3135 203a 2042 6574 7469 5461  |.|o15 : BettiTa
│ │ │ │ +0001e5c0: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │  0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e600: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001e5f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e640: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │ -0001e650: 6265 7474 6920 7265 7320 7072 756e 6520  betti res prune 
│ │ │ │ -0001e660: 5054 3120 2020 2020 2020 2020 2020 2020  PT1             
│ │ │ │ +0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001e640: 3136 203a 2062 6574 7469 2072 6573 2070  16 : betti res p
│ │ │ │ +0001e650: 7275 6e65 2050 5431 2020 2020 2020 2020  rune PT1        
│ │ │ │ +0001e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e680: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e6d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001e6e0: 2020 2020 3020 2031 2020 3220 2033 2020      0  1  2  3  
│ │ │ │ -0001e6f0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0001e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e710: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ -0001e720: 3d20 746f 7461 6c3a 2031 3820 3238 2033  = total: 18 28 3
│ │ │ │ -0001e730: 3920 3531 2036 3420 2020 2020 2020 2020  9 51 64         
│ │ │ │ +0001e6c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e6d0: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ +0001e6e0: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e710: 7c6f 3136 203d 2074 6f74 616c 3a20 3138  |o16 = total: 18
│ │ │ │ +0001e720: 2032 3820 3339 2035 3120 3634 2020 2020   28 39 51 64    
│ │ │ │ +0001e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e760: 0a7c 2020 2020 2020 2020 2020 313a 2031  .|          1: 1
│ │ │ │ -0001e770: 3820 3238 2033 3920 3531 2036 3420 2020  8 28 39 51 64   
│ │ │ │ +0001e750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001e760: 2031 3a20 3138 2032 3820 3339 2035 3120   1: 18 28 39 51 
│ │ │ │ +0001e770: 3634 2020 2020 2020 2020 2020 2020 2020  64              
│ │ │ │  0001e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001e790: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001e7f0: 3620 3a20 4265 7474 6954 616c 6c79 2020  6 : BettiTally  
│ │ │ │ +0001e7e0: 7c0a 7c6f 3136 203a 2042 6574 7469 5461  |.|o16 : BettiTa
│ │ │ │ +0001e7f0: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │  0001e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e830: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
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│ │ │ │  0001e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20  -------+.|i17 : 
│ │ │ │ -0001e880: 6265 7474 6920 7265 7320 7072 756e 6520  betti res prune 
│ │ │ │ -0001e890: 5054 2020 2020 2020 2020 2020 2020 2020  PT              
│ │ │ │ +0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001e870: 3137 203a 2062 6574 7469 2072 6573 2070  17 : betti res p
│ │ │ │ +0001e880: 7275 6e65 2050 5420 2020 2020 2020 2020  rune PT         
│ │ │ │ +0001e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e8b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e900: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001e910: 2020 2020 3020 2031 2020 3220 2020 3320      0  1  2   3 
│ │ │ │ -0001e920: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ -0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e940: 2020 2020 2020 2020 207c 0a7c 6f31 3720           |.|o17 
│ │ │ │ -0001e950: 3d20 746f 7461 6c3a 2033 3120 3535 2038  = total: 31 55 8
│ │ │ │ -0001e960: 3720 3132 3720 3137 3520 2020 2020 2020  7 127 175       
│ │ │ │ +0001e8f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e900: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ +0001e910: 2020 2033 2020 2034 2020 2020 2020 2020     3   4        
│ │ │ │ +0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e940: 7c6f 3137 203d 2074 6f74 616c 3a20 3331  |o17 = total: 31
│ │ │ │ +0001e950: 2035 3520 3837 2031 3237 2031 3735 2020   55 87 127 175  
│ │ │ │ +0001e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e990: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ -0001e9a0: 3320 3234 2033 3920 2035 3820 2038 3120  3 24 39  58  81 
│ │ │ │ +0001e980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001e990: 2030 3a20 3133 2032 3420 3339 2020 3538   0: 13 24 39  58
│ │ │ │ +0001e9a0: 2020 3831 2020 2020 2020 2020 2020 2020    81            
│ │ │ │  0001e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001e9e0: 2020 313a 2031 3820 3331 2034 3820 2036    1: 18 31 48  6
│ │ │ │ -0001e9f0: 3920 2039 3420 2020 2020 2020 2020 2020  9  94           
│ │ │ │ +0001e9c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e9d0: 2020 2020 2020 2031 3a20 3138 2033 3120         1: 18 31 
│ │ │ │ +0001e9e0: 3438 2020 3639 2020 3934 2020 2020 2020  48  69  94      
│ │ │ │ +0001e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001ea10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea60: 207c 0a7c 6f31 3720 3a20 4265 7474 6954   |.|o17 : BettiT
│ │ │ │ -0001ea70: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │ +0001ea50: 2020 2020 2020 7c0a 7c6f 3137 203a 2042        |.|o17 : B
│ │ │ │ +0001ea60: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
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│ │ │ │  0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eaa0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001ea90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │  0001eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ead0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0001eaf0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
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│ │ │ │ +0001ed60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001ed70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001ed80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001ed90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001eda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001edb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
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│ │ │ │ -0001f000: 2063 6f6d 7061 7265 7320 746f 2073 6565   compares to see
│ │ │ │ -0001f010: 2077 6865 7468 6572 2074 6869 7320 6973   whether this is
│ │ │ │ -0001f020: 2065 7175 616c 2074 6f0a 706f 287a 2f32   equal to.po(z/2
│ │ │ │ -0001f030: 2d31 2f32 292e 2041 7672 616d 6f76 2c20  -1/2). Avramov, 
│ │ │ │ -0001f040: 5365 6365 6c65 616e 7520 616e 6420 5a68  Seceleanu and Zh
│ │ │ │ -0001f050: 656e 6720 6861 7665 2070 726f 7665 6e20  eng have proven 
│ │ │ │ -0001f060: 7468 6174 2069 6620 7468 6520 6964 6561  that if the idea
│ │ │ │ -0001f070: 6c20 6f66 0a71 7561 6472 6174 6963 206c  l of.quadratic l
│ │ │ │ -0001f080: 6561 6469 6e67 2066 6f72 6d73 206f 6620  eading forms of 
│ │ │ │ -0001f090: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -0001f0a0: 7365 6374 696f 6e20 6f66 2063 6f64 696d  section of codim
│ │ │ │ -0001f0b0: 656e 7369 6f6e 2063 2067 656e 6572 6174  ension c generat
│ │ │ │ -0001f0c0: 6520 616e 0a69 6465 616c 206f 6620 636f  e an.ideal of co
│ │ │ │ -0001f0d0: 6469 6d65 6e73 696f 6e20 6174 206c 6561  dimension at lea
│ │ │ │ -0001f0e0: 7374 2063 2d31 2c20 7468 656e 2074 6865  st c-1, then the
│ │ │ │ -0001f0f0: 2042 6574 7469 206e 756d 6265 7273 206f   Betti numbers o
│ │ │ │ -0001f100: 6620 616e 7920 6d6f 6475 6c65 2067 726f  f any module gro
│ │ │ │ -0001f110: 772c 0a65 7665 6e74 7561 6c6c 792c 2061  w,.eventually, a
│ │ │ │ -0001f120: 7320 6120 7369 6e67 6c65 2070 6f6c 796e  s a single polyn
│ │ │ │ -0001f130: 6f6d 6961 6c20 2869 6e73 7465 6164 206f  omial (instead o
│ │ │ │ -0001f140: 6620 7265 7175 6972 696e 6720 7365 7061  f requiring sepa
│ │ │ │ -0001f150: 7261 7465 2070 6f6c 796e 6f6d 6961 6c73  rate polynomials
│ │ │ │ -0001f160: 0a66 6f72 2065 7665 6e20 616e 6420 6f64  .for even and od
│ │ │ │ -0001f170: 6420 7465 726d 732e 2920 5468 6973 2073  d terms.) This s
│ │ │ │ -0001f180: 6372 6970 7420 6368 6563 6b73 2074 6865  cript checks the
│ │ │ │ -0001f190: 2072 6573 756c 7420 696e 2074 6865 2068   result in the h
│ │ │ │ -0001f1a0: 6f6d 6f67 656e 656f 7573 2063 6173 650a  omogeneous case.
│ │ │ │ -0001f1b0: 2869 6e20 7768 6963 6820 6361 7365 2074  (in which case t
│ │ │ │ -0001f1c0: 6865 2063 6f6e 6469 7469 6f6e 2069 7320  he condition is 
│ │ │ │ -0001f1d0: 6e65 6365 7373 6172 7920 616e 6420 7375  necessary and su
│ │ │ │ -0001f1e0: 6666 6963 6965 6e74 2e29 0a0a 2b2d 2d2d  fficient.)..+---
│ │ │ │ +0001eda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +0001edb0: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +0001edc0: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +0001edd0: 2020 2020 2020 2020 2870 2c74 2920 3d20          (p,t) = 
│ │ │ │ +0001ede0: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ +0001edf0: 616c 204d 0a20 202a 2049 6e70 7574 733a  al M.  * Inputs:
│ │ │ │ +0001ee00: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ +0001ee10: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +0001ee20: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +0001ee30: 2c2c 206d 6f64 756c 6520 6f76 6572 2061  ,, module over a
│ │ │ │ +0001ee40: 2063 6f6d 706c 6574 650a 2020 2020 2020   complete.      
│ │ │ │ +0001ee50: 2020 696e 7465 7273 6563 7469 6f6e 0a20    intersection. 
│ │ │ │ +0001ee60: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0001ee70: 2020 2a20 702c 2061 202a 6e6f 7465 2072    * p, a *note r
│ │ │ │ +0001ee80: 696e 6720 656c 656d 656e 743a 2028 4d61  ing element: (Ma
│ │ │ │ +0001ee90: 6361 756c 6179 3244 6f63 2952 696e 6745  caulay2Doc)RingE
│ │ │ │ +0001eea0: 6c65 6d65 6e74 2c2c 2070 287a 293d 7065  lement,, p(z)=pe
│ │ │ │ +0001eeb0: 287a 2f32 292c 0a20 2020 2020 2020 2077  (z/2),.        w
│ │ │ │ +0001eec0: 6865 7265 2070 6520 6973 2074 6865 2048  here pe is the H
│ │ │ │ +0001eed0: 696c 6265 7274 2070 6f6c 7920 6f66 2045  ilbert poly of E
│ │ │ │ +0001eee0: 7874 5e7b 6576 656e 7d28 4d2c 6b29 0a20  xt^{even}(M,k). 
│ │ │ │ +0001eef0: 2020 2020 202a 2074 2c20 6120 2a6e 6f74       * t, a *not
│ │ │ │ +0001ef00: 6520 426f 6f6c 6561 6e20 7661 6c75 653a  e Boolean value:
│ │ │ │ +0001ef10: 2028 4d61 6361 756c 6179 3244 6f63 2942   (Macaulay2Doc)B
│ │ │ │ +0001ef20: 6f6f 6c65 616e 2c2c 2074 7275 6520 6966  oolean,, true if
│ │ │ │ +0001ef30: 2074 6865 2065 7665 6e20 616e 640a 2020   the even and.  
│ │ │ │ +0001ef40: 2020 2020 2020 6f64 6420 706f 6c79 6e6f        odd polyno
│ │ │ │ +0001ef50: 6d69 616c 7320 6d61 7463 6820 746f 2066  mials match to f
│ │ │ │ +0001ef60: 6f72 6d20 6f6e 6520 706f 6c79 6e6f 6d69  orm one polynomi
│ │ │ │ +0001ef70: 616c 0a0a 4465 7363 7269 7074 696f 6e0a  al..Description.
│ │ │ │ +0001ef80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d  ===========..Com
│ │ │ │ +0001ef90: 7075 7465 7320 7468 6520 4869 6c62 6572  putes the Hilber
│ │ │ │ +0001efa0: 7420 706f 6c79 6e6f 6d69 616c 7320 7065  t polynomials pe
│ │ │ │ +0001efb0: 287a 292c 2070 6f28 7a29 206f 6620 6576  (z), po(z) of ev
│ │ │ │ +0001efc0: 656e 4578 744d 6f64 756c 6520 616e 640a  enExtModule and.
│ │ │ │ +0001efd0: 6f64 6445 7874 4d6f 6475 6c65 2e20 4974  oddExtModule. It
│ │ │ │ +0001efe0: 2072 6574 7572 6e73 2070 6528 7a2f 3229   returns pe(z/2)
│ │ │ │ +0001eff0: 2c20 616e 6420 636f 6d70 6172 6573 2074  , and compares t
│ │ │ │ +0001f000: 6f20 7365 6520 7768 6574 6865 7220 7468  o see whether th
│ │ │ │ +0001f010: 6973 2069 7320 6571 7561 6c20 746f 0a70  is is equal to.p
│ │ │ │ +0001f020: 6f28 7a2f 322d 312f 3229 2e20 4176 7261  o(z/2-1/2). Avra
│ │ │ │ +0001f030: 6d6f 762c 2053 6563 656c 6561 6e75 2061  mov, Seceleanu a
│ │ │ │ +0001f040: 6e64 205a 6865 6e67 2068 6176 6520 7072  nd Zheng have pr
│ │ │ │ +0001f050: 6f76 656e 2074 6861 7420 6966 2074 6865  oven that if the
│ │ │ │ +0001f060: 2069 6465 616c 206f 660a 7175 6164 7261   ideal of.quadra
│ │ │ │ +0001f070: 7469 6320 6c65 6164 696e 6720 666f 726d  tic leading form
│ │ │ │ +0001f080: 7320 6f66 2061 2063 6f6d 706c 6574 6520  s of a complete 
│ │ │ │ +0001f090: 696e 7465 7273 6563 7469 6f6e 206f 6620  intersection of 
│ │ │ │ +0001f0a0: 636f 6469 6d65 6e73 696f 6e20 6320 6765  codimension c ge
│ │ │ │ +0001f0b0: 6e65 7261 7465 2061 6e0a 6964 6561 6c20  nerate an.ideal 
│ │ │ │ +0001f0c0: 6f66 2063 6f64 696d 656e 7369 6f6e 2061  of codimension a
│ │ │ │ +0001f0d0: 7420 6c65 6173 7420 632d 312c 2074 6865  t least c-1, the
│ │ │ │ +0001f0e0: 6e20 7468 6520 4265 7474 6920 6e75 6d62  n the Betti numb
│ │ │ │ +0001f0f0: 6572 7320 6f66 2061 6e79 206d 6f64 756c  ers of any modul
│ │ │ │ +0001f100: 6520 6772 6f77 2c0a 6576 656e 7475 616c  e grow,.eventual
│ │ │ │ +0001f110: 6c79 2c20 6173 2061 2073 696e 676c 6520  ly, as a single 
│ │ │ │ +0001f120: 706f 6c79 6e6f 6d69 616c 2028 696e 7374  polynomial (inst
│ │ │ │ +0001f130: 6561 6420 6f66 2072 6571 7569 7269 6e67  ead of requiring
│ │ │ │ +0001f140: 2073 6570 6172 6174 6520 706f 6c79 6e6f   separate polyno
│ │ │ │ +0001f150: 6d69 616c 730a 666f 7220 6576 656e 2061  mials.for even a
│ │ │ │ +0001f160: 6e64 206f 6464 2074 6572 6d73 2e29 2054  nd odd terms.) T
│ │ │ │ +0001f170: 6869 7320 7363 7269 7074 2063 6865 636b  his script check
│ │ │ │ +0001f180: 7320 7468 6520 7265 7375 6c74 2069 6e20  s the result in 
│ │ │ │ +0001f190: 7468 6520 686f 6d6f 6765 6e65 6f75 7320  the homogeneous 
│ │ │ │ +0001f1a0: 6361 7365 0a28 696e 2077 6869 6368 2063  case.(in which c
│ │ │ │ +0001f1b0: 6173 6520 7468 6520 636f 6e64 6974 696f  ase the conditio
│ │ │ │ +0001f1c0: 6e20 6973 206e 6563 6573 7361 7279 2061  n is necessary a
│ │ │ │ +0001f1d0: 6e64 2073 7566 6669 6369 656e 742e 290a  nd sufficient.).
│ │ │ │ +0001f1e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001f1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f220: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 313d  -----+.|i1 : R1=
│ │ │ │ -0001f230: 5a5a 2f31 3031 5b61 2c62 2c63 5d2f 6964  ZZ/101[a,b,c]/id
│ │ │ │ -0001f240: 6561 6c28 615e 322c 625e 322c 635e 3529  eal(a^2,b^2,c^5)
│ │ │ │ -0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f260: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +0001f220: 3a20 5231 3d5a 5a2f 3130 315b 612c 622c  : R1=ZZ/101[a,b,
│ │ │ │ +0001f230: 635d 2f69 6465 616c 2861 5e32 2c62 5e32  c]/ideal(a^2,b^2
│ │ │ │ +0001f240: 2c63 5e35 2920 2020 2020 2020 2020 2020  ,c^5)           
│ │ │ │ +0001f250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f290: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001f2a0: 203d 2052 3120 2020 2020 2020 2020 2020   = R1           
│ │ │ │ +0001f290: 7c0a 7c6f 3120 3d20 5231 2020 2020 2020  |.|o1 = R1      
│ │ │ │ +0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f310: 207c 0a7c 6f31 203a 2051 756f 7469 656e   |.|o1 : Quotien
│ │ │ │ -0001f320: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +0001f300: 2020 2020 2020 7c0a 7c6f 3120 3a20 5175        |.|o1 : Qu
│ │ │ │ +0001f310: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +0001f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001f340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f380: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ -0001f390: 323d 5a5a 2f31 3031 5b61 2c62 2c63 5d2f  2=ZZ/101[a,b,c]/
│ │ │ │ -0001f3a0: 6964 6561 6c28 615e 332c 625e 3329 2020  ideal(a^3,b^3)  
│ │ │ │ -0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001f380: 3220 3a20 5232 3d5a 5a2f 3130 315b 612c  2 : R2=ZZ/101[a,
│ │ │ │ +0001f390: 622c 635d 2f69 6465 616c 2861 5e33 2c62  b,c]/ideal(a^3,b
│ │ │ │ +0001f3a0: 5e33 2920 2020 2020 2020 2020 2020 2020  ^3)             
│ │ │ │ +0001f3b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f400: 6f32 203d 2052 3220 2020 2020 2020 2020  o2 = R2         
│ │ │ │ +0001f3f0: 2020 7c0a 7c6f 3220 3d20 5232 2020 2020    |.|o2 = R2    
│ │ │ │ +0001f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001f420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f470: 2020 207c 0a7c 6f32 203a 2051 756f 7469     |.|o2 : Quoti
│ │ │ │ -0001f480: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0001f460: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +0001f470: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0001f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f4b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001f4a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0001f4f0: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ -0001f500: 6961 6c20 636f 6b65 7220 7261 6e64 6f6d  ial coker random
│ │ │ │ -0001f510: 2852 315e 7b30 2c31 7d2c 5231 5e7b 333a  (R1^{0,1},R1^{3:
│ │ │ │ -0001f520: 2d31 7d29 7c0a 7c20 2020 2020 2020 2020  -1})|.|         
│ │ │ │ +0001f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0001f4e0: 7c69 3320 3a20 6578 7449 734f 6e65 506f  |i3 : extIsOnePo
│ │ │ │ +0001f4f0: 6c79 6e6f 6d69 616c 2063 6f6b 6572 2072  lynomial coker r
│ │ │ │ +0001f500: 616e 646f 6d28 5231 5e7b 302c 317d 2c52  andom(R1^{0,1},R
│ │ │ │ +0001f510: 315e 7b33 3a2d 317d 297c 0a7c 2020 2020  1^{3:-1})|.|    
│ │ │ │ +0001f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001f560: 0a7c 2020 2020 2020 3120 3220 2020 3120  .|      1 2   1 
│ │ │ │ +0001f550: 2020 2020 7c0a 7c20 2020 2020 2031 2032      |.|      1 2
│ │ │ │ +0001f560: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f590: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0001f5a0: 3d20 282d 7a20 202d 202d 7a20 2b20 332c  = (-z  - -z + 3,
│ │ │ │ -0001f5b0: 2074 7275 6529 2020 2020 2020 2020 2020   true)          
│ │ │ │ -0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5d0: 2020 2020 207c 0a7c 2020 2020 2020 3220       |.|      2 
│ │ │ │ -0001f5e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0001f580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f590: 0a7c 6f33 203d 2028 2d7a 2020 2d20 2d7a  .|o3 = (-z  - -z
│ │ │ │ +0001f5a0: 202b 2033 2c20 7472 7565 2920 2020 2020   + 3, true)     
│ │ │ │ +0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f5c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f5d0: 2020 2032 2020 2020 2032 2020 2020 2020     2     2      
│ │ │ │ +0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f610: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001f600: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f640: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -0001f650: 203a 2053 6571 7565 6e63 6520 2020 2020   : Sequence     
│ │ │ │ +0001f640: 7c0a 7c6f 3320 3a20 5365 7175 656e 6365  |.|o3 : Sequence
│ │ │ │ +0001f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f680: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f670: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6c0: 2d2b 0a7c 6934 203a 2065 7874 4973 4f6e  -+.|i4 : extIsOn
│ │ │ │ -0001f6d0: 6550 6f6c 796e 6f6d 6961 6c20 636f 6b65  ePolynomial coke
│ │ │ │ -0001f6e0: 7220 7261 6e64 6f6d 2852 325e 7b30 2c31  r random(R2^{0,1
│ │ │ │ -0001f6f0: 7d2c 5232 5e7b 333a 2d31 7d29 7c0a 7c20  },R2^{3:-1})|.| 
│ │ │ │ +0001f6b0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6578  ------+.|i4 : ex
│ │ │ │ +0001f6c0: 7449 734f 6e65 506f 6c79 6e6f 6d69 616c  tIsOnePolynomial
│ │ │ │ +0001f6d0: 2063 6f6b 6572 2072 616e 646f 6d28 5232   coker random(R2
│ │ │ │ +0001f6e0: 5e7b 302c 317d 2c52 325e 7b33 3a2d 317d  ^{0,1},R2^{3:-1}
│ │ │ │ +0001f6f0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │  0001f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f730: 2020 2020 2020 207c 0a7c 6f34 203d 2028         |.|o4 = (
│ │ │ │ -0001f740: 337a 202d 2032 2c20 6661 6c73 6529 2020  3z - 2, false)  
│ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001f730: 3420 3d20 2833 7a20 2d20 322c 2066 616c  4 = (3z - 2, fal
│ │ │ │ +0001f740: 7365 2920 2020 2020 2020 2020 2020 2020  se)             
│ │ │ │  0001f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f770: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f760: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f7b0: 6f34 203a 2053 6571 7565 6e63 6520 2020  o4 : Sequence   
│ │ │ │ +0001f7a0: 2020 7c0a 7c6f 3420 3a20 5365 7175 656e    |.|o4 : Sequen
│ │ │ │ +0001f7b0: 6365 2020 2020 2020 2020 2020 2020 2020  ce              
│ │ │ │  0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f820: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ -0001f830: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -0001f840: 7465 2065 7665 6e45 7874 4d6f 6475 6c65  te evenExtModule
│ │ │ │ -0001f850: 3a20 6576 656e 4578 744d 6f64 756c 652c  : evenExtModule,
│ │ │ │ -0001f860: 202d 2d20 6576 656e 2070 6172 7420 6f66   -- even part of
│ │ │ │ -0001f870: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ -0001f880: 2061 0a20 2020 2063 6f6d 706c 6574 6520   a.    complete 
│ │ │ │ -0001f890: 696e 7465 7273 6563 7469 6f6e 2061 7320  intersection as 
│ │ │ │ -0001f8a0: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ -0001f8b0: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ -0001f8c0: 202a 6e6f 7465 206f 6464 4578 744d 6f64   *note oddExtMod
│ │ │ │ -0001f8d0: 756c 653a 206f 6464 4578 744d 6f64 756c  ule: oddExtModul
│ │ │ │ -0001f8e0: 652c 202d 2d20 6f64 6420 7061 7274 206f  e, -- odd part o
│ │ │ │ -0001f8f0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -0001f900: 7220 6120 636f 6d70 6c65 7465 0a20 2020  r a complete.   
│ │ │ │ -0001f910: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ -0001f920: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -0001f930: 6f70 6572 6174 6f72 2072 696e 670a 0a57  operator ring..W
│ │ │ │ -0001f940: 6179 7320 746f 2075 7365 2065 7874 4973  ays to use extIs
│ │ │ │ -0001f950: 4f6e 6550 6f6c 796e 6f6d 6961 6c3a 0a3d  OnePolynomial:.=
│ │ │ │ +0001f810: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +0001f820: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +0001f830: 2a20 2a6e 6f74 6520 6576 656e 4578 744d  * *note evenExtM
│ │ │ │ +0001f840: 6f64 756c 653a 2065 7665 6e45 7874 4d6f  odule: evenExtMo
│ │ │ │ +0001f850: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061  dule, -- even pa
│ │ │ │ +0001f860: 7274 206f 6620 4578 745e 2a28 4d2c 6b29  rt of Ext^*(M,k)
│ │ │ │ +0001f870: 206f 7665 7220 610a 2020 2020 636f 6d70   over a.    comp
│ │ │ │ +0001f880: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +0001f890: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ +0001f8a0: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ +0001f8b0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445  g.  * *note oddE
│ │ │ │ +0001f8c0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874  xtModule: oddExt
│ │ │ │ +0001f8d0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070  Module, -- odd p
│ │ │ │ +0001f8e0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ +0001f8f0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +0001f900: 650a 2020 2020 696e 7465 7273 6563 7469  e.    intersecti
│ │ │ │ +0001f910: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ +0001f920: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +0001f930: 6e67 0a0a 5761 7973 2074 6f20 7573 6520  ng..Ways to use 
│ │ │ │ +0001f940: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ +0001f950: 616c 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  al:.============
│ │ │ │  0001f960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001f970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -0001f980: 2020 2a20 2265 7874 4973 4f6e 6550 6f6c    * "extIsOnePol
│ │ │ │ -0001f990: 796e 6f6d 6961 6c28 4d6f 6475 6c65 2922  ynomial(Module)"
│ │ │ │ -0001f9a0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0001f9b0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0001f9c0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0001f9d0: 6563 7420 2a6e 6f74 6520 6578 7449 734f  ect *note extIsO
│ │ │ │ -0001f9e0: 6e65 506f 6c79 6e6f 6d69 616c 3a20 6578  nePolynomial: ex
│ │ │ │ -0001f9f0: 7449 734f 6e65 506f 6c79 6e6f 6d69 616c  tIsOnePolynomial
│ │ │ │ -0001fa00: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0001fa10: 686f 640a 6675 6e63 7469 6f6e 3a20 284d  hod.function: (M
│ │ │ │ -0001fa20: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0001fa30: 6f64 4675 6e63 7469 6f6e 2c2e 0a1f 0a46  odFunction,....F
│ │ │ │ -0001fa40: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0001fa50: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0001fa60: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0001fa70: 2045 7874 4d6f 6475 6c65 2c20 4e65 7874   ExtModule, Next
│ │ │ │ -0001fa80: 3a20 4578 744d 6f64 756c 6544 6174 612c  : ExtModuleData,
│ │ │ │ -0001fa90: 2050 7265 763a 2065 7874 4973 4f6e 6550   Prev: extIsOneP
│ │ │ │ -0001faa0: 6f6c 796e 6f6d 6961 6c2c 2055 703a 2054  olynomial, Up: T
│ │ │ │ -0001fab0: 6f70 0a0a 4578 744d 6f64 756c 6520 2d2d  op..ExtModule --
│ │ │ │ -0001fac0: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ -0001fad0: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -0001fae0: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
│ │ │ │ -0001faf0: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ -0001fb00: 746f 7220 7269 6e67 0a2a 2a2a 2a2a 2a2a  tor ring.*******
│ │ │ │ +0001f970: 3d3d 3d0a 0a20 202a 2022 6578 7449 734f  ===..  * "extIsO
│ │ │ │ +0001f980: 6e65 506f 6c79 6e6f 6d69 616c 284d 6f64  nePolynomial(Mod
│ │ │ │ +0001f990: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ +0001f9a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0001f9b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0001f9c0: 6520 6f62 6a65 6374 202a 6e6f 7465 2065  e object *note e
│ │ │ │ +0001f9d0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ +0001f9e0: 6c3a 2065 7874 4973 4f6e 6550 6f6c 796e  l: extIsOnePolyn
│ │ │ │ +0001f9f0: 6f6d 6961 6c2c 2069 7320 6120 2a6e 6f74  omial, is a *not
│ │ │ │ +0001fa00: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ +0001fa10: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
│ │ │ │ +0001fa20: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0001fa30: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0001fa40: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0001fa50: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0001fa60: 4e6f 6465 3a20 4578 744d 6f64 756c 652c  Node: ExtModule,
│ │ │ │ +0001fa70: 204e 6578 743a 2045 7874 4d6f 6475 6c65   Next: ExtModule
│ │ │ │ +0001fa80: 4461 7461 2c20 5072 6576 3a20 6578 7449  Data, Prev: extI
│ │ │ │ +0001fa90: 734f 6e65 506f 6c79 6e6f 6d69 616c 2c20  sOnePolynomial, 
│ │ │ │ +0001faa0: 5570 3a20 546f 700a 0a45 7874 4d6f 6475  Up: Top..ExtModu
│ │ │ │ +0001fab0: 6c65 202d 2d20 4578 745e 2a28 4d2c 6b29  le -- Ext^*(M,k)
│ │ │ │ +0001fac0: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ +0001fad0: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ +0001fae0: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ +0001faf0: 6f70 6572 6174 6f72 2072 696e 670a 2a2a  operator ring.**
│ │ │ │ +0001fb00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fb10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fb20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fb30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fb40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001fb50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -0001fb60: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -0001fb70: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -0001fb80: 2020 2020 2020 4520 3d20 4578 744d 6f64        E = ExtMod
│ │ │ │ -0001fb90: 756c 6520 4d0a 2020 2a20 496e 7075 7473  ule M.  * Inputs
│ │ │ │ -0001fba0: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ -0001fbb0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0001fbc0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0001fbd0: 652c 2c20 6f76 6572 2061 2063 6f6d 706c  e,, over a compl
│ │ │ │ -0001fbe0: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ -0001fbf0: 0a20 2020 2020 2020 2072 696e 670a 2020  .        ring.  
│ │ │ │ -0001fc00: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0001fc10: 202a 2045 2c20 6120 2a6e 6f74 6520 6d6f   * E, a *note mo
│ │ │ │ -0001fc20: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0001fc30: 446f 6329 4d6f 6475 6c65 2c2c 206f 7665  Doc)Module,, ove
│ │ │ │ -0001fc40: 7220 6120 706f 6c79 6e6f 6d69 616c 2072  r a polynomial r
│ │ │ │ -0001fc50: 696e 6720 7769 7468 0a20 2020 2020 2020  ing with.       
│ │ │ │ -0001fc60: 2067 656e 7320 696e 2065 7665 6e20 6465   gens in even de
│ │ │ │ -0001fc70: 6772 6565 0a0a 4465 7363 7269 7074 696f  gree..Descriptio
│ │ │ │ -0001fc80: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a55  n.===========..U
│ │ │ │ -0001fc90: 7365 7320 636f 6465 206f 6620 4176 7261  ses code of Avra
│ │ │ │ -0001fca0: 6d6f 762d 4772 6179 736f 6e20 6465 7363  mov-Grayson desc
│ │ │ │ -0001fcb0: 7269 6265 6420 696e 204d 6163 6175 6c61  ribed in Macaula
│ │ │ │ -0001fcc0: 7932 2062 6f6f 6b0a 0a2b 2d2d 2d2d 2d2d  y2 book..+------
│ │ │ │ +0001fb50: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +0001fb60: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +0001fb70: 3a20 0a20 2020 2020 2020 2045 203d 2045  : .        E = E
│ │ │ │ +0001fb80: 7874 4d6f 6475 6c65 204d 0a20 202a 2049  xtModule M.  * I
│ │ │ │ +0001fb90: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ +0001fba0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0001fbb0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0001fbc0: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120  Module,, over a 
│ │ │ │ +0001fbd0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +0001fbe0: 6374 696f 6e0a 2020 2020 2020 2020 7269  ction.        ri
│ │ │ │ +0001fbf0: 6e67 0a20 202a 204f 7574 7075 7473 3a0a  ng.  * Outputs:.
│ │ │ │ +0001fc00: 2020 2020 2020 2a20 452c 2061 202a 6e6f        * E, a *no
│ │ │ │ +0001fc10: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ +0001fc20: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ +0001fc30: 2c20 6f76 6572 2061 2070 6f6c 796e 6f6d  , over a polynom
│ │ │ │ +0001fc40: 6961 6c20 7269 6e67 2077 6974 680a 2020  ial ring with.  
│ │ │ │ +0001fc50: 2020 2020 2020 6765 6e73 2069 6e20 6576        gens in ev
│ │ │ │ +0001fc60: 656e 2064 6567 7265 650a 0a44 6573 6372  en degree..Descr
│ │ │ │ +0001fc70: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0001fc80: 3d3d 0a0a 5573 6573 2063 6f64 6520 6f66  ==..Uses code of
│ │ │ │ +0001fc90: 2041 7672 616d 6f76 2d47 7261 7973 6f6e   Avramov-Grayson
│ │ │ │ +0001fca0: 2064 6573 6372 6962 6564 2069 6e20 4d61   described in Ma
│ │ │ │ +0001fcb0: 6361 756c 6179 3220 626f 6f6b 0a0a 2b2d  caulay2 book..+-
│ │ │ │ +0001fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fd00: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20  ----+.|i1 : kk= 
│ │ │ │ -0001fd10: 5a5a 2f31 3031 2020 2020 2020 2020 2020  ZZ/101          
│ │ │ │ +0001fcf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0001fd00: 206b 6b3d 205a 5a2f 3130 3120 2020 2020   kk= ZZ/101     
│ │ │ │ +0001fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001fd30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fd80: 7c6f 3120 3d20 6b6b 2020 2020 2020 2020  |o1 = kk        
│ │ │ │ +0001fd70: 2020 207c 0a7c 6f31 203d 206b 6b20 2020     |.|o1 = kk   
│ │ │ │ +0001fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001fdb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fdf0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -0001fe00: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0001fde0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001fdf0: 6f31 203a 2051 756f 7469 656e 7452 696e  o1 : QuotientRin
│ │ │ │ +0001fe00: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001fe20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001fe30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fe60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fe70: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b  --+.|i2 : S = kk
│ │ │ │ -0001fe80: 5b78 2c79 2c7a 5d20 2020 2020 2020 2020  [x,y,z]         
│ │ │ │ +0001fe60: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ +0001fe70: 203d 206b 6b5b 782c 792c 7a5d 2020 2020   = kk[x,y,z]    
│ │ │ │ +0001fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001feb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fee0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001fef0: 3220 3d20 5320 2020 2020 2020 2020 2020  2 = S           
│ │ │ │ +0001fee0: 207c 0a7c 6f32 203d 2053 2020 2020 2020   |.|o2 = S      
│ │ │ │ +0001fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ff10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001ff20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff60: 2020 2020 2020 7c0a 7c6f 3220 3a20 506f        |.|o2 : Po
│ │ │ │ -0001ff70: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +0001ff50: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001ff60: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0001ff70: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  0001ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ffa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001ff90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ffd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ffe0: 2b0a 7c69 3320 3a20 4931 203d 2069 6465  +.|i3 : I1 = ide
│ │ │ │ -0001fff0: 616c 2022 7833 7922 2020 2020 2020 2020  al "x3y"        
│ │ │ │ +0001ffd0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 3120  -----+.|i3 : I1 
│ │ │ │ +0001ffe0: 3d20 6964 6561 6c20 2278 3379 2220 2020  = ideal "x3y"   
│ │ │ │ +0001fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00020020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020050: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020060: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +00020040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020050: 0a7c 2020 2020 2020 2020 2020 2020 3320  .|            3 
│ │ │ │ +00020060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020090: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -000200a0: 6465 616c 2878 2079 2920 2020 2020 2020  deal(x y)       
│ │ │ │ +00020080: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00020090: 3320 3d20 6964 6561 6c28 7820 7929 2020  3 = ideal(x y)  
│ │ │ │ +000200a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000200c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000200d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020110: 207c 0a7c 6f33 203a 2049 6465 616c 206f   |.|o3 : Ideal o
│ │ │ │ -00020120: 6620 5320 2020 2020 2020 2020 2020 2020  f S             
│ │ │ │ +00020100: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964        |.|o3 : Id
│ │ │ │ +00020110: 6561 6c20 6f66 2053 2020 2020 2020 2020  eal of S        
│ │ │ │ +00020120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020150: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00020140: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00020190: 203a 2052 3120 3d20 532f 4931 2020 2020   : R1 = S/I1    
│ │ │ │ +00020180: 2b0a 7c69 3420 3a20 5231 203d 2053 2f49  +.|i4 : R1 = S/I
│ │ │ │ +00020190: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  000201a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000201b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000201c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020200: 2020 2020 207c 0a7c 6f34 203d 2052 3120       |.|o4 = R1 
│ │ │ │ +000201f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +00020200: 3d20 5231 2020 2020 2020 2020 2020 2020  = R1            
│ │ │ │  00020210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020240: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00020230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00020240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020280: 0a7c 6f34 203a 2051 756f 7469 656e 7452  .|o4 : QuotientR
│ │ │ │ -00020290: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00020270: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74      |.|o4 : Quot
│ │ │ │ +00020280: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00020290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000202a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000202b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -00020300: 204d 3120 3d20 5231 5e31 2f69 6465 616c   M1 = R1^1/ideal
│ │ │ │ -00020310: 2878 5e32 2920 2020 2020 2020 2020 2020  (x^2)           
│ │ │ │ -00020320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020330: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000202f0: 7c69 3520 3a20 4d31 203d 2052 315e 312f  |i5 : M1 = R1^1/
│ │ │ │ +00020300: 6964 6561 6c28 785e 3229 2020 2020 2020  ideal(x^2)      
│ │ │ │ +00020310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020320: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020370: 2020 207c 0a7c 6f35 203d 2063 6f6b 6572     |.|o5 = coker
│ │ │ │ -00020380: 6e65 6c20 7c20 7832 207c 2020 2020 2020  nel | x2 |      
│ │ │ │ +00020360: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ +00020370: 636f 6b65 726e 656c 207c 2078 3220 7c20  cokernel | x2 | 
│ │ │ │ +00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000203a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000203b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000203e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020400: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00020410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020420: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00020430: 3a20 5231 2d6d 6f64 756c 652c 2071 756f  : R1-module, quo
│ │ │ │ -00020440: 7469 656e 7420 6f66 2052 3120 2020 2020  tient of R1     
│ │ │ │ -00020450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020460: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00020400: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00020410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020420: 0a7c 6f35 203a 2052 312d 6d6f 6475 6c65  .|o5 : R1-module
│ │ │ │ +00020430: 2c20 7175 6f74 6965 6e74 206f 6620 5231  , quotient of R1
│ │ │ │ +00020440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020450: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00020460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204a0: 2d2d 2d2d 2b0a 7c69 3620 3a20 6265 7474  ----+.|i6 : bett
│ │ │ │ -000204b0: 6920 7265 7320 284d 312c 204c 656e 6774  i res (M1, Lengt
│ │ │ │ -000204c0: 684c 696d 6974 203d 3e35 2920 2020 2020  hLimit =>5)     
│ │ │ │ -000204d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000204e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00020490: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +000204a0: 2062 6574 7469 2072 6573 2028 4d31 2c20   betti res (M1, 
│ │ │ │ +000204b0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3529  LengthLimit =>5)
│ │ │ │ +000204c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000204d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000204e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000204f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020520: 7c20 2020 2020 2020 2020 2020 2030 2031  |            0 1
│ │ │ │ -00020530: 2032 2033 2034 2035 2020 2020 2020 2020   2 3 4 5        
│ │ │ │ +00020510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020520: 2020 3020 3120 3220 3320 3420 3520 2020    0 1 2 3 4 5   
│ │ │ │ +00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020550: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -00020560: 203d 2074 6f74 616c 3a20 3120 3120 3120   = total: 1 1 1 
│ │ │ │ -00020570: 3120 3120 3120 2020 2020 2020 2020 2020  1 1 1           
│ │ │ │ -00020580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020590: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000205a0: 2020 2020 303a 2031 202e 202e 202e 202e      0: 1 . . . .
│ │ │ │ -000205b0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -000205c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000205d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000205e0: 2031 3a20 2e20 3120 2e20 2e20 2e20 2e20   1: . 1 . . . . 
│ │ │ │ +00020550: 7c0a 7c6f 3620 3d20 746f 7461 6c3a 2031  |.|o6 = total: 1
│ │ │ │ +00020560: 2031 2031 2031 2031 2031 2020 2020 2020   1 1 1 1 1      
│ │ │ │ +00020570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020590: 2020 2020 2020 2020 2030 3a20 3120 2e20           0: 1 . 
│ │ │ │ +000205a0: 2e20 2e20 2e20 2e20 2020 2020 2020 2020  . . . .         
│ │ │ │ +000205b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000205d0: 2020 2020 2020 313a 202e 2031 202e 202e        1: . 1 . .
│ │ │ │ +000205e0: 202e 202e 2020 2020 2020 2020 2020 2020   . .            
│ │ │ │  000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020610: 2020 7c0a 7c20 2020 2020 2020 2020 323a    |.|         2:
│ │ │ │ -00020620: 202e 202e 2031 202e 202e 202e 2020 2020   . . 1 . . .    
│ │ │ │ +00020600: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00020610: 2020 2032 3a20 2e20 2e20 3120 2e20 2e20     2: . . 1 . . 
│ │ │ │ +00020620: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │  00020630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020650: 0a7c 2020 2020 2020 2020 2033 3a20 2e20  .|         3: . 
│ │ │ │ -00020660: 2e20 2e20 3120 2e20 2e20 2020 2020 2020  . . 1 . .       
│ │ │ │ +00020640: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020650: 333a 202e 202e 202e 2031 202e 202e 2020  3: . . . 1 . .  
│ │ │ │ +00020660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00020690: 2020 2020 2020 2020 343a 202e 202e 202e          4: . . .
│ │ │ │ -000206a0: 202e 2031 202e 2020 2020 2020 2020 2020   . 1 .          
│ │ │ │ -000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000206c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000206d0: 2020 2020 2035 3a20 2e20 2e20 2e20 2e20       5: . . . . 
│ │ │ │ -000206e0: 2e20 3120 2020 2020 2020 2020 2020 2020  . 1             
│ │ │ │ -000206f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020700: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020680: 207c 0a7c 2020 2020 2020 2020 2034 3a20   |.|         4: 
│ │ │ │ +00020690: 2e20 2e20 2e20 2e20 3120 2e20 2020 2020  . . . . 1 .     
│ │ │ │ +000206a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000206c0: 7c20 2020 2020 2020 2020 353a 202e 202e  |         5: . .
│ │ │ │ +000206d0: 202e 202e 202e 2031 2020 2020 2020 2020   . . . 1        
│ │ │ │ +000206e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00020700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020740: 2020 207c 0a7c 6f36 203a 2042 6574 7469     |.|o6 : Betti
│ │ │ │ -00020750: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +00020730: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +00020740: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020780: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00020770: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00020780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000207a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000207c0: 6937 203a 2045 203d 2045 7874 4d6f 6475  i7 : E = ExtModu
│ │ │ │ -000207d0: 6c65 204d 3120 2020 2020 2020 2020 2020  le M1           
│ │ │ │ -000207e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000207f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000207b0: 2d2d 2b0a 7c69 3720 3a20 4520 3d20 4578  --+.|i7 : E = Ex
│ │ │ │ +000207c0: 744d 6f64 756c 6520 4d31 2020 2020 2020  tModule M1      
│ │ │ │ +000207d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000207e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000207f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020830: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00020840: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00020820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00020830: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00020840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020870: 2020 2020 7c0a 7c6f 3720 3d20 286b 6b5b      |.|o7 = (kk[
│ │ │ │ -00020880: 5820 5d29 2020 2020 2020 2020 2020 2020  X ])            
│ │ │ │ +00020860: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +00020870: 2028 6b6b 5b58 205d 2920 2020 2020 2020   (kk[X ])       
│ │ │ │ +00020880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208b0: 207c 0a7c 2020 2020 2020 2020 2020 3020   |.|          0 
│ │ │ │ +000208a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000208b0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  000208c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000208d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000208f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000208e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000208f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020920: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00020930: 203a 206b 6b5b 5820 5d2d 6d6f 6475 6c65   : kk[X ]-module
│ │ │ │ -00020940: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ -00020950: 7b30 2e2e 317d 2020 2020 2020 2020 2020  {0..1}          
│ │ │ │ -00020960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00020970: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ +00020920: 7c0a 7c6f 3720 3a20 6b6b 5b58 205d 2d6d  |.|o7 : kk[X ]-m
│ │ │ │ +00020930: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ +00020940: 7265 6573 207b 302e 2e31 7d20 2020 2020  rees {0..1}     
│ │ │ │ +00020950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020960: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +00020970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00020990: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000209b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000209c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209e0: 2d2d 2b0a 7c69 3820 3a20 6170 706c 7928  --+.|i8 : apply(
│ │ │ │ -000209f0: 746f 4c69 7374 2830 2e2e 3130 292c 2069  toList(0..10), i
│ │ │ │ -00020a00: 2d3e 6869 6c62 6572 7446 756e 6374 696f  ->hilbertFunctio
│ │ │ │ -00020a10: 6e28 692c 2045 2929 2020 2020 2020 207c  n(i, E))       |
│ │ │ │ -00020a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000209d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2061  -------+.|i8 : a
│ │ │ │ +000209e0: 7070 6c79 2874 6f4c 6973 7428 302e 2e31  pply(toList(0..1
│ │ │ │ +000209f0: 3029 2c20 692d 3e68 696c 6265 7274 4675  0), i->hilbertFu
│ │ │ │ +00020a00: 6e63 7469 6f6e 2869 2c20 4529 2920 2020  nction(i, E))   
│ │ │ │ +00020a10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00020a60: 3820 3d20 7b31 2c20 312c 2031 2c20 312c  8 = {1, 1, 1, 1,
│ │ │ │ -00020a70: 2031 2c20 312c 2031 2c20 312c 2031 2c20   1, 1, 1, 1, 1, 
│ │ │ │ -00020a80: 312c 2031 7d20 2020 2020 2020 2020 2020  1, 1}           
│ │ │ │ -00020a90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020a50: 207c 0a7c 6f38 203d 207b 312c 2031 2c20   |.|o8 = {1, 1, 
│ │ │ │ +00020a60: 312c 2031 2c20 312c 2031 2c20 312c 2031  1, 1, 1, 1, 1, 1
│ │ │ │ +00020a70: 2c20 312c 2031 2c20 317d 2020 2020 2020  , 1, 1, 1}      
│ │ │ │ +00020a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020a90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00020aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ad0: 2020 2020 2020 7c0a 7c6f 3820 3a20 4c69        |.|o8 : Li
│ │ │ │ -00020ae0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +00020ac0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00020ad0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00020b00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00020b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020b50: 2b0a 7c69 3920 3a20 4565 7665 6e20 3d20  +.|i9 : Eeven = 
│ │ │ │ -00020b60: 6576 656e 4578 744d 6f64 756c 6528 4d31  evenExtModule(M1
│ │ │ │ -00020b70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00020b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020b40: 2d2d 2d2d 2d2b 0a7c 6939 203a 2045 6576  -----+.|i9 : Eev
│ │ │ │ +00020b50: 656e 203d 2065 7665 6e45 7874 4d6f 6475  en = evenExtModu
│ │ │ │ +00020b60: 6c65 284d 3129 2020 2020 2020 2020 2020  le(M1)          
│ │ │ │ +00020b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00020b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020bd0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00020bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020bc0: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +00020bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c00: 2020 2020 2020 207c 0a7c 6f39 203d 2028         |.|o9 = (
│ │ │ │ -00020c10: 6b6b 5b58 205d 2920 2020 2020 2020 2020  kk[X ])         
│ │ │ │ +00020bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00020c00: 3920 3d20 286b 6b5b 5820 5d29 2020 2020  9 = (kk[X ])    
│ │ │ │ +00020c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00020c50: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00020c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020c40: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00020c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00020c70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020cc0: 7c6f 3920 3a20 6b6b 5b58 205d 2d6d 6f64  |o9 : kk[X ]-mod
│ │ │ │ -00020cd0: 756c 652c 2066 7265 6520 2020 2020 2020  ule, free       
│ │ │ │ +00020cb0: 2020 207c 0a7c 6f39 203a 206b 6b5b 5820     |.|o9 : kk[X 
│ │ │ │ +00020cc0: 5d2d 6d6f 6475 6c65 2c20 6672 6565 2020  ]-module, free  
│ │ │ │ +00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020d00: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00020cf0: 7c0a 7c20 2020 2020 2020 2020 3020 2020  |.|         0   
│ │ │ │ +00020d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00020d20: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d70: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 6170  -----+.|i10 : ap
│ │ │ │ -00020d80: 706c 7928 746f 4c69 7374 2830 2e2e 3529  ply(toList(0..5)
│ │ │ │ -00020d90: 2c20 692d 3e68 696c 6265 7274 4675 6e63  , i->hilbertFunc
│ │ │ │ -00020da0: 7469 6f6e 2869 2c20 4565 7665 6e29 2920  tion(i, Eeven)) 
│ │ │ │ -00020db0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00020d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ +00020d70: 203a 2061 7070 6c79 2874 6f4c 6973 7428   : apply(toList(
│ │ │ │ +00020d80: 302e 2e35 292c 2069 2d3e 6869 6c62 6572  0..5), i->hilber
│ │ │ │ +00020d90: 7446 756e 6374 696f 6e28 692c 2045 6576  tFunction(i, Eev
│ │ │ │ +00020da0: 656e 2929 2020 207c 0a7c 2020 2020 2020  en))   |.|      
│ │ │ │ +00020db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020df0: 0a7c 6f31 3020 3d20 7b31 2c20 312c 2031  .|o10 = {1, 1, 1
│ │ │ │ -00020e00: 2c20 312c 2031 2c20 317d 2020 2020 2020  , 1, 1, 1}      
│ │ │ │ +00020de0: 2020 2020 7c0a 7c6f 3130 203d 207b 312c      |.|o10 = {1,
│ │ │ │ +00020df0: 2031 2c20 312c 2031 2c20 312c 2031 7d20   1, 1, 1, 1, 1} 
│ │ │ │ +00020e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00020e20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00020e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e60: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -00020e70: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00020e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020e60: 7c6f 3130 203a 204c 6973 7420 2020 2020  |o10 : List     
│ │ │ │ +00020e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ea0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00020e90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020ee0: 2d2d 2d2b 0a7c 6931 3120 3a20 456f 6464  ---+.|i11 : Eodd
│ │ │ │ -00020ef0: 203d 206f 6464 4578 744d 6f64 756c 6528   = oddExtModule(
│ │ │ │ -00020f00: 4d31 2920 2020 2020 2020 2020 2020 2020  M1)             
│ │ │ │ -00020f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00020ed0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +00020ee0: 2045 6f64 6420 3d20 6f64 6445 7874 4d6f   Eodd = oddExtMo
│ │ │ │ +00020ef0: 6475 6c65 284d 3129 2020 2020 2020 2020  dule(M1)        
│ │ │ │ +00020f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020f10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020f60: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00020f50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00020f60: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  00020f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f90: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -00020fa0: 203d 2028 6b6b 5b58 205d 2920 2020 2020   = (kk[X ])     
│ │ │ │ +00020f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020f90: 0a7c 6f31 3120 3d20 286b 6b5b 5820 5d29  .|o11 = (kk[X ])
│ │ │ │ +00020fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020fd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00020fe0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00020fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00020fd0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +00020fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021010: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021000: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021050: 207c 0a7c 6f31 3120 3a20 6b6b 5b58 205d   |.|o11 : kk[X ]
│ │ │ │ -00021060: 2d6d 6f64 756c 652c 2066 7265 6520 2020  -module, free   
│ │ │ │ +00021040: 2020 2020 2020 7c0a 7c6f 3131 203a 206b        |.|o11 : k
│ │ │ │ +00021050: 6b5b 5820 5d2d 6d6f 6475 6c65 2c20 6672  k[X ]-module, fr
│ │ │ │ +00021060: 6565 2020 2020 2020 2020 2020 2020 2020  ee              
│ │ │ │  00021070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021090: 7c20 2020 2020 2020 2020 2030 2020 2020  |          0    
│ │ │ │ +00021080: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021090: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  000210a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000210c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000210d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021100: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │ -00021110: 2061 7070 6c79 2874 6f4c 6973 7428 302e   apply(toList(0.
│ │ │ │ -00021120: 2e35 292c 2069 2d3e 6869 6c62 6572 7446  .5), i->hilbertF
│ │ │ │ -00021130: 756e 6374 696f 6e28 692c 2045 6f64 6429  unction(i, Eodd)
│ │ │ │ -00021140: 2920 2020 207c 0a7c 2020 2020 2020 2020  )    |.|        
│ │ │ │ +000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00021100: 6931 3220 3a20 6170 706c 7928 746f 4c69  i12 : apply(toLi
│ │ │ │ +00021110: 7374 2830 2e2e 3529 2c20 692d 3e68 696c  st(0..5), i->hil
│ │ │ │ +00021120: 6265 7274 4675 6e63 7469 6f6e 2869 2c20  bertFunction(i, 
│ │ │ │ +00021130: 456f 6464 2929 2020 2020 7c0a 7c20 2020  Eodd))    |.|   
│ │ │ │ +00021140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021180: 2020 7c0a 7c6f 3132 203d 207b 312c 2031    |.|o12 = {1, 1
│ │ │ │ -00021190: 2c20 312c 2031 2c20 312c 2031 7d20 2020  , 1, 1, 1, 1}   
│ │ │ │ +00021170: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +00021180: 7b31 2c20 312c 2031 2c20 312c 2031 2c20  {1, 1, 1, 1, 1, 
│ │ │ │ +00021190: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │  000211a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000211c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000211b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000211c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000211d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000211e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021200: 3132 203a 204c 6973 7420 2020 2020 2020  12 : List       
│ │ │ │ +000211f0: 207c 0a7c 6f31 3220 3a20 4c69 7374 2020   |.|o12 : List  
│ │ │ │ +00021200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021230: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00021220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021230: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00021240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021270: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2075  ------+.|i13 : u
│ │ │ │ -00021280: 7365 2053 2020 2020 2020 2020 2020 2020  se S            
│ │ │ │ +00021260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00021270: 3320 3a20 7573 6520 5320 2020 2020 2020  3 : use S       
│ │ │ │ +00021280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000212a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000212b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000212c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000212d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212f0: 7c0a 7c6f 3133 203d 2053 2020 2020 2020  |.|o13 = S      
│ │ │ │ +000212e0: 2020 2020 207c 0a7c 6f31 3320 3d20 5320       |.|o13 = S 
│ │ │ │ +000212f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021320: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00021330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021360: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -00021370: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -00021380: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00021390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00021350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021360: 0a7c 6f31 3320 3a20 506f 6c79 6e6f 6d69  .|o13 : Polynomi
│ │ │ │ +00021370: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +00021380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021390: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000213a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000213b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000213c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000213d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000213e0: 2d2d 2d2d 2b0a 7c69 3134 203a 2049 3220  ----+.|i14 : I2 
│ │ │ │ -000213f0: 3d20 6964 6561 6c22 7833 2c79 7a22 2020  = ideal"x3,yz"  
│ │ │ │ +000213d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420  ---------+.|i14 
│ │ │ │ +000213e0: 3a20 4932 203d 2069 6465 616c 2278 332c  : I2 = ideal"x3,
│ │ │ │ +000213f0: 797a 2220 2020 2020 2020 2020 2020 2020  yz"             
│ │ │ │  00021400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021420: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00021410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021460: 7c20 2020 2020 2020 2020 2020 2020 2033  |              3
│ │ │ │ +00021450: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021460: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021490: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000214a0: 3420 3d20 6964 6561 6c20 2878 202c 2079  4 = ideal (x , y
│ │ │ │ -000214b0: 2a7a 2920 2020 2020 2020 2020 2020 2020  *z)             
│ │ │ │ -000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021490: 7c0a 7c6f 3134 203d 2069 6465 616c 2028  |.|o14 = ideal (
│ │ │ │ +000214a0: 7820 2c20 792a 7a29 2020 2020 2020 2020  x , y*z)        
│ │ │ │ +000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000214c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000214d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021510: 2020 2020 207c 0a7c 6f31 3420 3a20 4964       |.|o14 : Id
│ │ │ │ -00021520: 6561 6c20 6f66 2053 2020 2020 2020 2020  eal of S        
│ │ │ │ +00021500: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ +00021510: 203a 2049 6465 616c 206f 6620 5320 2020   : Ideal of S   
│ │ │ │ +00021520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021550: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00021540: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00021550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00021590: 0a7c 6931 3520 3a20 5232 203d 2053 2f49  .|i15 : R2 = S/I
│ │ │ │ -000215a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00021580: 2d2d 2d2d 2b0a 7c69 3135 203a 2052 3220  ----+.|i15 : R2 
│ │ │ │ +00021590: 3d20 532f 4932 2020 2020 2020 2020 2020  = S/I2          
│ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000215c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000215d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021600: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ -00021610: 3d20 5232 2020 2020 2020 2020 2020 2020  = R2            
│ │ │ │ +000215f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021600: 7c6f 3135 203d 2052 3220 2020 2020 2020  |o15 = R2       
│ │ │ │ +00021610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021640: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021680: 2020 207c 0a7c 6f31 3520 3a20 5175 6f74     |.|o15 : Quot
│ │ │ │ -00021690: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00021670: 2020 2020 2020 2020 7c0a 7c6f 3135 203a          |.|o15 :
│ │ │ │ +00021680: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000216b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000216c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000216d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000216e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000216f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00021700: 6931 3620 3a20 4d32 203d 2052 325e 312f  i16 : M2 = R2^1/
│ │ │ │ -00021710: 6964 6561 6c22 7832 2c79 2c7a 2220 2020  ideal"x2,y,z"   
│ │ │ │ -00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021730: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000216f0: 2d2d 2b0a 7c69 3136 203a 204d 3220 3d20  --+.|i16 : M2 = 
│ │ │ │ +00021700: 5232 5e31 2f69 6465 616c 2278 322c 792c  R2^1/ideal"x2,y,
│ │ │ │ +00021710: 7a22 2020 2020 2020 2020 2020 2020 2020  z"              
│ │ │ │ +00021720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021770: 2020 2020 2020 207c 0a7c 6f31 3620 3d20         |.|o16 = 
│ │ │ │ -00021780: 636f 6b65 726e 656c 207c 2078 3220 7920  cokernel | x2 y 
│ │ │ │ -00021790: 7a20 7c20 2020 2020 2020 2020 2020 2020  z |             
│ │ │ │ -000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00021770: 3136 203d 2063 6f6b 6572 6e65 6c20 7c20  16 = cokernel | 
│ │ │ │ +00021780: 7832 2079 207a 207c 2020 2020 2020 2020  x2 y z |        
│ │ │ │ +00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000217b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00021800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021810: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00021820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021830: 7c6f 3136 203a 2052 322d 6d6f 6475 6c65  |o16 : R2-module
│ │ │ │ -00021840: 2c20 7175 6f74 6965 6e74 206f 6620 5232  , quotient of R2
│ │ │ │ +000217e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000217f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021800: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00021810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021820: 2020 207c 0a7c 6f31 3620 3a20 5232 2d6d     |.|o16 : R2-m
│ │ │ │ +00021830: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +00021840: 6f66 2052 3220 2020 2020 2020 2020 2020  of R2           
│ │ │ │  00021850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021860: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00021860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00021870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000218a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ -000218b0: 2062 6574 7469 2072 6573 2028 4d32 2c20   betti res (M2, 
│ │ │ │ -000218c0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3130  LengthLimit =>10
│ │ │ │ -000218d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -000218e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000218a0: 6931 3720 3a20 6265 7474 6920 7265 7320  i17 : betti res 
│ │ │ │ +000218b0: 284d 322c 204c 656e 6774 684c 696d 6974  (M2, LengthLimit
│ │ │ │ +000218c0: 203d 3e31 3029 2020 2020 2020 2020 2020   =>10)          
│ │ │ │ +000218d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000218f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021920: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00021930: 2020 3020 3120 3220 3320 3420 2035 2020    0 1 2 3 4  5  
│ │ │ │ -00021940: 3620 2037 2020 3820 2039 2031 3020 2020  6  7  8  9 10   
│ │ │ │ -00021950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021960: 0a7c 6f31 3720 3d20 746f 7461 6c3a 2031  .|o17 = total: 1
│ │ │ │ -00021970: 2033 2035 2037 2039 2031 3120 3133 2031   3 5 7 9 11 13 1
│ │ │ │ -00021980: 3520 3137 2031 3920 3231 2020 2020 2020  5 17 19 21      
│ │ │ │ -00021990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000219a0: 2020 2020 2020 2020 2030 3a20 3120 3220           0: 1 2 
│ │ │ │ -000219b0: 3220 3220 3220 2032 2020 3220 2032 2020  2 2 2  2  2  2  
│ │ │ │ -000219c0: 3220 2032 2020 3220 2020 2020 2020 2020  2  2  2         
│ │ │ │ -000219d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000219e0: 2020 2020 2020 313a 202e 2031 2033 2034        1: . 1 3 4
│ │ │ │ -000219f0: 2034 2020 3420 2034 2020 3420 2034 2020   4  4  4  4  4  
│ │ │ │ -00021a00: 3420 2034 2020 2020 2020 2020 2020 2020  4  4            
│ │ │ │ -00021a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00021a20: 2020 2032 3a20 2e20 2e20 2e20 3120 3320     2: . . . 1 3 
│ │ │ │ -00021a30: 2034 2020 3420 2034 2020 3420 2034 2020   4  4  4  4  4  
│ │ │ │ -00021a40: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00021a50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00021a60: 333a 202e 202e 202e 202e 202e 2020 3120  3: . . . . .  1 
│ │ │ │ -00021a70: 2033 2020 3420 2034 2020 3420 2034 2020   3  4  4  4  4  
│ │ │ │ -00021a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a90: 7c0a 7c20 2020 2020 2020 2020 2034 3a20  |.|          4: 
│ │ │ │ -00021aa0: 2e20 2e20 2e20 2e20 2e20 202e 2020 2e20  . . . . .  .  . 
│ │ │ │ -00021ab0: 2031 2020 3320 2034 2020 3420 2020 2020   1  3  4  4     
│ │ │ │ -00021ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021ad0: 2020 2020 2020 2020 2020 353a 202e 202e            5: . .
│ │ │ │ -00021ae0: 202e 202e 202e 2020 2e20 202e 2020 2e20   . . .  .  .  . 
│ │ │ │ -00021af0: 202e 2020 3120 2033 2020 2020 2020 2020   .  1  3        
│ │ │ │ -00021b00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021910: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021920: 2020 2020 2020 2030 2031 2032 2033 2034         0 1 2 3 4
│ │ │ │ +00021930: 2020 3520 2036 2020 3720 2038 2020 3920    5  6  7  8  9 
│ │ │ │ +00021940: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │ +00021950: 2020 2020 7c0a 7c6f 3137 203d 2074 6f74      |.|o17 = tot
│ │ │ │ +00021960: 616c 3a20 3120 3320 3520 3720 3920 3131  al: 1 3 5 7 9 11
│ │ │ │ +00021970: 2031 3320 3135 2031 3720 3139 2032 3120   13 15 17 19 21 
│ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021990: 207c 0a7c 2020 2020 2020 2020 2020 303a   |.|          0:
│ │ │ │ +000219a0: 2031 2032 2032 2032 2032 2020 3220 2032   1 2 2 2 2  2  2
│ │ │ │ +000219b0: 2020 3220 2032 2020 3220 2032 2020 2020    2  2  2  2    
│ │ │ │ +000219c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000219d0: 7c20 2020 2020 2020 2020 2031 3a20 2e20  |          1: . 
│ │ │ │ +000219e0: 3120 3320 3420 3420 2034 2020 3420 2034  1 3 4 4  4  4  4
│ │ │ │ +000219f0: 2020 3420 2034 2020 3420 2020 2020 2020    4  4  4       
│ │ │ │ +00021a00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021a10: 2020 2020 2020 2020 323a 202e 202e 202e          2: . . .
│ │ │ │ +00021a20: 2031 2033 2020 3420 2034 2020 3420 2034   1 3  4  4  4  4
│ │ │ │ +00021a30: 2020 3420 2034 2020 2020 2020 2020 2020    4  4          
│ │ │ │ +00021a40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021a50: 2020 2020 2033 3a20 2e20 2e20 2e20 2e20       3: . . . . 
│ │ │ │ +00021a60: 2e20 2031 2020 3320 2034 2020 3420 2034  .  1  3  4  4  4
│ │ │ │ +00021a70: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00021a80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021a90: 2020 343a 202e 202e 202e 202e 202e 2020    4: . . . . .  
│ │ │ │ +00021aa0: 2e20 202e 2020 3120 2033 2020 3420 2034  .  .  1  3  4  4
│ │ │ │ +00021ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ac0: 2020 7c0a 7c20 2020 2020 2020 2020 2035    |.|          5
│ │ │ │ +00021ad0: 3a20 2e20 2e20 2e20 2e20 2e20 202e 2020  : . . . . .  .  
│ │ │ │ +00021ae0: 2e20 202e 2020 2e20 2031 2020 3320 2020  .  .  .  1  3   
│ │ │ │ +00021af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b40: 2020 2020 2020 207c 0a7c 6f31 3720 3a20         |.|o17 : 
│ │ │ │ -00021b50: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00021b30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00021b40: 3137 203a 2042 6574 7469 5461 6c6c 7920  17 : BettiTally 
│ │ │ │ +00021b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021b70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00021b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021bc0: 2d2b 0a7c 6931 3820 3a20 4520 3d20 4578  -+.|i18 : E = Ex
│ │ │ │ -00021bd0: 744d 6f64 756c 6520 4d32 2020 2020 2020  tModule M2      
│ │ │ │ +00021bb0: 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a 2045  ------+.|i18 : E
│ │ │ │ +00021bc0: 203d 2045 7874 4d6f 6475 6c65 204d 3220   = ExtModule M2 
│ │ │ │ +00021bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021c00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00021bf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00021c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c50: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ -00021c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c70: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ -00021c80: 2028 6b6b 5b58 202e 2e58 205d 2920 2020   (kk[X ..X ])   
│ │ │ │ +00021c30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00021c40: 2020 2020 2038 2020 2020 2020 2020 2020       8          
│ │ │ │ +00021c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021c60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021c70: 6f31 3820 3d20 286b 6b5b 5820 2e2e 5820  o18 = (kk[X ..X 
│ │ │ │ +00021c80: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │  00021c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021cb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00021cc0: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ +00021ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021cb0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00021cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00021ce0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021d30: 0a7c 6f31 3820 3a20 6b6b 5b58 202e 2e58  .|o18 : kk[X ..X
│ │ │ │ -00021d40: 205d 2d6d 6f64 756c 652c 2066 7265 652c   ]-module, free,
│ │ │ │ -00021d50: 2064 6567 7265 6573 207b 302e 2e31 2c20   degrees {0..1, 
│ │ │ │ -00021d60: 323a 312c 2033 3a32 2c20 337d 7c0a 7c20  2:1, 3:2, 3}|.| 
│ │ │ │ -00021d70: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00021d20: 2020 2020 7c0a 7c6f 3138 203a 206b 6b5b      |.|o18 : kk[
│ │ │ │ +00021d30: 5820 2e2e 5820 5d2d 6d6f 6475 6c65 2c20  X ..X ]-module, 
│ │ │ │ +00021d40: 6672 6565 2c20 6465 6772 6565 7320 7b30  free, degrees {0
│ │ │ │ +00021d50: 2e2e 312c 2032 3a31 2c20 333a 322c 2033  ..1, 2:1, 3:2, 3
│ │ │ │ +00021d60: 7d7c 0a7c 2020 2020 2020 2020 2020 3020  }|.|          0 
│ │ │ │ +00021d70: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  00021d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021da0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00021d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021da0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00021db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021de0: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2061  ------+.|i19 : a
│ │ │ │ -00021df0: 7070 6c79 2874 6f4c 6973 7428 302e 2e31  pply(toList(0..1
│ │ │ │ -00021e00: 3029 2c20 692d 3e68 696c 6265 7274 4675  0), i->hilbertFu
│ │ │ │ -00021e10: 6e63 7469 6f6e 2869 2c20 4529 2920 2020  nction(i, E))   
│ │ │ │ -00021e20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00021de0: 3920 3a20 6170 706c 7928 746f 4c69 7374  9 : apply(toList
│ │ │ │ +00021df0: 2830 2e2e 3130 292c 2069 2d3e 6869 6c62  (0..10), i->hilb
│ │ │ │ +00021e00: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ +00021e10: 2929 2020 2020 2020 7c0a 7c20 2020 2020  ))      |.|     
│ │ │ │ +00021e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e60: 7c0a 7c6f 3139 203d 207b 312c 2033 2c20  |.|o19 = {1, 3, 
│ │ │ │ -00021e70: 352c 2037 2c20 392c 2031 312c 2031 332c  5, 7, 9, 11, 13,
│ │ │ │ -00021e80: 2031 352c 2031 372c 2031 392c 2032 317d   15, 17, 19, 21}
│ │ │ │ -00021e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021e50: 2020 2020 207c 0a7c 6f31 3920 3d20 7b31       |.|o19 = {1
│ │ │ │ +00021e60: 2c20 332c 2035 2c20 372c 2039 2c20 3131  , 3, 5, 7, 9, 11
│ │ │ │ +00021e70: 2c20 3133 2c20 3135 2c20 3137 2c20 3139  , 13, 15, 17, 19
│ │ │ │ +00021e80: 2c20 3231 7d20 2020 2020 2020 2020 2020  , 21}           
│ │ │ │ +00021e90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00021ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ed0: 2020 2020 2020 2020 2020 7c0a 7c6f 3139            |.|o19
│ │ │ │ -00021ee0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00021ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021ed0: 0a7c 6f31 3920 3a20 4c69 7374 2020 2020  .|o19 : List    
│ │ │ │ +00021ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00021f00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00021f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021f50: 2d2d 2d2d 2b0a 7c69 3230 203a 2045 6576  ----+.|i20 : Eev
│ │ │ │ -00021f60: 656e 203d 2065 7665 6e45 7874 4d6f 6475  en = evenExtModu
│ │ │ │ -00021f70: 6c65 204d 3220 2020 2020 2020 2020 2020  le M2           
│ │ │ │ -00021f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00021f40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ +00021f50: 3a20 4565 7665 6e20 3d20 6576 656e 4578  : Eeven = evenEx
│ │ │ │ +00021f60: 744d 6f64 756c 6520 4d32 2020 2020 2020  tModule M2      
│ │ │ │ +00021f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021f80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021fd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00021fe0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00021fc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021fd0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022000: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00022010: 3020 3d20 286b 6b5b 5820 2e2e 5820 5d29  0 = (kk[X ..X ])
│ │ │ │ +00022000: 7c0a 7c6f 3230 203d 2028 6b6b 5b58 202e  |.|o20 = (kk[X .
│ │ │ │ +00022010: 2e58 205d 2920 2020 2020 2020 2020 2020  .X ])           
│ │ │ │  00022020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022040: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00022050: 2020 2020 2020 3020 2020 3120 2020 2020        0   1     
│ │ │ │ +00022030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022040: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ +00022050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022070: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220c0: 2020 7c0a 7c6f 3230 203a 206b 6b5b 5820    |.|o20 : kk[X 
│ │ │ │ -000220d0: 2e2e 5820 5d2d 6d6f 6475 6c65 2c20 6672  ..X ]-module, fr
│ │ │ │ -000220e0: 6565 2c20 6465 6772 6565 7320 7b30 2e2e  ee, degrees {0..
│ │ │ │ -000220f0: 312c 2032 3a31 7d20 2020 2020 2020 207c  1, 2:1}        |
│ │ │ │ -00022100: 0a7c 2020 2020 2020 2020 2020 3020 2020  .|          0   
│ │ │ │ -00022110: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000220b0: 2020 2020 2020 207c 0a7c 6f32 3020 3a20         |.|o20 : 
│ │ │ │ +000220c0: 6b6b 5b58 202e 2e58 205d 2d6d 6f64 756c  kk[X ..X ]-modul
│ │ │ │ +000220d0: 652c 2066 7265 652c 2064 6567 7265 6573  e, free, degrees
│ │ │ │ +000220e0: 207b 302e 2e31 2c20 323a 317d 2020 2020   {0..1, 2:1}    
│ │ │ │ +000220f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022100: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │ +00022110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022130: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00022130: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022170: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120  ---------+.|i21 
│ │ │ │ -00022180: 3a20 6170 706c 7928 746f 4c69 7374 2830  : apply(toList(0
│ │ │ │ -00022190: 2e2e 3529 2c20 692d 3e68 696c 6265 7274  ..5), i->hilbert
│ │ │ │ -000221a0: 4675 6e63 7469 6f6e 2869 2c20 4565 7665  Function(i, Eeve
│ │ │ │ -000221b0: 6e29 2920 2020 7c0a 7c20 2020 2020 2020  n))   |.|       
│ │ │ │ +00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00022170: 7c69 3231 203a 2061 7070 6c79 2874 6f4c  |i21 : apply(toL
│ │ │ │ +00022180: 6973 7428 302e 2e35 292c 2069 2d3e 6869  ist(0..5), i->hi
│ │ │ │ +00022190: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
│ │ │ │ +000221a0: 2045 6576 656e 2929 2020 207c 0a7c 2020   Eeven))   |.|  
│ │ │ │ +000221b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221f0: 2020 207c 0a7c 6f32 3120 3d20 7b31 2c20     |.|o21 = {1, 
│ │ │ │ -00022200: 352c 2039 2c20 3133 2c20 3137 2c20 3231  5, 9, 13, 17, 21
│ │ │ │ -00022210: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -00022220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022230: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000221e0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d          |.|o21 =
│ │ │ │ +000221f0: 207b 312c 2035 2c20 392c 2031 332c 2031   {1, 5, 9, 13, 1
│ │ │ │ +00022200: 372c 2032 317d 2020 2020 2020 2020 2020  7, 21}          
│ │ │ │ +00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022270: 6f32 3120 3a20 4c69 7374 2020 2020 2020  o21 : List      
│ │ │ │ +00022260: 2020 7c0a 7c6f 3231 203a 204c 6973 7420    |.|o21 : List 
│ │ │ │ +00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000222a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000222b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000222c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000222d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000222e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20  -------+.|i22 : 
│ │ │ │ -000222f0: 456f 6464 203d 206f 6464 4578 744d 6f64  Eodd = oddExtMod
│ │ │ │ -00022300: 756c 6520 4d32 2020 2020 2020 2020 2020  ule M2          
│ │ │ │ -00022310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022320: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000222d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000222e0: 3232 203a 2045 6f64 6420 3d20 6f64 6445  22 : Eodd = oddE
│ │ │ │ +000222f0: 7874 4d6f 6475 6c65 204d 3220 2020 2020  xtModule M2     
│ │ │ │ +00022300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00022320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022360: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00022370: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +00022350: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00022360: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000223a0: 7c6f 3232 203d 2028 6b6b 5b58 202e 2e58  |o22 = (kk[X ..X
│ │ │ │ -000223b0: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │ +00022390: 2020 207c 0a7c 6f32 3220 3d20 286b 6b5b     |.|o22 = (kk[
│ │ │ │ +000223a0: 5820 2e2e 5820 5d29 2020 2020 2020 2020  X ..X ])        
│ │ │ │ +000223b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000223e0: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +000223d0: 7c0a 7c20 2020 2020 2020 2020 2020 3020  |.|           0 
│ │ │ │ +000223e0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  000223f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022450: 2020 2020 207c 0a7c 6f32 3220 3a20 6b6b       |.|o22 : kk
│ │ │ │ -00022460: 5b58 202e 2e58 205d 2d6d 6f64 756c 652c  [X ..X ]-module,
│ │ │ │ -00022470: 2066 7265 652c 2064 6567 7265 6573 207b   free, degrees {
│ │ │ │ -00022480: 333a 302c 2031 7d20 2020 2020 2020 2020  3:0, 1}         
│ │ │ │ -00022490: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ -000224a0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00022440: 2020 2020 2020 2020 2020 7c0a 7c6f 3232            |.|o22
│ │ │ │ +00022450: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f   : kk[X ..X ]-mo
│ │ │ │ +00022460: 6475 6c65 2c20 6672 6565 2c20 6465 6772  dule, free, degr
│ │ │ │ +00022470: 6565 7320 7b33 3a30 2c20 317d 2020 2020  ees {3:0, 1}    
│ │ │ │ +00022480: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022490: 2020 2020 3020 2020 3120 2020 2020 2020      0   1       
│ │ │ │ +000224a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000224b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000224d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000224c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000224d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000224e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000224f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022510: 3233 203a 2061 7070 6c79 2874 6f4c 6973  23 : apply(toLis
│ │ │ │ -00022520: 7428 302e 2e35 292c 2069 2d3e 6869 6c62  t(0..5), i->hilb
│ │ │ │ -00022530: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ -00022540: 6f64 6429 2920 2020 207c 0a7c 2020 2020  odd))    |.|    
│ │ │ │ +00022500: 2d2b 0a7c 6932 3320 3a20 6170 706c 7928  -+.|i23 : apply(
│ │ │ │ +00022510: 746f 4c69 7374 2830 2e2e 3529 2c20 692d  toList(0..5), i-
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│ │ │ │ +00022530: 2869 2c20 456f 6464 2929 2020 2020 7c0a  (i, Eodd))    |.
│ │ │ │ +00022540: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00022550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022580: 2020 2020 2020 7c0a 7c6f 3233 203d 207b        |.|o23 = {
│ │ │ │ -00022590: 332c 2037 2c20 3131 2c20 3135 2c20 3139  3, 7, 11, 15, 19
│ │ │ │ -000225a0: 2c20 3233 7d20 2020 2020 2020 2020 2020  , 23}           
│ │ │ │ -000225b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00022570: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00022580: 3320 3d20 7b33 2c20 372c 2031 312c 2031  3 = {3, 7, 11, 1
│ │ │ │ +00022590: 352c 2031 392c 2032 337d 2020 2020 2020  5, 19, 23}      
│ │ │ │ +000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000225b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022600: 7c0a 7c6f 3233 203a 204c 6973 7420 2020  |.|o23 : List   
│ │ │ │ +000225f0: 2020 2020 207c 0a7c 6f32 3320 3a20 4c69       |.|o23 : Li
│ │ │ │ +00022600: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022630: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022630: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00022640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022670: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565  ----------+..See
│ │ │ │ -00022680: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -00022690: 2020 2a20 2a6e 6f74 6520 6576 656e 4578    * *note evenEx
│ │ │ │ -000226a0: 744d 6f64 756c 653a 2065 7665 6e45 7874  tModule: evenExt
│ │ │ │ -000226b0: 4d6f 6475 6c65 2c20 2d2d 2065 7665 6e20  Module, -- even 
│ │ │ │ -000226c0: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ -000226d0: 6b29 206f 7665 7220 610a 2020 2020 636f  k) over a.    co
│ │ │ │ -000226e0: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ -000226f0: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -00022700: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -00022710: 696e 670a 2020 2a20 2a6e 6f74 6520 6f64  ing.  * *note od
│ │ │ │ -00022720: 6445 7874 4d6f 6475 6c65 3a20 6f64 6445  dExtModule: oddE
│ │ │ │ -00022730: 7874 4d6f 6475 6c65 2c20 2d2d 206f 6464  xtModule, -- odd
│ │ │ │ -00022740: 2070 6172 7420 6f66 2045 7874 5e2a 284d   part of Ext^*(M
│ │ │ │ -00022750: 2c6b 2920 6f76 6572 2061 2063 6f6d 706c  ,k) over a compl
│ │ │ │ -00022760: 6574 650a 2020 2020 696e 7465 7273 6563  ete.    intersec
│ │ │ │ -00022770: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ -00022780: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ -00022790: 7269 6e67 0a0a 5761 7973 2074 6f20 7573  ring..Ways to us
│ │ │ │ -000227a0: 6520 4578 744d 6f64 756c 653a 0a3d 3d3d  e ExtModule:.===
│ │ │ │ -000227b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000227c0: 3d3d 3d0a 0a20 202a 2022 4578 744d 6f64  ===..  * "ExtMod
│ │ │ │ -000227d0: 756c 6528 4d6f 6475 6c65 2922 0a0a 466f  ule(Module)"..Fo
│ │ │ │ -000227e0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -000227f0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00022800: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00022810: 2a6e 6f74 6520 4578 744d 6f64 756c 653a  *note ExtModule:
│ │ │ │ -00022820: 2045 7874 4d6f 6475 6c65 2c20 6973 2061   ExtModule, is a
│ │ │ │ -00022830: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ -00022840: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61  nction:.(Macaula
│ │ │ │ -00022850: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -00022860: 7469 6f6e 2c2e 0a1f 0a46 696c 653a 2043  tion,....File: C
│ │ │ │ -00022870: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00022880: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -00022890: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f  nfo, Node: ExtMo
│ │ │ │ -000228a0: 6475 6c65 4461 7461 2c20 4e65 7874 3a20  duleData, Next: 
│ │ │ │ -000228b0: 6578 7456 7343 6f68 6f6d 6f6c 6f67 792c  extVsCohomology,
│ │ │ │ -000228c0: 2050 7265 763a 2045 7874 4d6f 6475 6c65   Prev: ExtModule
│ │ │ │ -000228d0: 2c20 5570 3a20 546f 700a 0a45 7874 4d6f  , Up: Top..ExtMo
│ │ │ │ -000228e0: 6475 6c65 4461 7461 202d 2d20 4576 656e  duleData -- Even
│ │ │ │ -000228f0: 2061 6e64 206f 6464 2045 7874 206d 6f64   and odd Ext mod
│ │ │ │ -00022900: 756c 6573 2061 6e64 2074 6865 6972 2072  ules and their r
│ │ │ │ -00022910: 6567 756c 6172 6974 790a 2a2a 2a2a 2a2a  egularity.******
│ │ │ │ +00022660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00022670: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +00022680: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065  ===..  * *note e
│ │ │ │ +00022690: 7665 6e45 7874 4d6f 6475 6c65 3a20 6576  venExtModule: ev
│ │ │ │ +000226a0: 656e 4578 744d 6f64 756c 652c 202d 2d20  enExtModule, -- 
│ │ │ │ +000226b0: 6576 656e 2070 6172 7420 6f66 2045 7874  even part of Ext
│ │ │ │ +000226c0: 5e2a 284d 2c6b 2920 6f76 6572 2061 0a20  ^*(M,k) over a. 
│ │ │ │ +000226d0: 2020 2063 6f6d 706c 6574 6520 696e 7465     complete inte
│ │ │ │ +000226e0: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
│ │ │ │ +000226f0: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ +00022700: 746f 7220 7269 6e67 0a20 202a 202a 6e6f  tor ring.  * *no
│ │ │ │ +00022710: 7465 206f 6464 4578 744d 6f64 756c 653a  te oddExtModule:
│ │ │ │ +00022720: 206f 6464 4578 744d 6f64 756c 652c 202d   oddExtModule, -
│ │ │ │ +00022730: 2d20 6f64 6420 7061 7274 206f 6620 4578  - odd part of Ex
│ │ │ │ +00022740: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ +00022750: 636f 6d70 6c65 7465 0a20 2020 2069 6e74  complete.    int
│ │ │ │ +00022760: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ +00022770: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +00022780: 6174 6f72 2072 696e 670a 0a57 6179 7320  ator ring..Ways 
│ │ │ │ +00022790: 746f 2075 7365 2045 7874 4d6f 6475 6c65  to use ExtModule
│ │ │ │ +000227a0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +000227b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245  ========..  * "E
│ │ │ │ +000227c0: 7874 4d6f 6475 6c65 284d 6f64 756c 6529  xtModule(Module)
│ │ │ │ +000227d0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +000227e0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +000227f0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00022800: 6a65 6374 202a 6e6f 7465 2045 7874 4d6f  ject *note ExtMo
│ │ │ │ +00022810: 6475 6c65 3a20 4578 744d 6f64 756c 652c  dule: ExtModule,
│ │ │ │ +00022820: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +00022830: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ +00022840: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +00022850: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ +00022860: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +00022870: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +00022880: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +00022890: 4578 744d 6f64 756c 6544 6174 612c 204e  ExtModuleData, N
│ │ │ │ +000228a0: 6578 743a 2065 7874 5673 436f 686f 6d6f  ext: extVsCohomo
│ │ │ │ +000228b0: 6c6f 6779 2c20 5072 6576 3a20 4578 744d  logy, Prev: ExtM
│ │ │ │ +000228c0: 6f64 756c 652c 2055 703a 2054 6f70 0a0a  odule, Up: Top..
│ │ │ │ +000228d0: 4578 744d 6f64 756c 6544 6174 6120 2d2d  ExtModuleData --
│ │ │ │ +000228e0: 2045 7665 6e20 616e 6420 6f64 6420 4578   Even and odd Ex
│ │ │ │ +000228f0: 7420 6d6f 6475 6c65 7320 616e 6420 7468  t modules and th
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│ │ │ │ +00022910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00022920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00022930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00022940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00022950: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ -00022960: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ -00022970: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00022980: 204c 203d 2045 7874 4d6f 6475 6c65 4461   L = ExtModuleDa
│ │ │ │ -00022990: 7461 204d 0a20 202a 2049 6e70 7574 733a  ta M.  * Inputs:
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│ │ │ │ -000229b0: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ -000229c0: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ -000229d0: 2c2c 204d 6f64 756c 6520 6f76 6572 2061  ,, Module over a
│ │ │ │ -000229e0: 2063 6f6d 706c 6574 650a 2020 2020 2020   complete.      
│ │ │ │ -000229f0: 2020 696e 7465 7273 6563 7469 6f6e 2053    intersection S
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│ │ │ │ -00022a10: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -00022a20: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00022a30: 3244 6f63 294c 6973 742c 2c20 4c20 3d20  2Doc)List,, L = 
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│ │ │ │ -00022a60: 6f64 756c 652c 2072 6567 302c 2072 6567  odule, reg0, reg
│ │ │ │ -00022a70: 315c 7d0a 0a44 6573 6372 6970 7469 6f6e  1\}..Description
│ │ │ │ -00022a80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5375  .===========..Su
│ │ │ │ -00022a90: 7070 6f73 6520 7468 6174 204d 2069 7320  ppose that M is 
│ │ │ │ -00022aa0: 6120 6d6f 6475 6c65 206f 7665 7220 6120  a module over a 
│ │ │ │ -00022ab0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ -00022ac0: 6374 696f 6e20 5220 736f 2074 6861 740a  ction R so that.
│ │ │ │ -00022ad0: 0a45 203a 3d20 4578 744d 6f64 756c 6520  .E := ExtModule 
│ │ │ │ -00022ae0: 4d0a 0a69 7320 6120 6d6f 6475 6c65 2067  M..is a module g
│ │ │ │ -00022af0: 656e 6572 6174 6564 2069 6e20 6465 6772  enerated in degr
│ │ │ │ -00022b00: 6565 7320 3e3d 3020 6f76 6572 2061 2070  ees >=0 over a p
│ │ │ │ -00022b10: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2054  olynomial ring T
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│ │ │ │ -00022b30: 6567 7265 6520 322c 2061 6e64 0a0a 4530  egree 2, and..E0
│ │ │ │ -00022b40: 203a 3d20 6576 656e 4578 744d 6f64 756c   := evenExtModul
│ │ │ │ -00022b50: 6520 4d20 616e 6420 4531 203a 3d20 6f64  e M and E1 := od
│ │ │ │ -00022b60: 6445 7874 4d6f 6475 6c65 204d 0a0a 6172  dExtModule M..ar
│ │ │ │ -00022b70: 6520 6d6f 6475 6c65 7320 6765 6e65 7261  e modules genera
│ │ │ │ -00022b80: 7465 6420 696e 2064 6567 7265 6520 3e3d  ted in degree >=
│ │ │ │ -00022b90: 2030 206f 7665 7220 6120 706f 6c79 6e6f   0 over a polyno
│ │ │ │ -00022ba0: 6d69 616c 2072 696e 6720 5420 7769 7468  mial ring T with
│ │ │ │ -00022bb0: 2067 656e 6572 6174 6f72 730a 696e 2064   generators.in d
│ │ │ │ -00022bc0: 6567 7265 6520 312e 0a0a 5468 6520 7363  egree 1...The sc
│ │ │ │ -00022bd0: 7269 7074 2072 6574 7572 6e73 0a0a 4c20  ript returns..L 
│ │ │ │ -00022be0: 3d20 5c7b 4530 2c45 312c 2072 6567 756c  = \{E0,E1, regul
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│ │ │ │ -00022c00: 7269 7479 2045 315c 7d0a 0a61 6e64 2070  rity E1\}..and p
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│ │ │ │ -00022c20: 6966 207c 7265 6730 2d72 6567 317c 3e31  if |reg0-reg1|>1
│ │ │ │ -00022c30: 2e0a 0a49 6620 7765 2073 6574 2072 203d  ...If we set r =
│ │ │ │ -00022c40: 206d 6178 2832 2a72 6567 302c 2031 2b32   max(2*reg0, 1+2
│ │ │ │ -00022c50: 2a72 6567 3129 2c20 616e 6420 4620 6973  *reg1), and F is
│ │ │ │ -00022c60: 2061 2072 6573 6f6c 7574 696f 6e20 6f66   a resolution of
│ │ │ │ -00022c70: 204d 2c20 7468 656e 2063 6f6b 6572 0a46   M, then coker.F
│ │ │ │ -00022c80: 2e64 645f 7b28 722b 3129 7d20 6973 2074  .dd_{(r+1)} is t
│ │ │ │ -00022c90: 6865 2066 6972 7374 2073 7a79 6779 206d  he first szygy m
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│ │ │ │ -00022cc0: 6576 656e 4578 744d 6f64 756c 650a 4d20  evenExtModule.M 
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│ │ │ │ -00022d30: 6f72 2072 2c20 6275 7420 7468 6174 2773  or r, but that's
│ │ │ │ -00022d40: 206e 6f74 0a61 6c77 6179 7320 7468 6520   not.always the 
│ │ │ │ -00022d50: 7361 6d65 2e0a 0a54 6865 2072 6567 756c  same...The regul
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│ │ │ │ -00022d70: 7665 6e20 616e 6420 6f64 6420 4578 7420  ven and odd Ext 
│ │ │ │ -00022d80: 6d6f 6475 6c65 7320 2a63 616e 2a20 6469  modules *can* di
│ │ │ │ -00022d90: 6666 6572 2062 7920 6d6f 7265 2074 6861  ffer by more tha
│ │ │ │ -00022da0: 6e20 312e 0a41 6e20 6578 616d 706c 6520  n 1..An example 
│ │ │ │ -00022db0: 6361 6e20 6265 2070 726f 6475 6365 6420  can be produced 
│ │ │ │ -00022dc0: 7769 7468 2073 6574 5261 6e64 6f6d 5365  with setRandomSe
│ │ │ │ -00022dd0: 6564 2030 2053 203d 205a 5a2f 3130 315b  ed 0 S = ZZ/101[
│ │ │ │ -00022de0: 612c 622c 632c 645d 2066 660a 3d6d 6174  a,b,c,d] ff.=mat
│ │ │ │ -00022df0: 7269 7822 6134 2c62 342c 6334 2c64 3422  rix"a4,b4,c4,d4"
│ │ │ │ -00022e00: 2052 203d 2053 2f69 6465 616c 2066 6620   R = S/ideal ff 
│ │ │ │ -00022e10: 4e20 3d20 636f 6b65 7220 7261 6e64 6f6d  N = coker random
│ │ │ │ -00022e20: 2852 5e7b 302c 317d 2c20 525e 7b20 2d31  (R^{0,1}, R^{ -1
│ │ │ │ -00022e30: 2c2d 322c 2d33 2c2d 347d 290a 2d2d 6769  ,-2,-3,-4}).--gi
│ │ │ │ -00022e40: 7665 7320 7265 6720 4578 745e 6576 656e  ves reg Ext^even
│ │ │ │ -00022e50: 203d 2034 2c20 7265 6720 4578 745e 6f64   = 4, reg Ext^od
│ │ │ │ -00022e60: 6420 3d20 3320 4c20 3d20 4578 744d 6f64  d = 3 L = ExtMod
│ │ │ │ -00022e70: 756c 6544 6174 6120 4e3b 2062 7574 2074  uleData N; but t
│ │ │ │ -00022e80: 616b 6573 2073 6f6d 650a 7469 6d65 2074  akes some.time t
│ │ │ │ -00022e90: 6f20 636f 6d70 7574 652e 0a0a 0a0a 2b2d  o compute.....+-
│ │ │ │ +00022940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ +00022950: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ +00022960: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +00022970: 2020 2020 2020 4c20 3d20 4578 744d 6f64        L = ExtMod
│ │ │ │ +00022980: 756c 6544 6174 6120 4d0a 2020 2a20 496e  uleData M.  * In
│ │ │ │ +00022990: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c  puts:.      * M,
│ │ │ │ +000229a0: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ +000229b0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +000229c0: 6f64 756c 652c 2c20 4d6f 6475 6c65 206f  odule,, Module o
│ │ │ │ +000229d0: 7665 7220 6120 636f 6d70 6c65 7465 0a20  ver a complete. 
│ │ │ │ +000229e0: 2020 2020 2020 2069 6e74 6572 7365 6374         intersect
│ │ │ │ +000229f0: 696f 6e20 530a 2020 2a20 4f75 7470 7574  ion S.  * Output
│ │ │ │ +00022a00: 733a 0a20 2020 2020 202a 204c 2c20 6120  s:.      * L, a 
│ │ │ │ +00022a10: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +00022a20: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +00022a30: 204c 203d 205c 7b65 7665 6e45 7874 4d6f   L = \{evenExtMo
│ │ │ │ +00022a40: 6475 6c65 2c0a 2020 2020 2020 2020 6f64  dule,.        od
│ │ │ │ +00022a50: 6445 7874 4d6f 6475 6c65 2c20 7265 6730  dExtModule, reg0
│ │ │ │ +00022a60: 2c20 7265 6731 5c7d 0a0a 4465 7363 7269  , reg1\}..Descri
│ │ │ │ +00022a70: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00022a80: 3d0a 0a53 7570 706f 7365 2074 6861 7420  =..Suppose that 
│ │ │ │ +00022a90: 4d20 6973 2061 206d 6f64 756c 6520 6f76  M is a module ov
│ │ │ │ +00022aa0: 6572 2061 2063 6f6d 706c 6574 6520 696e  er a complete in
│ │ │ │ +00022ab0: 7465 7273 6563 7469 6f6e 2052 2073 6f20  tersection R so 
│ │ │ │ +00022ac0: 7468 6174 0a0a 4520 3a3d 2045 7874 4d6f  that..E := ExtMo
│ │ │ │ +00022ad0: 6475 6c65 204d 0a0a 6973 2061 206d 6f64  dule M..is a mod
│ │ │ │ +00022ae0: 756c 6520 6765 6e65 7261 7465 6420 696e  ule generated in
│ │ │ │ +00022af0: 2064 6567 7265 6573 203e 3d30 206f 7665   degrees >=0 ove
│ │ │ │ +00022b00: 7220 6120 706f 6c79 6e6f 6d69 616c 2072  r a polynomial r
│ │ │ │ +00022b10: 696e 6720 5427 2067 656e 6572 6174 6564  ing T' generated
│ │ │ │ +00022b20: 2069 6e0a 6465 6772 6565 2032 2c20 616e   in.degree 2, an
│ │ │ │ +00022b30: 640a 0a45 3020 3a3d 2065 7665 6e45 7874  d..E0 := evenExt
│ │ │ │ +00022b40: 4d6f 6475 6c65 204d 2061 6e64 2045 3120  Module M and E1 
│ │ │ │ +00022b50: 3a3d 206f 6464 4578 744d 6f64 756c 6520  := oddExtModule 
│ │ │ │ +00022b60: 4d0a 0a61 7265 206d 6f64 756c 6573 2067  M..are modules g
│ │ │ │ +00022b70: 656e 6572 6174 6564 2069 6e20 6465 6772  enerated in degr
│ │ │ │ +00022b80: 6565 203e 3d20 3020 6f76 6572 2061 2070  ee >= 0 over a p
│ │ │ │ +00022b90: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2054  olynomial ring T
│ │ │ │ +00022ba0: 2077 6974 6820 6765 6e65 7261 746f 7273   with generators
│ │ │ │ +00022bb0: 0a69 6e20 6465 6772 6565 2031 2e0a 0a54  .in degree 1...T
│ │ │ │ +00022bc0: 6865 2073 6372 6970 7420 7265 7475 726e  he script return
│ │ │ │ +00022bd0: 730a 0a4c 203d 205c 7b45 302c 4531 2c20  s..L = \{E0,E1, 
│ │ │ │ +00022be0: 7265 6775 6c61 7269 7479 2045 302c 2072  regularity E0, r
│ │ │ │ +00022bf0: 6567 756c 6172 6974 7920 4531 5c7d 0a0a  egularity E1\}..
│ │ │ │ +00022c00: 616e 6420 7072 696e 7473 2061 206d 6573  and prints a mes
│ │ │ │ +00022c10: 7361 6765 2069 6620 7c72 6567 302d 7265  sage if |reg0-re
│ │ │ │ +00022c20: 6731 7c3e 312e 0a0a 4966 2077 6520 7365  g1|>1...If we se
│ │ │ │ +00022c30: 7420 7220 3d20 6d61 7828 322a 7265 6730  t r = max(2*reg0
│ │ │ │ +00022c40: 2c20 312b 322a 7265 6731 292c 2061 6e64  , 1+2*reg1), and
│ │ │ │ +00022c50: 2046 2069 7320 6120 7265 736f 6c75 7469   F is a resoluti
│ │ │ │ +00022c60: 6f6e 206f 6620 4d2c 2074 6865 6e20 636f  on of M, then co
│ │ │ │ +00022c70: 6b65 720a 462e 6464 5f7b 2872 2b31 297d  ker.F.dd_{(r+1)}
│ │ │ │ +00022c80: 2069 7320 7468 6520 6669 7273 7420 737a   is the first sz
│ │ │ │ +00022c90: 7967 7920 6d6f 6475 6c65 206f 6620 4d20  ygy module of M 
│ │ │ │ +00022ca0: 7375 6368 2074 6861 7420 7265 6775 6c61  such that regula
│ │ │ │ +00022cb0: 7269 7479 2065 7665 6e45 7874 4d6f 6475  rity evenExtModu
│ │ │ │ +00022cc0: 6c65 0a4d 203d 3020 414e 4420 7265 6775  le.M =0 AND regu
│ │ │ │ +00022cd0: 6c61 7269 7479 206f 6464 4578 744d 6f64  larity oddExtMod
│ │ │ │ +00022ce0: 756c 6520 4d20 3d30 0a0a 5765 2068 6176  ule M =0..We hav
│ │ │ │ +00022cf0: 6520 6265 656e 2075 7369 6e67 2072 6567  e been using reg
│ │ │ │ +00022d00: 756c 6172 6974 7920 4578 744d 6f64 756c  ularity ExtModul
│ │ │ │ +00022d10: 6520 4d20 6173 2061 2073 7562 7374 6974  e M as a substit
│ │ │ │ +00022d20: 7574 6520 666f 7220 722c 2062 7574 2074  ute for r, but t
│ │ │ │ +00022d30: 6861 7427 7320 6e6f 740a 616c 7761 7973  hat's not.always
│ │ │ │ +00022d40: 2074 6865 2073 616d 652e 0a0a 5468 6520   the same...The 
│ │ │ │ +00022d50: 7265 6775 6c61 7269 7469 6573 206f 6620  regularities of 
│ │ │ │ +00022d60: 7468 6520 6576 656e 2061 6e64 206f 6464  the even and odd
│ │ │ │ +00022d70: 2045 7874 206d 6f64 756c 6573 202a 6361   Ext modules *ca
│ │ │ │ +00022d80: 6e2a 2064 6966 6665 7220 6279 206d 6f72  n* differ by mor
│ │ │ │ +00022d90: 6520 7468 616e 2031 2e0a 416e 2065 7861  e than 1..An exa
│ │ │ │ +00022da0: 6d70 6c65 2063 616e 2062 6520 7072 6f64  mple can be prod
│ │ │ │ +00022db0: 7563 6564 2077 6974 6820 7365 7452 616e  uced with setRan
│ │ │ │ +00022dc0: 646f 6d53 6565 6420 3020 5320 3d20 5a5a  domSeed 0 S = ZZ
│ │ │ │ +00022dd0: 2f31 3031 5b61 2c62 2c63 2c64 5d20 6666  /101[a,b,c,d] ff
│ │ │ │ +00022de0: 0a3d 6d61 7472 6978 2261 342c 6234 2c63  .=matrix"a4,b4,c
│ │ │ │ +00022df0: 342c 6434 2220 5220 3d20 532f 6964 6561  4,d4" R = S/idea
│ │ │ │ +00022e00: 6c20 6666 204e 203d 2063 6f6b 6572 2072  l ff N = coker r
│ │ │ │ +00022e10: 616e 646f 6d28 525e 7b30 2c31 7d2c 2052  andom(R^{0,1}, R
│ │ │ │ +00022e20: 5e7b 202d 312c 2d32 2c2d 332c 2d34 7d29  ^{ -1,-2,-3,-4})
│ │ │ │ +00022e30: 0a2d 2d67 6976 6573 2072 6567 2045 7874  .--gives reg Ext
│ │ │ │ +00022e40: 5e65 7665 6e20 3d20 342c 2072 6567 2045  ^even = 4, reg E
│ │ │ │ +00022e50: 7874 5e6f 6464 203d 2033 204c 203d 2045  xt^odd = 3 L = E
│ │ │ │ +00022e60: 7874 4d6f 6475 6c65 4461 7461 204e 3b20  xtModuleData N; 
│ │ │ │ +00022e70: 6275 7420 7461 6b65 7320 736f 6d65 0a74  but takes some.t
│ │ │ │ +00022e80: 696d 6520 746f 2063 6f6d 7075 7465 2e0a  ime to compute..
│ │ │ │ +00022e90: 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  00022ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00022ed0: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d  .|i1 : setRandom
│ │ │ │ -00022ee0: 5365 6564 2031 3030 2020 2020 2020 2020  Seed 100        
│ │ │ │ -00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00022ec0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7365 7452  ----+.|i1 : setR
│ │ │ │ +00022ed0: 616e 646f 6d53 6565 6420 3130 3020 2020  andomSeed 100   
│ │ │ │ +00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f30: 2020 2020 207c 0a7c 6f31 203d 2031 3030       |.|o1 = 100
│ │ │ │ +00022f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00022f30: 3d20 3130 3020 2020 2020 2020 2020 2020  = 100           
│ │ │ │  00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00022f50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00022fa0: 203a 2053 203d 205a 5a2f 3130 315b 612c   : S = ZZ/101[a,
│ │ │ │ -00022fb0: 622c 632c 645d 3b20 2020 2020 2020 2020  b,c,d];         
│ │ │ │ -00022fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00022fd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00022f90: 2b0a 7c69 3220 3a20 5320 3d20 5a5a 2f31  +.|i2 : S = ZZ/1
│ │ │ │ +00022fa0: 3031 5b61 2c62 2c63 2c64 5d3b 2020 2020  01[a,b,c,d];    
│ │ │ │ +00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022fc0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00022fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023000: 2d2b 0a7c 6933 203a 2066 203d 206d 6170  -+.|i3 : f = map
│ │ │ │ -00023010: 2853 5e31 2c20 535e 342c 2028 692c 6a29  (S^1, S^4, (i,j)
│ │ │ │ -00023020: 202d 3e20 535f 6a5e 3329 2020 2020 2020   -> S_j^3)      
│ │ │ │ -00023030: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022ff0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6620  ------+.|i3 : f 
│ │ │ │ +00023000: 3d20 6d61 7028 535e 312c 2053 5e34 2c20  = map(S^1, S^4, 
│ │ │ │ +00023010: 2869 2c6a 2920 2d3e 2053 5f6a 5e33 2920  (i,j) -> S_j^3) 
│ │ │ │ +00023020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023060: 2020 2020 2020 207c 0a7c 6f33 203d 207c         |.|o3 = |
│ │ │ │ -00023070: 2061 3320 6233 2063 3320 6433 207c 2020   a3 b3 c3 d3 |  
│ │ │ │ -00023080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023050: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00023060: 3320 3d20 7c20 6133 2062 3320 6333 2064  3 = | a3 b3 c3 d
│ │ │ │ +00023070: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │ +00023080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000230a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000230d0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -000230e0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -000230f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023100: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -00023110: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -00023120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023130: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000230c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000230d0: 2020 3120 2020 2020 2034 2020 2020 2020    1      4      
│ │ │ │ +000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000230f0: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ +00023100: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +00023110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023120: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00023130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023160: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5220  ------+.|i4 : R 
│ │ │ │ -00023170: 3d20 532f 6964 6561 6c20 663b 2020 2020  = S/ideal f;    
│ │ │ │ -00023180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023190: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00023150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00023160: 203a 2052 203d 2053 2f69 6465 616c 2066   : R = S/ideal f
│ │ │ │ +00023170: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00023180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023190: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000231a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000231b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000231d0: 3520 3a20 4d20 3d20 525e 312f 6964 6561  5 : M = R^1/idea
│ │ │ │ -000231e0: 6c22 6162 322b 6364 3222 3b20 2020 2020  l"ab2+cd2";     
│ │ │ │ -000231f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023200: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000231c0: 2d2b 0a7c 6935 203a 204d 203d 2052 5e31  -+.|i5 : M = R^1
│ │ │ │ +000231d0: 2f69 6465 616c 2261 6232 2b63 6432 223b  /ideal"ab2+cd2";
│ │ │ │ +000231e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000231f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023230: 2d2d 2b0a 7c69 3620 3a20 6265 7474 6920  --+.|i6 : betti 
│ │ │ │ -00023240: 2846 203d 2072 6573 284d 2c20 4c65 6e67  (F = res(M, Leng
│ │ │ │ -00023250: 7468 4c69 6d69 7420 3d3e 2035 2929 2020  thLimit => 5))  
│ │ │ │ -00023260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023220: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2062  -------+.|i6 : b
│ │ │ │ +00023230: 6574 7469 2028 4620 3d20 7265 7328 4d2c  etti (F = res(M,
│ │ │ │ +00023240: 204c 656e 6774 684c 696d 6974 203d 3e20   LengthLimit => 
│ │ │ │ +00023250: 3529 2920 2020 2020 2020 7c0a 7c20 2020  5))       |.|   
│ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000232a0: 2020 2020 2020 2030 2031 2032 2020 3320         0 1 2  3 
│ │ │ │ -000232b0: 2034 2020 3520 2020 2020 2020 2020 2020   4  5           
│ │ │ │ -000232c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -000232d0: 203d 2074 6f74 616c 3a20 3120 3120 3520   = total: 1 1 5 
│ │ │ │ -000232e0: 3136 2033 3520 3634 2020 2020 2020 2020  16 35 64        
│ │ │ │ -000232f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00023300: 7c20 2020 2020 2020 2020 303a 2031 202e  |         0: 1 .
│ │ │ │ -00023310: 202e 2020 2e20 202e 2020 2e20 2020 2020   .  .  .  .     
│ │ │ │ -00023320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023330: 207c 0a7c 2020 2020 2020 2020 2031 3a20   |.|         1: 
│ │ │ │ -00023340: 2e20 2e20 2e20 202e 2020 2e20 202e 2020  . . .  .  .  .  
│ │ │ │ -00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00023370: 323a 202e 2031 202e 2020 2e20 202e 2020  2: . 1 .  .  .  
│ │ │ │ -00023380: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -00023390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000233a0: 2020 2033 3a20 2e20 2e20 3120 202e 2020     3: . . 1  .  
│ │ │ │ -000233b0: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ -000233c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000233d0: 2020 2020 2020 343a 202e 202e 2033 2020        4: . . 3  
│ │ │ │ -000233e0: 3820 2035 2020 2e20 2020 2020 2020 2020  8  5  .         
│ │ │ │ -000233f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023400: 2020 2020 2020 2020 2035 3a20 2e20 2e20           5: . . 
│ │ │ │ -00023410: 3120 2038 2032 3520 3332 2020 2020 2020  1  8 25 32      
│ │ │ │ -00023420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023430: 7c0a 7c20 2020 2020 2020 2020 363a 202e  |.|         6: .
│ │ │ │ -00023440: 202e 202e 2020 2e20 2035 2033 3220 2020   . .  .  5 32   
│ │ │ │ -00023450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023460: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023290: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +000232a0: 3220 2033 2020 3420 2035 2020 2020 2020  2  3  4  5      
│ │ │ │ +000232b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232c0: 7c0a 7c6f 3620 3d20 746f 7461 6c3a 2031  |.|o6 = total: 1
│ │ │ │ +000232d0: 2031 2035 2031 3620 3335 2036 3420 2020   1 5 16 35 64   
│ │ │ │ +000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232f0: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ +00023300: 3a20 3120 2e20 2e20 202e 2020 2e20 202e  : 1 . .  .  .  .
│ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023330: 2020 313a 202e 202e 202e 2020 2e20 202e    1: . . .  .  .
│ │ │ │ +00023340: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +00023350: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00023360: 2020 2020 2032 3a20 2e20 3120 2e20 202e       2: . 1 .  .
│ │ │ │ +00023370: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +00023380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023390: 2020 2020 2020 2020 333a 202e 202e 2031          3: . . 1
│ │ │ │ +000233a0: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ +000233b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000233c0: 0a7c 2020 2020 2020 2020 2034 3a20 2e20  .|         4: . 
│ │ │ │ +000233d0: 2e20 3320 2038 2020 3520 202e 2020 2020  . 3  8  5  .    
│ │ │ │ +000233e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233f0: 2020 7c0a 7c20 2020 2020 2020 2020 353a    |.|         5:
│ │ │ │ +00023400: 202e 202e 2031 2020 3820 3235 2033 3220   . . 1  8 25 32 
│ │ │ │ +00023410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023430: 2036 3a20 2e20 2e20 2e20 202e 2020 3520   6: . . .  .  5 
│ │ │ │ +00023440: 3332 2020 2020 2020 2020 2020 2020 2020  32              
│ │ │ │ +00023450: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00023460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023490: 2020 2020 2020 7c0a 7c6f 3620 3a20 4265        |.|o6 : Be
│ │ │ │ -000234a0: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ -000234b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00023480: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00023490: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ +000234a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000234b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000234c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000234d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000234e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000234f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00023500: 3720 3a20 4520 3d20 4578 744d 6f64 756c  7 : E = ExtModul
│ │ │ │ -00023510: 6544 6174 6120 4d3b 2020 2020 2020 2020  eData M;        
│ │ │ │ -00023520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023530: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000234f0: 2d2b 0a7c 6937 203a 2045 203d 2045 7874  -+.|i7 : E = Ext
│ │ │ │ +00023500: 4d6f 6475 6c65 4461 7461 204d 3b20 2020  ModuleData M;   
│ │ │ │ +00023510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023520: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023560: 2d2d 2b0a 7c69 3820 3a20 455f 3220 2020  --+.|i8 : E_2   
│ │ │ │ +00023550: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2045  -------+.|i8 : E
│ │ │ │ +00023560: 5f32 2020 2020 2020 2020 2020 2020 2020  _2              
│ │ │ │  00023570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023580: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000235a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235c0: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ -000235d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000235b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000235c0: 6f38 203d 2032 2020 2020 2020 2020 2020  o8 = 2          
│ │ │ │ +000235d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000235e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000235f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00023600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00023630: 7c69 3920 3a20 455f 3320 2020 2020 2020  |i9 : E_3       
│ │ │ │ +00023620: 2d2d 2d2b 0a7c 6939 203a 2045 5f33 2020  ---+.|i9 : E_3  
│ │ │ │ +00023630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023690: 2020 2020 7c0a 7c6f 3920 3d20 3120 2020      |.|o9 = 1   
│ │ │ │ +00023680: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +00023690: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  000236a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000236b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000236c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000236d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000236e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000236f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ -00023700: 203a 2072 203d 206d 6178 2832 2a45 5f32   : r = max(2*E_2
│ │ │ │ -00023710: 2c32 2a45 5f33 2b31 2920 2020 2020 2020  ,2*E_3+1)       
│ │ │ │ -00023720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000236e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000236f0: 0a7c 6931 3020 3a20 7220 3d20 6d61 7828  .|i10 : r = max(
│ │ │ │ +00023700: 322a 455f 322c 322a 455f 332b 3129 2020  2*E_2,2*E_3+1)  
│ │ │ │ +00023710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023720: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00023730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023760: 7c0a 7c6f 3130 203d 2034 2020 2020 2020  |.|o10 = 4      
│ │ │ │ +00023750: 2020 2020 207c 0a7c 6f31 3020 3d20 3420       |.|o10 = 4 
│ │ │ │ +00023760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023790: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00023790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000237a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000237b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000237c0: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2045  ------+.|i11 : E
│ │ │ │ -000237d0: 7220 3d20 4578 744d 6f64 756c 6544 6174  r = ExtModuleDat
│ │ │ │ -000237e0: 6120 636f 6b65 7220 462e 6464 5f72 3b20  a coker F.dd_r; 
│ │ │ │ -000237f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000237b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000237c0: 3120 3a20 4572 203d 2045 7874 4d6f 6475  1 : Er = ExtModu
│ │ │ │ +000237d0: 6c65 4461 7461 2063 6f6b 6572 2046 2e64  leData coker F.d
│ │ │ │ +000237e0: 645f 723b 2020 2020 2020 2020 2020 7c0a  d_r;          |.
│ │ │ │ +000237f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00023800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00023830: 3132 203a 2072 6567 756c 6172 6974 7920  12 : regularity 
│ │ │ │ -00023840: 4572 5f30 2020 2020 2020 2020 2020 2020  Er_0            
│ │ │ │ -00023850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023820: 2d2b 0a7c 6931 3220 3a20 7265 6775 6c61  -+.|i12 : regula
│ │ │ │ +00023830: 7269 7479 2045 725f 3020 2020 2020 2020  rity Er_0       
│ │ │ │ +00023840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023890: 2020 7c0a 7c6f 3132 203d 2030 2020 2020    |.|o12 = 0    
│ │ │ │ +00023880: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +00023890: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  000238a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000238b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000238c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000238d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000238e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000238f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ -00023900: 2072 6567 756c 6172 6974 7920 4572 5f31   regularity Er_1
│ │ │ │ +000238e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000238f0: 6931 3320 3a20 7265 6775 6c61 7269 7479  i13 : regularity
│ │ │ │ +00023900: 2045 725f 3120 2020 2020 2020 2020 2020   Er_1           
│ │ │ │  00023910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023920: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023920: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00023930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00023960: 7c6f 3133 203d 2030 2020 2020 2020 2020  |o13 = 0        
│ │ │ │ +00023950: 2020 207c 0a7c 6f31 3320 3d20 3020 2020     |.|o13 = 0   
│ │ │ │ +00023960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023990: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00023980: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00023990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000239a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000239b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000239c0: 2d2d 2d2d 2b0a 7c69 3134 203a 2072 6567  ----+.|i14 : reg
│ │ │ │ -000239d0: 756c 6172 6974 7920 6576 656e 4578 744d  ularity evenExtM
│ │ │ │ -000239e0: 6f64 756c 6528 636f 6b65 7220 462e 6464  odule(coker F.dd
│ │ │ │ -000239f0: 5f28 722d 3129 297c 0a7c 2020 2020 2020  _(r-1))|.|      
│ │ │ │ +000239b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420  ---------+.|i14 
│ │ │ │ +000239c0: 3a20 7265 6775 6c61 7269 7479 2065 7665  : regularity eve
│ │ │ │ +000239d0: 6e45 7874 4d6f 6475 6c65 2863 6f6b 6572  nExtModule(coker
│ │ │ │ +000239e0: 2046 2e64 645f 2872 2d31 2929 7c0a 7c20   F.dd_(r-1))|.| 
│ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a20: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -00023a30: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +00023a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023a20: 0a7c 6f31 3420 3d20 3120 2020 2020 2020  .|o14 = 1       
│ │ │ │ +00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00023a50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00023a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023a90: 2b0a 7c69 3135 203a 2066 6620 3d20 662a  +.|i15 : ff = f*
│ │ │ │ -00023aa0: 7261 6e64 6f6d 2873 6f75 7263 6520 662c  random(source f,
│ │ │ │ -00023ab0: 2073 6f75 7263 6520 6629 3b20 2020 2020   source f);     
│ │ │ │ -00023ac0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023a80: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 6666  -----+.|i15 : ff
│ │ │ │ +00023a90: 203d 2066 2a72 616e 646f 6d28 736f 7572   = f*random(sour
│ │ │ │ +00023aa0: 6365 2066 2c20 736f 7572 6365 2066 293b  ce f, source f);
│ │ │ │ +00023ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00023ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00023b00: 2020 2020 2020 2031 2020 2020 2020 3420         1      4 
│ │ │ │ -00023b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b20: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ -00023b30: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -00023b40: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00023b50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00023ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023af0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00023b00: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00023b10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023b20: 7c6f 3135 203a 204d 6174 7269 7820 5320  |o15 : Matrix S 
│ │ │ │ +00023b30: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00023b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00023b90: 0a7c 6931 3620 3a20 6d61 7472 6978 4661  .|i16 : matrixFa
│ │ │ │ -00023ba0: 6374 6f72 697a 6174 696f 6e28 6666 2c20  ctorization(ff, 
│ │ │ │ -00023bb0: 636f 6b65 7220 462e 6464 5f28 722b 3129  coker F.dd_(r+1)
│ │ │ │ -00023bc0: 293b 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  );|.+-----------
│ │ │ │ +00023b80: 2d2d 2d2d 2b0a 7c69 3136 203a 206d 6174  ----+.|i16 : mat
│ │ │ │ +00023b90: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +00023ba0: 2866 662c 2063 6f6b 6572 2046 2e64 645f  (ff, coker F.dd_
│ │ │ │ +00023bb0: 2872 2b31 2929 3b7c 0a2b 2d2d 2d2d 2d2d  (r+1));|.+------
│ │ │ │ +00023bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023bf0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2073 7563  -----+..This suc
│ │ │ │ -00023c00: 6365 6564 732c 2062 7574 2077 6520 636f  ceeds, but we co
│ │ │ │ -00023c10: 756c 6420 6765 7420 616e 2065 7272 6f72  uld get an error
│ │ │ │ -00023c20: 2066 726f 6d0a 0a6d 6174 7269 7846 6163   from..matrixFac
│ │ │ │ -00023c30: 746f 7269 7a61 7469 6f6e 2866 662c 2063  torization(ff, c
│ │ │ │ -00023c40: 6f6b 6572 2046 2e64 645f 7229 0a0a 6966  oker F.dd_r)..if
│ │ │ │ -00023c50: 206f 6e65 206f 6620 7468 6520 4349 206f   one of the CI o
│ │ │ │ -00023c60: 7065 7261 746f 7273 2077 6572 6520 6e6f  perators were no
│ │ │ │ -00023c70: 7420 7375 726a 6563 7469 7665 2e0a 0a43  t surjective...C
│ │ │ │ -00023c80: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4578  aveat.======..Ex
│ │ │ │ -00023c90: 744d 6f64 756c 6520 6372 6561 7465 7320  tModule creates 
│ │ │ │ -00023ca0: 6120 7269 6e67 2069 6e73 6964 6520 7468  a ring inside th
│ │ │ │ -00023cb0: 6520 7363 7269 7074 2c20 736f 2069 6620  e script, so if 
│ │ │ │ -00023cc0: 6974 2773 2072 756e 2074 7769 6365 2079  it's run twice y
│ │ │ │ -00023cd0: 6f75 2067 6574 0a6d 6f64 756c 6573 206f  ou get.modules o
│ │ │ │ -00023ce0: 7665 7220 6469 6666 6572 656e 7420 7269  ver different ri
│ │ │ │ -00023cf0: 6e67 732e 2054 6869 7320 7368 6f75 6c64  ngs. This should
│ │ │ │ -00023d00: 2062 6520 6368 616e 6765 642e 0a0a 5365   be changed...Se
│ │ │ │ -00023d10: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -00023d20: 0a20 202a 202a 6e6f 7465 2045 7874 4d6f  .  * *note ExtMo
│ │ │ │ -00023d30: 6475 6c65 3a20 4578 744d 6f64 756c 652c  dule: ExtModule,
│ │ │ │ -00023d40: 202d 2d20 4578 745e 2a28 4d2c 6b29 206f   -- Ext^*(M,k) o
│ │ │ │ -00023d50: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ -00023d60: 6e74 6572 7365 6374 696f 6e20 6173 0a20  ntersection as. 
│ │ │ │ -00023d70: 2020 206d 6f64 756c 6520 6f76 6572 2043     module over C
│ │ │ │ -00023d80: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ -00023d90: 2020 2a20 2a6e 6f74 6520 6576 656e 4578    * *note evenEx
│ │ │ │ -00023da0: 744d 6f64 756c 653a 2065 7665 6e45 7874  tModule: evenExt
│ │ │ │ -00023db0: 4d6f 6475 6c65 2c20 2d2d 2065 7665 6e20  Module, -- even 
│ │ │ │ -00023dc0: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ -00023dd0: 6b29 206f 7665 7220 610a 2020 2020 636f  k) over a.    co
│ │ │ │ -00023de0: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ -00023df0: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -00023e00: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -00023e10: 696e 670a 2020 2a20 2a6e 6f74 6520 6f64  ing.  * *note od
│ │ │ │ -00023e20: 6445 7874 4d6f 6475 6c65 3a20 6f64 6445  dExtModule: oddE
│ │ │ │ -00023e30: 7874 4d6f 6475 6c65 2c20 2d2d 206f 6464  xtModule, -- odd
│ │ │ │ -00023e40: 2070 6172 7420 6f66 2045 7874 5e2a 284d   part of Ext^*(M
│ │ │ │ -00023e50: 2c6b 2920 6f76 6572 2061 2063 6f6d 706c  ,k) over a compl
│ │ │ │ -00023e60: 6574 650a 2020 2020 696e 7465 7273 6563  ete.    intersec
│ │ │ │ -00023e70: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ -00023e80: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ -00023e90: 7269 6e67 0a0a 5761 7973 2074 6f20 7573  ring..Ways to us
│ │ │ │ -00023ea0: 6520 4578 744d 6f64 756c 6544 6174 613a  e ExtModuleData:
│ │ │ │ -00023eb0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00023ec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00023ed0: 2022 4578 744d 6f64 756c 6544 6174 6128   "ExtModuleData(
│ │ │ │ -00023ee0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ -00023ef0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00023f00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00023f10: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00023f20: 6520 4578 744d 6f64 756c 6544 6174 613a  e ExtModuleData:
│ │ │ │ -00023f30: 2045 7874 4d6f 6475 6c65 4461 7461 2c20   ExtModuleData, 
│ │ │ │ -00023f40: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -00023f50: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -00023f60: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -00023f70: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ -00023f80: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00023f90: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00023fa0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2065  ns.info, Node: e
│ │ │ │ -00023fb0: 7874 5673 436f 686f 6d6f 6c6f 6779 2c20  xtVsCohomology, 
│ │ │ │ -00023fc0: 4e65 7874 3a20 6669 6e69 7465 4265 7474  Next: finiteBett
│ │ │ │ -00023fd0: 694e 756d 6265 7273 2c20 5072 6576 3a20  iNumbers, Prev: 
│ │ │ │ -00023fe0: 4578 744d 6f64 756c 6544 6174 612c 2055  ExtModuleData, U
│ │ │ │ -00023ff0: 703a 2054 6f70 0a0a 6578 7456 7343 6f68  p: Top..extVsCoh
│ │ │ │ -00024000: 6f6d 6f6c 6f67 7920 2d2d 2063 6f6d 7061  omology -- compa
│ │ │ │ -00024010: 7265 7320 4578 745f 5328 4d2c 6b29 2061  res Ext_S(M,k) a
│ │ │ │ -00024020: 7320 6578 7465 7269 6f72 206d 6f64 756c  s exterior modul
│ │ │ │ -00024030: 6520 7769 7468 2063 6f68 2074 6162 6c65  e with coh table
│ │ │ │ -00024040: 206f 6620 7368 6561 6620 4578 745f 5228   of sheaf Ext_R(
│ │ │ │ -00024050: 4d2c 6b29 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  M,k).***********
│ │ │ │ +00023be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869  ----------+..Thi
│ │ │ │ +00023bf0: 7320 7375 6363 6565 6473 2c20 6275 7420  s succeeds, but 
│ │ │ │ +00023c00: 7765 2063 6f75 6c64 2067 6574 2061 6e20  we could get an 
│ │ │ │ +00023c10: 6572 726f 7220 6672 6f6d 0a0a 6d61 7472  error from..matr
│ │ │ │ +00023c20: 6978 4661 6374 6f72 697a 6174 696f 6e28  ixFactorization(
│ │ │ │ +00023c30: 6666 2c20 636f 6b65 7220 462e 6464 5f72  ff, coker F.dd_r
│ │ │ │ +00023c40: 290a 0a69 6620 6f6e 6520 6f66 2074 6865  )..if one of the
│ │ │ │ +00023c50: 2043 4920 6f70 6572 6174 6f72 7320 7765   CI operators we
│ │ │ │ +00023c60: 7265 206e 6f74 2073 7572 6a65 6374 6976  re not surjectiv
│ │ │ │ +00023c70: 652e 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  e...Caveat.=====
│ │ │ │ +00023c80: 3d0a 0a45 7874 4d6f 6475 6c65 2063 7265  =..ExtModule cre
│ │ │ │ +00023c90: 6174 6573 2061 2072 696e 6720 696e 7369  ates a ring insi
│ │ │ │ +00023ca0: 6465 2074 6865 2073 6372 6970 742c 2073  de the script, s
│ │ │ │ +00023cb0: 6f20 6966 2069 7427 7320 7275 6e20 7477  o if it's run tw
│ │ │ │ +00023cc0: 6963 6520 796f 7520 6765 740a 6d6f 6475  ice you get.modu
│ │ │ │ +00023cd0: 6c65 7320 6f76 6572 2064 6966 6665 7265  les over differe
│ │ │ │ +00023ce0: 6e74 2072 696e 6773 2e20 5468 6973 2073  nt rings. This s
│ │ │ │ +00023cf0: 686f 756c 6420 6265 2063 6861 6e67 6564  hould be changed
│ │ │ │ +00023d00: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ +00023d10: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00023d20: 4578 744d 6f64 756c 653a 2045 7874 4d6f  ExtModule: ExtMo
│ │ │ │ +00023d30: 6475 6c65 2c20 2d2d 2045 7874 5e2a 284d  dule, -- Ext^*(M
│ │ │ │ +00023d40: 2c6b 2920 6f76 6572 2061 2063 6f6d 706c  ,k) over a compl
│ │ │ │ +00023d50: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ +00023d60: 2061 730a 2020 2020 6d6f 6475 6c65 206f   as.    module o
│ │ │ │ +00023d70: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ +00023d80: 7269 6e67 0a20 202a 202a 6e6f 7465 2065  ring.  * *note e
│ │ │ │ +00023d90: 7665 6e45 7874 4d6f 6475 6c65 3a20 6576  venExtModule: ev
│ │ │ │ +00023da0: 656e 4578 744d 6f64 756c 652c 202d 2d20  enExtModule, -- 
│ │ │ │ +00023db0: 6576 656e 2070 6172 7420 6f66 2045 7874  even part of Ext
│ │ │ │ +00023dc0: 5e2a 284d 2c6b 2920 6f76 6572 2061 0a20  ^*(M,k) over a. 
│ │ │ │ +00023dd0: 2020 2063 6f6d 706c 6574 6520 696e 7465     complete inte
│ │ │ │ +00023de0: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
│ │ │ │ +00023df0: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ +00023e00: 746f 7220 7269 6e67 0a20 202a 202a 6e6f  tor ring.  * *no
│ │ │ │ +00023e10: 7465 206f 6464 4578 744d 6f64 756c 653a  te oddExtModule:
│ │ │ │ +00023e20: 206f 6464 4578 744d 6f64 756c 652c 202d   oddExtModule, -
│ │ │ │ +00023e30: 2d20 6f64 6420 7061 7274 206f 6620 4578  - odd part of Ex
│ │ │ │ +00023e40: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ +00023e50: 636f 6d70 6c65 7465 0a20 2020 2069 6e74  complete.    int
│ │ │ │ +00023e60: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ +00023e70: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +00023e80: 6174 6f72 2072 696e 670a 0a57 6179 7320  ator ring..Ways 
│ │ │ │ +00023e90: 746f 2075 7365 2045 7874 4d6f 6475 6c65  to use ExtModule
│ │ │ │ +00023ea0: 4461 7461 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  Data:.==========
│ │ │ │ +00023eb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00023ec0: 0a0a 2020 2a20 2245 7874 4d6f 6475 6c65  ..  * "ExtModule
│ │ │ │ +00023ed0: 4461 7461 284d 6f64 756c 6529 220a 0a46  Data(Module)"..F
│ │ │ │ +00023ee0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +00023ef0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +00023f00: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +00023f10: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65   *note ExtModule
│ │ │ │ +00023f20: 4461 7461 3a20 4578 744d 6f64 756c 6544  Data: ExtModuleD
│ │ │ │ +00023f30: 6174 612c 2069 7320 6120 2a6e 6f74 6520  ata, is a *note 
│ │ │ │ +00023f40: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +00023f50: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00023f60: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +00023f70: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +00023f80: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +00023f90: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +00023fa0: 6465 3a20 6578 7456 7343 6f68 6f6d 6f6c  de: extVsCohomol
│ │ │ │ +00023fb0: 6f67 792c 204e 6578 743a 2066 696e 6974  ogy, Next: finit
│ │ │ │ +00023fc0: 6542 6574 7469 4e75 6d62 6572 732c 2050  eBettiNumbers, P
│ │ │ │ +00023fd0: 7265 763a 2045 7874 4d6f 6475 6c65 4461  rev: ExtModuleDa
│ │ │ │ +00023fe0: 7461 2c20 5570 3a20 546f 700a 0a65 7874  ta, Up: Top..ext
│ │ │ │ +00023ff0: 5673 436f 686f 6d6f 6c6f 6779 202d 2d20  VsCohomology -- 
│ │ │ │ +00024000: 636f 6d70 6172 6573 2045 7874 5f53 284d  compares Ext_S(M
│ │ │ │ +00024010: 2c6b 2920 6173 2065 7874 6572 696f 7220  ,k) as exterior 
│ │ │ │ +00024020: 6d6f 6475 6c65 2077 6974 6820 636f 6820  module with coh 
│ │ │ │ +00024030: 7461 626c 6520 6f66 2073 6865 6166 2045  table of sheaf E
│ │ │ │ +00024040: 7874 5f52 284d 2c6b 290a 2a2a 2a2a 2a2a  xt_R(M,k).******
│ │ │ │ +00024050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00024090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000240a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000240b0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -000240c0: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -000240d0: 200a 2020 2020 2020 2020 2845 2c54 2920   .        (E,T) 
│ │ │ │ -000240e0: 3d20 6578 7456 7343 6f68 6f6d 6f6c 6f67  = extVsCohomolog
│ │ │ │ -000240f0: 7928 6666 2c4e 290a 2020 2a20 496e 7075  y(ff,N).  * Inpu
│ │ │ │ -00024100: 7473 3a0a 2020 2020 2020 2a20 6666 2c20  ts:.      * ff, 
│ │ │ │ -00024110: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -00024120: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -00024130: 7472 6978 2c2c 2072 6567 756c 6172 2073  trix,, regular s
│ │ │ │ -00024140: 6571 7565 6e63 6520 696e 2061 0a20 2020  equence in a.   
│ │ │ │ -00024150: 2020 2020 2072 6567 756c 6172 2072 696e       regular rin
│ │ │ │ -00024160: 6720 530a 2020 2020 2020 2a20 4e2c 2061  g S.      * N, a
│ │ │ │ -00024170: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ -00024180: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ -00024190: 756c 652c 2c20 6772 6164 6564 206d 6f64  ule,, graded mod
│ │ │ │ -000241a0: 756c 6520 6f76 6572 2052 203d 0a20 2020  ule over R =.   
│ │ │ │ -000241b0: 2020 2020 2053 2f69 6465 616c 2866 6629       S/ideal(ff)
│ │ │ │ -000241c0: 2028 7573 7561 6c6c 7920 6120 6869 6768   (usually a high
│ │ │ │ -000241d0: 2073 797a 7967 7929 0a20 202a 204f 7574   syzygy).  * Out
│ │ │ │ -000241e0: 7075 7473 3a0a 2020 2020 2020 2a20 452c  puts:.      * E,
│ │ │ │ -000241f0: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ -00024200: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -00024210: 6f64 756c 652c 2c20 0a20 2020 2020 202a  odule,, .      *
│ │ │ │ -00024220: 2054 2c20 6120 2a6e 6f74 6520 6d6f 6475   T, a *note modu
│ │ │ │ -00024230: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00024240: 6329 4d6f 6475 6c65 2c2c 2045 7874 2061  c)Module,, Ext a
│ │ │ │ -00024250: 6e64 2054 6f72 2061 7320 6578 7465 7269  nd Tor as exteri
│ │ │ │ -00024260: 6f72 0a20 2020 2020 2020 206d 6f64 756c  or.        modul
│ │ │ │ -00024270: 6573 0a0a 4465 7363 7269 7074 696f 6e0a  es..Description.
│ │ │ │ -00024280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976  ===========..Giv
│ │ │ │ -00024290: 656e 2061 206d 6174 7269 7820 6666 2063  en a matrix ff c
│ │ │ │ -000242a0: 6f6e 7461 696e 696e 6720 6120 7265 6775  ontaining a regu
│ │ │ │ -000242b0: 6c61 7220 7365 7175 656e 6365 2069 6e20  lar sequence in 
│ │ │ │ -000242c0: 6120 706f 6c79 6e6f 6d69 616c 2072 696e  a polynomial rin
│ │ │ │ -000242d0: 6720 5320 6f76 6572 206b 2c0a 7365 7420  g S over k,.set 
│ │ │ │ -000242e0: 5220 3d20 532f 2869 6465 616c 2066 6629  R = S/(ideal ff)
│ │ │ │ -000242f0: 2e20 4966 204e 2069 7320 6120 6772 6164  . If N is a grad
│ │ │ │ -00024300: 6564 2052 2d6d 6f64 756c 652c 2061 6e64  ed R-module, and
│ │ │ │ -00024310: 204d 2069 7320 7468 6520 6d6f 6475 6c65   M is the module
│ │ │ │ -00024320: 204e 2072 6567 6172 6465 640a 6173 2061   N regarded.as a
│ │ │ │ -00024330: 6e20 532d 6d6f 6475 6c65 2c20 7468 6520  n S-module, the 
│ │ │ │ -00024340: 7363 7269 7074 2072 6574 7572 6e73 2045  script returns E
│ │ │ │ -00024350: 203d 2045 7874 5f53 284d 2c6b 2920 616e   = Ext_S(M,k) an
│ │ │ │ -00024360: 6420 5420 3d20 546f 725e 5328 4d2c 6b29  d T = Tor^S(M,k)
│ │ │ │ -00024370: 2061 7320 6d6f 6475 6c65 730a 6f76 6572   as modules.over
│ │ │ │ -00024380: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ -00024390: 6562 7261 2e0a 0a54 6865 2073 6372 6970  ebra...The scrip
│ │ │ │ -000243a0: 7420 7072 696e 7473 2074 6865 2054 6174  t prints the Tat
│ │ │ │ -000243b0: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ -000243c0: 453b 2061 6e64 2074 6865 2063 6f68 6f6d  E; and the cohom
│ │ │ │ -000243d0: 6f6c 6f67 7920 7461 626c 6520 6f66 2074  ology table of t
│ │ │ │ -000243e0: 6865 0a73 6865 6166 2061 7373 6f63 6961  he.sheaf associa
│ │ │ │ -000243f0: 7465 6420 746f 2045 7874 5f52 284e 2c6b  ted to Ext_R(N,k
│ │ │ │ -00024400: 2920 6f76 6572 2074 6865 2072 696e 6720  ) over the ring 
│ │ │ │ -00024410: 6f66 2043 4920 6f70 6572 6174 6f72 732c  of CI operators,
│ │ │ │ -00024420: 2077 6869 6368 2069 7320 610a 706f 6c79   which is a.poly
│ │ │ │ -00024430: 6e6f 6d69 616c 2072 696e 6720 6f76 6572  nomial ring over
│ │ │ │ -00024440: 206b 206f 6e20 6320 7661 7269 6162 6c65   k on c variable
│ │ │ │ -00024450: 732e 0a0a 5468 6520 6f75 7470 7574 2063  s...The output c
│ │ │ │ -00024460: 616e 2062 6520 7573 6564 2074 6f20 2873  an be used to (s
│ │ │ │ -00024470: 6f6d 6574 696d 6573 2920 6368 6563 6b20  ometimes) check 
│ │ │ │ -00024480: 7768 6574 6865 7220 7468 6520 7375 626d  whether the subm
│ │ │ │ -00024490: 6f64 756c 6520 6f66 2045 7874 5f53 284d  odule of Ext_S(M
│ │ │ │ -000244a0: 2c6b 290a 6765 6e65 7261 7465 6420 696e  ,k).generated in
│ │ │ │ -000244b0: 2064 6567 7265 6520 3020 7370 6c69 7473   degree 0 splits
│ │ │ │ -000244c0: 2028 6173 2061 6e20 6578 7465 7269 6f72   (as an exterior
│ │ │ │ -000244d0: 206d 6f64 756c 650a 0a2b 2d2d 2d2d 2d2d   module..+------
│ │ │ │ +000240a0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +000240b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +000240c0: 7361 6765 3a20 0a20 2020 2020 2020 2028  sage: .        (
│ │ │ │ +000240d0: 452c 5429 203d 2065 7874 5673 436f 686f  E,T) = extVsCoho
│ │ │ │ +000240e0: 6d6f 6c6f 6779 2866 662c 4e29 0a20 202a  mology(ff,N).  *
│ │ │ │ +000240f0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00024100: 2066 662c 2061 202a 6e6f 7465 206d 6174   ff, a *note mat
│ │ │ │ +00024110: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ +00024120: 6f63 294d 6174 7269 782c 2c20 7265 6775  oc)Matrix,, regu
│ │ │ │ +00024130: 6c61 7220 7365 7175 656e 6365 2069 6e20  lar sequence in 
│ │ │ │ +00024140: 610a 2020 2020 2020 2020 7265 6775 6c61  a.        regula
│ │ │ │ +00024150: 7220 7269 6e67 2053 0a20 2020 2020 202a  r ring S.      *
│ │ │ │ +00024160: 204e 2c20 6120 2a6e 6f74 6520 6d6f 6475   N, a *note modu
│ │ │ │ +00024170: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +00024180: 6329 4d6f 6475 6c65 2c2c 2067 7261 6465  c)Module,, grade
│ │ │ │ +00024190: 6420 6d6f 6475 6c65 206f 7665 7220 5220  d module over R 
│ │ │ │ +000241a0: 3d0a 2020 2020 2020 2020 532f 6964 6561  =.        S/idea
│ │ │ │ +000241b0: 6c28 6666 2920 2875 7375 616c 6c79 2061  l(ff) (usually a
│ │ │ │ +000241c0: 2068 6967 6820 7379 7a79 6779 290a 2020   high syzygy).  
│ │ │ │ +000241d0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +000241e0: 202a 2045 2c20 6120 2a6e 6f74 6520 6d6f   * E, a *note mo
│ │ │ │ +000241f0: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ +00024200: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020  Doc)Module,, .  
│ │ │ │ +00024210: 2020 2020 2a20 542c 2061 202a 6e6f 7465      * T, a *note
│ │ │ │ +00024220: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00024230: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00024240: 4578 7420 616e 6420 546f 7220 6173 2065  Ext and Tor as e
│ │ │ │ +00024250: 7874 6572 696f 720a 2020 2020 2020 2020  xterior.        
│ │ │ │ +00024260: 6d6f 6475 6c65 730a 0a44 6573 6372 6970  modules..Descrip
│ │ │ │ +00024270: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00024280: 0a0a 4769 7665 6e20 6120 6d61 7472 6978  ..Given a matrix
│ │ │ │ +00024290: 2066 6620 636f 6e74 6169 6e69 6e67 2061   ff containing a
│ │ │ │ +000242a0: 2072 6567 756c 6172 2073 6571 7565 6e63   regular sequenc
│ │ │ │ +000242b0: 6520 696e 2061 2070 6f6c 796e 6f6d 6961  e in a polynomia
│ │ │ │ +000242c0: 6c20 7269 6e67 2053 206f 7665 7220 6b2c  l ring S over k,
│ │ │ │ +000242d0: 0a73 6574 2052 203d 2053 2f28 6964 6561  .set R = S/(idea
│ │ │ │ +000242e0: 6c20 6666 292e 2049 6620 4e20 6973 2061  l ff). If N is a
│ │ │ │ +000242f0: 2067 7261 6465 6420 522d 6d6f 6475 6c65   graded R-module
│ │ │ │ +00024300: 2c20 616e 6420 4d20 6973 2074 6865 206d  , and M is the m
│ │ │ │ +00024310: 6f64 756c 6520 4e20 7265 6761 7264 6564  odule N regarded
│ │ │ │ +00024320: 0a61 7320 616e 2053 2d6d 6f64 756c 652c  .as an S-module,
│ │ │ │ +00024330: 2074 6865 2073 6372 6970 7420 7265 7475   the script retu
│ │ │ │ +00024340: 726e 7320 4520 3d20 4578 745f 5328 4d2c  rns E = Ext_S(M,
│ │ │ │ +00024350: 6b29 2061 6e64 2054 203d 2054 6f72 5e53  k) and T = Tor^S
│ │ │ │ +00024360: 284d 2c6b 2920 6173 206d 6f64 756c 6573  (M,k) as modules
│ │ │ │ +00024370: 0a6f 7665 7220 616e 2065 7874 6572 696f  .over an exterio
│ │ │ │ +00024380: 7220 616c 6765 6272 612e 0a0a 5468 6520  r algebra...The 
│ │ │ │ +00024390: 7363 7269 7074 2070 7269 6e74 7320 7468  script prints th
│ │ │ │ +000243a0: 6520 5461 7465 2072 6573 6f6c 7574 696f  e Tate resolutio
│ │ │ │ +000243b0: 6e20 6f66 2045 3b20 616e 6420 7468 6520  n of E; and the 
│ │ │ │ +000243c0: 636f 686f 6d6f 6c6f 6779 2074 6162 6c65  cohomology table
│ │ │ │ +000243d0: 206f 6620 7468 650a 7368 6561 6620 6173   of the.sheaf as
│ │ │ │ +000243e0: 736f 6369 6174 6564 2074 6f20 4578 745f  sociated to Ext_
│ │ │ │ +000243f0: 5228 4e2c 6b29 206f 7665 7220 7468 6520  R(N,k) over the 
│ │ │ │ +00024400: 7269 6e67 206f 6620 4349 206f 7065 7261  ring of CI opera
│ │ │ │ +00024410: 746f 7273 2c20 7768 6963 6820 6973 2061  tors, which is a
│ │ │ │ +00024420: 0a70 6f6c 796e 6f6d 6961 6c20 7269 6e67  .polynomial ring
│ │ │ │ +00024430: 206f 7665 7220 6b20 6f6e 2063 2076 6172   over k on c var
│ │ │ │ +00024440: 6961 626c 6573 2e0a 0a54 6865 206f 7574  iables...The out
│ │ │ │ +00024450: 7075 7420 6361 6e20 6265 2075 7365 6420  put can be used 
│ │ │ │ +00024460: 746f 2028 736f 6d65 7469 6d65 7329 2063  to (sometimes) c
│ │ │ │ +00024470: 6865 636b 2077 6865 7468 6572 2074 6865  heck whether the
│ │ │ │ +00024480: 2073 7562 6d6f 6475 6c65 206f 6620 4578   submodule of Ex
│ │ │ │ +00024490: 745f 5328 4d2c 6b29 0a67 656e 6572 6174  t_S(M,k).generat
│ │ │ │ +000244a0: 6564 2069 6e20 6465 6772 6565 2030 2073  ed in degree 0 s
│ │ │ │ +000244b0: 706c 6974 7320 2861 7320 616e 2065 7874  plits (as an ext
│ │ │ │ +000244c0: 6572 696f 7220 6d6f 6475 6c65 0a0a 2b2d  erior module..+-
│ │ │ │ +000244d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000244e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000244f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024510: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ -00024520: 203d 205a 5a2f 3130 315b 612c 622c 635d   = ZZ/101[a,b,c]
│ │ │ │ +00024500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00024510: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b61  1 : S = ZZ/101[a
│ │ │ │ +00024520: 2c62 2c63 5d20 2020 2020 2020 2020 2020  ,b,c]           
│ │ │ │  00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024590: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ +00024580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024590: 3120 3d20 5320 2020 2020 2020 2020 2020  1 = S           
│ │ │ │  000245a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000245c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000245d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024610: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -00024620: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00024600: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024610: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ +00024620: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00024630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024650: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024640: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00024650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024690: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2066  -------+.|i2 : f
│ │ │ │ -000246a0: 6620 3d20 6d61 7472 6978 2022 6132 2c62  f = matrix "a2,b
│ │ │ │ -000246b0: 322c 6332 2220 2020 2020 2020 2020 2020  2,c2"           
│ │ │ │ -000246c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00024690: 3220 3a20 6666 203d 206d 6174 7269 7820  2 : ff = matrix 
│ │ │ │ +000246a0: 2261 322c 6232 2c63 3222 2020 2020 2020  "a2,b2,c2"      
│ │ │ │ +000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000246c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000246d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024710: 2020 2020 2020 207c 0a7c 6f32 203d 207c         |.|o2 = |
│ │ │ │ -00024720: 2061 3220 6232 2063 3220 7c20 2020 2020   a2 b2 c2 |     
│ │ │ │ +00024700: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024710: 3220 3d20 7c20 6132 2062 3220 6332 207c  2 = | a2 b2 c2 |
│ │ │ │ +00024720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024750: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024790: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000247a0: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +00024780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024790: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +000247a0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247d0: 2020 2020 2020 207c 0a7c 6f32 203a 204d         |.|o2 : M
│ │ │ │ -000247e0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +000247c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000247d0: 3220 3a20 4d61 7472 6978 2053 2020 3c2d  2 : Matrix S  <-
│ │ │ │ +000247e0: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │  000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024810: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024800: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00024810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024850: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2052  -------+.|i3 : R
│ │ │ │ -00024860: 203d 2053 2f28 6964 6561 6c20 6666 2920   = S/(ideal ff) 
│ │ │ │ +00024840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00024850: 3320 3a20 5220 3d20 532f 2869 6465 616c  3 : R = S/(ideal
│ │ │ │ +00024860: 2066 6629 2020 2020 2020 2020 2020 2020   ff)            
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024890: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000248a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000248b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248d0: 2020 2020 2020 207c 0a7c 6f33 203d 2052         |.|o3 = R
│ │ │ │ +000248c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000248d0: 3320 3d20 5220 2020 2020 2020 2020 2020  3 = R           
│ │ │ │  000248e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000248f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024910: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024900: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024950: 2020 2020 2020 207c 0a7c 6f33 203a 2051         |.|o3 : Q
│ │ │ │ -00024960: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024950: 3320 3a20 5175 6f74 6965 6e74 5269 6e67  3 : QuotientRing
│ │ │ │ +00024960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00024990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000249a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000249b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000249c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000249d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 204e  -------+.|i4 : N
│ │ │ │ -000249e0: 203d 2068 6967 6853 797a 7967 7928 525e   = highSyzygy(R^
│ │ │ │ -000249f0: 312f 6964 6561 6c28 612a 622c 6329 2920  1/ideal(a*b,c)) 
│ │ │ │ -00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000249c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000249d0: 3420 3a20 4e20 3d20 6869 6768 5379 7a79  4 : N = highSyzy
│ │ │ │ +000249e0: 6779 2852 5e31 2f69 6465 616c 2861 2a62  gy(R^1/ideal(a*b
│ │ │ │ +000249f0: 2c63 2929 2020 2020 2020 2020 2020 2020  ,c))            
│ │ │ │ +00024a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a50: 2020 2020 2020 207c 0a7c 6f34 203d 2063         |.|o4 = c
│ │ │ │ -00024a60: 6f6b 6572 6e65 6c20 7b34 7d20 7c20 6320  okernel {4} | c 
│ │ │ │ -00024a70: 2d61 6220 3020 3020 3020 2030 2020 3020  -ab 0 0 0  0  0 
│ │ │ │ -00024a80: 2030 2030 2030 2030 2020 3020 3020 2030   0 0 0 0  0 0  0
│ │ │ │ -00024a90: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024aa0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024ab0: 6320 2020 6220 6120 3020 2030 2020 3020  c   b a 0  0  0 
│ │ │ │ -00024ac0: 2030 2030 2030 2030 2020 3020 3020 2030   0 0 0 0  0 0  0
│ │ │ │ -00024ad0: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024ae0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024af0: 3020 2020 6320 3020 2d62 2061 2020 3020  0   c 0 -b a  0 
│ │ │ │ -00024b00: 2030 2030 2030 2030 2020 3020 3020 2030   0 0 0 0  0 0  0
│ │ │ │ -00024b10: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024b20: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024b30: 3020 2020 3020 6320 3020 202d 6220 2d61  0   0 c 0  -b -a
│ │ │ │ -00024b40: 2030 2030 2030 2030 2020 3020 3020 2030   0 0 0 0  0 0  0
│ │ │ │ -00024b50: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024b60: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024b70: 3020 2020 3020 3020 6320 2030 2020 3020  0   0 0 c  0  0 
│ │ │ │ -00024b80: 2062 2061 2030 2030 2020 3020 3020 2030   b a 0 0  0 0  0
│ │ │ │ -00024b90: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024ba0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024bb0: 3020 2020 3020 3020 3020 2063 2020 3020  0   0 0 0  c  0 
│ │ │ │ -00024bc0: 2030 2062 2030 2030 2020 3020 2d61 2030   0 b 0 0  0 -a 0
│ │ │ │ -00024bd0: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024be0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024bf0: 3020 2020 3020 3020 3020 2030 2020 6320  0   0 0 0  0  c 
│ │ │ │ -00024c00: 2030 2030 2030 2030 2020 3020 6220 2030   0 0 0 0  0 b  0
│ │ │ │ -00024c10: 2020 6120 3020 7c7c 0a7c 2020 2020 2020    a 0 ||.|      
│ │ │ │ -00024c20: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024c30: 3020 2020 3020 3020 3020 2030 2020 3020  0   0 0 0  0  0 
│ │ │ │ -00024c40: 2063 2030 2062 202d 6120 3020 3020 2030   c 0 b -a 0 0  0
│ │ │ │ -00024c50: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024c60: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024c70: 3020 2020 3020 3020 3020 2030 2020 3020  0   0 0 0  0  0 
│ │ │ │ -00024c80: 2030 2063 2030 2062 2020 6120 3020 2030   0 c 0 b  a 0  0
│ │ │ │ -00024c90: 2020 3020 3020 7c7c 0a7c 2020 2020 2020    0 0 ||.|      
│ │ │ │ -00024ca0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024cb0: 3020 2020 3020 3020 3020 2030 2020 3020  0   0 0 0  0  0 
│ │ │ │ -00024cc0: 2030 2030 2030 2030 2020 6220 6320 202d   0 0 0 0  b c  -
│ │ │ │ -00024cd0: 6120 3020 3020 7c7c 0a7c 2020 2020 2020  a 0 0 ||.|      
│ │ │ │ -00024ce0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ -00024cf0: 3020 2020 3020 3020 3020 2030 2020 3020  0   0 0 0  0  0 
│ │ │ │ -00024d00: 2030 2030 2030 2030 2020 3020 3020 2062   0 0 0 0  0 0  b
│ │ │ │ -00024d10: 2020 6320 6120 7c7c 0a7c 2020 2020 2020    c a ||.|      
│ │ │ │ +00024a40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024a50: 3420 3d20 636f 6b65 726e 656c 207b 347d  4 = cokernel {4}
│ │ │ │ +00024a60: 207c 2063 202d 6162 2030 2030 2030 2020   | c -ab 0 0 0  
│ │ │ │ +00024a70: 3020 2030 2020 3020 3020 3020 3020 2030  0  0  0 0 0 0  0
│ │ │ │ +00024a80: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024a90: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024aa0: 207c 2030 2063 2020 2062 2061 2030 2020   | 0 c   b a 0  
│ │ │ │ +00024ab0: 3020 2030 2020 3020 3020 3020 3020 2030  0  0  0 0 0 0  0
│ │ │ │ +00024ac0: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024ad0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024ae0: 207c 2030 2030 2020 2063 2030 202d 6220   | 0 0   c 0 -b 
│ │ │ │ +00024af0: 6120 2030 2020 3020 3020 3020 3020 2030  a  0  0 0 0 0  0
│ │ │ │ +00024b00: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024b10: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024b20: 207c 2030 2030 2020 2030 2063 2030 2020   | 0 0   0 c 0  
│ │ │ │ +00024b30: 2d62 202d 6120 3020 3020 3020 3020 2030  -b -a 0 0 0 0  0
│ │ │ │ +00024b40: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024b50: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024b60: 207c 2030 2030 2020 2030 2030 2063 2020   | 0 0   0 0 c  
│ │ │ │ +00024b70: 3020 2030 2020 6220 6120 3020 3020 2030  0  0  b a 0 0  0
│ │ │ │ +00024b80: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024b90: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024ba0: 207c 2030 2030 2020 2030 2030 2030 2020   | 0 0   0 0 0  
│ │ │ │ +00024bb0: 6320 2030 2020 3020 6220 3020 3020 2030  c  0  0 b 0 0  0
│ │ │ │ +00024bc0: 202d 6120 3020 2030 2030 207c 7c0a 7c20   -a 0  0 0 ||.| 
│ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024be0: 207c 2030 2030 2020 2030 2030 2030 2020   | 0 0   0 0 0  
│ │ │ │ +00024bf0: 3020 2063 2020 3020 3020 3020 3020 2030  0  c  0 0 0 0  0
│ │ │ │ +00024c00: 2062 2020 3020 2061 2030 207c 7c0a 7c20   b  0  a 0 ||.| 
│ │ │ │ +00024c10: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024c20: 207c 2030 2030 2020 2030 2030 2030 2020   | 0 0   0 0 0  
│ │ │ │ +00024c30: 3020 2030 2020 6320 3020 6220 2d61 2030  0  0  c 0 b -a 0
│ │ │ │ +00024c40: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024c50: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024c60: 207c 2030 2030 2020 2030 2030 2030 2020   | 0 0   0 0 0  
│ │ │ │ +00024c70: 3020 2030 2020 3020 6320 3020 6220 2061  0  0  0 c 0 b  a
│ │ │ │ +00024c80: 2030 2020 3020 2030 2030 207c 7c0a 7c20   0  0  0 0 ||.| 
│ │ │ │ +00024c90: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024ca0: 207c 2030 2030 2020 2030 2030 2030 2020   | 0 0   0 0 0  
│ │ │ │ +00024cb0: 3020 2030 2020 3020 3020 3020 3020 2062  0  0  0 0 0 0  b
│ │ │ │ +00024cc0: 2063 2020 2d61 2030 2030 207c 7c0a 7c20   c  -a 0 0 ||.| 
│ │ │ │ +00024cd0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +00024ce0: 207c 2030 2030 2020 2030 2030 2030 2020   | 0 0   0 0 0  
│ │ │ │ +00024cf0: 3020 2030 2020 3020 3020 3020 3020 2030  0  0  0 0 0 0  0
│ │ │ │ +00024d00: 2030 2020 6220 2063 2061 207c 7c0a 7c20   0  b  c a ||.| 
│ │ │ │ +00024d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00024d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d70: 2020 2020 2020 3131 2020 2020 2020 2020        11        
│ │ │ │ -00024d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d90: 2020 2020 2020 207c 0a7c 6f34 203a 2052         |.|o4 : R
│ │ │ │ -00024da0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00024db0: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ -00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024dd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00024d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d60: 2020 2020 2020 2020 2020 2031 3120 2020             11   
│ │ │ │ +00024d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024d90: 3420 3a20 522d 6d6f 6475 6c65 2c20 7175  4 : R-module, qu
│ │ │ │ +00024da0: 6f74 6965 6e74 206f 6620 5220 2020 2020  otient of R     
│ │ │ │ +00024db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024dc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00024dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024e10: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2045  -------+.|i5 : E
│ │ │ │ -00024e20: 203d 2065 7874 5673 436f 686f 6d6f 6c6f   = extVsCohomolo
│ │ │ │ -00024e30: 6779 2866 662c 6869 6768 5379 7a79 6779  gy(ff,highSyzygy
│ │ │ │ -00024e40: 204e 293b 2020 2020 2020 2020 2020 2020   N);            
│ │ │ │ -00024e50: 2020 2020 2020 207c 0a7c 5461 7465 2052         |.|Tate R
│ │ │ │ -00024e60: 6573 6f6c 7574 696f 6e20 6f66 2045 7874  esolution of Ext
│ │ │ │ -00024e70: 5f53 284d 2c6b 2920 6173 2065 7874 6572  _S(M,k) as exter
│ │ │ │ -00024e80: 696f 7220 6d6f 6475 6c65 3a20 2020 2020  ior module:     
│ │ │ │ -00024e90: 2020 2020 2020 207c 0a7c 4e6f 7465 2074         |.|Note t
│ │ │ │ -00024ea0: 6861 7420 6d61 7073 2067 6f20 6c65 6674  hat maps go left
│ │ │ │ -00024eb0: 2074 6f20 7269 6768 7420 2020 2020 2020   to right       
│ │ │ │ -00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ed0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00024ee0: 202d 3131 202d 3130 2020 2d39 202d 3820   -11 -10  -9 -8 
│ │ │ │ -00024ef0: 2d37 202d 3620 2d35 202d 3420 2d33 202d  -7 -6 -5 -4 -3 -
│ │ │ │ -00024f00: 3220 202d 3120 2020 2020 2020 2020 2020  2  -1           
│ │ │ │ -00024f10: 2020 2020 2020 207c 0a7c 746f 7461 6c3a         |.|total:
│ │ │ │ -00024f20: 2031 3938 2031 3436 2031 3032 2036 3620   198 146 102 66 
│ │ │ │ -00024f30: 3338 2031 3820 2039 2031 3620 3336 2036  38 18  9 16 36 6
│ │ │ │ -00024f40: 3420 3130 3020 2020 2020 2020 2020 2020  4 100           
│ │ │ │ -00024f50: 2020 2020 2020 207c 0a7c 2020 2020 383a         |.|    8:
│ │ │ │ -00024f60: 2031 3036 2020 3739 2020 3536 2033 3720   106  79  56 37 
│ │ │ │ -00024f70: 3232 2031 3120 2034 2020 3120 2031 2020  22 11  4  1  1  
│ │ │ │ -00024f80: 3120 2020 3120 2020 2020 2020 2020 2020  1   1           
│ │ │ │ -00024f90: 2020 2020 2020 207c 0a7c 2020 2020 393a         |.|    9:
│ │ │ │ -00024fa0: 2020 3932 2020 3637 2020 3436 2032 3920    92  67  46 29 
│ │ │ │ -00024fb0: 3136 2020 3720 2032 2020 2e20 202e 2020  16  7  2  .  .  
│ │ │ │ -00024fc0: 2e20 2020 2e20 2020 2020 2020 2020 2020  .   .           
│ │ │ │ -00024fd0: 2020 2020 2020 207c 0a7c 2020 2031 303a         |.|   10:
│ │ │ │ -00024fe0: 2020 202e 2020 202e 2020 202e 2020 2e20     .   .   .  . 
│ │ │ │ -00024ff0: 202e 2020 2e20 202e 2020 3520 3134 2032   .  .  .  5 14 2
│ │ │ │ -00025000: 3720 2034 3420 2020 2020 2020 2020 2020  7  44           
│ │ │ │ -00025010: 2020 2020 2020 207c 0a7c 2020 2031 313a         |.|   11:
│ │ │ │ -00025020: 2020 202e 2020 202e 2020 202e 2020 2e20     .   .   .  . 
│ │ │ │ -00025030: 202e 2020 2e20 2033 2031 3020 3231 2033   .  .  3 10 21 3
│ │ │ │ -00025040: 3620 2035 3520 2020 2020 2020 2020 2020  6  55           
│ │ │ │ -00025050: 2020 2020 2020 207c 0a7c 2d2d 2d20 2020         |.|---   
│ │ │ │ +00024e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00024e10: 3520 3a20 4520 3d20 6578 7456 7343 6f68  5 : E = extVsCoh
│ │ │ │ +00024e20: 6f6d 6f6c 6f67 7928 6666 2c68 6967 6853  omology(ff,highS
│ │ │ │ +00024e30: 797a 7967 7920 4e29 3b20 2020 2020 2020  yzygy N);       
│ │ │ │ +00024e40: 2020 2020 2020 2020 2020 2020 7c0a 7c54              |.|T
│ │ │ │ +00024e50: 6174 6520 5265 736f 6c75 7469 6f6e 206f  ate Resolution o
│ │ │ │ +00024e60: 6620 4578 745f 5328 4d2c 6b29 2061 7320  f Ext_S(M,k) as 
│ │ │ │ +00024e70: 6578 7465 7269 6f72 206d 6f64 756c 653a  exterior module:
│ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2020 7c0a 7c4e              |.|N
│ │ │ │ +00024e90: 6f74 6520 7468 6174 206d 6170 7320 676f  ote that maps go
│ │ │ │ +00024ea0: 206c 6566 7420 746f 2072 6967 6874 2020   left to right  
│ │ │ │ +00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024ed0: 2020 2020 2020 2d31 3120 2d31 3020 202d        -11 -10  -
│ │ │ │ +00024ee0: 3920 2d38 202d 3720 2d36 202d 3520 2d34  9 -8 -7 -6 -5 -4
│ │ │ │ +00024ef0: 202d 3320 2d32 2020 2d31 2020 2020 2020   -3 -2  -1      
│ │ │ │ +00024f00: 2020 2020 2020 2020 2020 2020 7c0a 7c74              |.|t
│ │ │ │ +00024f10: 6f74 616c 3a20 3139 3820 3134 3620 3130  otal: 198 146 10
│ │ │ │ +00024f20: 3220 3636 2033 3820 3138 2020 3920 3136  2 66 38 18  9 16
│ │ │ │ +00024f30: 2033 3620 3634 2031 3030 2020 2020 2020   36 64 100      
│ │ │ │ +00024f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024f50: 2020 2038 3a20 3130 3620 2037 3920 2035     8: 106  79  5
│ │ │ │ +00024f60: 3620 3337 2032 3220 3131 2020 3420 2031  6 37 22 11  4  1
│ │ │ │ +00024f70: 2020 3120 2031 2020 2031 2020 2020 2020    1  1   1      
│ │ │ │ +00024f80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024f90: 2020 2039 3a20 2039 3220 2036 3720 2034     9:  92  67  4
│ │ │ │ +00024fa0: 3620 3239 2031 3620 2037 2020 3220 202e  6 29 16  7  2  .
│ │ │ │ +00024fb0: 2020 2e20 202e 2020 202e 2020 2020 2020    .  .   .      
│ │ │ │ +00024fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024fd0: 2020 3130 3a20 2020 2e20 2020 2e20 2020    10:   .   .   
│ │ │ │ +00024fe0: 2e20 202e 2020 2e20 202e 2020 2e20 2035  .  .  .  .  .  5
│ │ │ │ +00024ff0: 2031 3420 3237 2020 3434 2020 2020 2020   14 27  44      
│ │ │ │ +00025000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00025010: 2020 3131 3a20 2020 2e20 2020 2e20 2020    11:   .   .   
│ │ │ │ +00025020: 2e20 202e 2020 2e20 202e 2020 3320 3130  .  .  .  .  3 10
│ │ │ │ +00025030: 2032 3120 3336 2020 3535 2020 2020 2020   21 36  55      
│ │ │ │ +00025040: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00025050: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025090: 2020 2020 2020 207c 0a7c 436f 686f 6d6f         |.|Cohomo
│ │ │ │ -000250a0: 6c6f 6779 2074 6162 6c65 206f 6620 6576  logy table of ev
│ │ │ │ -000250b0: 656e 4578 744d 6f64 756c 6520 4d3a 2020  enExtModule M:  
│ │ │ │ -000250c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250d0: 2020 2020 2020 207c 0a7c 2020 202d 3520         |.|   -5 
│ │ │ │ -000250e0: 2d34 202d 3320 2d32 202d 3120 2030 2020  -4 -3 -2 -1  0  
│ │ │ │ -000250f0: 3120 2032 2020 3320 2034 2020 2035 2020  1  2  3  4   5  
│ │ │ │ -00025100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025110: 2020 2020 2020 207c 0a7c 323a 2033 3620         |.|2: 36 
│ │ │ │ -00025120: 3231 2031 3020 2033 2020 2e20 202e 2020  21 10  3  .  .  
│ │ │ │ -00025130: 2e20 202e 2020 2e20 202e 2020 202e 2020  .  .  .  .   .  
│ │ │ │ -00025140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025150: 2020 2020 2020 207c 0a7c 313a 2020 2e20         |.|1:  . 
│ │ │ │ -00025160: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -00025170: 2e20 202e 2020 2e20 202e 2020 202e 2020  .  .  .  .   .  
│ │ │ │ -00025180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025190: 2020 2020 2020 207c 0a7c 303a 2020 3120         |.|0:  1 
│ │ │ │ -000251a0: 2031 2020 3120 2032 2020 3720 3136 2032   1  1  2  7 16 2
│ │ │ │ -000251b0: 3920 3436 2036 3720 3932 2031 3231 2020  9 46 67 92 121  
│ │ │ │ -000251c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251d0: 2020 2020 2020 207c 0a7c 2d2d 2d20 2020         |.|---   
│ │ │ │ +00025080: 2020 2020 2020 2020 2020 2020 7c0a 7c43              |.|C
│ │ │ │ +00025090: 6f68 6f6d 6f6c 6f67 7920 7461 626c 6520  ohomology table 
│ │ │ │ +000250a0: 6f66 2065 7665 6e45 7874 4d6f 6475 6c65  of evenExtModule
│ │ │ │ +000250b0: 204d 3a20 2020 2020 2020 2020 2020 2020   M:             
│ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000250d0: 2020 2d35 202d 3420 2d33 202d 3220 2d31    -5 -4 -3 -2 -1
│ │ │ │ +000250e0: 2020 3020 2031 2020 3220 2033 2020 3420    0  1  2  3  4 
│ │ │ │ +000250f0: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +00025110: 3a20 3336 2032 3120 3130 2020 3320 202e  : 36 21 10  3  .
│ │ │ │ +00025120: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00025130: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +00025140: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +00025150: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ +00025160: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00025170: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +00025180: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00025190: 3a20 2031 2020 3120 2031 2020 3220 2037  :  1  1  1  2  7
│ │ │ │ +000251a0: 2031 3620 3239 2034 3620 3637 2039 3220   16 29 46 67 92 
│ │ │ │ +000251b0: 3132 3120 2020 2020 2020 2020 2020 2020  121             
│ │ │ │ +000251c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +000251d0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  000251e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025210: 2020 2020 2020 207c 0a7c 436f 686f 6d6f         |.|Cohomo
│ │ │ │ -00025220: 6c6f 6779 2074 6162 6c65 206f 6620 6f64  logy table of od
│ │ │ │ -00025230: 6445 7874 4d6f 6475 6c65 204d 3a20 2020  dExtModule M:   
│ │ │ │ -00025240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025250: 2020 2020 2020 207c 0a7c 2020 202d 3520         |.|   -5 
│ │ │ │ -00025260: 2d34 202d 3320 2d32 202d 3120 2030 2020  -4 -3 -2 -1  0  
│ │ │ │ -00025270: 3120 2032 2020 3320 2020 3420 2020 3520  1  2  3   4   5 
│ │ │ │ -00025280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025290: 2020 2020 2020 207c 0a7c 323a 2032 3820         |.|2: 28 
│ │ │ │ -000252a0: 3135 2020 3620 2031 2020 2e20 202e 2020  15  6  1  .  .  
│ │ │ │ -000252b0: 2e20 202e 2020 2e20 2020 2e20 2020 2e20  .  .  .   .   . 
│ │ │ │ -000252c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252d0: 2020 2020 2020 207c 0a7c 313a 2020 2e20         |.|1:  . 
│ │ │ │ -000252e0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -000252f0: 2e20 202e 2020 2e20 2020 2e20 2020 2e20  .  .  .   .   . 
│ │ │ │ -00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025310: 2020 2020 2020 207c 0a7c 303a 2020 3120         |.|0:  1 
│ │ │ │ -00025320: 2031 2020 3120 2034 2031 3120 3232 2033   1  1  4 11 22 3
│ │ │ │ -00025330: 3720 3536 2037 3920 3130 3620 3133 3720  7 56 79 106 137 
│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00025200: 2020 2020 2020 2020 2020 2020 7c0a 7c43              |.|C
│ │ │ │ +00025210: 6f68 6f6d 6f6c 6f67 7920 7461 626c 6520  ohomology table 
│ │ │ │ +00025220: 6f66 206f 6464 4578 744d 6f64 756c 6520  of oddExtModule 
│ │ │ │ +00025230: 4d3a 2020 2020 2020 2020 2020 2020 2020  M:              
│ │ │ │ +00025240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00025250: 2020 2d35 202d 3420 2d33 202d 3220 2d31    -5 -4 -3 -2 -1
│ │ │ │ +00025260: 2020 3020 2031 2020 3220 2033 2020 2034    0  1  2  3   4
│ │ │ │ +00025270: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │ +00025280: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +00025290: 3a20 3238 2031 3520 2036 2020 3120 202e  : 28 15  6  1  .
│ │ │ │ +000252a0: 2020 2e20 202e 2020 2e20 202e 2020 202e    .  .  .  .   .
│ │ │ │ +000252b0: 2020 202e 2020 2020 2020 2020 2020 2020     .            
│ │ │ │ +000252c0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +000252d0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ +000252e0: 2020 2e20 202e 2020 2e20 202e 2020 202e    .  .  .  .   .
│ │ │ │ +000252f0: 2020 202e 2020 2020 2020 2020 2020 2020     .            
│ │ │ │ +00025300: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00025310: 3a20 2031 2020 3120 2031 2020 3420 3131  :  1  1  1  4 11
│ │ │ │ +00025320: 2032 3220 3337 2035 3620 3739 2031 3036   22 37 56 79 106
│ │ │ │ +00025330: 2031 3337 2020 2020 2020 2020 2020 2020   137            
│ │ │ │ +00025340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00025350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025390: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -000253a0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -000253b0: 202a 6e6f 7465 2068 6967 6853 797a 7967   *note highSyzyg
│ │ │ │ -000253c0: 793a 2068 6967 6853 797a 7967 792c 202d  y: highSyzygy, -
│ │ │ │ -000253d0: 2d20 5265 7475 726e 7320 6120 7379 7a79  - Returns a syzy
│ │ │ │ -000253e0: 6779 206d 6f64 756c 6520 6f6e 6520 6265  gy module one be
│ │ │ │ -000253f0: 796f 6e64 2074 6865 0a20 2020 2072 6567  yond the.    reg
│ │ │ │ -00025400: 756c 6172 6974 7920 6f66 2045 7874 284d  ularity of Ext(M
│ │ │ │ -00025410: 2c6b 290a 2020 2a20 2a6e 6f74 6520 6578  ,k).  * *note ex
│ │ │ │ -00025420: 7465 7269 6f72 4578 744d 6f64 756c 653a  teriorExtModule:
│ │ │ │ -00025430: 2065 7874 6572 696f 7245 7874 4d6f 6475   exteriorExtModu
│ │ │ │ -00025440: 6c65 2c20 2d2d 2045 7874 284d 2c6b 2920  le, -- Ext(M,k) 
│ │ │ │ -00025450: 6f72 2045 7874 284d 2c4e 2920 6173 2061  or Ext(M,N) as a
│ │ │ │ -00025460: 0a20 2020 206d 6f64 756c 6520 6f76 6572  .    module over
│ │ │ │ -00025470: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ -00025480: 6562 7261 0a0a 5761 7973 2074 6f20 7573  ebra..Ways to us
│ │ │ │ -00025490: 6520 6578 7456 7343 6f68 6f6d 6f6c 6f67  e extVsCohomolog
│ │ │ │ -000254a0: 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  y:.=============
│ │ │ │ -000254b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -000254c0: 0a20 202a 2022 6578 7456 7343 6f68 6f6d  .  * "extVsCohom
│ │ │ │ -000254d0: 6f6c 6f67 7928 4d61 7472 6978 2c4d 6f64  ology(Matrix,Mod
│ │ │ │ -000254e0: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ -000254f0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00025500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00025510: 6520 6f62 6a65 6374 202a 6e6f 7465 2065  e object *note e
│ │ │ │ -00025520: 7874 5673 436f 686f 6d6f 6c6f 6779 3a20  xtVsCohomology: 
│ │ │ │ -00025530: 6578 7456 7343 6f68 6f6d 6f6c 6f67 792c  extVsCohomology,
│ │ │ │ -00025540: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -00025550: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ -00025560: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00025570: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ -00025580: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -00025590: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -000255a0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ -000255b0: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -000255c0: 7273 2c20 4e65 7874 3a20 6672 6565 4578  rs, Next: freeEx
│ │ │ │ -000255d0: 7465 7269 6f72 5375 6d6d 616e 642c 2050  teriorSummand, P
│ │ │ │ -000255e0: 7265 763a 2065 7874 5673 436f 686f 6d6f  rev: extVsCohomo
│ │ │ │ -000255f0: 6c6f 6779 2c20 5570 3a20 546f 700a 0a66  logy, Up: Top..f
│ │ │ │ -00025600: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -00025610: 7320 2d2d 2062 6574 7469 206e 756d 6265  s -- betti numbe
│ │ │ │ -00025620: 7273 206f 6620 6669 6e69 7465 2072 6573  rs of finite res
│ │ │ │ -00025630: 6f6c 7574 696f 6e20 636f 6d70 7574 6564  olution computed
│ │ │ │ -00025640: 2066 726f 6d20 6120 6d61 7472 6978 2066   from a matrix f
│ │ │ │ -00025650: 6163 746f 7269 7a61 7469 6f6e 0a2a 2a2a  actorization.***
│ │ │ │ +00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00025390: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +000253a0: 0a0a 2020 2a20 2a6e 6f74 6520 6869 6768  ..  * *note high
│ │ │ │ +000253b0: 5379 7a79 6779 3a20 6869 6768 5379 7a79  Syzygy: highSyzy
│ │ │ │ +000253c0: 6779 2c20 2d2d 2052 6574 7572 6e73 2061  gy, -- Returns a
│ │ │ │ +000253d0: 2073 797a 7967 7920 6d6f 6475 6c65 206f   syzygy module o
│ │ │ │ +000253e0: 6e65 2062 6579 6f6e 6420 7468 650a 2020  ne beyond the.  
│ │ │ │ +000253f0: 2020 7265 6775 6c61 7269 7479 206f 6620    regularity of 
│ │ │ │ +00025400: 4578 7428 4d2c 6b29 0a20 202a 202a 6e6f  Ext(M,k).  * *no
│ │ │ │ +00025410: 7465 2065 7874 6572 696f 7245 7874 4d6f  te exteriorExtMo
│ │ │ │ +00025420: 6475 6c65 3a20 6578 7465 7269 6f72 4578  dule: exteriorEx
│ │ │ │ +00025430: 744d 6f64 756c 652c 202d 2d20 4578 7428  tModule, -- Ext(
│ │ │ │ +00025440: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29  M,k) or Ext(M,N)
│ │ │ │ +00025450: 2061 7320 610a 2020 2020 6d6f 6475 6c65   as a.    module
│ │ │ │ +00025460: 206f 7665 7220 616e 2065 7874 6572 696f   over an exterio
│ │ │ │ +00025470: 7220 616c 6765 6272 610a 0a57 6179 7320  r algebra..Ways 
│ │ │ │ +00025480: 746f 2075 7365 2065 7874 5673 436f 686f  to use extVsCoho
│ │ │ │ +00025490: 6d6f 6c6f 6779 3a0a 3d3d 3d3d 3d3d 3d3d  mology:.========
│ │ │ │ +000254a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000254b0: 3d3d 3d3d 0a0a 2020 2a20 2265 7874 5673  ====..  * "extVs
│ │ │ │ +000254c0: 436f 686f 6d6f 6c6f 6779 284d 6174 7269  Cohomology(Matri
│ │ │ │ +000254d0: 782c 4d6f 6475 6c65 2922 0a0a 466f 7220  x,Module)"..For 
│ │ │ │ +000254e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +000254f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00025500: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +00025510: 6f74 6520 6578 7456 7343 6f68 6f6d 6f6c  ote extVsCohomol
│ │ │ │ +00025520: 6f67 793a 2065 7874 5673 436f 686f 6d6f  ogy: extVsCohomo
│ │ │ │ +00025530: 6c6f 6779 2c20 6973 2061 202a 6e6f 7465  logy, is a *note
│ │ │ │ +00025540: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00025550: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00025560: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +00025570: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00025580: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +00025590: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +000255a0: 6f64 653a 2066 696e 6974 6542 6574 7469  ode: finiteBetti
│ │ │ │ +000255b0: 4e75 6d62 6572 732c 204e 6578 743a 2066  Numbers, Next: f
│ │ │ │ +000255c0: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ +000255d0: 6e64 2c20 5072 6576 3a20 6578 7456 7343  nd, Prev: extVsC
│ │ │ │ +000255e0: 6f68 6f6d 6f6c 6f67 792c 2055 703a 2054  ohomology, Up: T
│ │ │ │ +000255f0: 6f70 0a0a 6669 6e69 7465 4265 7474 694e  op..finiteBettiN
│ │ │ │ +00025600: 756d 6265 7273 202d 2d20 6265 7474 6920  umbers -- betti 
│ │ │ │ +00025610: 6e75 6d62 6572 7320 6f66 2066 696e 6974  numbers of finit
│ │ │ │ +00025620: 6520 7265 736f 6c75 7469 6f6e 2063 6f6d  e resolution com
│ │ │ │ +00025630: 7075 7465 6420 6672 6f6d 2061 206d 6174  puted from a mat
│ │ │ │ +00025640: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ +00025650: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │  00025660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000256a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000256b0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -000256c0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -000256d0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -000256e0: 2020 204c 203d 2066 696e 6974 6542 6574     L = finiteBet
│ │ │ │ -000256f0: 7469 4e75 6d62 6572 7320 4d46 0a20 202a  tiNumbers MF.  *
│ │ │ │ -00025700: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00025710: 204d 462c 2061 202a 6e6f 7465 206c 6973   MF, a *note lis
│ │ │ │ -00025720: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -00025730: 294c 6973 742c 2c20 4c69 7374 206f 6620  )List,, List of 
│ │ │ │ -00025740: 4861 7368 5461 626c 6573 2061 7320 636f  HashTables as co
│ │ │ │ -00025750: 6d70 7574 6564 0a20 2020 2020 2020 2062  mputed.        b
│ │ │ │ -00025760: 7920 226d 6174 7269 7846 6163 746f 7269  y "matrixFactori
│ │ │ │ -00025770: 7a61 7469 6f6e 220a 2020 2a20 4f75 7470  zation".  * Outp
│ │ │ │ -00025780: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ -00025790: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -000257a0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -000257b0: 2c2c 204c 6973 7420 6f66 2062 6574 7469  ,, List of betti
│ │ │ │ -000257c0: 206e 756d 6265 7273 0a0a 4465 7363 7269   numbers..Descri
│ │ │ │ -000257d0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -000257e0: 3d0a 0a55 7365 7320 7468 6520 7261 6e6b  =..Uses the rank
│ │ │ │ -000257f0: 7320 6f66 2074 6865 2042 206d 6174 7269  s of the B matri
│ │ │ │ -00025800: 6365 7320 696e 2061 206d 6174 7269 7820  ces in a matrix 
│ │ │ │ -00025810: 6661 6374 6f72 697a 6174 696f 6e20 666f  factorization fo
│ │ │ │ -00025820: 7220 6120 6d6f 6475 6c65 204d 206f 7665  r a module M ove
│ │ │ │ -00025830: 720a 532f 2866 5f31 2c2e 2e2c 665f 6329  r.S/(f_1,..,f_c)
│ │ │ │ -00025840: 2074 6f20 636f 6d70 7574 6520 7468 6520   to compute the 
│ │ │ │ -00025850: 6265 7474 6920 6e75 6d62 6572 7320 6f66  betti numbers of
│ │ │ │ -00025860: 2074 6865 206d 696e 696d 616c 2072 6573   the minimal res
│ │ │ │ -00025870: 6f6c 7574 696f 6e20 6f66 204d 206f 7665  olution of M ove
│ │ │ │ -00025880: 720a 532c 2077 6869 6368 2069 7320 7468  r.S, which is th
│ │ │ │ -00025890: 6520 7375 6d20 6f66 2074 6865 204b 6f73  e sum of the Kos
│ │ │ │ -000258a0: 7a75 6c20 636f 6d70 6c65 7865 7320 4b28  zul complexes K(
│ │ │ │ -000258b0: 665f 312e 2e66 5f7b 6a2d 317d 2920 7465  f_1..f_{j-1}) te
│ │ │ │ -000258c0: 6e73 6f72 6564 2077 6974 6820 4228 6a29  nsored with B(j)
│ │ │ │ -000258d0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ -000258e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000258f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -00025900: 3a20 7365 7452 616e 646f 6d53 6565 6420  : setRandomSeed 
│ │ │ │ -00025910: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00025920: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000256a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +000256b0: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +000256c0: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +000256d0: 2020 2020 2020 2020 4c20 3d20 6669 6e69          L = fini
│ │ │ │ +000256e0: 7465 4265 7474 694e 756d 6265 7273 204d  teBettiNumbers M
│ │ │ │ +000256f0: 460a 2020 2a20 496e 7075 7473 3a0a 2020  F.  * Inputs:.  
│ │ │ │ +00025700: 2020 2020 2a20 4d46 2c20 6120 2a6e 6f74      * MF, a *not
│ │ │ │ +00025710: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +00025720: 7932 446f 6329 4c69 7374 2c2c 204c 6973  y2Doc)List,, Lis
│ │ │ │ +00025730: 7420 6f66 2048 6173 6854 6162 6c65 7320  t of HashTables 
│ │ │ │ +00025740: 6173 2063 6f6d 7075 7465 640a 2020 2020  as computed.    
│ │ │ │ +00025750: 2020 2020 6279 2022 6d61 7472 6978 4661      by "matrixFa
│ │ │ │ +00025760: 6374 6f72 697a 6174 696f 6e22 0a20 202a  ctorization".  *
│ │ │ │ +00025770: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +00025780: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973  * L, a *note lis
│ │ │ │ +00025790: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +000257a0: 294c 6973 742c 2c20 4c69 7374 206f 6620  )List,, List of 
│ │ │ │ +000257b0: 6265 7474 6920 6e75 6d62 6572 730a 0a44  betti numbers..D
│ │ │ │ +000257c0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +000257d0: 3d3d 3d3d 3d3d 0a0a 5573 6573 2074 6865  ======..Uses the
│ │ │ │ +000257e0: 2072 616e 6b73 206f 6620 7468 6520 4220   ranks of the B 
│ │ │ │ +000257f0: 6d61 7472 6963 6573 2069 6e20 6120 6d61  matrices in a ma
│ │ │ │ +00025800: 7472 6978 2066 6163 746f 7269 7a61 7469  trix factorizati
│ │ │ │ +00025810: 6f6e 2066 6f72 2061 206d 6f64 756c 6520  on for a module 
│ │ │ │ +00025820: 4d20 6f76 6572 0a53 2f28 665f 312c 2e2e  M over.S/(f_1,..
│ │ │ │ +00025830: 2c66 5f63 2920 746f 2063 6f6d 7075 7465  ,f_c) to compute
│ │ │ │ +00025840: 2074 6865 2062 6574 7469 206e 756d 6265   the betti numbe
│ │ │ │ +00025850: 7273 206f 6620 7468 6520 6d69 6e69 6d61  rs of the minima
│ │ │ │ +00025860: 6c20 7265 736f 6c75 7469 6f6e 206f 6620  l resolution of 
│ │ │ │ +00025870: 4d20 6f76 6572 0a53 2c20 7768 6963 6820  M over.S, which 
│ │ │ │ +00025880: 6973 2074 6865 2073 756d 206f 6620 7468  is the sum of th
│ │ │ │ +00025890: 6520 4b6f 737a 756c 2063 6f6d 706c 6578  e Koszul complex
│ │ │ │ +000258a0: 6573 204b 2866 5f31 2e2e 665f 7b6a 2d31  es K(f_1..f_{j-1
│ │ │ │ +000258b0: 7d29 2074 656e 736f 7265 6420 7769 7468  }) tensored with
│ │ │ │ +000258c0: 2042 286a 290a 0a2b 2d2d 2d2d 2d2d 2d2d   B(j)..+--------
│ │ │ │ +000258d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000258e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000258f0: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d  .|i1 : setRandom
│ │ │ │ +00025900: 5365 6564 2030 2020 2020 2020 2020 2020  Seed 0          
│ │ │ │ +00025910: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00025920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00025950: 7c6f 3120 3d20 3020 2020 2020 2020 2020  |o1 = 0         
│ │ │ │ -00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00025940: 2020 207c 0a7c 6f31 203d 2030 2020 2020     |.|o1 = 0    
│ │ │ │ +00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025960: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00025970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000259a0: 2d2d 2b0a 7c69 3220 3a20 6b6b 203d 205a  --+.|i2 : kk = Z
│ │ │ │ -000259b0: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ -000259c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00025990: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 206b  -------+.|i2 : k
│ │ │ │ +000259a0: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020  k = ZZ/101      
│ │ │ │ +000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000259c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000259d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259f0: 2020 2020 2020 7c0a 7c6f 3220 3d20 6b6b        |.|o2 = kk
│ │ │ │ +000259e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000259f0: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │  00025a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00025a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a40: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00025a50: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ -00025a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00025a10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025a40: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ +00025a50: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00025a60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00025a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00025aa0: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62  |i3 : S = kk[a,b
│ │ │ │ -00025ab0: 2c75 2c76 5d20 2020 2020 2020 2020 2020  ,u,v]           
│ │ │ │ -00025ac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00025a90: 2d2d 2d2b 0a7c 6933 203a 2053 203d 206b  ---+.|i3 : S = k
│ │ │ │ +00025aa0: 6b5b 612c 622c 752c 765d 2020 2020 2020  k[a,b,u,v]      
│ │ │ │ +00025ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025af0: 2020 7c0a 7c6f 3320 3d20 5320 2020 2020    |.|o3 = S     
│ │ │ │ +00025ae0: 2020 2020 2020 207c 0a7c 6f33 203d 2053         |.|o3 = S
│ │ │ │ +00025af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00025b10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00025b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b40: 2020 2020 2020 7c0a 7c6f 3320 3a20 506f        |.|o3 : Po
│ │ │ │ -00025b50: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ -00025b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00025b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -00025ba0: 3a20 6666 203d 206d 6174 7269 7822 6175  : ff = matrix"au
│ │ │ │ -00025bb0: 2c62 7622 2020 2020 2020 2020 2020 2020  ,bv"            
│ │ │ │ -00025bc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025b30: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00025b40: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +00025b50: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00025b60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00025b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00025b90: 0a7c 6934 203a 2066 6620 3d20 6d61 7472  .|i4 : ff = matr
│ │ │ │ +00025ba0: 6978 2261 752c 6276 2220 2020 2020 2020  ix"au,bv"       
│ │ │ │ +00025bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00025bf0: 7c6f 3420 3d20 7c20 6175 2062 7620 7c20  |o4 = | au bv | 
│ │ │ │ -00025c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00025be0: 2020 207c 0a7c 6f34 203d 207c 2061 7520     |.|o4 = | au 
│ │ │ │ +00025bf0: 6276 207c 2020 2020 2020 2020 2020 2020  bv |            
│ │ │ │ +00025c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00025c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00025c50: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ -00025c60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00025c70: 3420 3a20 4d61 7472 6978 2053 2020 3c2d  4 : Matrix S  <-
│ │ │ │ -00025c80: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │ -00025c90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00025c30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00025c40: 2020 2020 2020 2031 2020 2020 2020 3220         1      2 
│ │ │ │ +00025c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c60: 207c 0a7c 6f34 203a 204d 6174 7269 7820   |.|o4 : Matrix 
│ │ │ │ +00025c70: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ +00025c80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025cc0: 2b0a 7c69 3520 3a20 5220 3d20 532f 6964  +.|i5 : R = S/id
│ │ │ │ -00025cd0: 6561 6c20 6666 2020 2020 2020 2020 2020  eal ff          
│ │ │ │ -00025ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00025cb0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2052 203d  -----+.|i5 : R =
│ │ │ │ +00025cc0: 2053 2f69 6465 616c 2066 6620 2020 2020   S/ideal ff     
│ │ │ │ +00025cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025ce0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d10: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ +00025d00: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +00025d10: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  00025d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00025d40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00025d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d60: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -00025d70: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ -00025d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00025d30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00025d60: 6f35 203a 2051 756f 7469 656e 7452 696e  o5 : QuotientRin
│ │ │ │ +00025d70: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00025d80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00025d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00025dc0: 3620 3a20 4d30 203d 2052 5e31 2f69 6465  6 : M0 = R^1/ide
│ │ │ │ -00025dd0: 616c 2261 2c62 2220 2020 2020 2020 2020  al"a,b"         
│ │ │ │ -00025de0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025db0: 2d2b 0a7c 6936 203a 204d 3020 3d20 525e  -+.|i6 : M0 = R^
│ │ │ │ +00025dc0: 312f 6964 6561 6c22 612c 6222 2020 2020  1/ideal"a,b"    
│ │ │ │ +00025dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e10: 7c0a 7c6f 3620 3d20 636f 6b65 726e 656c  |.|o6 = cokernel
│ │ │ │ -00025e20: 207c 2061 2062 207c 2020 2020 2020 2020   | a b |        
│ │ │ │ -00025e30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00025e00: 2020 2020 207c 0a7c 6f36 203d 2063 6f6b       |.|o6 = cok
│ │ │ │ +00025e10: 6572 6e65 6c20 7c20 6120 6220 7c20 2020  ernel | a b |   
│ │ │ │ +00025e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025e30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00025e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e80: 2020 2031 2020 2020 2020 2020 2020 7c0a     1          |.
│ │ │ │ -00025e90: 7c6f 3620 3a20 522d 6d6f 6475 6c65 2c20  |o6 : R-module, 
│ │ │ │ -00025ea0: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ -00025eb0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00025e50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00025e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025e70: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00025e80: 2020 207c 0a7c 6f36 203a 2052 2d6d 6f64     |.|o6 : R-mod
│ │ │ │ +00025e90: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ +00025ea0: 2052 2020 2020 2020 2020 2020 207c 0a2b   R           |.+
│ │ │ │ +00025eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025ee0: 2d2d 2b0a 7c69 3720 3a20 4620 3d20 7265  --+.|i7 : F = re
│ │ │ │ -00025ef0: 7328 4d30 2c20 4c65 6e67 7468 4c69 6d69  s(M0, LengthLimi
│ │ │ │ -00025f00: 7420 3d3e 3329 2020 2020 2020 7c0a 7c20  t =>3)      |.| 
│ │ │ │ +00025ed0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2046  -------+.|i7 : F
│ │ │ │ +00025ee0: 203d 2072 6573 284d 302c 204c 656e 6774   = res(M0, Lengt
│ │ │ │ +00025ef0: 684c 696d 6974 203d 3e33 2920 2020 2020  hLimit =>3)     
│ │ │ │ +00025f00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00025f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025f30: 2020 2020 2020 7c0a 7c20 2020 2020 2031        |.|      1
│ │ │ │ -00025f40: 2020 2020 2020 3220 2020 2020 2033 2020        2      3  
│ │ │ │ -00025f50: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -00025f60: 7c0a 7c6f 3720 3d20 5220 203c 2d2d 2052  |.|o7 = R  <-- R
│ │ │ │ -00025f70: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020    <-- R  <-- R  
│ │ │ │ -00025f80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00025f20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00025f30: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ +00025f40: 2020 3320 2020 2020 2034 2020 2020 2020    3      4      
│ │ │ │ +00025f50: 2020 2020 207c 0a7c 6f37 203d 2052 2020       |.|o7 = R  
│ │ │ │ +00025f60: 3c2d 2d20 5220 203c 2d2d 2052 2020 3c2d  <-- R  <-- R  <-
│ │ │ │ +00025f70: 2d20 5220 2020 2020 2020 2020 2020 207c  - R            |
│ │ │ │ +00025f80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025fb0: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ -00025fc0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -00025fd0: 2033 2020 2020 2020 2020 2020 2020 7c0a   3            |.
│ │ │ │ -00025fe0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00025ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026000: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -00026010: 4368 6169 6e43 6f6d 706c 6578 2020 2020  ChainComplex    
│ │ │ │ -00026020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026030: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00025fa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00025fb0: 2030 2020 2020 2020 3120 2020 2020 2032   0      1      2
│ │ │ │ +00025fc0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00025fd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00026000: 6f37 203a 2043 6861 696e 436f 6d70 6c65  o7 : ChainComple
│ │ │ │ +00026010: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +00026020: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00026030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00026060: 3820 3a20 4d20 3d20 636f 6b65 7220 462e  8 : M = coker F.
│ │ │ │ -00026070: 6464 5f33 3b20 2020 2020 2020 2020 2020  dd_3;           
│ │ │ │ -00026080: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00026050: 2d2b 0a7c 6938 203a 204d 203d 2063 6f6b  -+.|i8 : M = cok
│ │ │ │ +00026060: 6572 2046 2e64 645f 333b 2020 2020 2020  er F.dd_3;      
│ │ │ │ +00026070: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00026080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000260a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000260b0: 2b0a 7c69 3920 3a20 4d46 203d 206d 6174  +.|i9 : MF = mat
│ │ │ │ -000260c0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -000260d0: 2866 662c 4d29 3b20 2020 7c0a 2b2d 2d2d  (ff,M);   |.+---
│ │ │ │ +000260a0: 2d2d 2d2d 2d2b 0a7c 6939 203a 204d 4620  -----+.|i9 : MF 
│ │ │ │ +000260b0: 3d20 6d61 7472 6978 4661 6374 6f72 697a  = matrixFactoriz
│ │ │ │ +000260c0: 6174 696f 6e28 6666 2c4d 293b 2020 207c  ation(ff,M);   |
│ │ │ │ +000260d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000260e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000260f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026100: 2d2d 2d2d 2b0a 7c69 3130 203a 2062 6574  ----+.|i10 : bet
│ │ │ │ -00026110: 7469 2072 6573 2070 7573 6846 6f72 7761  ti res pushForwa
│ │ │ │ -00026120: 7264 286d 6170 2852 2c53 292c 4d29 7c0a  rd(map(R,S),M)|.
│ │ │ │ -00026130: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00026140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00026160: 2020 2020 2020 2020 3020 3120 3220 2020          0 1 2   
│ │ │ │ -00026170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026180: 2020 7c0a 7c6f 3130 203d 2074 6f74 616c    |.|o10 = total
│ │ │ │ -00026190: 3a20 3320 3520 3220 2020 2020 2020 2020  : 3 5 2         
│ │ │ │ -000261a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000261b0: 2020 2020 2020 2020 2032 3a20 3320 3420           2: 3 4 
│ │ │ │ -000261c0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -000261d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000261e0: 2020 2033 3a20 2e20 3120 3220 2020 2020     3: . 1 2     
│ │ │ │ -000261f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026200: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00026210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026220: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -00026230: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ -00026240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026250: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000260f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ +00026100: 3a20 6265 7474 6920 7265 7320 7075 7368  : betti res push
│ │ │ │ +00026110: 466f 7277 6172 6428 6d61 7028 522c 5329  Forward(map(R,S)
│ │ │ │ +00026120: 2c4d 297c 0a7c 2020 2020 2020 2020 2020  ,M)|.|          
│ │ │ │ +00026130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00026150: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ +00026160: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00026170: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +00026180: 746f 7461 6c3a 2033 2035 2032 2020 2020  total: 3 5 2    
│ │ │ │ +00026190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000261a0: 207c 0a7c 2020 2020 2020 2020 2020 323a   |.|          2:
│ │ │ │ +000261b0: 2033 2034 202e 2020 2020 2020 2020 2020   3 4 .          
│ │ │ │ +000261c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000261d0: 2020 2020 2020 2020 333a 202e 2031 2032          3: . 1 2
│ │ │ │ +000261e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000261f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026220: 0a7c 6f31 3020 3a20 4265 7474 6954 616c  .|o10 : BettiTal
│ │ │ │ +00026230: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ +00026240: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00026250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00026280: 7c69 3131 203a 2066 696e 6974 6542 6574  |i11 : finiteBet
│ │ │ │ -00026290: 7469 4e75 6d62 6572 7320 4d46 2020 2020  tiNumbers MF    
│ │ │ │ -000262a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026270: 2d2d 2d2b 0a7c 6931 3120 3a20 6669 6e69  ---+.|i11 : fini
│ │ │ │ +00026280: 7465 4265 7474 694e 756d 6265 7273 204d  teBettiNumbers M
│ │ │ │ +00026290: 4620 2020 2020 2020 2020 2020 207c 0a7c  F            |.|
│ │ │ │ +000262a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000262b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262d0: 2020 7c0a 7c6f 3131 203d 207b 332c 2035    |.|o11 = {3, 5
│ │ │ │ -000262e0: 2c20 327d 2020 2020 2020 2020 2020 2020  , 2}            
│ │ │ │ -000262f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000262c0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +000262d0: 7b33 2c20 352c 2032 7d20 2020 2020 2020  {3, 5, 2}       
│ │ │ │ +000262e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000262f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026320: 2020 2020 2020 7c0a 7c6f 3131 203a 204c        |.|o11 : L
│ │ │ │ -00026330: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00026340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026350: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00026360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026370: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -00026380: 203a 2069 6e66 696e 6974 6542 6574 7469   : infiniteBetti
│ │ │ │ -00026390: 4e75 6d62 6572 7328 4d46 2c35 2920 2020  Numbers(MF,5)   
│ │ │ │ -000263a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026310: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00026320: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │ +00026330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026340: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00026350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00026370: 0a7c 6931 3220 3a20 696e 6669 6e69 7465  .|i12 : infinite
│ │ │ │ +00026380: 4265 7474 694e 756d 6265 7273 284d 462c  BettiNumbers(MF,
│ │ │ │ +00026390: 3529 2020 2020 2020 207c 0a7c 2020 2020  5)       |.|    
│ │ │ │ +000263a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000263d0: 7c6f 3132 203d 207b 332c 2034 2c20 352c  |o12 = {3, 4, 5,
│ │ │ │ -000263e0: 2036 2c20 372c 2038 7d20 2020 2020 2020   6, 7, 8}       
│ │ │ │ -000263f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000263c0: 2020 207c 0a7c 6f31 3220 3d20 7b33 2c20     |.|o12 = {3, 
│ │ │ │ +000263d0: 342c 2035 2c20 362c 2037 2c20 387d 2020  4, 5, 6, 7, 8}  
│ │ │ │ +000263e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000263f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026420: 2020 7c0a 7c6f 3132 203a 204c 6973 7420    |.|o12 : List 
│ │ │ │ +00026410: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ +00026420: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │  00026430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026440: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00026440: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00026450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026470: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2062  ------+.|i13 : b
│ │ │ │ -00026480: 6574 7469 2072 6573 2028 4d2c 204c 656e  etti res (M, Len
│ │ │ │ -00026490: 6774 684c 696d 6974 203d 3e20 3529 2020  gthLimit => 5)  
│ │ │ │ -000264a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000264d0: 2020 2020 2020 2020 2020 3020 3120 3220            0 1 2 
│ │ │ │ -000264e0: 3320 3420 3520 2020 2020 2020 2020 2020  3 4 5           
│ │ │ │ -000264f0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -00026500: 616c 3a20 3320 3420 3520 3620 3720 3820  al: 3 4 5 6 7 8 
│ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00026520: 7c20 2020 2020 2020 2020 2032 3a20 3320  |          2: 3 
│ │ │ │ -00026530: 3420 3520 3620 3720 3820 2020 2020 2020  4 5 6 7 8       
│ │ │ │ -00026540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00026470: 3320 3a20 6265 7474 6920 7265 7320 284d  3 : betti res (M
│ │ │ │ +00026480: 2c20 4c65 6e67 7468 4c69 6d69 7420 3d3e  , LengthLimit =>
│ │ │ │ +00026490: 2035 2920 207c 0a7c 2020 2020 2020 2020   5)  |.|        
│ │ │ │ +000264a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000264b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000264c0: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ +000264d0: 2031 2032 2033 2034 2035 2020 2020 2020   1 2 3 4 5      
│ │ │ │ +000264e0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ +000264f0: 3d20 746f 7461 6c3a 2033 2034 2035 2036  = total: 3 4 5 6
│ │ │ │ +00026500: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ +00026510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026520: 323a 2033 2034 2035 2036 2037 2038 2020  2: 3 4 5 6 7 8  
│ │ │ │ +00026530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00026540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026570: 2020 7c0a 7c6f 3133 203a 2042 6574 7469    |.|o13 : Betti
│ │ │ │ -00026580: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ -00026590: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00026560: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ +00026570: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00026580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026590: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000265a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000265c0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ -000265d0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -000265e0: 2a6e 6f74 6520 6d61 7472 6978 4661 6374  *note matrixFact
│ │ │ │ -000265f0: 6f72 697a 6174 696f 6e3a 206d 6174 7269  orization: matri
│ │ │ │ -00026600: 7846 6163 746f 7269 7a61 7469 6f6e 2c20  xFactorization, 
│ │ │ │ -00026610: 2d2d 204d 6170 7320 696e 2061 2068 6967  -- Maps in a hig
│ │ │ │ -00026620: 6865 720a 2020 2020 636f 6469 6d65 6e73  her.    codimens
│ │ │ │ -00026630: 696f 6e20 6d61 7472 6978 2066 6163 746f  ion matrix facto
│ │ │ │ -00026640: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f  rization.  * *no
│ │ │ │ -00026650: 7465 2069 6e66 696e 6974 6542 6574 7469  te infiniteBetti
│ │ │ │ -00026660: 4e75 6d62 6572 733a 2069 6e66 696e 6974  Numbers: infinit
│ │ │ │ -00026670: 6542 6574 7469 4e75 6d62 6572 732c 202d  eBettiNumbers, -
│ │ │ │ -00026680: 2d20 6265 7474 6920 6e75 6d62 6572 7320  - betti numbers 
│ │ │ │ -00026690: 6f66 0a20 2020 2066 696e 6974 6520 7265  of.    finite re
│ │ │ │ -000266a0: 736f 6c75 7469 6f6e 2063 6f6d 7075 7465  solution compute
│ │ │ │ -000266b0: 6420 6672 6f6d 2061 206d 6174 7269 7820  d from a matrix 
│ │ │ │ -000266c0: 6661 6374 6f72 697a 6174 696f 6e0a 0a57  factorization..W
│ │ │ │ -000266d0: 6179 7320 746f 2075 7365 2066 696e 6974  ays to use finit
│ │ │ │ -000266e0: 6542 6574 7469 4e75 6d62 6572 733a 0a3d  eBettiNumbers:.=
│ │ │ │ +000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +000265c0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +000265d0: 0a20 202a 202a 6e6f 7465 206d 6174 7269  .  * *note matri
│ │ │ │ +000265e0: 7846 6163 746f 7269 7a61 7469 6f6e 3a20  xFactorization: 
│ │ │ │ +000265f0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +00026600: 696f 6e2c 202d 2d20 4d61 7073 2069 6e20  ion, -- Maps in 
│ │ │ │ +00026610: 6120 6869 6768 6572 0a20 2020 2063 6f64  a higher.    cod
│ │ │ │ +00026620: 696d 656e 7369 6f6e 206d 6174 7269 7820  imension matrix 
│ │ │ │ +00026630: 6661 6374 6f72 697a 6174 696f 6e0a 2020  factorization.  
│ │ │ │ +00026640: 2a20 2a6e 6f74 6520 696e 6669 6e69 7465  * *note infinite
│ │ │ │ +00026650: 4265 7474 694e 756d 6265 7273 3a20 696e  BettiNumbers: in
│ │ │ │ +00026660: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ +00026670: 7273 2c20 2d2d 2062 6574 7469 206e 756d  rs, -- betti num
│ │ │ │ +00026680: 6265 7273 206f 660a 2020 2020 6669 6e69  bers of.    fini
│ │ │ │ +00026690: 7465 2072 6573 6f6c 7574 696f 6e20 636f  te resolution co
│ │ │ │ +000266a0: 6d70 7574 6564 2066 726f 6d20 6120 6d61  mputed from a ma
│ │ │ │ +000266b0: 7472 6978 2066 6163 746f 7269 7a61 7469  trix factorizati
│ │ │ │ +000266c0: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520  on..Ways to use 
│ │ │ │ +000266d0: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ +000266e0: 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  rs:.============
│ │ │ │  000266f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00026700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00026710: 2020 2a20 2266 696e 6974 6542 6574 7469    * "finiteBetti
│ │ │ │ -00026720: 4e75 6d62 6572 7328 4c69 7374 2922 0a0a  Numbers(List)"..
│ │ │ │ -00026730: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00026740: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00026750: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00026760: 7420 2a6e 6f74 6520 6669 6e69 7465 4265  t *note finiteBe
│ │ │ │ -00026770: 7474 694e 756d 6265 7273 3a20 6669 6e69  ttiNumbers: fini
│ │ │ │ -00026780: 7465 4265 7474 694e 756d 6265 7273 2c20  teBettiNumbers, 
│ │ │ │ -00026790: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -000267a0: 640a 6675 6e63 7469 6f6e 3a20 284d 6163  d.function: (Mac
│ │ │ │ -000267b0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -000267c0: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ -000267d0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -000267e0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -000267f0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2066  ns.info, Node: f
│ │ │ │ -00026800: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ -00026810: 6e64 2c20 4e65 7874 3a20 4772 6164 696e  nd, Next: Gradin
│ │ │ │ -00026820: 672c 2050 7265 763a 2066 696e 6974 6542  g, Prev: finiteB
│ │ │ │ -00026830: 6574 7469 4e75 6d62 6572 732c 2055 703a  ettiNumbers, Up:
│ │ │ │ -00026840: 2054 6f70 0a0a 6672 6565 4578 7465 7269   Top..freeExteri
│ │ │ │ -00026850: 6f72 5375 6d6d 616e 6420 2d2d 2066 696e  orSummand -- fin
│ │ │ │ -00026860: 6420 7468 6520 6672 6565 2073 756d 6d61  d the free summa
│ │ │ │ -00026870: 6e64 7320 6f66 2061 206d 6f64 756c 6520  nds of a module 
│ │ │ │ -00026880: 6f76 6572 2061 6e20 6578 7465 7269 6f72  over an exterior
│ │ │ │ -00026890: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a   algebra.*******
│ │ │ │ +00026700: 3d3d 3d0a 0a20 202a 2022 6669 6e69 7465  ===..  * "finite
│ │ │ │ +00026710: 4265 7474 694e 756d 6265 7273 284c 6973  BettiNumbers(Lis
│ │ │ │ +00026720: 7429 220a 0a46 6f72 2074 6865 2070 726f  t)"..For the pro
│ │ │ │ +00026730: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +00026740: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00026750: 6f62 6a65 6374 202a 6e6f 7465 2066 696e  object *note fin
│ │ │ │ +00026760: 6974 6542 6574 7469 4e75 6d62 6572 733a  iteBettiNumbers:
│ │ │ │ +00026770: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ +00026780: 6572 732c 2069 7320 6120 2a6e 6f74 6520  ers, is a *note 
│ │ │ │ +00026790: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a  method.function:
│ │ │ │ +000267a0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +000267b0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +000267c0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +000267d0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +000267e0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +000267f0: 6465 3a20 6672 6565 4578 7465 7269 6f72  de: freeExterior
│ │ │ │ +00026800: 5375 6d6d 616e 642c 204e 6578 743a 2047  Summand, Next: G
│ │ │ │ +00026810: 7261 6469 6e67 2c20 5072 6576 3a20 6669  rading, Prev: fi
│ │ │ │ +00026820: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ +00026830: 2c20 5570 3a20 546f 700a 0a66 7265 6545  , Up: Top..freeE
│ │ │ │ +00026840: 7874 6572 696f 7253 756d 6d61 6e64 202d  xteriorSummand -
│ │ │ │ +00026850: 2d20 6669 6e64 2074 6865 2066 7265 6520  - find the free 
│ │ │ │ +00026860: 7375 6d6d 616e 6473 206f 6620 6120 6d6f  summands of a mo
│ │ │ │ +00026870: 6475 6c65 206f 7665 7220 616e 2065 7874  dule over an ext
│ │ │ │ +00026880: 6572 696f 7220 616c 6765 6272 610a 2a2a  erior algebra.**
│ │ │ │ +00026890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000268a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000268b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000268c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000268d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000268e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ -000268f0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ -00026900: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00026910: 2020 2020 4620 3d20 6672 6565 4578 7465      F = freeExte
│ │ │ │ -00026920: 7269 6f72 5375 6d6d 616e 6420 4d0a 2020  riorSummand M.  
│ │ │ │ -00026930: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00026940: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64  * M, a *note mod
│ │ │ │ -00026950: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -00026960: 6f63 294d 6f64 756c 652c 2c20 6f76 6572  oc)Module,, over
│ │ │ │ -00026970: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ -00026980: 6562 7261 0a20 202a 204f 7574 7075 7473  ebra.  * Outputs
│ │ │ │ -00026990: 3a0a 2020 2020 2020 2a20 462c 2061 202a  :.      * F, a *
│ │ │ │ -000269a0: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -000269b0: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -000269c0: 782c 2c20 4d61 7020 6672 6f6d 2061 2066  x,, Map from a f
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│ │ │ │ +00026980: 7470 7574 733a 0a20 2020 2020 202a 2046  tputs:.      * F
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│ │ │ │  00026a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00026a60: 203a 206b 6b3d 205a 5a2f 3130 3120 2020   : kk= ZZ/101   
│ │ │ │ +00026a50: 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a 2f31  +.|i1 : kk= ZZ/1
│ │ │ │ +00026a60: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │  00026a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026a80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00026a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ac0: 2020 2020 2020 207c 0a7c 6f31 203d 206b         |.|o1 = k
│ │ │ │ -00026ad0: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00026ac0: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +00026ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00026af0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b30: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ -00026b40: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
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│ │ │ │ -00026b60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
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│ │ │ │ +00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026b60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00026b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
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│ │ │ │ +00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00026c10: 203d 2045 2020 2020 2020 2020 2020 2020   = E            
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│ │ │ │ -00026c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00026d00: 7b2d 317d 2020 2020 2020 2020 2020 7c0a  {-1}          |.
│ │ │ │ +00026d10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -00026dc0: 2020 2020 2020 2020 207b 317d 207c 2030           {1} | 0
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│ │ │ │  00026ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00026f60: 7c0a 7c20 2020 2020 7b31 7d20 7c20 3020  |.|     {1} | 0 
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│ │ │ │ +00027190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
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│ │ │ │ +000271b0: 2066 7265 6545 7874 6572 696f 7253 756d   freeExteriorSum
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│ │ │ │ +000271e0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
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│ │ │ │ +00027250: 672c 204e 6578 743a 2068 662c 2050 7265  g, Next: hf, Pre
│ │ │ │ +00027260: 763a 2066 7265 6545 7874 6572 696f 7253  v: freeExteriorS
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│ │ │ │ +00027280: 0a47 7261 6469 6e67 202d 2d20 4f70 7469  .Grading -- Opti
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│ │ │ │ +000272b0: 4578 740a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  Ext.************
│ │ │ │  000272c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000272d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000272e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000272f0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -00027300: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -00027310: 200a 2020 2020 2020 2020 4569 7365 6e62   .        Eisenb
│ │ │ │ -00027320: 7564 5368 616d 6173 6854 6f74 616c 284d  udShamashTotal(M
│ │ │ │ -00027330: 6261 722c 4772 6164 696e 6720 3d3e 2032  bar,Grading => 2
│ │ │ │ -00027340: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -00027350: 2020 2020 2a20 4368 6563 6b2c 2061 6e20      * Check, an 
│ │ │ │ -00027360: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ -00027370: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ -00027380: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a  , ..Description.
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│ │ │ │ -000273a0: 4772 6164 696e 6720 3d3e 312c 2074 6865  Grading =>1, the
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│ │ │ │ -000273e0: 6675 6c20 696e 2074 6865 0a70 6163 6b61  ful in the.packa
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│ │ │ │ -00027410: 2020 2a20 2a6e 6f74 6520 4569 7365 6e62    * *note Eisenb
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│ │ │ │ -00027430: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
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│ │ │ │ -00027460: 2020 746f 7461 6c20 4578 740a 2020 2a20    total Ext.  * 
│ │ │ │ -00027470: 2a6e 6f74 6520 6e65 7745 7874 3a20 6e65  *note newExt: ne
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│ │ │ │ -000274b0: 2020 2020 496e 7465 7273 6563 7469 6f6e      Intersection
│ │ │ │ -000274c0: 0a0a 4675 6e63 7469 6f6e 7320 7769 7468  ..Functions with
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│ │ │ │ -000274f0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +000272e0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +000272f0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +00027300: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ +00027310: 6973 656e 6275 6453 6861 6d61 7368 546f  isenbudShamashTo
│ │ │ │ +00027320: 7461 6c28 4d62 6172 2c47 7261 6469 6e67  tal(Mbar,Grading
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│ │ │ │ +00027380: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00027390: 0a0a 6966 2047 7261 6469 6e67 203d 3e31  ..if Grading =>1
│ │ │ │ +000273a0: 2c20 7468 656e 2074 6865 206f 7574 7075  , then the outpu
│ │ │ │ +000273b0: 7420 6973 2063 6f6e 7665 7274 6564 2074  t is converted t
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│ │ │ │ +000273d0: 2c20 7573 6566 756c 2069 6e20 7468 650a  , useful in the.
│ │ │ │ +000273e0: 7061 636b 6167 6520 436c 6966 666f 7264  package Clifford
│ │ │ │ +000273f0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +00027400: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2045  ===..  * *note E
│ │ │ │ +00027410: 6973 656e 6275 6453 6861 6d61 7368 546f  isenbudShamashTo
│ │ │ │ +00027420: 7461 6c3a 2045 6973 656e 6275 6453 6861  tal: EisenbudSha
│ │ │ │ +00027430: 6d61 7368 546f 7461 6c2c 202d 2d20 5072  mashTotal, -- Pr
│ │ │ │ +00027440: 6563 7572 736f 7220 636f 6d70 6c65 7820  ecursor complex 
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│ │ │ │ +000274e0: 6164 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d  ading:.=========
│ │ │ │ +000274f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00027500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027520: 3d0a 0a20 202a 2022 4569 7365 6e62 7564  =..  * "Eisenbud
│ │ │ │ -00027530: 5368 616d 6173 6854 6f74 616c 282e 2e2e  ShamashTotal(...
│ │ │ │ -00027540: 2c47 7261 6469 6e67 3d3e 2e2e 2e29 2220  ,Grading=>...)" 
│ │ │ │ -00027550: 2d2d 2073 6565 202a 6e6f 7465 2045 6973  -- see *note Eis
│ │ │ │ -00027560: 656e 6275 6453 6861 6d61 7368 546f 7461  enbudShamashTota
│ │ │ │ -00027570: 6c3a 0a20 2020 2045 6973 656e 6275 6453  l:.    EisenbudS
│ │ │ │ -00027580: 6861 6d61 7368 546f 7461 6c2c 202d 2d20  hamashTotal, -- 
│ │ │ │ -00027590: 5072 6563 7572 736f 7220 636f 6d70 6c65  Precursor comple
│ │ │ │ -000275a0: 7820 6f66 2074 6f74 616c 2045 7874 0a20  x of total Ext. 
│ │ │ │ -000275b0: 202a 2022 6e65 7745 7874 282e 2e2e 2c47   * "newExt(...,G
│ │ │ │ -000275c0: 7261 6469 6e67 3d3e 2e2e 2e29 2220 2d2d  rading=>...)" --
│ │ │ │ -000275d0: 2073 6565 202a 6e6f 7465 206e 6577 4578   see *note newEx
│ │ │ │ -000275e0: 743a 206e 6577 4578 742c 202d 2d20 476c  t: newExt, -- Gl
│ │ │ │ -000275f0: 6f62 616c 2045 7874 2066 6f72 0a20 2020  obal Ext for.   
│ │ │ │ -00027600: 206d 6f64 756c 6573 206f 7665 7220 6120   modules over a 
│ │ │ │ -00027610: 636f 6d70 6c65 7465 2049 6e74 6572 7365  complete Interse
│ │ │ │ -00027620: 6374 696f 6e0a 0a46 6f72 2074 6865 2070  ction..For the p
│ │ │ │ -00027630: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00027640: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00027650: 6520 6f62 6a65 6374 202a 6e6f 7465 2047  e object *note G
│ │ │ │ -00027660: 7261 6469 6e67 3a20 4772 6164 696e 672c  rading: Grading,
│ │ │ │ -00027670: 2069 7320 6120 2a6e 6f74 6520 7379 6d62   is a *note symb
│ │ │ │ -00027680: 6f6c 3a20 284d 6163 6175 6c61 7932 446f  ol: (Macaulay2Do
│ │ │ │ -00027690: 6329 5379 6d62 6f6c 2c2e 0a1f 0a46 696c  c)Symbol,....Fil
│ │ │ │ -000276a0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -000276b0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -000276c0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2068  ns.info, Node: h
│ │ │ │ -000276d0: 662c 204e 6578 743a 2068 664d 6f64 756c  f, Next: hfModul
│ │ │ │ -000276e0: 6541 7345 7874 2c20 5072 6576 3a20 4772  eAsExt, Prev: Gr
│ │ │ │ -000276f0: 6164 696e 672c 2055 703a 2054 6f70 0a0a  ading, Up: Top..
│ │ │ │ -00027700: 6866 202d 2d20 436f 6d70 7574 6573 2074  hf -- Computes t
│ │ │ │ -00027710: 6865 2068 696c 6265 7274 2066 756e 6374  he hilbert funct
│ │ │ │ -00027720: 696f 6e20 696e 2061 2072 616e 6765 206f  ion in a range o
│ │ │ │ -00027730: 6620 6465 6772 6565 730a 2a2a 2a2a 2a2a  f degrees.******
│ │ │ │ +00027510: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 6973  ======..  * "Eis
│ │ │ │ +00027520: 656e 6275 6453 6861 6d61 7368 546f 7461  enbudShamashTota
│ │ │ │ +00027530: 6c28 2e2e 2e2c 4772 6164 696e 673d 3e2e  l(...,Grading=>.
│ │ │ │ +00027540: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ +00027550: 6520 4569 7365 6e62 7564 5368 616d 6173  e EisenbudShamas
│ │ │ │ +00027560: 6854 6f74 616c 3a0a 2020 2020 4569 7365  hTotal:.    Eise
│ │ │ │ +00027570: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ +00027580: 2c20 2d2d 2050 7265 6375 7273 6f72 2063  , -- Precursor c
│ │ │ │ +00027590: 6f6d 706c 6578 206f 6620 746f 7461 6c20  omplex of total 
│ │ │ │ +000275a0: 4578 740a 2020 2a20 226e 6577 4578 7428  Ext.  * "newExt(
│ │ │ │ +000275b0: 2e2e 2e2c 4772 6164 696e 673d 3e2e 2e2e  ...,Grading=>...
│ │ │ │ +000275c0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ +000275d0: 6e65 7745 7874 3a20 6e65 7745 7874 2c20  newExt: newExt, 
│ │ │ │ +000275e0: 2d2d 2047 6c6f 6261 6c20 4578 7420 666f  -- Global Ext fo
│ │ │ │ +000275f0: 720a 2020 2020 6d6f 6475 6c65 7320 6f76  r.    modules ov
│ │ │ │ +00027600: 6572 2061 2063 6f6d 706c 6574 6520 496e  er a complete In
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│ │ │ │ +00027620: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00027630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00027650: 6f74 6520 4772 6164 696e 673a 2047 7261  ote Grading: Gra
│ │ │ │ +00027660: 6469 6e67 2c20 6973 2061 202a 6e6f 7465  ding, is a *note
│ │ │ │ +00027670: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
│ │ │ │ +00027680: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a  ay2Doc)Symbol,..
│ │ │ │ +00027690: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +000276a0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
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│ │ │ │ +000276c0: 6465 3a20 6866 2c20 4e65 7874 3a20 6866  de: hf, Next: hf
│ │ │ │ +000276d0: 4d6f 6475 6c65 4173 4578 742c 2050 7265  ModuleAsExt, Pre
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│ │ │ │ +000276f0: 546f 700a 0a68 6620 2d2d 2043 6f6d 7075  Top..hf -- Compu
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│ │ │ │ +00027710: 6675 6e63 7469 6f6e 2069 6e20 6120 7261  function in a ra
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│ │ │ │ +00027730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027770: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -00027780: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ -00027790: 653a 200a 2020 2020 2020 2020 4820 3d20  e: .        H = 
│ │ │ │ -000277a0: 6866 2873 2c50 290a 2020 2a20 496e 7075  hf(s,P).  * Inpu
│ │ │ │ -000277b0: 7473 3a0a 2020 2020 2020 2a20 732c 2061  ts:.      * s, a
│ │ │ │ -000277c0: 202a 6e6f 7465 2073 6571 7565 6e63 653a   *note sequence:
│ │ │ │ -000277d0: 2028 4d61 6361 756c 6179 3244 6f63 2953   (Macaulay2Doc)S
│ │ │ │ -000277e0: 6571 7565 6e63 652c 2c20 6f72 204c 6973  equence,, or Lis
│ │ │ │ -000277f0: 740a 2020 2020 2020 2a20 502c 2061 202a  t.      * P, a *
│ │ │ │ -00027800: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00027810: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00027820: 652c 2c20 6772 6164 6564 206d 6f64 756c  e,, graded modul
│ │ │ │ -00027830: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ -00027840: 2020 2020 202a 2048 2c20 6120 2a6e 6f74       * H, a *not
│ │ │ │ -00027850: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ -00027860: 7932 446f 6329 4c69 7374 2c2c 200a 0a57  y2Doc)List,, ..W
│ │ │ │ -00027870: 6179 7320 746f 2075 7365 2068 663a 0a3d  ays to use hf:.=
│ │ │ │ -00027880: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00027890: 2020 2a20 2268 6628 4c69 7374 2c4d 6f64    * "hf(List,Mod
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│ │ │ │ -000278b0: 7175 656e 6365 2c4d 6f64 756c 6529 220a  quence,Module)".
│ │ │ │ -000278c0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -000278d0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -000278e0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
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│ │ │ │ -00027900: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -00027910: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ -00027920: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00027930: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ -00027940: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -00027950: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -00027960: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ -00027970: 6866 4d6f 6475 6c65 4173 4578 742c 204e  hfModuleAsExt, N
│ │ │ │ -00027980: 6578 743a 2068 6967 6853 797a 7967 792c  ext: highSyzygy,
│ │ │ │ -00027990: 2050 7265 763a 2068 662c 2055 703a 2054   Prev: hf, Up: T
│ │ │ │ -000279a0: 6f70 0a0a 6866 4d6f 6475 6c65 4173 4578  op..hfModuleAsEx
│ │ │ │ -000279b0: 7420 2d2d 2070 7265 6469 6374 2062 6574  t -- predict bet
│ │ │ │ -000279c0: 7469 206e 756d 6265 7273 206f 6620 6d6f  ti numbers of mo
│ │ │ │ -000279d0: 6475 6c65 4173 4578 7428 4d2c 5229 0a2a  duleAsExt(M,R).*
│ │ │ │ +00027760: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ +00027770: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ +00027780: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00027790: 2048 203d 2068 6628 732c 5029 0a20 202a   H = hf(s,P).  *
│ │ │ │ +000277a0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +000277b0: 2073 2c20 6120 2a6e 6f74 6520 7365 7175   s, a *note sequ
│ │ │ │ +000277c0: 656e 6365 3a20 284d 6163 6175 6c61 7932  ence: (Macaulay2
│ │ │ │ +000277d0: 446f 6329 5365 7175 656e 6365 2c2c 206f  Doc)Sequence,, o
│ │ │ │ +000277e0: 7220 4c69 7374 0a20 2020 2020 202a 2050  r List.      * P
│ │ │ │ +000277f0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +00027800: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00027810: 4d6f 6475 6c65 2c2c 2067 7261 6465 6420  Module,, graded 
│ │ │ │ +00027820: 6d6f 6475 6c65 0a20 202a 204f 7574 7075  module.  * Outpu
│ │ │ │ +00027830: 7473 3a0a 2020 2020 2020 2a20 482c 2061  ts:.      * H, a
│ │ │ │ +00027840: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ +00027850: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ +00027860: 2c20 0a0a 5761 7973 2074 6f20 7573 6520  , ..Ways to use 
│ │ │ │ +00027870: 6866 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  hf:.============
│ │ │ │ +00027880: 3d3d 3d0a 0a20 202a 2022 6866 284c 6973  ===..  * "hf(Lis
│ │ │ │ +00027890: 742c 4d6f 6475 6c65 2922 0a20 202a 2022  t,Module)".  * "
│ │ │ │ +000278a0: 6866 2853 6571 7565 6e63 652c 4d6f 6475  hf(Sequence,Modu
│ │ │ │ +000278b0: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ +000278c0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +000278d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +000278e0: 206f 626a 6563 7420 2a6e 6f74 6520 6866   object *note hf
│ │ │ │ +000278f0: 3a20 6866 2c20 6973 2061 202a 6e6f 7465  : hf, is a *note
│ │ │ │ +00027900: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00027910: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00027920: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +00027930: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00027940: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +00027950: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +00027960: 6f64 653a 2068 664d 6f64 756c 6541 7345  ode: hfModuleAsE
│ │ │ │ +00027970: 7874 2c20 4e65 7874 3a20 6869 6768 5379  xt, Next: highSy
│ │ │ │ +00027980: 7a79 6779 2c20 5072 6576 3a20 6866 2c20  zygy, Prev: hf, 
│ │ │ │ +00027990: 5570 3a20 546f 700a 0a68 664d 6f64 756c  Up: Top..hfModul
│ │ │ │ +000279a0: 6541 7345 7874 202d 2d20 7072 6564 6963  eAsExt -- predic
│ │ │ │ +000279b0: 7420 6265 7474 6920 6e75 6d62 6572 7320  t betti numbers 
│ │ │ │ +000279c0: 6f66 206d 6f64 756c 6541 7345 7874 284d  of moduleAsExt(M
│ │ │ │ +000279d0: 2c52 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ,R).************
│ │ │ │  000279e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000279f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027a10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ -00027a20: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ -00027a30: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -00027a40: 2020 7365 7120 3d20 6866 4d6f 6475 6c65    seq = hfModule
│ │ │ │ -00027a50: 4173 4578 7428 6e75 6d56 616c 7565 732c  AsExt(numValues,
│ │ │ │ -00027a60: 4d2c 6e75 6d67 656e 7352 290a 2020 2a20  M,numgensR).  * 
│ │ │ │ -00027a70: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00027a80: 6e75 6d56 616c 7565 732c 2061 6e20 2a6e  numValues, an *n
│ │ │ │ -00027a90: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -00027aa0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -00027ab0: 6e75 6d62 6572 206f 6620 7661 6c75 6573  number of values
│ │ │ │ -00027ac0: 2074 6f0a 2020 2020 2020 2020 636f 6d70   to.        comp
│ │ │ │ -00027ad0: 7574 650a 2020 2020 2020 2a20 4d2c 2061  ute.      * M, a
│ │ │ │ -00027ae0: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ -00027af0: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ -00027b00: 756c 652c 2c20 6d6f 6475 6c65 206f 7665  ule,, module ove
│ │ │ │ -00027b10: 7220 7468 6520 7269 6e67 206f 660a 2020  r the ring of.  
│ │ │ │ -00027b20: 2020 2020 2020 6f70 6572 6174 6f72 730a        operators.
│ │ │ │ -00027b30: 2020 2020 2020 2a20 6e75 6d67 656e 7352        * numgensR
│ │ │ │ -00027b40: 2c20 616e 202a 6e6f 7465 2069 6e74 6567  , an *note integ
│ │ │ │ -00027b50: 6572 3a20 284d 6163 6175 6c61 7932 446f  er: (Macaulay2Do
│ │ │ │ -00027b60: 6329 5a5a 2c2c 206e 756d 6265 7220 6f66  c)ZZ,, number of
│ │ │ │ -00027b70: 2067 656e 6572 6174 6f72 7320 6f66 0a20   generators of. 
│ │ │ │ -00027b80: 2020 2020 2020 2074 6865 2074 6172 6765         the targe
│ │ │ │ -00027b90: 7420 7269 6e67 0a20 202a 204f 7574 7075  t ring.  * Outpu
│ │ │ │ -00027ba0: 7473 3a0a 2020 2020 2020 2a20 7365 712c  ts:.      * seq,
│ │ │ │ -00027bb0: 2061 202a 6e6f 7465 2073 6571 7565 6e63   a *note sequenc
│ │ │ │ -00027bc0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00027bd0: 2953 6571 7565 6e63 652c 2c20 7365 7175  )Sequence,, sequ
│ │ │ │ -00027be0: 656e 6365 206f 6620 6e75 6d56 616c 7565  ence of numValue
│ │ │ │ -00027bf0: 730a 2020 2020 2020 2020 696e 7465 6765  s.        intege
│ │ │ │ -00027c00: 7273 2c20 7468 6520 6578 7065 6374 6564  rs, the expected
│ │ │ │ -00027c10: 2074 6f74 616c 2042 6574 7469 206e 756d   total Betti num
│ │ │ │ -00027c20: 6265 7273 0a0a 4465 7363 7269 7074 696f  bers..Descriptio
│ │ │ │ -00027c30: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a47  n.===========..G
│ │ │ │ -00027c40: 6976 656e 2061 206d 6f64 756c 6520 4d20  iven a module M 
│ │ │ │ -00027c50: 6f76 6572 2074 6865 2072 696e 6720 6f66  over the ring of
│ │ │ │ -00027c60: 206f 7065 7261 746f 7273 2024 6b5b 785f   operators $k[x_
│ │ │ │ -00027c70: 312e 2e78 5f63 5d24 2c20 7468 6520 6361  1..x_c]$, the ca
│ │ │ │ -00027c80: 6c6c 2024 4e20 3d0a 6d6f 6475 6c65 4173  ll $N =.moduleAs
│ │ │ │ -00027c90: 4578 7428 4d2c 5229 2420 7072 6f64 7563  Ext(M,R)$ produc
│ │ │ │ -00027ca0: 6573 2061 206d 6f64 756c 6520 4e20 6f76  es a module N ov
│ │ │ │ -00027cb0: 6572 2074 6865 2072 696e 6720 5220 7768  er the ring R wh
│ │ │ │ -00027cc0: 6f73 6520 6578 7420 6d6f 6475 6c65 2069  ose ext module i
│ │ │ │ -00027cd0: 7320 7468 650a 6578 7465 7269 6f72 2061  s the.exterior a
│ │ │ │ -00027ce0: 6c67 6562 7261 206f 6e20 6e3d 6e75 6d67  lgebra on n=numg
│ │ │ │ -00027cf0: 656e 7352 2067 656e 6572 6174 6f72 7320  ensR generators 
│ │ │ │ -00027d00: 7465 6e73 6f72 6564 2077 6974 6820 4d2e  tensored with M.
│ │ │ │ -00027d10: 2054 6869 7320 7363 7269 7074 2063 6f6d   This script com
│ │ │ │ -00027d20: 7075 7465 730a 6e75 6d56 616c 7565 7320  putes.numValues 
│ │ │ │ -00027d30: 7661 6c75 6573 206f 6620 7468 6520 4869  values of the Hi
│ │ │ │ -00027d40: 6c62 6572 7420 6675 6e63 7469 6f6e 206f  lbert function o
│ │ │ │ -00027d50: 6620 2424 204d 205c 6f74 696d 6573 205c  f $$ M \otimes \
│ │ │ │ -00027d60: 7765 6467 6520 6b5e 6e2c 2024 2420 7768  wedge k^n, $$ wh
│ │ │ │ -00027d70: 6963 680a 7368 6f75 6c64 2062 6520 6571  ich.should be eq
│ │ │ │ -00027d80: 7561 6c20 746f 2074 6865 2074 6f74 616c  ual to the total
│ │ │ │ -00027d90: 2062 6574 7469 206e 756d 6265 7273 206f   betti numbers o
│ │ │ │ -00027da0: 6620 4e2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  f N...+---------
│ │ │ │ +00027a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00027a10: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ +00027a20: 3d0a 0a20 202a 2055 7361 6765 3a20 0a20  =..  * Usage: . 
│ │ │ │ +00027a30: 2020 2020 2020 2073 6571 203d 2068 664d         seq = hfM
│ │ │ │ +00027a40: 6f64 756c 6541 7345 7874 286e 756d 5661  oduleAsExt(numVa
│ │ │ │ +00027a50: 6c75 6573 2c4d 2c6e 756d 6765 6e73 5229  lues,M,numgensR)
│ │ │ │ +00027a60: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00027a70: 2020 202a 206e 756d 5661 6c75 6573 2c20     * numValues, 
│ │ │ │ +00027a80: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +00027a90: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00027aa0: 5a5a 2c2c 206e 756d 6265 7220 6f66 2076  ZZ,, number of v
│ │ │ │ +00027ab0: 616c 7565 7320 746f 0a20 2020 2020 2020  alues to.       
│ │ │ │ +00027ac0: 2063 6f6d 7075 7465 0a20 2020 2020 202a   compute.      *
│ │ │ │ +00027ad0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ +00027ae0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +00027af0: 6329 4d6f 6475 6c65 2c2c 206d 6f64 756c  c)Module,, modul
│ │ │ │ +00027b00: 6520 6f76 6572 2074 6865 2072 696e 6720  e over the ring 
│ │ │ │ +00027b10: 6f66 0a20 2020 2020 2020 206f 7065 7261  of.        opera
│ │ │ │ +00027b20: 746f 7273 0a20 2020 2020 202a 206e 756d  tors.      * num
│ │ │ │ +00027b30: 6765 6e73 522c 2061 6e20 2a6e 6f74 6520  gensR, an *note 
│ │ │ │ +00027b40: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +00027b50: 6179 3244 6f63 295a 5a2c 2c20 6e75 6d62  ay2Doc)ZZ,, numb
│ │ │ │ +00027b60: 6572 206f 6620 6765 6e65 7261 746f 7273  er of generators
│ │ │ │ +00027b70: 206f 660a 2020 2020 2020 2020 7468 6520   of.        the 
│ │ │ │ +00027b80: 7461 7267 6574 2072 696e 670a 2020 2a20  target ring.  * 
│ │ │ │ +00027b90: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00027ba0: 2073 6571 2c20 6120 2a6e 6f74 6520 7365   seq, a *note se
│ │ │ │ +00027bb0: 7175 656e 6365 3a20 284d 6163 6175 6c61  quence: (Macaula
│ │ │ │ +00027bc0: 7932 446f 6329 5365 7175 656e 6365 2c2c  y2Doc)Sequence,,
│ │ │ │ +00027bd0: 2073 6571 7565 6e63 6520 6f66 206e 756d   sequence of num
│ │ │ │ +00027be0: 5661 6c75 6573 0a20 2020 2020 2020 2069  Values.        i
│ │ │ │ +00027bf0: 6e74 6567 6572 732c 2074 6865 2065 7870  ntegers, the exp
│ │ │ │ +00027c00: 6563 7465 6420 746f 7461 6c20 4265 7474  ected total Bett
│ │ │ │ +00027c10: 6920 6e75 6d62 6572 730a 0a44 6573 6372  i numbers..Descr
│ │ │ │ +00027c20: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00027c30: 3d3d 0a0a 4769 7665 6e20 6120 6d6f 6475  ==..Given a modu
│ │ │ │ +00027c40: 6c65 204d 206f 7665 7220 7468 6520 7269  le M over the ri
│ │ │ │ +00027c50: 6e67 206f 6620 6f70 6572 6174 6f72 7320  ng of operators 
│ │ │ │ +00027c60: 246b 5b78 5f31 2e2e 785f 635d 242c 2074  $k[x_1..x_c]$, t
│ │ │ │ +00027c70: 6865 2063 616c 6c20 244e 203d 0a6d 6f64  he call $N =.mod
│ │ │ │ +00027c80: 756c 6541 7345 7874 284d 2c52 2924 2070  uleAsExt(M,R)$ p
│ │ │ │ +00027c90: 726f 6475 6365 7320 6120 6d6f 6475 6c65  roduces a module
│ │ │ │ +00027ca0: 204e 206f 7665 7220 7468 6520 7269 6e67   N over the ring
│ │ │ │ +00027cb0: 2052 2077 686f 7365 2065 7874 206d 6f64   R whose ext mod
│ │ │ │ +00027cc0: 756c 6520 6973 2074 6865 0a65 7874 6572  ule is the.exter
│ │ │ │ +00027cd0: 696f 7220 616c 6765 6272 6120 6f6e 206e  ior algebra on n
│ │ │ │ +00027ce0: 3d6e 756d 6765 6e73 5220 6765 6e65 7261  =numgensR genera
│ │ │ │ +00027cf0: 746f 7273 2074 656e 736f 7265 6420 7769  tors tensored wi
│ │ │ │ +00027d00: 7468 204d 2e20 5468 6973 2073 6372 6970  th M. This scrip
│ │ │ │ +00027d10: 7420 636f 6d70 7574 6573 0a6e 756d 5661  t computes.numVa
│ │ │ │ +00027d20: 6c75 6573 2076 616c 7565 7320 6f66 2074  lues values of t
│ │ │ │ +00027d30: 6865 2048 696c 6265 7274 2066 756e 6374  he Hilbert funct
│ │ │ │ +00027d40: 696f 6e20 6f66 2024 2420 4d20 5c6f 7469  ion of $$ M \oti
│ │ │ │ +00027d50: 6d65 7320 5c77 6564 6765 206b 5e6e 2c20  mes \wedge k^n, 
│ │ │ │ +00027d60: 2424 2077 6869 6368 0a73 686f 756c 6420  $$ which.should 
│ │ │ │ +00027d70: 6265 2065 7175 616c 2074 6f20 7468 6520  be equal to the 
│ │ │ │ +00027d80: 746f 7461 6c20 6265 7474 6920 6e75 6d62  total betti numb
│ │ │ │ +00027d90: 6572 7320 6f66 204e 2e0a 0a2b 2d2d 2d2d  ers of N...+----
│ │ │ │ +00027da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027dd0: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 203d  ----+.|i1 : kk =
│ │ │ │ -00027de0: 205a 5a2f 3130 313b 2020 2020 2020 2020   ZZ/101;        
│ │ │ │ -00027df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027e00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00027dd0: 206b 6b20 3d20 5a5a 2f31 3031 3b20 2020   kk = ZZ/101;   
│ │ │ │ +00027de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027df0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e30: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20  ----+.|i2 : S = 
│ │ │ │ -00027e40: 6b6b 5b61 2c62 2c63 5d3b 2020 2020 2020  kk[a,b,c];      
│ │ │ │ -00027e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027e60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027e20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +00027e30: 2053 203d 206b 6b5b 612c 622c 635d 3b20   S = kk[a,b,c]; 
│ │ │ │ +00027e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e50: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e90: 2d2d 2d2d 2b0a 7c69 3320 3a20 6666 203d  ----+.|i3 : ff =
│ │ │ │ -00027ea0: 206d 6174 7269 787b 7b61 5e34 2c20 625e   matrix{{a^4, b^
│ │ │ │ -00027eb0: 342c 635e 347d 7d3b 2020 2020 2020 2020  4,c^4}};        
│ │ │ │ -00027ec0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00027e80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00027e90: 2066 6620 3d20 6d61 7472 6978 7b7b 615e   ff = matrix{{a^
│ │ │ │ +00027ea0: 342c 2062 5e34 2c63 5e34 7d7d 3b20 2020  4, b^4,c^4}};   
│ │ │ │ +00027eb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00027f00: 2020 2020 3120 2020 2020 2033 2020 2020      1      3    
│ │ │ │ -00027f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f20: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472      |.|o3 : Matr
│ │ │ │ -00027f30: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ -00027f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027ee0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00027ef0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00027f00: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00027f10: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00027f20: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +00027f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027f80: 2d2d 2d2d 2b0a 7c69 3420 3a20 5220 3d20  ----+.|i4 : R = 
│ │ │ │ -00027f90: 532f 6964 6561 6c20 6666 3b20 2020 2020  S/ideal ff;     
│ │ │ │ -00027fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027f70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00027f80: 2052 203d 2053 2f69 6465 616c 2066 663b   R = S/ideal ff;
│ │ │ │ +00027f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027fa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00027fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027fe0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 7320  ----+.|i5 : Ops 
│ │ │ │ -00027ff0: 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 5f33  = kk[x_1,x_2,x_3
│ │ │ │ -00028000: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ -00028010: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027fd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +00027fe0: 204f 7073 203d 206b 6b5b 785f 312c 785f   Ops = kk[x_1,x_
│ │ │ │ +00027ff0: 322c 785f 335d 3b20 2020 2020 2020 2020  2,x_3];         
│ │ │ │ +00028000: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028040: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d4d 203d  ----+.|i6 : MM =
│ │ │ │ -00028050: 204f 7073 5e31 2f28 785f 312a 6964 6561   Ops^1/(x_1*idea
│ │ │ │ -00028060: 6c28 785f 325e 322c 785f 3329 293b 2020  l(x_2^2,x_3));  
│ │ │ │ -00028070: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028030: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +00028040: 204d 4d20 3d20 4f70 735e 312f 2878 5f31   MM = Ops^1/(x_1
│ │ │ │ +00028050: 2a69 6465 616c 2878 5f32 5e32 2c78 5f33  *ideal(x_2^2,x_3
│ │ │ │ +00028060: 2929 3b20 2020 2020 207c 0a2b 2d2d 2d2d  ));      |.+----
│ │ │ │ +00028070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280a0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4e20 3d20  ----+.|i7 : N = 
│ │ │ │ -000280b0: 6d6f 6475 6c65 4173 4578 7428 4d4d 2c52  moduleAsExt(MM,R
│ │ │ │ -000280c0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -000280d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028090: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ +000280a0: 204e 203d 206d 6f64 756c 6541 7345 7874   N = moduleAsExt
│ │ │ │ +000280b0: 284d 4d2c 5229 3b20 2020 2020 2020 2020  (MM,R);         
│ │ │ │ +000280c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000280d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028100: 2d2d 2d2d 2b0a 7c69 3820 3a20 6265 7474  ----+.|i8 : bett
│ │ │ │ -00028110: 6920 7265 7328 204e 2c20 4c65 6e67 7468  i res( N, Length
│ │ │ │ -00028120: 4c69 6d69 7420 3d3e 2031 3029 2020 2020  Limit => 10)    
│ │ │ │ -00028130: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000280f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +00028100: 2062 6574 7469 2072 6573 2820 4e2c 204c   betti res( N, L
│ │ │ │ +00028110: 656e 6774 684c 696d 6974 203d 3e20 3130  engthLimit => 10
│ │ │ │ +00028120: 2920 2020 2020 2020 207c 0a7c 2020 2020  )        |.|    
│ │ │ │ +00028130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028160: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00028170: 2020 2020 3020 2031 2020 3220 2033 2020      0  1  2  3  
│ │ │ │ -00028180: 3420 2035 2020 3620 2037 2020 3820 2039  4  5  6  7  8  9
│ │ │ │ -00028190: 2031 3020 7c0a 7c6f 3820 3d20 746f 7461   10 |.|o8 = tota
│ │ │ │ -000281a0: 6c3a 2033 3620 3237 2032 3920 3331 2033  l: 36 27 29 31 3
│ │ │ │ -000281b0: 3320 3335 2033 3720 3339 2034 3120 3433  3 35 37 39 41 43
│ │ │ │ -000281c0: 2034 3520 7c0a 7c20 2020 2020 2020 202d   45 |.|        -
│ │ │ │ -000281d0: 363a 2031 3820 2036 2020 2e20 202e 2020  6: 18  6  .  .  
│ │ │ │ -000281e0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -000281f0: 2020 2e20 7c0a 7c20 2020 2020 2020 202d    . |.|        -
│ │ │ │ -00028200: 353a 2020 2e20 202e 2020 2e20 202e 2020  5:  .  .  .  .  
│ │ │ │ -00028210: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00028220: 2020 2e20 7c0a 7c20 2020 2020 2020 202d    . |.|        -
│ │ │ │ -00028230: 343a 2031 3820 3231 2032 3120 2037 2020  4: 18 21 21  7  
│ │ │ │ -00028240: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00028250: 2020 2e20 7c0a 7c20 2020 2020 2020 202d    . |.|        -
│ │ │ │ -00028260: 333a 2020 2e20 202e 2020 2e20 202e 2020  3:  .  .  .  .  
│ │ │ │ -00028270: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00028280: 2020 2e20 7c0a 7c20 2020 2020 2020 202d    . |.|        -
│ │ │ │ -00028290: 323a 2020 2e20 202e 2020 3820 3234 2032  2:  .  .  8 24 2
│ │ │ │ -000282a0: 3420 2038 2020 2e20 202e 2020 2e20 202e  4  8  .  .  .  .
│ │ │ │ -000282b0: 2020 2e20 7c0a 7c20 2020 2020 2020 202d    . |.|        -
│ │ │ │ -000282c0: 313a 2020 2e20 202e 2020 2e20 202e 2020  1:  .  .  .  .  
│ │ │ │ -000282d0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -000282e0: 2020 2e20 7c0a 7c20 2020 2020 2020 2020    . |.|         
│ │ │ │ -000282f0: 303a 2020 2e20 202e 2020 2e20 202e 2020  0:  .  .  .  .  
│ │ │ │ -00028300: 3920 3237 2032 3720 2039 2020 2e20 202e  9 27 27  9  .  .
│ │ │ │ -00028310: 2020 2e20 7c0a 7c20 2020 2020 2020 2020    . |.|         
│ │ │ │ -00028320: 313a 2020 2e20 202e 2020 2e20 202e 2020  1:  .  .  .  .  
│ │ │ │ -00028330: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00028340: 2020 2e20 7c0a 7c20 2020 2020 2020 2020    . |.|         
│ │ │ │ -00028350: 323a 2020 2e20 202e 2020 2e20 202e 2020  2:  .  .  .  .  
│ │ │ │ -00028360: 2e20 202e 2031 3020 3330 2033 3020 3130  .  . 10 30 30 10
│ │ │ │ -00028370: 2020 2e20 7c0a 7c20 2020 2020 2020 2020    . |.|         
│ │ │ │ -00028380: 333a 2020 2e20 202e 2020 2e20 202e 2020  3:  .  .  .  .  
│ │ │ │ -00028390: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -000283a0: 2020 2e20 7c0a 7c20 2020 2020 2020 2020    . |.|         
│ │ │ │ -000283b0: 343a 2020 2e20 202e 2020 2e20 202e 2020  4:  .  .  .  .  
│ │ │ │ -000283c0: 2e20 202e 2020 2e20 202e 2031 3120 3333  .  .  .  . 11 33
│ │ │ │ -000283d0: 2033 3320 7c0a 7c20 2020 2020 2020 2020   33 |.|         
│ │ │ │ -000283e0: 353a 2020 2e20 202e 2020 2e20 202e 2020  5:  .  .  .  .  
│ │ │ │ -000283f0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00028400: 2020 2e20 7c0a 7c20 2020 2020 2020 2020    . |.|         
│ │ │ │ -00028410: 363a 2020 2e20 202e 2020 2e20 202e 2020  6:  .  .  .  .  
│ │ │ │ -00028420: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00028430: 2031 3220 7c0a 7c20 2020 2020 2020 2020   12 |.|         
│ │ │ │ +00028150: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028160: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ +00028170: 2020 3320 2034 2020 3520 2036 2020 3720    3  4  5  6  7 
│ │ │ │ +00028180: 2038 2020 3920 3130 207c 0a7c 6f38 203d   8  9 10 |.|o8 =
│ │ │ │ +00028190: 2074 6f74 616c 3a20 3336 2032 3720 3239   total: 36 27 29
│ │ │ │ +000281a0: 2033 3120 3333 2033 3520 3337 2033 3920   31 33 35 37 39 
│ │ │ │ +000281b0: 3431 2034 3320 3435 207c 0a7c 2020 2020  41 43 45 |.|    
│ │ │ │ +000281c0: 2020 2020 2d36 3a20 3138 2020 3620 202e      -6: 18  6  .
│ │ │ │ +000281d0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +000281e0: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +000281f0: 2020 2020 2d35 3a20 202e 2020 2e20 202e      -5:  .  .  .
│ │ │ │ +00028200: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00028210: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +00028220: 2020 2020 2d34 3a20 3138 2032 3120 3231      -4: 18 21 21
│ │ │ │ +00028230: 2020 3720 202e 2020 2e20 202e 2020 2e20    7  .  .  .  . 
│ │ │ │ +00028240: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +00028250: 2020 2020 2d33 3a20 202e 2020 2e20 202e      -3:  .  .  .
│ │ │ │ +00028260: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00028270: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +00028280: 2020 2020 2d32 3a20 202e 2020 2e20 2038      -2:  .  .  8
│ │ │ │ +00028290: 2032 3420 3234 2020 3820 202e 2020 2e20   24 24  8  .  . 
│ │ │ │ +000282a0: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +000282b0: 2020 2020 2d31 3a20 202e 2020 2e20 202e      -1:  .  .  .
│ │ │ │ +000282c0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +000282d0: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +000282e0: 2020 2020 2030 3a20 202e 2020 2e20 202e       0:  .  .  .
│ │ │ │ +000282f0: 2020 2e20 2039 2032 3720 3237 2020 3920    .  9 27 27  9 
│ │ │ │ +00028300: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +00028310: 2020 2020 2031 3a20 202e 2020 2e20 202e       1:  .  .  .
│ │ │ │ +00028320: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00028330: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +00028340: 2020 2020 2032 3a20 202e 2020 2e20 202e       2:  .  .  .
│ │ │ │ +00028350: 2020 2e20 202e 2020 2e20 3130 2033 3020    .  .  . 10 30 
│ │ │ │ +00028360: 3330 2031 3020 202e 207c 0a7c 2020 2020  30 10  . |.|    
│ │ │ │ +00028370: 2020 2020 2033 3a20 202e 2020 2e20 202e       3:  .  .  .
│ │ │ │ +00028380: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00028390: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +000283a0: 2020 2020 2034 3a20 202e 2020 2e20 202e       4:  .  .  .
│ │ │ │ +000283b0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +000283c0: 3131 2033 3320 3333 207c 0a7c 2020 2020  11 33 33 |.|    
│ │ │ │ +000283d0: 2020 2020 2035 3a20 202e 2020 2e20 202e       5:  .  .  .
│ │ │ │ +000283e0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +000283f0: 202e 2020 2e20 202e 207c 0a7c 2020 2020   .  .  . |.|    
│ │ │ │ +00028400: 2020 2020 2036 3a20 202e 2020 2e20 202e       6:  .  .  .
│ │ │ │ +00028410: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00028420: 202e 2020 2e20 3132 207c 0a7c 2020 2020   .  . 12 |.|    
│ │ │ │ +00028430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028460: 2020 2020 7c0a 7c6f 3820 3a20 4265 7474      |.|o8 : Bett
│ │ │ │ -00028470: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -00028480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028490: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028450: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ +00028460: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028480: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00028490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284c0: 2d2d 2d2d 2b0a 7c69 3920 3a20 6866 4d6f  ----+.|i9 : hfMo
│ │ │ │ -000284d0: 6475 6c65 4173 4578 7428 3132 2c4d 4d2c  duleAsExt(12,MM,
│ │ │ │ -000284e0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │ -000284f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000284b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ +000284c0: 2068 664d 6f64 756c 6541 7345 7874 2831   hfModuleAsExt(1
│ │ │ │ +000284d0: 322c 4d4d 2c33 2920 2020 2020 2020 2020  2,MM,3)         
│ │ │ │ +000284e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000284f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028520: 2020 2020 7c0a 7c6f 3920 3d20 2832 332c      |.|o9 = (23,
│ │ │ │ -00028530: 2032 352c 2032 372c 2032 392c 2033 312c   25, 27, 29, 31,
│ │ │ │ -00028540: 2033 332c 2033 352c 2033 372c 2033 392c   33, 35, 37, 39,
│ │ │ │ -00028550: 2034 3129 7c0a 7c20 2020 2020 2020 2020   41)|.|         
│ │ │ │ +00028510: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +00028520: 2028 3233 2c20 3235 2c20 3237 2c20 3239   (23, 25, 27, 29
│ │ │ │ +00028530: 2c20 3331 2c20 3333 2c20 3335 2c20 3337  , 31, 33, 35, 37
│ │ │ │ +00028540: 2c20 3339 2c20 3431 297c 0a7c 2020 2020  , 39, 41)|.|    
│ │ │ │ +00028550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028580: 2020 2020 7c0a 7c6f 3920 3a20 5365 7175      |.|o9 : Sequ
│ │ │ │ -00028590: 656e 6365 2020 2020 2020 2020 2020 2020  ence            
│ │ │ │ -000285a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028570: 2020 2020 2020 2020 207c 0a7c 6f39 203a           |.|o9 :
│ │ │ │ +00028580: 2053 6571 7565 6e63 6520 2020 2020 2020   Sequence       
│ │ │ │ +00028590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000285b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000285c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285e0: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ -000285f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00028600: 6f74 6520 6d6f 6475 6c65 4173 4578 743a  ote moduleAsExt:
│ │ │ │ -00028610: 206d 6f64 756c 6541 7345 7874 2c20 2d2d   moduleAsExt, --
│ │ │ │ -00028620: 2046 696e 6420 6120 6d6f 6475 6c65 2077   Find a module w
│ │ │ │ -00028630: 6974 6820 6769 7665 6e20 6173 796d 7074  ith given asympt
│ │ │ │ -00028640: 6f74 6963 0a20 2020 2072 6573 6f6c 7574  otic.    resolut
│ │ │ │ -00028650: 696f 6e0a 0a57 6179 7320 746f 2075 7365  ion..Ways to use
│ │ │ │ -00028660: 2068 664d 6f64 756c 6541 7345 7874 3a0a   hfModuleAsExt:.
│ │ │ │ -00028670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00028680: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00028690: 2268 664d 6f64 756c 6541 7345 7874 285a  "hfModuleAsExt(Z
│ │ │ │ -000286a0: 5a2c 4d6f 6475 6c65 2c5a 5a29 220a 0a46  Z,Module,ZZ)"..F
│ │ │ │ -000286b0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -000286c0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -000286d0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -000286e0: 202a 6e6f 7465 2068 664d 6f64 756c 6541   *note hfModuleA
│ │ │ │ -000286f0: 7345 7874 3a20 6866 4d6f 6475 6c65 4173  sExt: hfModuleAs
│ │ │ │ -00028700: 4578 742c 2069 7320 6120 2a6e 6f74 6520  Ext, is a *note 
│ │ │ │ -00028710: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ -00028720: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ -00028730: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ -00028740: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ -00028750: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -00028760: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ -00028770: 6465 3a20 6869 6768 5379 7a79 6779 2c20  de: highSyzygy, 
│ │ │ │ -00028780: 4e65 7874 3a20 684d 6170 732c 2050 7265  Next: hMaps, Pre
│ │ │ │ -00028790: 763a 2068 664d 6f64 756c 6541 7345 7874  v: hfModuleAsExt
│ │ │ │ -000287a0: 2c20 5570 3a20 546f 700a 0a68 6967 6853  , Up: Top..highS
│ │ │ │ -000287b0: 797a 7967 7920 2d2d 2052 6574 7572 6e73  yzygy -- Returns
│ │ │ │ -000287c0: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ -000287d0: 206f 6e65 2062 6579 6f6e 6420 7468 6520   one beyond the 
│ │ │ │ -000287e0: 7265 6775 6c61 7269 7479 206f 6620 4578  regularity of Ex
│ │ │ │ -000287f0: 7428 4d2c 6b29 0a2a 2a2a 2a2a 2a2a 2a2a  t(M,k).*********
│ │ │ │ +000285d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ +000285e0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +000285f0: 202a 202a 6e6f 7465 206d 6f64 756c 6541   * *note moduleA
│ │ │ │ +00028600: 7345 7874 3a20 6d6f 6475 6c65 4173 4578  sExt: moduleAsEx
│ │ │ │ +00028610: 742c 202d 2d20 4669 6e64 2061 206d 6f64  t, -- Find a mod
│ │ │ │ +00028620: 756c 6520 7769 7468 2067 6976 656e 2061  ule with given a
│ │ │ │ +00028630: 7379 6d70 746f 7469 630a 2020 2020 7265  symptotic.    re
│ │ │ │ +00028640: 736f 6c75 7469 6f6e 0a0a 5761 7973 2074  solution..Ways t
│ │ │ │ +00028650: 6f20 7573 6520 6866 4d6f 6475 6c65 4173  o use hfModuleAs
│ │ │ │ +00028660: 4578 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  Ext:.===========
│ │ │ │ +00028670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00028680: 0a20 202a 2022 6866 4d6f 6475 6c65 4173  .  * "hfModuleAs
│ │ │ │ +00028690: 4578 7428 5a5a 2c4d 6f64 756c 652c 5a5a  Ext(ZZ,Module,ZZ
│ │ │ │ +000286a0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +000286b0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +000286c0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +000286d0: 626a 6563 7420 2a6e 6f74 6520 6866 4d6f  bject *note hfMo
│ │ │ │ +000286e0: 6475 6c65 4173 4578 743a 2068 664d 6f64  duleAsExt: hfMod
│ │ │ │ +000286f0: 756c 6541 7345 7874 2c20 6973 2061 202a  uleAsExt, is a *
│ │ │ │ +00028700: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00028710: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +00028720: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00028730: 6f6e 2c2e 0a1f 0a46 696c 653a 2043 6f6d  on,....File: Com
│ │ │ │ +00028740: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +00028750: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ +00028760: 6f2c 204e 6f64 653a 2068 6967 6853 797a  o, Node: highSyz
│ │ │ │ +00028770: 7967 792c 204e 6578 743a 2068 4d61 7073  ygy, Next: hMaps
│ │ │ │ +00028780: 2c20 5072 6576 3a20 6866 4d6f 6475 6c65  , Prev: hfModule
│ │ │ │ +00028790: 4173 4578 742c 2055 703a 2054 6f70 0a0a  AsExt, Up: Top..
│ │ │ │ +000287a0: 6869 6768 5379 7a79 6779 202d 2d20 5265  highSyzygy -- Re
│ │ │ │ +000287b0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ +000287c0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ +000287d0: 2074 6865 2072 6567 756c 6172 6974 7920   the regularity 
│ │ │ │ +000287e0: 6f66 2045 7874 284d 2c6b 290a 2a2a 2a2a  of Ext(M,k).****
│ │ │ │ +000287f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00028800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00028810: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00028820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00028830: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00028840: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ -00028850: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ -00028860: 3a20 0a20 2020 2020 2020 204d 203d 2068  : .        M = h
│ │ │ │ -00028870: 6967 6853 797a 7967 7920 4d30 0a20 202a  ighSyzygy M0.  *
│ │ │ │ -00028880: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00028890: 204d 302c 2061 202a 6e6f 7465 206d 6f64   M0, a *note mod
│ │ │ │ -000288a0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -000288b0: 6f63 294d 6f64 756c 652c 2c20 6f76 6572  oc)Module,, over
│ │ │ │ -000288c0: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -000288d0: 7273 6563 7469 6f6e 0a20 2020 2020 2020  rsection.       
│ │ │ │ -000288e0: 2072 696e 670a 2020 2a20 2a6e 6f74 6520   ring.  * *note 
│ │ │ │ -000288f0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ -00028900: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ -00028910: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ -00028920: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ -00028930: 7574 732c 3a0a 2020 2020 2020 2a20 4f70  uts,:.      * Op
│ │ │ │ -00028940: 7469 6d69 736d 203d 3e20 2e2e 2e2c 2064  timism => ..., d
│ │ │ │ -00028950: 6566 6175 6c74 2076 616c 7565 2030 0a20  efault value 0. 
│ │ │ │ -00028960: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00028970: 2020 2a20 4d2c 2061 202a 6e6f 7465 206d    * M, a *note m
│ │ │ │ -00028980: 6f64 756c 653a 2028 4d61 6361 756c 6179  odule: (Macaulay
│ │ │ │ -00028990: 3244 6f63 294d 6f64 756c 652c 2c20 6120  2Doc)Module,, a 
│ │ │ │ -000289a0: 7379 7a79 6779 206d 6f64 756c 6520 6f66  syzygy module of
│ │ │ │ -000289b0: 204d 300a 0a44 6573 6372 6970 7469 6f6e   M0..Description
│ │ │ │ -000289c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120  .===========..A 
│ │ │ │ -000289d0: 2268 6967 6820 7379 7a79 6779 2220 6f76  "high syzygy" ov
│ │ │ │ -000289e0: 6572 2061 2063 6f6d 706c 6574 6520 696e  er a complete in
│ │ │ │ -000289f0: 7465 7273 6563 7469 6f6e 2069 7320 6f6e  tersection is on
│ │ │ │ -00028a00: 6520 7375 6368 2074 6861 7420 6765 6e65  e such that gene
│ │ │ │ -00028a10: 7261 6c0a 6369 2d6f 7065 7261 746f 7273  ral.ci-operators
│ │ │ │ -00028a20: 2068 6176 6520 7370 6c69 7420 6b65 726e   have split kern
│ │ │ │ -00028a30: 656c 7320 7768 656e 2061 7070 6c69 6564  els when applied
│ │ │ │ -00028a40: 2072 6563 7572 7369 7665 6c79 206f 6e20   recursively on 
│ │ │ │ -00028a50: 636f 7379 7a79 6779 2063 6861 696e 7320  cosyzygy chains 
│ │ │ │ -00028a60: 6f66 0a70 7265 7669 6f75 7320 6b65 726e  of.previous kern
│ │ │ │ -00028a70: 656c 732e 0a0a 4966 2070 203d 206d 6642  els...If p = mfB
│ │ │ │ -00028a80: 6f75 6e64 204d 302c 2074 6865 6e20 6869  ound M0, then hi
│ │ │ │ -00028a90: 6768 5379 7a79 6779 204d 3020 7265 7475  ghSyzygy M0 retu
│ │ │ │ -00028aa0: 726e 7320 7468 6520 702d 7468 2073 797a  rns the p-th syz
│ │ │ │ -00028ab0: 7967 7920 6f66 204d 302e 2028 6966 2046  ygy of M0. (if F
│ │ │ │ -00028ac0: 2069 7320 610a 7265 736f 6c75 7469 6f6e   is a.resolution
│ │ │ │ -00028ad0: 206f 6620 4d20 7468 6973 2069 7320 7468   of M this is th
│ │ │ │ -00028ae0: 6520 636f 6b65 726e 656c 206f 6620 462e  e cokernel of F.
│ │ │ │ -00028af0: 6464 5f7b 702b 317d 292e 204f 7074 696d  dd_{p+1}). Optim
│ │ │ │ -00028b00: 6973 6d20 3d3e 2072 2061 7320 6f70 7469  ism => r as opti
│ │ │ │ -00028b10: 6f6e 616c 0a61 7267 756d 656e 742c 2068  onal.argument, h
│ │ │ │ -00028b20: 6967 6853 797a 7967 7928 4d30 2c4f 7074  ighSyzygy(M0,Opt
│ │ │ │ -00028b30: 696d 6973 6d3d 3e72 2920 7265 7475 726e  imism=>r) return
│ │ │ │ -00028b40: 7320 7468 6520 2870 2d72 292d 7468 2073  s the (p-r)-th s
│ │ │ │ -00028b50: 797a 7967 792e 2054 6865 2073 6372 6970  yzygy. The scrip
│ │ │ │ -00028b60: 7420 6973 0a75 7365 6675 6c20 7769 7468  t is.useful with
│ │ │ │ -00028b70: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -00028b80: 7469 6f6e 2866 662c 2068 6967 6853 797a  tion(ff, highSyz
│ │ │ │ -00028b90: 7967 7920 4d30 292e 0a0a 2b2d 2d2d 2d2d  ygy M0)...+-----
│ │ │ │ +00028830: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +00028840: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +00028850: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00028860: 4d20 3d20 6869 6768 5379 7a79 6779 204d  M = highSyzygy M
│ │ │ │ +00028870: 300a 2020 2a20 496e 7075 7473 3a0a 2020  0.  * Inputs:.  
│ │ │ │ +00028880: 2020 2020 2a20 4d30 2c20 6120 2a6e 6f74      * M0, a *not
│ │ │ │ +00028890: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +000288a0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +000288b0: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ +000288c0: 2069 6e74 6572 7365 6374 696f 6e0a 2020   intersection.  
│ │ │ │ +000288d0: 2020 2020 2020 7269 6e67 0a20 202a 202a        ring.  * *
│ │ │ │ +000288e0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ +000288f0: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ +00028900: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ +00028910: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +00028920: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ +00028930: 202a 204f 7074 696d 6973 6d20 3d3e 202e   * Optimism => .
│ │ │ │ +00028940: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +00028950: 6520 300a 2020 2a20 4f75 7470 7574 733a  e 0.  * Outputs:
│ │ │ │ +00028960: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ +00028970: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +00028980: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +00028990: 2c2c 2061 2073 797a 7967 7920 6d6f 6475  ,, a syzygy modu
│ │ │ │ +000289a0: 6c65 206f 6620 4d30 0a0a 4465 7363 7269  le of M0..Descri
│ │ │ │ +000289b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +000289c0: 3d0a 0a41 2022 6869 6768 2073 797a 7967  =..A "high syzyg
│ │ │ │ +000289d0: 7922 206f 7665 7220 6120 636f 6d70 6c65  y" over a comple
│ │ │ │ +000289e0: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ +000289f0: 6973 206f 6e65 2073 7563 6820 7468 6174  is one such that
│ │ │ │ +00028a00: 2067 656e 6572 616c 0a63 692d 6f70 6572   general.ci-oper
│ │ │ │ +00028a10: 6174 6f72 7320 6861 7665 2073 706c 6974  ators have split
│ │ │ │ +00028a20: 206b 6572 6e65 6c73 2077 6865 6e20 6170   kernels when ap
│ │ │ │ +00028a30: 706c 6965 6420 7265 6375 7273 6976 656c  plied recursivel
│ │ │ │ +00028a40: 7920 6f6e 2063 6f73 797a 7967 7920 6368  y on cosyzygy ch
│ │ │ │ +00028a50: 6169 6e73 206f 660a 7072 6576 696f 7573  ains of.previous
│ │ │ │ +00028a60: 206b 6572 6e65 6c73 2e0a 0a49 6620 7020   kernels...If p 
│ │ │ │ +00028a70: 3d20 6d66 426f 756e 6420 4d30 2c20 7468  = mfBound M0, th
│ │ │ │ +00028a80: 656e 2068 6967 6853 797a 7967 7920 4d30  en highSyzygy M0
│ │ │ │ +00028a90: 2072 6574 7572 6e73 2074 6865 2070 2d74   returns the p-t
│ │ │ │ +00028aa0: 6820 7379 7a79 6779 206f 6620 4d30 2e20  h syzygy of M0. 
│ │ │ │ +00028ab0: 2869 6620 4620 6973 2061 0a72 6573 6f6c  (if F is a.resol
│ │ │ │ +00028ac0: 7574 696f 6e20 6f66 204d 2074 6869 7320  ution of M this 
│ │ │ │ +00028ad0: 6973 2074 6865 2063 6f6b 6572 6e65 6c20  is the cokernel 
│ │ │ │ +00028ae0: 6f66 2046 2e64 645f 7b70 2b31 7d29 2e20  of F.dd_{p+1}). 
│ │ │ │ +00028af0: 4f70 7469 6d69 736d 203d 3e20 7220 6173  Optimism => r as
│ │ │ │ +00028b00: 206f 7074 696f 6e61 6c0a 6172 6775 6d65   optional.argume
│ │ │ │ +00028b10: 6e74 2c20 6869 6768 5379 7a79 6779 284d  nt, highSyzygy(M
│ │ │ │ +00028b20: 302c 4f70 7469 6d69 736d 3d3e 7229 2072  0,Optimism=>r) r
│ │ │ │ +00028b30: 6574 7572 6e73 2074 6865 2028 702d 7229  eturns the (p-r)
│ │ │ │ +00028b40: 2d74 6820 7379 7a79 6779 2e20 5468 6520  -th syzygy. The 
│ │ │ │ +00028b50: 7363 7269 7074 2069 730a 7573 6566 756c  script is.useful
│ │ │ │ +00028b60: 2077 6974 6820 6d61 7472 6978 4661 6374   with matrixFact
│ │ │ │ +00028b70: 6f72 697a 6174 696f 6e28 6666 2c20 6869  orization(ff, hi
│ │ │ │ +00028b80: 6768 5379 7a79 6779 204d 3029 2e0a 0a2b  ghSyzygy M0)...+
│ │ │ │ +00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028bd0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365  ------+.|i1 : se
│ │ │ │ -00028be0: 7452 616e 646f 6d53 6565 6420 3130 3020  tRandomSeed 100 
│ │ │ │ +00028bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00028bd0: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ +00028be0: 2031 3030 2020 2020 2020 2020 2020 2020   100            
│ │ │ │  00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00028c00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c50: 2020 7c0a 7c6f 3120 3d20 3130 3020 2020    |.|o1 = 100   
│ │ │ │ +00028c40: 2020 2020 2020 207c 0a7c 6f31 203d 2031         |.|o1 = 1
│ │ │ │ +00028c50: 3030 2020 2020 2020 2020 2020 2020 2020  00              
│ │ │ │  00028c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00028c80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00028c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00028cd0: 7c69 3220 3a20 5320 3d20 5a5a 2f31 3031  |i2 : S = ZZ/101
│ │ │ │ -00028ce0: 5b78 2c79 2c7a 5d20 2020 2020 2020 2020  [x,y,z]         
│ │ │ │ +00028cc0: 2d2d 2d2b 0a7c 6932 203a 2053 203d 205a  ---+.|i2 : S = Z
│ │ │ │ +00028cd0: 5a2f 3130 315b 782c 792c 7a5d 2020 2020  Z/101[x,y,z]    
│ │ │ │ +00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028d00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00028d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d40: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00028d50: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │ +00028d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028d40: 0a7c 6f32 203d 2053 2020 2020 2020 2020  .|o2 = S        
│ │ │ │ +00028d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028dc0: 2020 2020 2020 7c0a 7c6f 3220 3a20 506f        |.|o2 : Po
│ │ │ │ -00028dd0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +00028db0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00028dc0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +00028dd0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00028de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028df0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00028e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028e40: 2d2d 2b0a 7c69 3320 3a20 6620 3d20 6d61  --+.|i3 : f = ma
│ │ │ │ -00028e50: 7472 6978 2278 332c 7933 2b78 332c 7a33  trix"x3,y3+x3,z3
│ │ │ │ -00028e60: 2b78 332b 7933 2220 2020 2020 2020 2020  +x3+y3"         
│ │ │ │ -00028e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00028e30: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066  -------+.|i3 : f
│ │ │ │ +00028e40: 203d 206d 6174 7269 7822 7833 2c79 332b   = matrix"x3,y3+
│ │ │ │ +00028e50: 7833 2c7a 332b 7833 2b79 3322 2020 2020  x3,z3+x3+y3"    
│ │ │ │ +00028e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028eb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028ec0: 7c6f 3320 3d20 7c20 7833 2078 332b 7933  |o3 = | x3 x3+y3
│ │ │ │ -00028ed0: 2078 332b 7933 2b7a 3320 7c20 2020 2020   x3+y3+z3 |     
│ │ │ │ +00028eb0: 2020 207c 0a7c 6f33 203d 207c 2078 3320     |.|o3 = | x3 
│ │ │ │ +00028ec0: 7833 2b79 3320 7833 2b79 332b 7a33 207c  x3+y3 x3+y3+z3 |
│ │ │ │ +00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028ef0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00028f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00028f40: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00028f50: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00028f80: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +00028f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028f30: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +00028f40: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00028f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028f70: 6f33 203a 204d 6174 7269 7820 5320 203c  o3 : Matrix S  <
│ │ │ │ +00028f80: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  00028f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00028fa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ff0: 2d2d 2d2d 2b0a 7c69 3420 3a20 6666 203d  ----+.|i4 : ff =
│ │ │ │ -00029000: 2066 2a72 616e 646f 6d28 736f 7572 6365   f*random(source
│ │ │ │ -00029010: 2066 2c20 736f 7572 6365 2066 2920 2020   f, source f)   
│ │ │ │ -00029020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029030: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00028fe0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00028ff0: 2066 6620 3d20 662a 7261 6e64 6f6d 2873   ff = f*random(s
│ │ │ │ +00029000: 6f75 7263 6520 662c 2073 6f75 7263 6520  ource f, source 
│ │ │ │ +00029010: 6629 2020 2020 2020 2020 2020 2020 2020  f)              
│ │ │ │ +00029020: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029070: 7c0a 7c6f 3420 3d20 7c20 3130 7833 2d32  |.|o4 = | 10x3-2
│ │ │ │ -00029080: 3279 332d 347a 3320 2d32 3078 332d 3230  2y3-4z3 -20x3-20
│ │ │ │ -00029090: 7933 2d36 7a33 202d 3237 7833 2d34 3179  y3-6z3 -27x3-41y
│ │ │ │ -000290a0: 332b 7a33 207c 2020 2020 2020 2020 7c0a  3+z3 |        |.
│ │ │ │ -000290b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00029060: 2020 2020 207c 0a7c 6f34 203d 207c 2031       |.|o4 = | 1
│ │ │ │ +00029070: 3078 332d 3232 7933 2d34 7a33 202d 3230  0x3-22y3-4z3 -20
│ │ │ │ +00029080: 7833 2d32 3079 332d 367a 3320 2d32 3778  x3-20y3-6z3 -27x
│ │ │ │ +00029090: 332d 3431 7933 2b7a 3320 7c20 2020 2020  3-41y3+z3 |     
│ │ │ │ +000290a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000290b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000290e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000290f0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00029100: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00029110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029120: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -00029130: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -00029140: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00029150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029160: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000290e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000290f0: 2031 2020 2020 2020 3320 2020 2020 2020   1      3       
│ │ │ │ +00029100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029120: 0a7c 6f34 203a 204d 6174 7269 7820 5320  .|o4 : Matrix S 
│ │ │ │ +00029130: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00029140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029150: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00029160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000291a0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 5220  ------+.|i5 : R 
│ │ │ │ -000291b0: 3d20 532f 6964 6561 6c20 6620 2020 2020  = S/ideal f     
│ │ │ │ +00029190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +000291a0: 203a 2052 203d 2053 2f69 6465 616c 2066   : R = S/ideal f
│ │ │ │ +000291b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000291d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000291e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029220: 2020 7c0a 7c6f 3520 3d20 5220 2020 2020    |.|o5 = R     
│ │ │ │ +00029210: 2020 2020 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │ +00029220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029260: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00029250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000292a0: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -000292b0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00029290: 2020 207c 0a7c 6f35 203a 2051 756f 7469     |.|o5 : Quoti
│ │ │ │ +000292a0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +000292b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000292c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000292d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000292d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000292e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000292f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029310: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -00029320: 3a20 4d30 203d 2052 5e31 2f69 6465 616c  : M0 = R^1/ideal
│ │ │ │ -00029330: 2278 327a 322c 7879 7a22 2020 2020 2020  "x2z2,xyz"      
│ │ │ │ -00029340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00029310: 0a7c 6936 203a 204d 3020 3d20 525e 312f  .|i6 : M0 = R^1/
│ │ │ │ +00029320: 6964 6561 6c22 7832 7a32 2c78 797a 2220  ideal"x2z2,xyz" 
│ │ │ │ +00029330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029340: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00029350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029390: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ -000293a0: 6b65 726e 656c 207c 2078 327a 3220 7879  kernel | x2z2 xy
│ │ │ │ -000293b0: 7a20 7c20 2020 2020 2020 2020 2020 2020  z |             
│ │ │ │ -000293c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029380: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00029390: 203d 2063 6f6b 6572 6e65 6c20 7c20 7832   = cokernel | x2
│ │ │ │ +000293a0: 7a32 2078 797a 207c 2020 2020 2020 2020  z2 xyz |        
│ │ │ │ +000293b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000293c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000293d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000293e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000293f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029410: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00029420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029430: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00029440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029450: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65  |.|o6 : R-module
│ │ │ │ -00029460: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +00029400: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029420: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +00029430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029440: 2020 2020 207c 0a7c 6f36 203a 2052 2d6d       |.|o6 : R-m
│ │ │ │ +00029450: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +00029460: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  00029470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00029490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00029480: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00029490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000294a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000294b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000294c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000294d0: 3720 3a20 6265 7474 6920 7265 7320 284d  7 : betti res (M
│ │ │ │ -000294e0: 302c 204c 656e 6774 684c 696d 6974 203d  0, LengthLimit =
│ │ │ │ -000294f0: 3e20 3729 2020 2020 2020 2020 2020 2020  > 7)            
│ │ │ │ -00029500: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000294c0: 2d2b 0a7c 6937 203a 2062 6574 7469 2072  -+.|i7 : betti r
│ │ │ │ +000294d0: 6573 2028 4d30 2c20 4c65 6e67 7468 4c69  es (M0, LengthLi
│ │ │ │ +000294e0: 6d69 7420 3d3e 2037 2920 2020 2020 2020  mit => 7)       
│ │ │ │ +000294f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00029550: 2020 2020 2020 2030 2031 2032 2020 3320         0 1 2  3 
│ │ │ │ -00029560: 2034 2020 3520 2036 2020 3720 2020 2020   4  5  6  7     
│ │ │ │ -00029570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029580: 2020 2020 2020 7c0a 7c6f 3720 3d20 746f        |.|o7 = to
│ │ │ │ -00029590: 7461 6c3a 2031 2032 2036 2031 3120 3138  tal: 1 2 6 11 18
│ │ │ │ -000295a0: 2032 3620 3336 2034 3720 2020 2020 2020   26 36 47       
│ │ │ │ -000295b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000295c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000295d0: 303a 2031 202e 202e 2020 2e20 202e 2020  0: 1 . .  .  .  
│ │ │ │ -000295e0: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -000295f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029600: 2020 7c0a 7c20 2020 2020 2020 2020 313a    |.|         1:
│ │ │ │ -00029610: 202e 202e 202e 2020 2e20 202e 2020 2e20   . . .  .  .  . 
│ │ │ │ -00029620: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ -00029630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029640: 7c0a 7c20 2020 2020 2020 2020 323a 202e  |.|         2: .
│ │ │ │ -00029650: 2031 202e 2020 2e20 202e 2020 2e20 202e   1 .  .  .  .  .
│ │ │ │ -00029660: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ -00029670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00029680: 7c20 2020 2020 2020 2020 333a 202e 2031  |         3: . 1
│ │ │ │ -00029690: 2036 2020 3620 202e 2020 2e20 202e 2020   6  6  .  .  .  
│ │ │ │ -000296a0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -000296b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000296c0: 2020 2020 2020 2020 343a 202e 202e 202e          4: . . .
│ │ │ │ -000296d0: 2020 3520 3138 2031 3420 202e 2020 2e20    5 18 14  .  . 
│ │ │ │ -000296e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000296f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00029700: 2020 2020 2020 353a 202e 202e 202e 2020        5: . . .  
│ │ │ │ -00029710: 2e20 202e 2031 3220 3336 2032 3520 2020  .  . 12 36 25   
│ │ │ │ -00029720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00029740: 2020 2020 363a 202e 202e 202e 2020 2e20      6: . . .  . 
│ │ │ │ -00029750: 202e 2020 2e20 202e 2032 3220 2020 2020   .  .  . 22     
│ │ │ │ -00029760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00029530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00029540: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +00029550: 3220 2033 2020 3420 2035 2020 3620 2037  2  3  4  5  6  7
│ │ │ │ +00029560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029570: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00029580: 203d 2074 6f74 616c 3a20 3120 3220 3620   = total: 1 2 6 
│ │ │ │ +00029590: 3131 2031 3820 3236 2033 3620 3437 2020  11 18 26 36 47  
│ │ │ │ +000295a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000295b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000295c0: 2020 2020 2030 3a20 3120 2e20 2e20 202e       0: 1 . .  .
│ │ │ │ +000295d0: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +000295e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000295f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029600: 2020 2031 3a20 2e20 2e20 2e20 202e 2020     1: . . .  .  
│ │ │ │ +00029610: 2e20 202e 2020 2e20 202e 2020 2020 2020  .  .  .  .      
│ │ │ │ +00029620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029630: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029640: 2032 3a20 2e20 3120 2e20 202e 2020 2e20   2: . 1 .  .  . 
│ │ │ │ +00029650: 202e 2020 2e20 202e 2020 2020 2020 2020   .  .  .        
│ │ │ │ +00029660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029670: 2020 207c 0a7c 2020 2020 2020 2020 2033     |.|         3
│ │ │ │ +00029680: 3a20 2e20 3120 3620 2036 2020 2e20 202e  : . 1 6  6  .  .
│ │ │ │ +00029690: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +000296a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000296b0: 207c 0a7c 2020 2020 2020 2020 2034 3a20   |.|         4: 
│ │ │ │ +000296c0: 2e20 2e20 2e20 2035 2031 3820 3134 2020  . . .  5 18 14  
│ │ │ │ +000296d0: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ +000296e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000296f0: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ +00029700: 2e20 2e20 202e 2020 2e20 3132 2033 3620  . .  .  . 12 36 
│ │ │ │ +00029710: 3235 2020 2020 2020 2020 2020 2020 2020  25              
│ │ │ │ +00029720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00029730: 2020 2020 2020 2020 2036 3a20 2e20 2e20           6: . . 
│ │ │ │ +00029740: 2e20 202e 2020 2e20 202e 2020 2e20 3232  .  .  .  .  . 22
│ │ │ │ +00029750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297b0: 2020 2020 7c0a 7c6f 3720 3a20 4265 7474      |.|o7 : Bett
│ │ │ │ -000297c0: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +000297a0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +000297b0: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +000297c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000297e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000297f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029830: 2b0a 7c69 3820 3a20 6d66 426f 756e 6420  +.|i8 : mfBound 
│ │ │ │ -00029840: 4d30 2020 2020 2020 2020 2020 2020 2020  M0              
│ │ │ │ +00029820: 2d2d 2d2d 2d2b 0a7c 6938 203a 206d 6642  -----+.|i8 : mfB
│ │ │ │ +00029830: 6f75 6e64 204d 3020 2020 2020 2020 2020  ound M0         
│ │ │ │ +00029840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00029870: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00029860: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00029870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000298b0: 3820 3d20 3320 2020 2020 2020 2020 2020  8 = 3           
│ │ │ │ +000298a0: 207c 0a7c 6f38 203d 2033 2020 2020 2020   |.|o8 = 3      
│ │ │ │ +000298b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000298d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000298e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000298f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029920: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ -00029930: 4d20 3d20 6265 7474 6920 7265 7320 6869  M = betti res hi
│ │ │ │ -00029940: 6768 5379 7a79 6779 204d 3020 2020 2020  ghSyzygy M0     
│ │ │ │ -00029950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029960: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00029910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00029920: 6939 203a 204d 203d 2062 6574 7469 2072  i9 : M = betti r
│ │ │ │ +00029930: 6573 2068 6967 6853 797a 7967 7920 4d30  es highSyzygy M0
│ │ │ │ +00029940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000299a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000299b0: 2020 2020 3020 2031 2020 3220 2033 2020      0  1  2  3  
│ │ │ │ -000299c0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -000299d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000299e0: 2020 7c0a 7c6f 3920 3d20 746f 7461 6c3a    |.|o9 = total:
│ │ │ │ -000299f0: 2031 3120 3138 2032 3620 3336 2034 3720   11 18 26 36 47 
│ │ │ │ +00029990: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000299a0: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ +000299b0: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +000299c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000299d0: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │ +000299e0: 6f74 616c 3a20 3131 2031 3820 3236 2033  otal: 11 18 26 3
│ │ │ │ +000299f0: 3620 3437 2020 2020 2020 2020 2020 2020  6 47            
│ │ │ │  00029a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a20: 7c0a 7c20 2020 2020 2020 2020 363a 2020  |.|         6:  
│ │ │ │ -00029a30: 3620 202e 2020 2e20 202e 2020 2e20 2020  6  .  .  .  .   
│ │ │ │ +00029a10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029a20: 2036 3a20 2036 2020 2e20 202e 2020 2e20   6:  6  .  .  . 
│ │ │ │ +00029a30: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  00029a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00029a60: 7c20 2020 2020 2020 2020 373a 2020 3520  |         7:  5 
│ │ │ │ -00029a70: 3138 2031 3420 202e 2020 2e20 2020 2020  18 14  .  .     
│ │ │ │ +00029a50: 2020 207c 0a7c 2020 2020 2020 2020 2037     |.|         7
│ │ │ │ +00029a60: 3a20 2035 2031 3820 3134 2020 2e20 202e  :  5 18 14  .  .
│ │ │ │ +00029a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029aa0: 2020 2020 2020 2020 383a 2020 2e20 202e          8:  .  .
│ │ │ │ -00029ab0: 2031 3220 3336 2032 3520 2020 2020 2020   12 36 25       
│ │ │ │ -00029ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ad0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00029ae0: 2020 2020 2020 393a 2020 2e20 202e 2020        9:  .  .  
│ │ │ │ -00029af0: 2e20 202e 2032 3220 2020 2020 2020 2020  .  . 22         
│ │ │ │ -00029b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00029a90: 207c 0a7c 2020 2020 2020 2020 2038 3a20   |.|         8: 
│ │ │ │ +00029aa0: 202e 2020 2e20 3132 2033 3620 3235 2020   .  . 12 36 25  
│ │ │ │ +00029ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ad0: 0a7c 2020 2020 2020 2020 2039 3a20 202e  .|         9:  .
│ │ │ │ +00029ae0: 2020 2e20 202e 2020 2e20 3232 2020 2020    .  .  . 22    
│ │ │ │ +00029af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00029b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b50: 2020 2020 2020 7c0a 7c6f 3920 3a20 4265        |.|o9 : Be
│ │ │ │ -00029b60: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +00029b40: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ +00029b50: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ +00029b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00029b80: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00029b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029bd0: 2d2d 2b0a 7c69 3130 203a 206e 6574 4c69  --+.|i10 : netLi
│ │ │ │ -00029be0: 7374 2042 5261 6e6b 7320 6d61 7472 6978  st BRanks matrix
│ │ │ │ -00029bf0: 4661 6374 6f72 697a 6174 696f 6e28 6666  Factorization(ff
│ │ │ │ -00029c00: 2c20 6869 6768 5379 7a79 6779 204d 3029  , highSyzygy M0)
│ │ │ │ -00029c10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00029bc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ +00029bd0: 6e65 744c 6973 7420 4252 616e 6b73 206d  netList BRanks m
│ │ │ │ +00029be0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +00029bf0: 6f6e 2866 662c 2068 6967 6853 797a 7967  on(ff, highSyzyg
│ │ │ │ +00029c00: 7920 4d30 297c 0a7c 2020 2020 2020 2020  y M0)|.|        
│ │ │ │ +00029c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00029c50: 7c20 2020 2020 202b 2d2b 2d2b 2020 2020  |      +-+-+    
│ │ │ │ +00029c40: 2020 207c 0a7c 2020 2020 2020 2b2d 2b2d     |.|      +-+-
│ │ │ │ +00029c50: 2b20 2020 2020 2020 2020 2020 2020 2020  +               
│ │ │ │  00029c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029c90: 3130 203d 207c 367c 367c 2020 2020 2020  10 = |6|6|      
│ │ │ │ +00029c80: 207c 0a7c 6f31 3020 3d20 7c36 7c36 7c20   |.|o10 = |6|6| 
│ │ │ │ +00029c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00029cd0: 2020 202b 2d2b 2d2b 2020 2020 2020 2020     +-+-+        
│ │ │ │ +00029cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029cc0: 0a7c 2020 2020 2020 2b2d 2b2d 2b20 2020  .|      +-+-+   
│ │ │ │ +00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00029d10: 207c 337c 367c 2020 2020 2020 2020 2020   |3|6|          
│ │ │ │ +00029cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00029d00: 2020 2020 2020 7c33 7c36 7c20 2020 2020        |3|6|     
│ │ │ │ +00029d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d40: 2020 2020 2020 7c0a 7c20 2020 2020 202b        |.|      +
│ │ │ │ -00029d50: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00029d30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029d40: 2020 2020 2b2d 2b2d 2b20 2020 2020 2020      +-+-+       
│ │ │ │ +00029d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d80: 2020 2020 7c0a 7c20 2020 2020 207c 327c      |.|      |2|
│ │ │ │ -00029d90: 367c 2020 2020 2020 2020 2020 2020 2020  6|              
│ │ │ │ +00029d70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00029d80: 2020 7c32 7c36 7c20 2020 2020 2020 2020    |2|6|         
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dc0: 2020 7c0a 7c20 2020 2020 202b 2d2b 2d2b    |.|      +-+-+
│ │ │ │ +00029db0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00029dc0: 2b2d 2b2d 2b20 2020 2020 2020 2020 2020  +-+-+           
│ │ │ │  00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00029df0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00029e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00029e40: 0a49 6e20 7468 6973 2063 6173 6520 6173  .In this case as
│ │ │ │ -00029e50: 2069 6e20 616c 6c20 6f74 6865 7273 2077   in all others w
│ │ │ │ -00029e60: 6520 6861 7665 2065 7861 6d69 6e65 642c  e have examined,
│ │ │ │ -00029e70: 2067 7265 6174 6572 2022 4f70 7469 6d69   greater "Optimi
│ │ │ │ -00029e80: 736d 2220 6973 206e 6f74 0a6a 7573 7469  sm" is not.justi
│ │ │ │ -00029e90: 6669 6564 2c20 616e 6420 7468 7573 206d  fied, and thus m
│ │ │ │ -00029ea0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00029eb0: 6f6e 2866 662c 2068 6967 6853 797a 7967  on(ff, highSyzyg
│ │ │ │ -00029ec0: 7928 4d30 2c20 4f70 7469 6d69 736d 3d3e  y(M0, Optimism=>
│ │ │ │ -00029ed0: 3129 293b 2077 6f75 6c64 0a70 726f 6475  1)); would.produ
│ │ │ │ -00029ee0: 6365 2061 6e20 6572 726f 722e 0a0a 4361  ce an error...Ca
│ │ │ │ -00029ef0: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a41 2062  veat.======..A b
│ │ │ │ -00029f00: 7567 2069 6e20 7468 6520 746f 7461 6c20  ug in the total 
│ │ │ │ -00029f10: 4578 7420 7363 7269 7074 206d 6561 6e73  Ext script means
│ │ │ │ -00029f20: 2074 6861 7420 7468 6520 6f64 6445 7874   that the oddExt
│ │ │ │ -00029f30: 4d6f 6475 6c65 2069 7320 736f 6d65 7469  Module is someti
│ │ │ │ -00029f40: 6d65 7320 7a65 726f 2c0a 616e 6420 7468  mes zero,.and th
│ │ │ │ -00029f50: 6973 2063 616e 2063 6175 7365 2061 2077  is can cause a w
│ │ │ │ -00029f60: 726f 6e67 2076 616c 7565 2074 6f20 6265  rong value to be
│ │ │ │ -00029f70: 2072 6574 7572 6e65 642e 0a0a 5365 6520   returned...See 
│ │ │ │ -00029f80: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -00029f90: 202a 202a 6e6f 7465 2065 7665 6e45 7874   * *note evenExt
│ │ │ │ -00029fa0: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -00029fb0: 6f64 756c 652c 202d 2d20 6576 656e 2070  odule, -- even p
│ │ │ │ -00029fc0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ -00029fd0: 2920 6f76 6572 2061 0a20 2020 2063 6f6d  ) over a.    com
│ │ │ │ -00029fe0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -00029ff0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -0002a000: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -0002a010: 6e67 0a20 202a 202a 6e6f 7465 206f 6464  ng.  * *note odd
│ │ │ │ -0002a020: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ -0002a030: 744d 6f64 756c 652c 202d 2d20 6f64 6420  tModule, -- odd 
│ │ │ │ -0002a040: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ -0002a050: 6b29 206f 7665 7220 6120 636f 6d70 6c65  k) over a comple
│ │ │ │ -0002a060: 7465 0a20 2020 2069 6e74 6572 7365 6374  te.    intersect
│ │ │ │ -0002a070: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -0002a080: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -0002a090: 696e 670a 2020 2a20 2a6e 6f74 6520 6d66  ing.  * *note mf
│ │ │ │ -0002a0a0: 426f 756e 643a 206d 6642 6f75 6e64 2c20  Bound: mfBound, 
│ │ │ │ -0002a0b0: 2d2d 2064 6574 6572 6d69 6e65 7320 686f  -- determines ho
│ │ │ │ -0002a0c0: 7720 6869 6768 2061 2073 797a 7967 7920  w high a syzygy 
│ │ │ │ -0002a0d0: 746f 2074 616b 6520 666f 720a 2020 2020  to take for.    
│ │ │ │ -0002a0e0: 226d 6174 7269 7846 6163 746f 7269 7a61  "matrixFactoriza
│ │ │ │ -0002a0f0: 7469 6f6e 220a 2020 2a20 2a6e 6f74 6520  tion".  * *note 
│ │ │ │ -0002a100: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -0002a110: 696f 6e3a 206d 6174 7269 7846 6163 746f  ion: matrixFacto
│ │ │ │ -0002a120: 7269 7a61 7469 6f6e 2c20 2d2d 204d 6170  rization, -- Map
│ │ │ │ -0002a130: 7320 696e 2061 2068 6967 6865 720a 2020  s in a higher.  
│ │ │ │ -0002a140: 2020 636f 6469 6d65 6e73 696f 6e20 6d61    codimension ma
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│ │ │ │ -0002a170: 6869 6768 5379 7a79 6779 3a0a 3d3d 3d3d  highSyzygy:.====
│ │ │ │ -0002a180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002a190: 3d3d 3d0a 0a20 202a 2022 6869 6768 5379  ===..  * "highSy
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│ │ │ │ -0002a1f0: 793a 2068 6967 6853 797a 7967 792c 2069  y: highSyzygy, i
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│ │ │ │ -0002a290: 743a 2048 6f6d 5769 7468 436f 6d70 6f6e  t: HomWithCompon
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│ │ │ │ +00029e80: 6a75 7374 6966 6965 642c 2061 6e64 2074  justified, and t
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│ │ │ │ +00029eb0: 5379 7a79 6779 284d 302c 204f 7074 696d  Syzygy(M0, Optim
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│ │ │ │ +00029ed0: 7072 6f64 7563 6520 616e 2065 7272 6f72  produce an error
│ │ │ │ +00029ee0: 2e0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d  ...Caveat.======
│ │ │ │ +00029ef0: 0a0a 4120 6275 6720 696e 2074 6865 2074  ..A bug in the t
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│ │ │ │ +00029f10: 6d65 616e 7320 7468 6174 2074 6865 206f  means that the o
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│ │ │ │ +00029f60: 746f 2062 6520 7265 7475 726e 6564 2e0a  to be returned..
│ │ │ │ +00029f70: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
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│ │ │ │ +00029fa0: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065  nExtModule, -- e
│ │ │ │ +00029fb0: 7665 6e20 7061 7274 206f 6620 4578 745e  ven part of Ext^
│ │ │ │ +00029fc0: 2a28 4d2c 6b29 206f 7665 7220 610a 2020  *(M,k) over a.  
│ │ │ │ +00029fd0: 2020 636f 6d70 6c65 7465 2069 6e74 6572    complete inter
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│ │ │ │ +0002a000: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74  or ring.  * *not
│ │ │ │ +0002a010: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20  e oddExtModule: 
│ │ │ │ +0002a020: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d  oddExtModule, --
│ │ │ │ +0002a030: 206f 6464 2070 6172 7420 6f66 2045 7874   odd part of Ext
│ │ │ │ +0002a040: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063  ^*(M,k) over a c
│ │ │ │ +0002a050: 6f6d 706c 6574 650a 2020 2020 696e 7465  omplete.    inte
│ │ │ │ +0002a060: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
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│ │ │ │ +0002a080: 746f 7220 7269 6e67 0a20 202a 202a 6e6f  tor ring.  * *no
│ │ │ │ +0002a090: 7465 206d 6642 6f75 6e64 3a20 6d66 426f  te mfBound: mfBo
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│ │ │ │ +0002a0e0: 6f72 697a 6174 696f 6e22 0a20 202a 202a  orization".  * *
│ │ │ │ +0002a0f0: 6e6f 7465 206d 6174 7269 7846 6163 746f  note matrixFacto
│ │ │ │ +0002a100: 7269 7a61 7469 6f6e 3a20 6d61 7472 6978  rization: matrix
│ │ │ │ +0002a110: 4661 6374 6f72 697a 6174 696f 6e2c 202d  Factorization, -
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│ │ │ │ +0002a130: 6572 0a20 2020 2063 6f64 696d 656e 7369  er.    codimensi
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│ │ │ │ +0002a150: 697a 6174 696f 6e0a 0a57 6179 7320 746f  ization..Ways to
│ │ │ │ +0002a160: 2075 7365 2068 6967 6853 797a 7967 793a   use highSyzygy:
│ │ │ │ +0002a170: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0002a180: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268  ========..  * "h
│ │ │ │ +0002a190: 6967 6853 797a 7967 7928 4d6f 6475 6c65  ighSyzygy(Module
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│ │ │ │ +0002a1b0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +0002a1c0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
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│ │ │ │ +0002a1e0: 5379 7a79 6779 3a20 6869 6768 5379 7a79  Syzygy: highSyzy
│ │ │ │ +0002a1f0: 6779 2c20 6973 2061 202a 6e6f 7465 206d  gy, is a *note m
│ │ │ │ +0002a200: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ +0002a210: 6974 680a 6f70 7469 6f6e 733a 2028 4d61  ith.options: (Ma
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│ │ │ │ +0002a230: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
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│ │ │ │ +0002a250: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +0002a260: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
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│ │ │ │ +0002a280: 2c20 4e65 7874 3a20 486f 6d57 6974 6843  , Next: HomWithC
│ │ │ │ +0002a290: 6f6d 706f 6e65 6e74 732c 2050 7265 763a  omponents, Prev:
│ │ │ │ +0002a2a0: 2068 6967 6853 797a 7967 792c 2055 703a   highSyzygy, Up:
│ │ │ │ +0002a2b0: 2054 6f70 0a0a 684d 6170 7320 2d2d 206c   Top..hMaps -- l
│ │ │ │ +0002a2c0: 6973 7420 7468 6520 6d61 7073 2020 6828  ist the maps  h(
│ │ │ │ +0002a2d0: 7029 3a20 415f 3028 7029 2d2d 3e20 415f  p): A_0(p)--> A_
│ │ │ │ +0002a2e0: 3128 7029 2069 6e20 6120 6d61 7472 6978  1(p) in a matrix
│ │ │ │ +0002a2f0: 4661 6374 6f72 697a 6174 696f 6e0a 2a2a  Factorization.**
│ │ │ │ +0002a300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002a310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002a320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002a330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002a340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002a350: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ -0002a360: 3d3d 3d0a 0a20 202a 2055 7361 6765 3a20  ===..  * Usage: 
│ │ │ │ -0002a370: 0a20 2020 2020 2020 2068 4d61 7073 203d  .        hMaps =
│ │ │ │ -0002a380: 2068 4d61 7073 206d 660a 2020 2a20 496e   hMaps mf.  * In
│ │ │ │ -0002a390: 7075 7473 3a0a 2020 2020 2020 2a20 6d66  puts:.      * mf
│ │ │ │ -0002a3a0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -0002a3b0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -0002a3c0: 7374 2c2c 206f 7574 7075 7420 6f66 2061  st,, output of a
│ │ │ │ -0002a3d0: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0002a3e0: 7469 6f6e 0a20 2020 2020 2020 2063 6f6d  tion.        com
│ │ │ │ -0002a3f0: 7075 7461 7469 6f6e 0a20 202a 204f 7574  putation.  * Out
│ │ │ │ -0002a400: 7075 7473 3a0a 2020 2020 2020 2a20 684d  puts:.      * hM
│ │ │ │ -0002a410: 6170 732c 2061 202a 6e6f 7465 206c 6973  aps, a *note lis
│ │ │ │ -0002a420: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -0002a430: 294c 6973 742c 2c20 6c69 7374 206d 6174  )List,, list mat
│ │ │ │ -0002a440: 7269 6365 7320 2468 5f70 3a20 415f 3028  rices $h_p: A_0(
│ │ │ │ -0002a450: 7029 5c74 6f0a 2020 2020 2020 2020 415f  p)\to.        A_
│ │ │ │ -0002a460: 3128 7029 242e 2054 6865 2073 6f75 7263  1(p)$. The sourc
│ │ │ │ -0002a470: 6573 2061 6e64 2074 6172 6765 7473 206f  es and targets o
│ │ │ │ -0002a480: 6620 7468 6573 6520 6d61 7073 2068 6176  f these maps hav
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│ │ │ │ -0002a4a0: 0a20 2020 2020 2020 2042 5f73 2870 292e  .        B_s(p).
│ │ │ │ -0002a4b0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -0002a4c0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 6565 2074  =========..See t
│ │ │ │ -0002a4d0: 6865 2064 6f63 756d 656e 7461 7469 6f6e  he documentation
│ │ │ │ -0002a4e0: 2066 6f72 206d 6174 7269 7846 6163 746f   for matrixFacto
│ │ │ │ -0002a4f0: 7269 7a61 7469 6f6e 2066 6f72 2061 6e20  rization for an 
│ │ │ │ -0002a500: 6578 616d 706c 652e 0a0a 5365 6520 616c  example...See al
│ │ │ │ -0002a510: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -0002a520: 202a 6e6f 7465 206d 6174 7269 7846 6163   *note matrixFac
│ │ │ │ -0002a530: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472  torization: matr
│ │ │ │ -0002a540: 6978 4661 6374 6f72 697a 6174 696f 6e2c  ixFactorization,
│ │ │ │ -0002a550: 202d 2d20 4d61 7073 2069 6e20 6120 6869   -- Maps in a hi
│ │ │ │ -0002a560: 6768 6572 0a20 2020 2063 6f64 696d 656e  gher.    codimen
│ │ │ │ -0002a570: 7369 6f6e 206d 6174 7269 7820 6661 6374  sion matrix fact
│ │ │ │ -0002a580: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e  orization.  * *n
│ │ │ │ -0002a590: 6f74 6520 644d 6170 733a 2064 4d61 7073  ote dMaps: dMaps
│ │ │ │ -0002a5a0: 2c20 2d2d 206c 6973 7420 7468 6520 6d61  , -- list the ma
│ │ │ │ -0002a5b0: 7073 2020 6428 7029 3a41 5f31 2870 292d  ps  d(p):A_1(p)-
│ │ │ │ -0002a5c0: 2d3e 2041 5f30 2870 2920 696e 2061 0a20  -> A_0(p) in a. 
│ │ │ │ -0002a5d0: 2020 206d 6174 7269 7846 6163 746f 7269     matrixFactori
│ │ │ │ -0002a5e0: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465  zation.  * *note
│ │ │ │ -0002a5f0: 2042 5261 6e6b 733a 2042 5261 6e6b 732c   BRanks: BRanks,
│ │ │ │ -0002a600: 202d 2d20 7261 6e6b 7320 6f66 2074 6865   -- ranks of the
│ │ │ │ -0002a610: 206d 6f64 756c 6573 2042 5f69 2864 2920   modules B_i(d) 
│ │ │ │ -0002a620: 696e 2061 0a20 2020 206d 6174 7269 7846  in a.    matrixF
│ │ │ │ -0002a630: 6163 746f 7269 7a61 7469 6f6e 0a20 202a  actorization.  *
│ │ │ │ -0002a640: 202a 6e6f 7465 2062 4d61 7073 3a20 624d   *note bMaps: bM
│ │ │ │ -0002a650: 6170 732c 202d 2d20 6c69 7374 2074 6865  aps, -- list the
│ │ │ │ -0002a660: 206d 6170 7320 2064 5f70 3a42 5f31 2870   maps  d_p:B_1(p
│ │ │ │ -0002a670: 292d 2d3e 425f 3028 7029 2069 6e20 610a  )-->B_0(p) in a.
│ │ │ │ -0002a680: 2020 2020 6d61 7472 6978 4661 6374 6f72      matrixFactor
│ │ │ │ -0002a690: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74  ization.  * *not
│ │ │ │ -0002a6a0: 6520 7073 694d 6170 733a 2070 7369 4d61  e psiMaps: psiMa
│ │ │ │ -0002a6b0: 7073 2c20 2d2d 206c 6973 7420 7468 6520  ps, -- list the 
│ │ │ │ -0002a6c0: 6d61 7073 2020 7073 6928 7029 3a20 425f  maps  psi(p): B_
│ │ │ │ -0002a6d0: 3128 7029 202d 2d3e 2041 5f30 2870 2d31  1(p) --> A_0(p-1
│ │ │ │ -0002a6e0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ -0002a6f0: 7846 6163 746f 7269 7a61 7469 6f6e 0a0a  xFactorization..
│ │ │ │ -0002a700: 5761 7973 2074 6f20 7573 6520 684d 6170  Ways to use hMap
│ │ │ │ -0002a710: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
│ │ │ │ -0002a720: 3d3d 3d3d 3d0a 0a20 202a 2022 684d 6170  =====..  * "hMap
│ │ │ │ -0002a730: 7328 4c69 7374 2922 0a0a 466f 7220 7468  s(List)"..For th
│ │ │ │ -0002a740: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0002a750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0002a760: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0002a770: 6520 684d 6170 733a 2068 4d61 7073 2c20  e hMaps: hMaps, 
│ │ │ │ -0002a780: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -0002a790: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -0002a7a0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -0002a7b0: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ -0002a7c0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -0002a7d0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -0002a7e0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2048  ns.info, Node: H
│ │ │ │ -0002a7f0: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473  omWithComponents
│ │ │ │ -0002a800: 2c20 4e65 7874 3a20 696e 6669 6e69 7465  , Next: infinite
│ │ │ │ -0002a810: 4265 7474 694e 756d 6265 7273 2c20 5072  BettiNumbers, Pr
│ │ │ │ -0002a820: 6576 3a20 684d 6170 732c 2055 703a 2054  ev: hMaps, Up: T
│ │ │ │ -0002a830: 6f70 0a0a 486f 6d57 6974 6843 6f6d 706f  op..HomWithCompo
│ │ │ │ -0002a840: 6e65 6e74 7320 2d2d 2063 6f6d 7075 7465  nents -- compute
│ │ │ │ -0002a850: 7320 486f 6d2c 2070 7265 7365 7276 696e  s Hom, preservin
│ │ │ │ -0002a860: 6720 6469 7265 6374 2073 756d 2069 6e66  g direct sum inf
│ │ │ │ -0002a870: 6f72 6d61 7469 6f6e 0a2a 2a2a 2a2a 2a2a  ormation.*******
│ │ │ │ +0002a340: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +0002a350: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +0002a360: 6167 653a 200a 2020 2020 2020 2020 684d  age: .        hM
│ │ │ │ +0002a370: 6170 7320 3d20 684d 6170 7320 6d66 0a20  aps = hMaps mf. 
│ │ │ │ +0002a380: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +0002a390: 202a 206d 662c 2061 202a 6e6f 7465 206c   * mf, a *note l
│ │ │ │ +0002a3a0: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ +0002a3b0: 6f63 294c 6973 742c 2c20 6f75 7470 7574  oc)List,, output
│ │ │ │ +0002a3c0: 206f 6620 6120 6d61 7472 6978 4661 6374   of a matrixFact
│ │ │ │ +0002a3d0: 6f72 697a 6174 696f 6e0a 2020 2020 2020  orization.      
│ │ │ │ +0002a3e0: 2020 636f 6d70 7574 6174 696f 6e0a 2020    computation.  
│ │ │ │ +0002a3f0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +0002a400: 202a 2068 4d61 7073 2c20 6120 2a6e 6f74   * hMaps, a *not
│ │ │ │ +0002a410: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +0002a420: 7932 446f 6329 4c69 7374 2c2c 206c 6973  y2Doc)List,, lis
│ │ │ │ +0002a430: 7420 6d61 7472 6963 6573 2024 685f 703a  t matrices $h_p:
│ │ │ │ +0002a440: 2041 5f30 2870 295c 746f 0a20 2020 2020   A_0(p)\to.     
│ │ │ │ +0002a450: 2020 2041 5f31 2870 2924 2e20 5468 6520     A_1(p)$. The 
│ │ │ │ +0002a460: 736f 7572 6365 7320 616e 6420 7461 7267  sources and targ
│ │ │ │ +0002a470: 6574 7320 6f66 2074 6865 7365 206d 6170  ets of these map
│ │ │ │ +0002a480: 7320 6861 7665 2074 6865 2063 6f6d 706f  s have the compo
│ │ │ │ +0002a490: 6e65 6e74 730a 2020 2020 2020 2020 425f  nents.        B_
│ │ │ │ +0002a4a0: 7328 7029 2e0a 0a44 6573 6372 6970 7469  s(p)...Descripti
│ │ │ │ +0002a4b0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +0002a4c0: 5365 6520 7468 6520 646f 6375 6d65 6e74  See the document
│ │ │ │ +0002a4d0: 6174 696f 6e20 666f 7220 6d61 7472 6978  ation for matrix
│ │ │ │ +0002a4e0: 4661 6374 6f72 697a 6174 696f 6e20 666f  Factorization fo
│ │ │ │ +0002a4f0: 7220 616e 2065 7861 6d70 6c65 2e0a 0a53  r an example...S
│ │ │ │ +0002a500: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +0002a510: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ +0002a520: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ +0002a530: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0002a540: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ +0002a550: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ +0002a560: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ +0002a570: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ +0002a580: 202a 202a 6e6f 7465 2064 4d61 7073 3a20   * *note dMaps: 
│ │ │ │ +0002a590: 644d 6170 732c 202d 2d20 6c69 7374 2074  dMaps, -- list t
│ │ │ │ +0002a5a0: 6865 206d 6170 7320 2064 2870 293a 415f  he maps  d(p):A_
│ │ │ │ +0002a5b0: 3128 7029 2d2d 3e20 415f 3028 7029 2069  1(p)--> A_0(p) i
│ │ │ │ +0002a5c0: 6e20 610a 2020 2020 6d61 7472 6978 4661  n a.    matrixFa
│ │ │ │ +0002a5d0: 6374 6f72 697a 6174 696f 6e0a 2020 2a20  ctorization.  * 
│ │ │ │ +0002a5e0: 2a6e 6f74 6520 4252 616e 6b73 3a20 4252  *note BRanks: BR
│ │ │ │ +0002a5f0: 616e 6b73 2c20 2d2d 2072 616e 6b73 206f  anks, -- ranks o
│ │ │ │ +0002a600: 6620 7468 6520 6d6f 6475 6c65 7320 425f  f the modules B_
│ │ │ │ +0002a610: 6928 6429 2069 6e20 610a 2020 2020 6d61  i(d) in a.    ma
│ │ │ │ +0002a620: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0002a630: 6e0a 2020 2a20 2a6e 6f74 6520 624d 6170  n.  * *note bMap
│ │ │ │ +0002a640: 733a 2062 4d61 7073 2c20 2d2d 206c 6973  s: bMaps, -- lis
│ │ │ │ +0002a650: 7420 7468 6520 6d61 7073 2020 645f 703a  t the maps  d_p:
│ │ │ │ +0002a660: 425f 3128 7029 2d2d 3e42 5f30 2870 2920  B_1(p)-->B_0(p) 
│ │ │ │ +0002a670: 696e 2061 0a20 2020 206d 6174 7269 7846  in a.    matrixF
│ │ │ │ +0002a680: 6163 746f 7269 7a61 7469 6f6e 0a20 202a  actorization.  *
│ │ │ │ +0002a690: 202a 6e6f 7465 2070 7369 4d61 7073 3a20   *note psiMaps: 
│ │ │ │ +0002a6a0: 7073 694d 6170 732c 202d 2d20 6c69 7374  psiMaps, -- list
│ │ │ │ +0002a6b0: 2074 6865 206d 6170 7320 2070 7369 2870   the maps  psi(p
│ │ │ │ +0002a6c0: 293a 2042 5f31 2870 2920 2d2d 3e20 415f  ): B_1(p) --> A_
│ │ │ │ +0002a6d0: 3028 702d 3129 2069 6e20 610a 2020 2020  0(p-1) in a.    
│ │ │ │ +0002a6e0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0002a6f0: 696f 6e0a 0a57 6179 7320 746f 2075 7365  ion..Ways to use
│ │ │ │ +0002a700: 2068 4d61 7073 3a0a 3d3d 3d3d 3d3d 3d3d   hMaps:.========
│ │ │ │ +0002a710: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0002a720: 2268 4d61 7073 284c 6973 7429 220a 0a46  "hMaps(List)"..F
│ │ │ │ +0002a730: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +0002a740: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +0002a750: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +0002a760: 202a 6e6f 7465 2068 4d61 7073 3a20 684d   *note hMaps: hM
│ │ │ │ +0002a770: 6170 732c 2069 7320 6120 2a6e 6f74 6520  aps, is a *note 
│ │ │ │ +0002a780: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +0002a790: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +0002a7a0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +0002a7b0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +0002a7c0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +0002a7d0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +0002a7e0: 6465 3a20 486f 6d57 6974 6843 6f6d 706f  de: HomWithCompo
│ │ │ │ +0002a7f0: 6e65 6e74 732c 204e 6578 743a 2069 6e66  nents, Next: inf
│ │ │ │ +0002a800: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0002a810: 732c 2050 7265 763a 2068 4d61 7073 2c20  s, Prev: hMaps, 
│ │ │ │ +0002a820: 5570 3a20 546f 700a 0a48 6f6d 5769 7468  Up: Top..HomWith
│ │ │ │ +0002a830: 436f 6d70 6f6e 656e 7473 202d 2d20 636f  Components -- co
│ │ │ │ +0002a840: 6d70 7574 6573 2048 6f6d 2c20 7072 6573  mputes Hom, pres
│ │ │ │ +0002a850: 6572 7669 6e67 2064 6972 6563 7420 7375  erving direct su
│ │ │ │ +0002a860: 6d20 696e 666f 726d 6174 696f 6e0a 2a2a  m information.**
│ │ │ │ +0002a870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002a880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002a890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002a8a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002a8b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -0002a8c0: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -0002a8d0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -0002a8e0: 2020 2020 2020 4820 3d20 486f 6d28 4d2c        H = Hom(M,
│ │ │ │ -0002a8f0: 4e29 0a20 202a 2049 6e70 7574 733a 0a20  N).  * Inputs:. 
│ │ │ │ -0002a900: 2020 2020 202a 204d 2c20 6120 2a6e 6f74       * M, a *not
│ │ │ │ -0002a910: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ -0002a920: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ -0002a930: 200a 2020 2020 2020 2a20 4e2c 2061 202a   .      * N, a *
│ │ │ │ -0002a940: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0002a950: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0002a960: 652c 2c20 0a20 202a 204f 7574 7075 7473  e,, .  * Outputs
│ │ │ │ -0002a970: 3a0a 2020 2020 2020 2a20 482c 2061 202a  :.      * H, a *
│ │ │ │ -0002a980: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0002a990: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0002a9a0: 652c 2c20 0a0a 4465 7363 7269 7074 696f  e,, ..Descriptio
│ │ │ │ -0002a9b0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49  n.===========..I
│ │ │ │ -0002a9c0: 6620 4d20 616e 642f 6f72 204e 2061 7265  f M and/or N are
│ │ │ │ -0002a9d0: 2064 6972 6563 7420 7375 6d20 6d6f 6475   direct sum modu
│ │ │ │ -0002a9e0: 6c65 7320 2869 7344 6972 6563 7453 756d  les (isDirectSum
│ │ │ │ -0002a9f0: 204d 203d 3d20 7472 7565 2920 7468 656e   M == true) then
│ │ │ │ -0002aa00: 2048 2069 7320 7468 650a 6469 7265 6374   H is the.direct
│ │ │ │ -0002aa10: 2073 756d 206f 6620 7468 6520 486f 6d73   sum of the Homs
│ │ │ │ -0002aa20: 2062 6574 7765 656e 2074 6865 2063 6f6d   between the com
│ │ │ │ -0002aa30: 706f 6e65 6e74 732e 2054 6869 7320 5348  ponents. This SH
│ │ │ │ -0002aa40: 4f55 4c44 2062 6520 6275 696c 7420 696e  OULD be built in
│ │ │ │ -0002aa50: 746f 0a48 6f6d 284d 2c4e 292c 2062 7574  to.Hom(M,N), but
│ │ │ │ -0002aa60: 2069 736e 2774 2061 7320 6f66 204d 322c   isn't as of M2,
│ │ │ │ -0002aa70: 2076 2e20 312e 370a 0a53 6565 2061 6c73   v. 1.7..See als
│ │ │ │ -0002aa80: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -0002aa90: 2a6e 6f74 6520 7465 6e73 6f72 5769 7468  *note tensorWith
│ │ │ │ -0002aaa0: 436f 6d70 6f6e 656e 7473 3a20 7465 6e73  Components: tens
│ │ │ │ -0002aab0: 6f72 5769 7468 436f 6d70 6f6e 656e 7473  orWithComponents
│ │ │ │ -0002aac0: 2c20 2d2d 2066 6f72 6d73 2074 6865 2074  , -- forms the t
│ │ │ │ -0002aad0: 656e 736f 720a 2020 2020 7072 6f64 7563  ensor.    produc
│ │ │ │ -0002aae0: 742c 2070 7265 7365 7276 696e 6720 6469  t, preserving di
│ │ │ │ -0002aaf0: 7265 6374 2073 756d 2069 6e66 6f72 6d61  rect sum informa
│ │ │ │ -0002ab00: 7469 6f6e 0a20 202a 202a 6e6f 7465 2064  tion.  * *note d
│ │ │ │ -0002ab10: 7561 6c57 6974 6843 6f6d 706f 6e65 6e74  ualWithComponent
│ │ │ │ -0002ab20: 733a 2064 7561 6c57 6974 6843 6f6d 706f  s: dualWithCompo
│ │ │ │ -0002ab30: 6e65 6e74 732c 202d 2d20 6475 616c 206d  nents, -- dual m
│ │ │ │ -0002ab40: 6f64 756c 6520 7072 6573 6572 7669 6e67  odule preserving
│ │ │ │ -0002ab50: 0a20 2020 2064 6972 6563 7420 7375 6d20  .    direct sum 
│ │ │ │ -0002ab60: 696e 666f 726d 6174 696f 6e0a 0a57 6179  information..Way
│ │ │ │ -0002ab70: 7320 746f 2075 7365 2048 6f6d 5769 7468  s to use HomWith
│ │ │ │ -0002ab80: 436f 6d70 6f6e 656e 7473 3a0a 3d3d 3d3d  Components:.====
│ │ │ │ -0002ab90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002aba0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0002abb0: 2248 6f6d 5769 7468 436f 6d70 6f6e 656e  "HomWithComponen
│ │ │ │ -0002abc0: 7473 284d 6f64 756c 652c 4d6f 6475 6c65  ts(Module,Module
│ │ │ │ -0002abd0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ -0002abe0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -0002abf0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -0002ac00: 626a 6563 7420 2a6e 6f74 6520 486f 6d57  bject *note HomW
│ │ │ │ -0002ac10: 6974 6843 6f6d 706f 6e65 6e74 733a 2048  ithComponents: H
│ │ │ │ -0002ac20: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473  omWithComponents
│ │ │ │ -0002ac30: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0002ac40: 686f 640a 6675 6e63 7469 6f6e 3a20 284d  hod.function: (M
│ │ │ │ -0002ac50: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0002ac60: 6f64 4675 6e63 7469 6f6e 2c2e 0a1f 0a46  odFunction,....F
│ │ │ │ -0002ac70: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0002ac80: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0002ac90: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0002aca0: 2069 6e66 696e 6974 6542 6574 7469 4e75   infiniteBettiNu
│ │ │ │ -0002acb0: 6d62 6572 732c 204e 6578 743a 2069 734c  mbers, Next: isL
│ │ │ │ -0002acc0: 696e 6561 722c 2050 7265 763a 2048 6f6d  inear, Prev: Hom
│ │ │ │ -0002acd0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ -0002ace0: 5570 3a20 546f 700a 0a69 6e66 696e 6974  Up: Top..infinit
│ │ │ │ -0002acf0: 6542 6574 7469 4e75 6d62 6572 7320 2d2d  eBettiNumbers --
│ │ │ │ -0002ad00: 2062 6574 7469 206e 756d 6265 7273 206f   betti numbers o
│ │ │ │ -0002ad10: 6620 6669 6e69 7465 2072 6573 6f6c 7574  f finite resolut
│ │ │ │ -0002ad20: 696f 6e20 636f 6d70 7574 6564 2066 726f  ion computed fro
│ │ │ │ -0002ad30: 6d20 6120 6d61 7472 6978 2066 6163 746f  m a matrix facto
│ │ │ │ -0002ad40: 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a  rization.*******
│ │ │ │ +0002a8b0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +0002a8c0: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +0002a8d0: 3a20 0a20 2020 2020 2020 2048 203d 2048  : .        H = H
│ │ │ │ +0002a8e0: 6f6d 284d 2c4e 290a 2020 2a20 496e 7075  om(M,N).  * Inpu
│ │ │ │ +0002a8f0: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061  ts:.      * M, a
│ │ │ │ +0002a900: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ +0002a910: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ +0002a920: 756c 652c 2c20 0a20 2020 2020 202a 204e  ule,, .      * N
│ │ │ │ +0002a930: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0002a940: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0002a950: 4d6f 6475 6c65 2c2c 200a 2020 2a20 4f75  Module,, .  * Ou
│ │ │ │ +0002a960: 7470 7574 733a 0a20 2020 2020 202a 2048  tputs:.      * H
│ │ │ │ +0002a970: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0002a980: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0002a990: 4d6f 6475 6c65 2c2c 200a 0a44 6573 6372  Module,, ..Descr
│ │ │ │ +0002a9a0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0002a9b0: 3d3d 0a0a 4966 204d 2061 6e64 2f6f 7220  ==..If M and/or 
│ │ │ │ +0002a9c0: 4e20 6172 6520 6469 7265 6374 2073 756d  N are direct sum
│ │ │ │ +0002a9d0: 206d 6f64 756c 6573 2028 6973 4469 7265   modules (isDire
│ │ │ │ +0002a9e0: 6374 5375 6d20 4d20 3d3d 2074 7275 6529  ctSum M == true)
│ │ │ │ +0002a9f0: 2074 6865 6e20 4820 6973 2074 6865 0a64   then H is the.d
│ │ │ │ +0002aa00: 6972 6563 7420 7375 6d20 6f66 2074 6865  irect sum of the
│ │ │ │ +0002aa10: 2048 6f6d 7320 6265 7477 6565 6e20 7468   Homs between th
│ │ │ │ +0002aa20: 6520 636f 6d70 6f6e 656e 7473 2e20 5468  e components. Th
│ │ │ │ +0002aa30: 6973 2053 484f 554c 4420 6265 2062 7569  is SHOULD be bui
│ │ │ │ +0002aa40: 6c74 2069 6e74 6f0a 486f 6d28 4d2c 4e29  lt into.Hom(M,N)
│ │ │ │ +0002aa50: 2c20 6275 7420 6973 6e27 7420 6173 206f  , but isn't as o
│ │ │ │ +0002aa60: 6620 4d32 2c20 762e 2031 2e37 0a0a 5365  f M2, v. 1.7..Se
│ │ │ │ +0002aa70: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +0002aa80: 0a20 202a 202a 6e6f 7465 2074 656e 736f  .  * *note tenso
│ │ │ │ +0002aa90: 7257 6974 6843 6f6d 706f 6e65 6e74 733a  rWithComponents:
│ │ │ │ +0002aaa0: 2074 656e 736f 7257 6974 6843 6f6d 706f   tensorWithCompo
│ │ │ │ +0002aab0: 6e65 6e74 732c 202d 2d20 666f 726d 7320  nents, -- forms 
│ │ │ │ +0002aac0: 7468 6520 7465 6e73 6f72 0a20 2020 2070  the tensor.    p
│ │ │ │ +0002aad0: 726f 6475 6374 2c20 7072 6573 6572 7669  roduct, preservi
│ │ │ │ +0002aae0: 6e67 2064 6972 6563 7420 7375 6d20 696e  ng direct sum in
│ │ │ │ +0002aaf0: 666f 726d 6174 696f 6e0a 2020 2a20 2a6e  formation.  * *n
│ │ │ │ +0002ab00: 6f74 6520 6475 616c 5769 7468 436f 6d70  ote dualWithComp
│ │ │ │ +0002ab10: 6f6e 656e 7473 3a20 6475 616c 5769 7468  onents: dualWith
│ │ │ │ +0002ab20: 436f 6d70 6f6e 656e 7473 2c20 2d2d 2064  Components, -- d
│ │ │ │ +0002ab30: 7561 6c20 6d6f 6475 6c65 2070 7265 7365  ual module prese
│ │ │ │ +0002ab40: 7276 696e 670a 2020 2020 6469 7265 6374  rving.    direct
│ │ │ │ +0002ab50: 2073 756d 2069 6e66 6f72 6d61 7469 6f6e   sum information
│ │ │ │ +0002ab60: 0a0a 5761 7973 2074 6f20 7573 6520 486f  ..Ways to use Ho
│ │ │ │ +0002ab70: 6d57 6974 6843 6f6d 706f 6e65 6e74 733a  mWithComponents:
│ │ │ │ +0002ab80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0002ab90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002aba0: 0a20 202a 2022 486f 6d57 6974 6843 6f6d  .  * "HomWithCom
│ │ │ │ +0002abb0: 706f 6e65 6e74 7328 4d6f 6475 6c65 2c4d  ponents(Module,M
│ │ │ │ +0002abc0: 6f64 756c 6529 220a 0a46 6f72 2074 6865  odule)"..For the
│ │ │ │ +0002abd0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +0002abe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0002abf0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +0002ac00: 2048 6f6d 5769 7468 436f 6d70 6f6e 656e   HomWithComponen
│ │ │ │ +0002ac10: 7473 3a20 486f 6d57 6974 6843 6f6d 706f  ts: HomWithCompo
│ │ │ │ +0002ac20: 6e65 6e74 732c 2069 7320 6120 2a6e 6f74  nents, is a *not
│ │ │ │ +0002ac30: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ +0002ac40: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
│ │ │ │ +0002ac50: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0002ac60: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0002ac70: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0002ac80: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0002ac90: 4e6f 6465 3a20 696e 6669 6e69 7465 4265  Node: infiniteBe
│ │ │ │ +0002aca0: 7474 694e 756d 6265 7273 2c20 4e65 7874  ttiNumbers, Next
│ │ │ │ +0002acb0: 3a20 6973 4c69 6e65 6172 2c20 5072 6576  : isLinear, Prev
│ │ │ │ +0002acc0: 3a20 486f 6d57 6974 6843 6f6d 706f 6e65  : HomWithCompone
│ │ │ │ +0002acd0: 6e74 732c 2055 703a 2054 6f70 0a0a 696e  nts, Up: Top..in
│ │ │ │ +0002ace0: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ +0002acf0: 7273 202d 2d20 6265 7474 6920 6e75 6d62  rs -- betti numb
│ │ │ │ +0002ad00: 6572 7320 6f66 2066 696e 6974 6520 7265  ers of finite re
│ │ │ │ +0002ad10: 736f 6c75 7469 6f6e 2063 6f6d 7075 7465  solution compute
│ │ │ │ +0002ad20: 6420 6672 6f6d 2061 206d 6174 7269 7820  d from a matrix 
│ │ │ │ +0002ad30: 6661 6374 6f72 697a 6174 696f 6e0a 2a2a  factorization.**
│ │ │ │ +0002ad40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ad50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ad60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ad70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ad80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ad90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ada0: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ -0002adb0: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ -0002adc0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0002add0: 204c 203d 2066 696e 6974 6542 6574 7469   L = finiteBetti
│ │ │ │ -0002ade0: 4e75 6d62 6572 7320 284d 462c 6c65 6e29  Numbers (MF,len)
│ │ │ │ -0002adf0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -0002ae00: 2020 202a 204d 462c 2061 202a 6e6f 7465     * MF, a *note
│ │ │ │ -0002ae10: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -0002ae20: 3244 6f63 294c 6973 742c 2c20 4c69 7374  2Doc)List,, List
│ │ │ │ -0002ae30: 206f 6620 4861 7368 5461 626c 6573 2061   of HashTables a
│ │ │ │ -0002ae40: 7320 636f 6d70 7574 6564 0a20 2020 2020  s computed.     
│ │ │ │ -0002ae50: 2020 2062 7920 226d 6174 7269 7846 6163     by "matrixFac
│ │ │ │ -0002ae60: 746f 7269 7a61 7469 6f6e 220a 2020 2020  torization".    
│ │ │ │ -0002ae70: 2020 2a20 6c65 6e2c 2061 6e20 2a6e 6f74    * len, an *not
│ │ │ │ -0002ae80: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ -0002ae90: 756c 6179 3244 6f63 295a 5a2c 2c20 6c65  ulay2Doc)ZZ,, le
│ │ │ │ -0002aea0: 6e67 7468 206f 6620 6265 7474 6920 6e75  ngth of betti nu
│ │ │ │ -0002aeb0: 6d62 6572 0a20 2020 2020 2020 2073 6571  mber.        seq
│ │ │ │ -0002aec0: 7565 6e63 6520 746f 2070 726f 6475 6365  uence to produce
│ │ │ │ -0002aed0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -0002aee0: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -0002aef0: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -0002af00: 3244 6f63 294c 6973 742c 2c20 4c69 7374  2Doc)List,, List
│ │ │ │ -0002af10: 206f 6620 6265 7474 6920 6e75 6d62 6572   of betti number
│ │ │ │ -0002af20: 730a 0a44 6573 6372 6970 7469 6f6e 0a3d  s..Description.=
│ │ │ │ -0002af30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5573 6573  ==========..Uses
│ │ │ │ -0002af40: 2074 6865 2072 616e 6b73 206f 6620 7468   the ranks of th
│ │ │ │ -0002af50: 6520 4220 6d61 7472 6963 6573 2069 6e20  e B matrices in 
│ │ │ │ -0002af60: 6120 6d61 7472 6978 2066 6163 746f 7269  a matrix factori
│ │ │ │ -0002af70: 7a61 7469 6f6e 2066 6f72 2061 206d 6f64  zation for a mod
│ │ │ │ -0002af80: 756c 6520 4d20 6f76 6572 0a53 2f28 665f  ule M over.S/(f_
│ │ │ │ -0002af90: 312c 2e2e 2c66 5f63 2920 746f 2063 6f6d  1,..,f_c) to com
│ │ │ │ -0002afa0: 7075 7465 2074 6865 2062 6574 7469 206e  pute the betti n
│ │ │ │ -0002afb0: 756d 6265 7273 206f 6620 7468 6520 6d69  umbers of the mi
│ │ │ │ -0002afc0: 6e69 6d61 6c20 7265 736f 6c75 7469 6f6e  nimal resolution
│ │ │ │ -0002afd0: 206f 6620 4d20 6f76 6572 0a52 2c20 7768   of M over.R, wh
│ │ │ │ -0002afe0: 6963 6820 6973 2074 6865 2073 756d 206f  ich is the sum o
│ │ │ │ -0002aff0: 6620 7468 6520 6469 7669 6465 6420 706f  f the divided po
│ │ │ │ -0002b000: 7765 7220 616c 6765 6272 6173 206f 6e20  wer algebras on 
│ │ │ │ -0002b010: 632d 6a2b 3120 7661 7269 6162 6c65 7320  c-j+1 variables 
│ │ │ │ -0002b020: 7465 6e73 6f72 6564 0a77 6974 6820 4228  tensored.with B(
│ │ │ │ -0002b030: 6a29 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  j)...+----------
│ │ │ │ +0002ad90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ +0002ada0: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ +0002adb0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +0002adc0: 2020 2020 2020 4c20 3d20 6669 6e69 7465        L = finite
│ │ │ │ +0002add0: 4265 7474 694e 756d 6265 7273 2028 4d46  BettiNumbers (MF
│ │ │ │ +0002ade0: 2c6c 656e 290a 2020 2a20 496e 7075 7473  ,len).  * Inputs
│ │ │ │ +0002adf0: 3a0a 2020 2020 2020 2a20 4d46 2c20 6120  :.      * MF, a 
│ │ │ │ +0002ae00: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +0002ae10: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +0002ae20: 204c 6973 7420 6f66 2048 6173 6854 6162   List of HashTab
│ │ │ │ +0002ae30: 6c65 7320 6173 2063 6f6d 7075 7465 640a  les as computed.
│ │ │ │ +0002ae40: 2020 2020 2020 2020 6279 2022 6d61 7472          by "matr
│ │ │ │ +0002ae50: 6978 4661 6374 6f72 697a 6174 696f 6e22  ixFactorization"
│ │ │ │ +0002ae60: 0a20 2020 2020 202a 206c 656e 2c20 616e  .      * len, an
│ │ │ │ +0002ae70: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +0002ae80: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +0002ae90: 2c2c 206c 656e 6774 6820 6f66 2062 6574  ,, length of bet
│ │ │ │ +0002aea0: 7469 206e 756d 6265 720a 2020 2020 2020  ti number.      
│ │ │ │ +0002aeb0: 2020 7365 7175 656e 6365 2074 6f20 7072    sequence to pr
│ │ │ │ +0002aec0: 6f64 7563 650a 2020 2a20 4f75 7470 7574  oduce.  * Output
│ │ │ │ +0002aed0: 733a 0a20 2020 2020 202a 204c 2c20 6120  s:.      * L, a 
│ │ │ │ +0002aee0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +0002aef0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +0002af00: 204c 6973 7420 6f66 2062 6574 7469 206e   List of betti n
│ │ │ │ +0002af10: 756d 6265 7273 0a0a 4465 7363 7269 7074  umbers..Descript
│ │ │ │ +0002af20: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +0002af30: 0a55 7365 7320 7468 6520 7261 6e6b 7320  .Uses the ranks 
│ │ │ │ +0002af40: 6f66 2074 6865 2042 206d 6174 7269 6365  of the B matrice
│ │ │ │ +0002af50: 7320 696e 2061 206d 6174 7269 7820 6661  s in a matrix fa
│ │ │ │ +0002af60: 6374 6f72 697a 6174 696f 6e20 666f 7220  ctorization for 
│ │ │ │ +0002af70: 6120 6d6f 6475 6c65 204d 206f 7665 720a  a module M over.
│ │ │ │ +0002af80: 532f 2866 5f31 2c2e 2e2c 665f 6329 2074  S/(f_1,..,f_c) t
│ │ │ │ +0002af90: 6f20 636f 6d70 7574 6520 7468 6520 6265  o compute the be
│ │ │ │ +0002afa0: 7474 6920 6e75 6d62 6572 7320 6f66 2074  tti numbers of t
│ │ │ │ +0002afb0: 6865 206d 696e 696d 616c 2072 6573 6f6c  he minimal resol
│ │ │ │ +0002afc0: 7574 696f 6e20 6f66 204d 206f 7665 720a  ution of M over.
│ │ │ │ +0002afd0: 522c 2077 6869 6368 2069 7320 7468 6520  R, which is the 
│ │ │ │ +0002afe0: 7375 6d20 6f66 2074 6865 2064 6976 6964  sum of the divid
│ │ │ │ +0002aff0: 6564 2070 6f77 6572 2061 6c67 6562 7261  ed power algebra
│ │ │ │ +0002b000: 7320 6f6e 2063 2d6a 2b31 2076 6172 6961  s on c-j+1 varia
│ │ │ │ +0002b010: 626c 6573 2074 656e 736f 7265 640a 7769  bles tensored.wi
│ │ │ │ +0002b020: 7468 2042 286a 292e 0a0a 2b2d 2d2d 2d2d  th B(j)...+-----
│ │ │ │ +0002b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002b060: 6931 203a 2073 6574 5261 6e64 6f6d 5365  i1 : setRandomSe
│ │ │ │ -0002b070: 6564 2030 2020 2020 2020 2020 2020 2020  ed 0            
│ │ │ │ -0002b080: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002b050: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ +0002b060: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0b0: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ +0002b0a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ +0002b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002b0d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b100: 2d2d 2d2d 2d2b 0a7c 6932 203a 206b 6b20  -----+.|i2 : kk 
│ │ │ │ -0002b110: 3d20 5a5a 2f31 3031 2020 2020 2020 2020  = ZZ/101        
│ │ │ │ -0002b120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b150: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -0002b160: 206b 6b20 2020 2020 2020 2020 2020 2020   kk             
│ │ │ │ -0002b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b180: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +0002b100: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020  : kk = ZZ/101   
│ │ │ │ +0002b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b120: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b150: 7c6f 3220 3d20 6b6b 2020 2020 2020 2020  |o2 = kk        
│ │ │ │ +0002b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002b1b0: 6f32 203a 2051 756f 7469 656e 7452 696e  o2 : QuotientRin
│ │ │ │ -0002b1c0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -0002b1d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002b1a0: 2020 7c0a 7c6f 3220 3a20 5175 6f74 6965    |.|o2 : Quotie
│ │ │ │ +0002b1b0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0002b1c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b200: 2d2b 0a7c 6933 203a 2053 203d 206b 6b5b  -+.|i3 : S = kk[
│ │ │ │ -0002b210: 612c 622c 752c 765d 2020 2020 2020 2020  a,b,u,v]        
│ │ │ │ -0002b220: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002b1f0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5320  ------+.|i3 : S 
│ │ │ │ +0002b200: 3d20 6b6b 5b61 2c62 2c75 2c76 5d20 2020  = kk[a,b,u,v]   
│ │ │ │ +0002b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b220: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0002b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b250: 2020 2020 207c 0a7c 6f33 203d 2053 2020       |.|o3 = S  
│ │ │ │ +0002b240: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +0002b250: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │  0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b280: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2a0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -0002b2b0: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ -0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002b270: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b2a0: 7c6f 3320 3a20 506f 6c79 6e6f 6d69 616c  |o3 : Polynomial
│ │ │ │ +0002b2b0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0002b2c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002b300: 6934 203a 2066 6620 3d20 6d61 7472 6978  i4 : ff = matrix
│ │ │ │ -0002b310: 2261 752c 6276 2220 2020 2020 2020 2020  "au,bv"         
│ │ │ │ -0002b320: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002b2f0: 2d2d 2b0a 7c69 3420 3a20 6666 203d 206d  --+.|i4 : ff = m
│ │ │ │ +0002b300: 6174 7269 7822 6175 2c62 7622 2020 2020  atrix"au,bv"    
│ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b350: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -0002b360: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0002b370: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002b340: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20        |.|o4 = | 
│ │ │ │ +0002b350: 6175 2062 7620 7c20 2020 2020 2020 2020  au bv |         
│ │ │ │ +0002b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b370: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0002b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002b3b0: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ -0002b3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b3d0: 0a7c 6f34 203a 204d 6174 7269 7820 5320  .|o4 : Matrix S 
│ │ │ │ -0002b3e0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ -0002b3f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002b390: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002b3a0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +0002b3b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002b3c0: 2020 2020 7c0a 7c6f 3420 3a20 4d61 7472      |.|o4 : Matr
│ │ │ │ +0002b3d0: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b3f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b420: 2d2d 2d2b 0a7c 6935 203a 2052 203d 2053  ---+.|i5 : R = S
│ │ │ │ -0002b430: 2f69 6465 616c 2066 6620 2020 2020 2020  /ideal ff       
│ │ │ │ -0002b440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b410: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +0002b420: 5220 3d20 532f 6964 6561 6c20 6666 2020  R = S/ideal ff  
│ │ │ │ +0002b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b440: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b470: 2020 2020 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │ +0002b460: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002b470: 3520 3d20 5220 2020 2020 2020 2020 2020  5 = R           
│ │ │ │  0002b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b490: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4c0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -0002b4d0: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ -0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002b4c0: 7c0a 7c6f 3520 3a20 5175 6f74 6965 6e74  |.|o5 : Quotient
│ │ │ │ +0002b4d0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0002b4e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002b520: 0a7c 6936 203a 204d 3020 3d20 525e 312f  .|i6 : M0 = R^1/
│ │ │ │ -0002b530: 6964 6561 6c22 612c 6222 2020 2020 2020  ideal"a,b"      
│ │ │ │ -0002b540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002b510: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d30 203d  ----+.|i6 : M0 =
│ │ │ │ +0002b520: 2052 5e31 2f69 6465 616c 2261 2c62 2220   R^1/ideal"a,b" 
│ │ │ │ +0002b530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b540: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b570: 2020 207c 0a7c 6f36 203d 2063 6f6b 6572     |.|o6 = coker
│ │ │ │ -0002b580: 6e65 6c20 7c20 6120 6220 7c20 2020 2020  nel | a b |     
│ │ │ │ -0002b590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b560: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +0002b570: 636f 6b65 726e 656c 207c 2061 2062 207c  cokernel | a b |
│ │ │ │ +0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b590: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5e0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -0002b5f0: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -0002b600: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0002b610: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5d0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0002b5e0: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ +0002b5f0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +0002b600: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +0002b610: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0002b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b640: 2d2d 2d2d 2d2b 0a7c 6937 203a 2046 203d  -----+.|i7 : F =
│ │ │ │ -0002b650: 2072 6573 284d 302c 204c 656e 6774 684c   res(M0, LengthL
│ │ │ │ -0002b660: 696d 6974 203d 3e33 2920 2020 2020 207c  imit =>3)      |
│ │ │ │ -0002b670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b690: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002b6a0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ -0002b6b0: 3320 2020 2020 2034 2020 2020 2020 2020  3      4        
│ │ │ │ -0002b6c0: 2020 207c 0a7c 6f37 203d 2052 2020 3c2d     |.|o7 = R  <-
│ │ │ │ -0002b6d0: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20  - R  <-- R  <-- 
│ │ │ │ -0002b6e0: 5220 2020 2020 2020 2020 2020 207c 0a7c  R            |.|
│ │ │ │ +0002b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0002b640: 3a20 4620 3d20 7265 7328 4d30 2c20 4c65  : F = res(M0, Le
│ │ │ │ +0002b650: 6e67 7468 4c69 6d69 7420 3d3e 3329 2020  ngthLimit =>3)  
│ │ │ │ +0002b660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b690: 7c20 2020 2020 2031 2020 2020 2020 3220  |      1      2 
│ │ │ │ +0002b6a0: 2020 2020 2033 2020 2020 2020 3420 2020       3      4   
│ │ │ │ +0002b6b0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +0002b6c0: 5220 203c 2d2d 2052 2020 3c2d 2d20 5220  R  <-- R  <-- R 
│ │ │ │ +0002b6d0: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +0002b6e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b710: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ -0002b720: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ -0002b730: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0002b740: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002b710: 2020 2020 3020 2020 2020 2031 2020 2020      0      1    
│ │ │ │ +0002b720: 2020 3220 2020 2020 2033 2020 2020 2020    2      3      
│ │ │ │ +0002b730: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b760: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0002b770: 203a 2043 6861 696e 436f 6d70 6c65 7820   : ChainComplex 
│ │ │ │ -0002b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b790: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002b760: 7c0a 7c6f 3720 3a20 4368 6169 6e43 6f6d  |.|o7 : ChainCom
│ │ │ │ +0002b770: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │ +0002b780: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002b7c0: 0a7c 6938 203a 204d 203d 2063 6f6b 6572  .|i8 : M = coker
│ │ │ │ -0002b7d0: 2046 2e64 645f 333b 2020 2020 2020 2020   F.dd_3;        
│ │ │ │ -0002b7e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002b7b0: 2d2d 2d2d 2b0a 7c69 3820 3a20 4d20 3d20  ----+.|i8 : M = 
│ │ │ │ +0002b7c0: 636f 6b65 7220 462e 6464 5f33 3b20 2020  coker F.dd_3;   
│ │ │ │ +0002b7d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b7e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0002b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b810: 2d2d 2d2b 0a7c 6939 203a 204d 4620 3d20  ---+.|i9 : MF = 
│ │ │ │ -0002b820: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -0002b830: 696f 6e28 6666 2c4d 293b 2020 207c 0a2b  ion(ff,M);   |.+
│ │ │ │ +0002b800: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +0002b810: 4d46 203d 206d 6174 7269 7846 6163 746f  MF = matrixFacto
│ │ │ │ +0002b820: 7269 7a61 7469 6f6e 2866 662c 4d29 3b20  rization(ff,M); 
│ │ │ │ +0002b830: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0002b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b860: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -0002b870: 6265 7474 6920 7265 7320 7075 7368 466f  betti res pushFo
│ │ │ │ -0002b880: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -0002b890: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +0002b850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002b860: 3130 203a 2062 6574 7469 2072 6573 2070  10 : betti res p
│ │ │ │ +0002b870: 7573 6846 6f72 7761 7264 286d 6170 2852  ushForward(map(R
│ │ │ │ +0002b880: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020  ,S),M)|.|       
│ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b8c0: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ -0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8e0: 2020 2020 207c 0a7c 6f31 3020 3d20 746f       |.|o10 = to
│ │ │ │ -0002b8f0: 7461 6c3a 2033 2035 2032 2020 2020 2020  tal: 3 5 2      
│ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b910: 0a7c 2020 2020 2020 2020 2020 323a 2033  .|          2: 3
│ │ │ │ -0002b920: 2034 202e 2020 2020 2020 2020 2020 2020   4 .            
│ │ │ │ -0002b930: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002b940: 2020 2020 2020 333a 202e 2031 2032 2020        3: . 1 2  
│ │ │ │ -0002b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b960: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002b8b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002b8c0: 3020 3120 3220 2020 2020 2020 2020 2020  0 1 2           
│ │ │ │ +0002b8d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ +0002b8e0: 203d 2074 6f74 616c 3a20 3320 3520 3220   = total: 3 5 2 
│ │ │ │ +0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b910: 2032 3a20 3320 3420 2e20 2020 2020 2020   2: 3 4 .       
│ │ │ │ +0002b920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b930: 7c20 2020 2020 2020 2020 2033 3a20 2e20  |          3: . 
│ │ │ │ +0002b940: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │ +0002b950: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002b990: 6f31 3020 3a20 4265 7474 6954 616c 6c79  o10 : BettiTally
│ │ │ │ -0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002b980: 2020 7c0a 7c6f 3130 203a 2042 6574 7469    |.|o10 : Betti
│ │ │ │ +0002b990: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0002b9a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002b9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b9e0: 2d2b 0a7c 6931 3120 3a20 6669 6e69 7465  -+.|i11 : finite
│ │ │ │ -0002b9f0: 4265 7474 694e 756d 6265 7273 204d 4620  BettiNumbers MF 
│ │ │ │ -0002ba00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002b9d0: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2066  ------+.|i11 : f
│ │ │ │ +0002b9e0: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0002b9f0: 7320 4d46 2020 2020 2020 2020 2020 2020  s MF            
│ │ │ │ +0002ba00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba30: 2020 2020 207c 0a7c 6f31 3120 3d20 7b33       |.|o11 = {3
│ │ │ │ -0002ba40: 2c20 352c 2032 7d20 2020 2020 2020 2020  , 5, 2}         
│ │ │ │ -0002ba50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ba60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba80: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -0002ba90: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ -0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bab0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002ba20: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ +0002ba30: 203d 207b 332c 2035 2c20 327d 2020 2020   = {3, 5, 2}    
│ │ │ │ +0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ba80: 7c6f 3131 203a 204c 6973 7420 2020 2020  |o11 : List     
│ │ │ │ +0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002baa0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002bae0: 6931 3220 3a20 696e 6669 6e69 7465 4265  i12 : infiniteBe
│ │ │ │ -0002baf0: 7474 694e 756d 6265 7273 284d 462c 3529  ttiNumbers(MF,5)
│ │ │ │ -0002bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002bad0: 2d2d 2b0a 7c69 3132 203a 2069 6e66 696e  --+.|i12 : infin
│ │ │ │ +0002bae0: 6974 6542 6574 7469 4e75 6d62 6572 7328  iteBettiNumbers(
│ │ │ │ +0002baf0: 4d46 2c35 2920 2020 2020 2020 7c0a 7c20  MF,5)       |.| 
│ │ │ │ +0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb30: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c   |.|o12 = {3, 4,
│ │ │ │ -0002bb40: 2035 2c20 362c 2037 2c20 387d 2020 2020   5, 6, 7, 8}    
│ │ │ │ -0002bb50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002bb20: 2020 2020 2020 7c0a 7c6f 3132 203d 207b        |.|o12 = {
│ │ │ │ +0002bb30: 332c 2034 2c20 352c 2036 2c20 372c 2038  3, 4, 5, 6, 7, 8
│ │ │ │ +0002bb40: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002bb50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0002bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb80: 2020 2020 207c 0a7c 6f31 3220 3a20 4c69       |.|o12 : Li
│ │ │ │ -0002bb90: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ -0002bba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bbb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0002bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bbd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ -0002bbe0: 3a20 6265 7474 6920 7265 7320 284d 2c20  : betti res (M, 
│ │ │ │ -0002bbf0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2035  LengthLimit => 5
│ │ │ │ -0002bc00: 2920 207c 0a7c 2020 2020 2020 2020 2020  )  |.|          
│ │ │ │ +0002bb70: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ +0002bb80: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0002bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bba0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002bbd0: 7c69 3133 203a 2062 6574 7469 2072 6573  |i13 : betti res
│ │ │ │ +0002bbe0: 2028 4d2c 204c 656e 6774 684c 696d 6974   (M, LengthLimit
│ │ │ │ +0002bbf0: 203d 3e20 3529 2020 7c0a 7c20 2020 2020   => 5)  |.|     
│ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002bc30: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -0002bc40: 2032 2033 2034 2035 2020 2020 2020 2020   2 3 4 5        
│ │ │ │ -0002bc50: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -0002bc60: 746f 7461 6c3a 2033 2034 2035 2036 2037  total: 3 4 5 6 7
│ │ │ │ -0002bc70: 2038 2020 2020 2020 2020 2020 2020 2020   8              
│ │ │ │ -0002bc80: 207c 0a7c 2020 2020 2020 2020 2020 323a   |.|          2:
│ │ │ │ -0002bc90: 2033 2034 2035 2036 2037 2038 2020 2020   3 4 5 6 7 8    
│ │ │ │ -0002bca0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002bc20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002bc30: 2020 3020 3120 3220 3320 3420 3520 2020    0 1 2 3 4 5   
│ │ │ │ +0002bc40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002bc50: 3133 203d 2074 6f74 616c 3a20 3320 3420  13 = total: 3 4 
│ │ │ │ +0002bc60: 3520 3620 3720 3820 2020 2020 2020 2020  5 6 7 8         
│ │ │ │ +0002bc70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bc80: 2020 2032 3a20 3320 3420 3520 3620 3720     2: 3 4 5 6 7 
│ │ │ │ +0002bc90: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ +0002bca0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0002bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bcd0: 2020 2020 207c 0a7c 6f31 3320 3a20 4265       |.|o13 : Be
│ │ │ │ -0002bce0: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ -0002bcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bd00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0002bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002bd50: 6163 746f 7269 7a61 7469 6f6e 3a20 6d61  actorization: ma
│ │ │ │ -0002bd60: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
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│ │ │ │ -0002bd80: 6869 6768 6572 0a20 2020 2063 6f64 696d  higher.    codim
│ │ │ │ -0002bd90: 656e 7369 6f6e 206d 6174 7269 7820 6661  ension matrix fa
│ │ │ │ -0002bda0: 6374 6f72 697a 6174 696f 6e0a 2020 2a20  ctorization.  * 
│ │ │ │ -0002bdb0: 2a6e 6f74 6520 6669 6e69 7465 4265 7474  *note finiteBett
│ │ │ │ -0002bdc0: 694e 756d 6265 7273 3a20 6669 6e69 7465  iNumbers: finite
│ │ │ │ -0002bdd0: 4265 7474 694e 756d 6265 7273 2c20 2d2d  BettiNumbers, --
│ │ │ │ -0002bde0: 2062 6574 7469 206e 756d 6265 7273 206f   betti numbers o
│ │ │ │ -0002bdf0: 6620 6669 6e69 7465 0a20 2020 2072 6573  f finite.    res
│ │ │ │ -0002be00: 6f6c 7574 696f 6e20 636f 6d70 7574 6564  olution computed
│ │ │ │ -0002be10: 2066 726f 6d20 6120 6d61 7472 6978 2066   from a matrix f
│ │ │ │ -0002be20: 6163 746f 7269 7a61 7469 6f6e 0a0a 5761  actorization..Wa
│ │ │ │ -0002be30: 7973 2074 6f20 7573 6520 696e 6669 6e69  ys to use infini
│ │ │ │ -0002be40: 7465 4265 7474 694e 756d 6265 7273 3a0a  teBettiNumbers:.
│ │ │ │ +0002bcc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ +0002bcd0: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ +0002bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bcf0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002bd20: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0002bd30: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ +0002bd40: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0002bd50: 6e3a 206d 6174 7269 7846 6163 746f 7269  n: matrixFactori
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│ │ │ │ +0002bd70: 696e 2061 2068 6967 6865 720a 2020 2020  in a higher.    
│ │ │ │ +0002bd80: 636f 6469 6d65 6e73 696f 6e20 6d61 7472  codimension matr
│ │ │ │ +0002bd90: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ +0002bda0: 0a20 202a 202a 6e6f 7465 2066 696e 6974  .  * *note finit
│ │ │ │ +0002bdb0: 6542 6574 7469 4e75 6d62 6572 733a 2066  eBettiNumbers: f
│ │ │ │ +0002bdc0: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0002bdd0: 732c 202d 2d20 6265 7474 6920 6e75 6d62  s, -- betti numb
│ │ │ │ +0002bde0: 6572 7320 6f66 2066 696e 6974 650a 2020  ers of finite.  
│ │ │ │ +0002bdf0: 2020 7265 736f 6c75 7469 6f6e 2063 6f6d    resolution com
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│ │ │ │ +0002be20: 6e0a 0a57 6179 7320 746f 2075 7365 2069  n..Ways to use i
│ │ │ │ +0002be30: 6e66 696e 6974 6542 6574 7469 4e75 6d62  nfiniteBettiNumb
│ │ │ │ +0002be40: 6572 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ers:.===========
│ │ │ │  0002be50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002be60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002be70: 3d0a 0a20 202a 2022 696e 6669 6e69 7465  =..  * "infinite
│ │ │ │ -0002be80: 4265 7474 694e 756d 6265 7273 284c 6973  BettiNumbers(Lis
│ │ │ │ -0002be90: 742c 5a5a 2922 0a0a 466f 7220 7468 6520  t,ZZ)"..For the 
│ │ │ │ -0002bea0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -0002beb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -0002bec0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -0002bed0: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -0002bee0: 6265 7273 3a20 696e 6669 6e69 7465 4265  bers: infiniteBe
│ │ │ │ -0002bef0: 7474 694e 756d 6265 7273 2c20 6973 2061  ttiNumbers, is a
│ │ │ │ -0002bf00: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -0002bf10: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61  nction: (Macaula
│ │ │ │ -0002bf20: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -0002bf30: 7469 6f6e 2c2e 0a1f 0a46 696c 653a 2043  tion,....File: C
│ │ │ │ -0002bf40: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0002bf50: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0002bf60: 6e66 6f2c 204e 6f64 653a 2069 734c 696e  nfo, Node: isLin
│ │ │ │ -0002bf70: 6561 722c 204e 6578 743a 2069 7351 7561  ear, Next: isQua
│ │ │ │ -0002bf80: 7369 5265 6775 6c61 722c 2050 7265 763a  siRegular, Prev:
│ │ │ │ -0002bf90: 2069 6e66 696e 6974 6542 6574 7469 4e75   infiniteBettiNu
│ │ │ │ -0002bfa0: 6d62 6572 732c 2055 703a 2054 6f70 0a0a  mbers, Up: Top..
│ │ │ │ -0002bfb0: 6973 4c69 6e65 6172 202d 2d20 6368 6563  isLinear -- chec
│ │ │ │ -0002bfc0: 6b20 7768 6574 6865 7220 6d61 7472 6978  k whether matrix
│ │ │ │ -0002bfd0: 2065 6e74 7269 6573 2068 6176 6520 6465   entries have de
│ │ │ │ -0002bfe0: 6772 6565 2031 0a2a 2a2a 2a2a 2a2a 2a2a  gree 1.*********
│ │ │ │ +0002be60: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 6e66  ======..  * "inf
│ │ │ │ +0002be70: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0002be80: 7328 4c69 7374 2c5a 5a29 220a 0a46 6f72  s(List,ZZ)"..For
│ │ │ │ +0002be90: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +0002bea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002beb0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +0002bec0: 6e6f 7465 2069 6e66 696e 6974 6542 6574  note infiniteBet
│ │ │ │ +0002bed0: 7469 4e75 6d62 6572 733a 2069 6e66 696e  tiNumbers: infin
│ │ │ │ +0002bee0: 6974 6542 6574 7469 4e75 6d62 6572 732c  iteBettiNumbers,
│ │ │ │ +0002bef0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +0002bf00: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61  od.function: (Ma
│ │ │ │ +0002bf10: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +0002bf20: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ +0002bf30: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0002bf40: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0002bf50: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0002bf60: 6973 4c69 6e65 6172 2c20 4e65 7874 3a20  isLinear, Next: 
│ │ │ │ +0002bf70: 6973 5175 6173 6952 6567 756c 6172 2c20  isQuasiRegular, 
│ │ │ │ +0002bf80: 5072 6576 3a20 696e 6669 6e69 7465 4265  Prev: infiniteBe
│ │ │ │ +0002bf90: 7474 694e 756d 6265 7273 2c20 5570 3a20  ttiNumbers, Up: 
│ │ │ │ +0002bfa0: 546f 700a 0a69 734c 696e 6561 7220 2d2d  Top..isLinear --
│ │ │ │ +0002bfb0: 2063 6865 636b 2077 6865 7468 6572 206d   check whether m
│ │ │ │ +0002bfc0: 6174 7269 7820 656e 7472 6965 7320 6861  atrix entries ha
│ │ │ │ +0002bfd0: 7665 2064 6567 7265 6520 310a 2a2a 2a2a  ve degree 1.****
│ │ │ │ +0002bfe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002bff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -0002c020: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -0002c030: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -0002c040: 2020 2020 2020 6220 3d20 6973 4c69 6e65        b = isLine
│ │ │ │ -0002c050: 6172 204d 0a20 202a 2049 6e70 7574 733a  ar M.  * Inputs:
│ │ │ │ -0002c060: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
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│ │ │ │ -0002c080: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -0002c090: 2c2c 200a 2020 2a20 4f75 7470 7574 733a  ,, .  * Outputs:
│ │ │ │ -0002c0a0: 0a20 2020 2020 202a 2062 2c20 6120 2a6e  .      * b, a *n
│ │ │ │ -0002c0b0: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75  ote Boolean valu
│ │ │ │ -0002c0c0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
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│ │ │ │ -0002c0f0: 3d3d 3d3d 0a0a 4e6f 7465 2074 6861 7420  ====..Note that 
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│ │ │ │ -0002c120: 6361 6e20 7374 696c 6c20 6861 7665 2064  can still have d
│ │ │ │ -0002c130: 6966 6665 7265 6e74 2074 6172 6765 740a  ifferent target.
│ │ │ │ -0002c140: 6465 6772 6565 7320 2869 6e20 7768 6963  degrees (in whic
│ │ │ │ -0002c150: 6820 6361 7365 2074 6865 2063 6f6b 6572  h case the coker
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│ │ │ │ -0002c190: 6772 6565 2e29 0a0a 5761 7973 2074 6f20  gree.)..Ways to 
│ │ │ │ -0002c1a0: 7573 6520 6973 4c69 6e65 6172 3a0a 3d3d  use isLinear:.==
│ │ │ │ -0002c1b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002c1c0: 3d3d 3d0a 0a20 202a 2022 6973 4c69 6e65  ===..  * "isLine
│ │ │ │ -0002c1d0: 6172 284d 6174 7269 7829 220a 0a46 6f72  ar(Matrix)"..For
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│ │ │ │ -0002c1f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -0002c220: 734c 696e 6561 722c 2069 7320 6120 2a6e  sLinear, is a *n
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│ │ │ │ -0002c280: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
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│ │ │ │ -0002c2a0: 6567 756c 6172 2c20 4e65 7874 3a20 6973  egular, Next: is
│ │ │ │ -0002c2b0: 5374 6162 6c79 5472 6976 6961 6c2c 2050  StablyTrivial, P
│ │ │ │ -0002c2c0: 7265 763a 2069 734c 696e 6561 722c 2055  rev: isLinear, U
│ │ │ │ -0002c2d0: 703a 2054 6f70 0a0a 6973 5175 6173 6952  p: Top..isQuasiR
│ │ │ │ -0002c2e0: 6567 756c 6172 202d 2d20 7465 7374 7320  egular -- tests 
│ │ │ │ -0002c2f0: 6120 6d61 7472 6978 206f 7220 7365 7175  a matrix or sequ
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│ │ │ │ -0002c320: 7920 6f6e 2061 206d 6f64 756c 650a 2a2a  y on a module.**
│ │ │ │ +0002c010: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +0002c020: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +0002c030: 3a20 0a20 2020 2020 2020 2062 203d 2069  : .        b = i
│ │ │ │ +0002c040: 734c 696e 6561 7220 4d0a 2020 2a20 496e  sLinear M.  * In
│ │ │ │ +0002c050: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c  puts:.      * M,
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│ │ │ │ +0002c070: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0002c080: 6174 7269 782c 2c20 0a20 202a 204f 7574  atrix,, .  * Out
│ │ │ │ +0002c090: 7075 7473 3a0a 2020 2020 2020 2a20 622c  puts:.      * b,
│ │ │ │ +0002c0a0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
│ │ │ │ +0002c0b0: 2076 616c 7565 3a20 284d 6163 6175 6c61   value: (Macaula
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│ │ │ │ +0002c0d0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
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│ │ │ │ +0002c130: 7267 6574 0a64 6567 7265 6573 2028 696e  rget.degrees (in
│ │ │ │ +0002c140: 2077 6869 6368 2063 6173 6520 7468 6520   which case the 
│ │ │ │ +0002c150: 636f 6b65 726e 656c 2064 6563 6f6d 706f  cokernel decompo
│ │ │ │ +0002c160: 7365 7320 696e 746f 2061 2064 6972 6563  ses into a direc
│ │ │ │ +0002c170: 7420 7375 6d20 6279 2067 656e 6572 6174  t sum by generat
│ │ │ │ +0002c180: 6f72 0a64 6567 7265 652e 290a 0a57 6179  or.degree.)..Way
│ │ │ │ +0002c190: 7320 746f 2075 7365 2069 734c 696e 6561  s to use isLinea
│ │ │ │ +0002c1a0: 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r:.=============
│ │ │ │ +0002c1b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269  ========..  * "i
│ │ │ │ +0002c1c0: 734c 696e 6561 7228 4d61 7472 6978 2922  sLinear(Matrix)"
│ │ │ │ +0002c1d0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0002c1e0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0002c1f0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +0002c200: 6563 7420 2a6e 6f74 6520 6973 4c69 6e65  ect *note isLine
│ │ │ │ +0002c210: 6172 3a20 6973 4c69 6e65 6172 2c20 6973  ar: isLinear, is
│ │ │ │ +0002c220: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +0002c230: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ +0002c240: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
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│ │ │ │ +0002c260: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +0002c270: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0002c280: 2e69 6e66 6f2c 204e 6f64 653a 2069 7351  .info, Node: isQ
│ │ │ │ +0002c290: 7561 7369 5265 6775 6c61 722c 204e 6578  uasiRegular, Nex
│ │ │ │ +0002c2a0: 743a 2069 7353 7461 626c 7954 7269 7669  t: isStablyTrivi
│ │ │ │ +0002c2b0: 616c 2c20 5072 6576 3a20 6973 4c69 6e65  al, Prev: isLine
│ │ │ │ +0002c2c0: 6172 2c20 5570 3a20 546f 700a 0a69 7351  ar, Up: Top..isQ
│ │ │ │ +0002c2d0: 7561 7369 5265 6775 6c61 7220 2d2d 2074  uasiRegular -- t
│ │ │ │ +0002c2e0: 6573 7473 2061 206d 6174 7269 7820 6f72  ests a matrix or
│ │ │ │ +0002c2f0: 2073 6571 7565 6e63 6520 6f72 206c 6973   sequence or lis
│ │ │ │ +0002c300: 7420 666f 7220 7175 6173 692d 7265 6775  t for quasi-regu
│ │ │ │ +0002c310: 6c61 7269 7479 206f 6e20 6120 6d6f 6475  larity on a modu
│ │ │ │ +0002c320: 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  le.*************
│ │ │ │  0002c330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c380: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -0002c390: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ -0002c3a0: 653a 200a 2020 2020 2020 2020 7420 3d20  e: .        t = 
│ │ │ │ -0002c3b0: 6973 5175 6173 6952 6567 756c 6172 2866  isQuasiRegular(f
│ │ │ │ -0002c3c0: 662c 4d29 0a20 202a 2049 6e70 7574 733a  f,M).  * Inputs:
│ │ │ │ -0002c3d0: 0a20 2020 2020 202a 2066 662c 2061 202a  .      * ff, a *
│ │ │ │ -0002c3e0: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -0002c3f0: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -0002c400: 782c 2c20 0a20 2020 2020 202a 2066 662c  x,, .      * ff,
│ │ │ │ -0002c410: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ -0002c420: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ -0002c430: 742c 2c20 0a20 2020 2020 202a 2066 662c  t,, .      * ff,
│ │ │ │ -0002c440: 2061 202a 6e6f 7465 2073 6571 7565 6e63   a *note sequenc
│ │ │ │ -0002c450: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0002c460: 2953 6571 7565 6e63 652c 2c20 0a20 2020  )Sequence,, .   
│ │ │ │ -0002c470: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ -0002c480: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ -0002c490: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a  y2Doc)Module,, .
│ │ │ │ -0002c4a0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -0002c4b0: 2020 202a 2074 2c20 6120 2a6e 6f74 6520     * t, a *note 
│ │ │ │ -0002c4c0: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ -0002c4d0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ -0002c4e0: 6c65 616e 2c2c 200a 0a44 6573 6372 6970  lean,, ..Descrip
│ │ │ │ -0002c4f0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -0002c500: 0a0a 6666 2069 7320 7175 6173 692d 7265  ..ff is quasi-re
│ │ │ │ -0002c510: 6775 6c61 7220 6966 2074 6865 206c 656e  gular if the len
│ │ │ │ -0002c520: 6774 6820 6f66 2066 6620 6973 203c 3d20  gth of ff is <= 
│ │ │ │ -0002c530: 6469 6d20 4d20 616e 6420 7468 6520 616e  dim M and the an
│ │ │ │ -0002c540: 6e69 6869 6c61 746f 7220 6f66 2066 665f  nihilator of ff_
│ │ │ │ -0002c550: 690a 6f6e 204d 2f28 6666 5f30 2e2e 6666  i.on M/(ff_0..ff
│ │ │ │ -0002c560: 5f7b 2869 2d31 2929 7d4d 2068 6173 2066  _{(i-1))}M has f
│ │ │ │ -0002c570: 696e 6974 6520 6c65 6e67 7468 2066 6f72  inite length for
│ │ │ │ -0002c580: 2061 6c6c 2069 3d30 2e2e 286c 656e 6774   all i=0..(lengt
│ │ │ │ -0002c590: 6820 6666 292d 312e 0a0a 2b2d 2d2d 2d2d  h ff)-1...+-----
│ │ │ │ +0002c370: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ +0002c380: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ +0002c390: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0002c3a0: 2074 203d 2069 7351 7561 7369 5265 6775   t = isQuasiRegu
│ │ │ │ +0002c3b0: 6c61 7228 6666 2c4d 290a 2020 2a20 496e  lar(ff,M).  * In
│ │ │ │ +0002c3c0: 7075 7473 3a0a 2020 2020 2020 2a20 6666  puts:.      * ff
│ │ │ │ +0002c3d0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ +0002c3e0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0002c3f0: 4d61 7472 6978 2c2c 200a 2020 2020 2020  Matrix,, .      
│ │ │ │ +0002c400: 2a20 6666 2c20 6120 2a6e 6f74 6520 6c69  * ff, a *note li
│ │ │ │ +0002c410: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +0002c420: 6329 4c69 7374 2c2c 200a 2020 2020 2020  c)List,, .      
│ │ │ │ +0002c430: 2a20 6666 2c20 6120 2a6e 6f74 6520 7365  * ff, a *note se
│ │ │ │ +0002c440: 7175 656e 6365 3a20 284d 6163 6175 6c61  quence: (Macaula
│ │ │ │ +0002c450: 7932 446f 6329 5365 7175 656e 6365 2c2c  y2Doc)Sequence,,
│ │ │ │ +0002c460: 200a 2020 2020 2020 2a20 4d2c 2061 202a   .      * M, a *
│ │ │ │ +0002c470: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +0002c480: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +0002c490: 652c 2c20 0a20 202a 204f 7574 7075 7473  e,, .  * Outputs
│ │ │ │ +0002c4a0: 3a0a 2020 2020 2020 2a20 742c 2061 202a  :.      * t, a *
│ │ │ │ +0002c4b0: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ +0002c4c0: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ +0002c4d0: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465  c)Boolean,, ..De
│ │ │ │ +0002c4e0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0002c4f0: 3d3d 3d3d 3d0a 0a66 6620 6973 2071 7561  =====..ff is qua
│ │ │ │ +0002c500: 7369 2d72 6567 756c 6172 2069 6620 7468  si-regular if th
│ │ │ │ +0002c510: 6520 6c65 6e67 7468 206f 6620 6666 2069  e length of ff i
│ │ │ │ +0002c520: 7320 3c3d 2064 696d 204d 2061 6e64 2074  s <= dim M and t
│ │ │ │ +0002c530: 6865 2061 6e6e 6968 696c 6174 6f72 206f  he annihilator o
│ │ │ │ +0002c540: 6620 6666 5f69 0a6f 6e20 4d2f 2866 665f  f ff_i.on M/(ff_
│ │ │ │ +0002c550: 302e 2e66 665f 7b28 692d 3129 297d 4d20  0..ff_{(i-1))}M 
│ │ │ │ +0002c560: 6861 7320 6669 6e69 7465 206c 656e 6774  has finite lengt
│ │ │ │ +0002c570: 6820 666f 7220 616c 6c20 693d 302e 2e28  h for all i=0..(
│ │ │ │ +0002c580: 6c65 6e67 7468 2066 6629 2d31 2e0a 0a2b  length ff)-1...+
│ │ │ │ +0002c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c5c0: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d5a  ----+.|i1 : kk=Z
│ │ │ │ -0002c5d0: 5a2f 3130 313b 2020 2020 2020 2020 2020  Z/101;          
│ │ │ │ -0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002c5b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0002c5c0: 206b 6b3d 5a5a 2f31 3031 3b20 2020 2020   kk=ZZ/101;     
│ │ │ │ +0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002c620: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63  2 : S = kk[a,b,c
│ │ │ │ -0002c630: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ -0002c640: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002c610: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b  -+.|i2 : S = kk[
│ │ │ │ +0002c620: 612c 622c 635d 3b20 2020 2020 2020 2020  a,b,c];         
│ │ │ │ +0002c630: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c670: 2d2d 2d2d 2b0a 7c69 3320 3a20 4520 3d20  ----+.|i3 : E = 
│ │ │ │ -0002c680: 535e 312f 6964 6561 6c22 6162 222b 2b53  S^1/ideal"ab"++S
│ │ │ │ -0002c690: 5e31 2f69 6465 616c 2076 6172 7320 533b  ^1/ideal vars S;
│ │ │ │ -0002c6a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002c660: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0002c670: 2045 203d 2053 5e31 2f69 6465 616c 2261   E = S^1/ideal"a
│ │ │ │ +0002c680: 6222 2b2b 535e 312f 6964 6561 6c20 7661  b"++S^1/ideal va
│ │ │ │ +0002c690: 7273 2053 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d  rs S;|.+--------
│ │ │ │ +0002c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002c6d0: 3420 3a20 6631 203d 6d61 7472 6978 2261  4 : f1 =matrix"a
│ │ │ │ -0002c6e0: 223b 2020 2020 2020 2020 2020 2020 2020  ";              
│ │ │ │ -0002c6f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002c6c0: 2d2b 0a7c 6934 203a 2066 3120 3d6d 6174  -+.|i4 : f1 =mat
│ │ │ │ +0002c6d0: 7269 7822 6122 3b20 2020 2020 2020 2020  rix"a";         
│ │ │ │ +0002c6e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002c730: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -0002c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c750: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -0002c760: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -0002c770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002c710: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002c720: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +0002c730: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0002c740: 2020 2020 207c 0a7c 6f34 203a 204d 6174       |.|o4 : Mat
│ │ │ │ +0002c750: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0002c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c770: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c7a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -0002c7b0: 6632 203d 6d61 7472 6978 2261 2b62 2c63  f2 =matrix"a+b,c
│ │ │ │ -0002c7c0: 223b 2020 2020 2020 2020 2020 2020 2020  ";              
│ │ │ │ -0002c7d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002c7a0: 6935 203a 2066 3220 3d6d 6174 7269 7822  i5 : f2 =matrix"
│ │ │ │ +0002c7b0: 612b 622c 6322 3b20 2020 2020 2020 2020  a+b,c";         
│ │ │ │ +0002c7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c800: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002c810: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ -0002c820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002c830: 3520 3a20 4d61 7472 6978 2053 2020 3c2d  5 : Matrix S  <-
│ │ │ │ -0002c840: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │ -0002c850: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002c7f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c800: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +0002c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c820: 207c 0a7c 6f35 203a 204d 6174 7269 7820   |.|o5 : Matrix 
│ │ │ │ +0002c830: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ +0002c840: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c880: 2d2d 2d2d 2b0a 7c69 3620 3a20 6633 203d  ----+.|i6 : f3 =
│ │ │ │ -0002c890: 206d 6174 7269 7822 612b 6222 3b20 2020   matrix"a+b";   
│ │ │ │ -0002c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002c870: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0002c880: 2066 3320 3d20 6d61 7472 6978 2261 2b62   f3 = matrix"a+b
│ │ │ │ +0002c890: 223b 2020 2020 2020 2020 2020 2020 2020  ";              
│ │ │ │ +0002c8a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002c8e0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -0002c8f0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -0002c900: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -0002c910: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ -0002c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c930: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002c8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002c8e0: 2031 2020 2020 2020 3120 2020 2020 2020   1      1       
│ │ │ │ +0002c8f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c900: 6f36 203a 204d 6174 7269 7820 5320 203c  o6 : Matrix S  <
│ │ │ │ +0002c910: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +0002c920: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c960: 2b0a 7c69 3720 3a20 6634 203d 206d 6174  +.|i7 : f4 = mat
│ │ │ │ -0002c970: 7269 7822 612b 622c 2061 322b 6222 3b20  rix"a+b, a2+b"; 
│ │ │ │ -0002c980: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002c950: 2d2d 2d2d 2d2b 0a7c 6937 203a 2066 3420  -----+.|i7 : f4 
│ │ │ │ +0002c960: 3d20 6d61 7472 6978 2261 2b62 2c20 6132  = matrix"a+b, a2
│ │ │ │ +0002c970: 2b62 223b 2020 2020 2020 2020 2020 2020  +b";            
│ │ │ │ +0002c980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002c9c0: 2020 2020 2020 2020 3120 2020 2020 2032          1      2
│ │ │ │ -0002c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9e0: 2020 2020 7c0a 7c6f 3720 3a20 4d61 7472      |.|o7 : Matr
│ │ │ │ -0002c9f0: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ -0002ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca10: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002c9a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c9b0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0002c9c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002c9d0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0002c9e0: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ca00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002ca40: 3820 3a20 6973 5175 6173 6952 6567 756c  8 : isQuasiRegul
│ │ │ │ -0002ca50: 6172 2866 312c 4529 2020 2020 2020 2020  ar(f1,E)        
│ │ │ │ -0002ca60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002ca30: 2d2b 0a7c 6938 203a 2069 7351 7561 7369  -+.|i8 : isQuasi
│ │ │ │ +0002ca40: 5265 6775 6c61 7228 6631 2c45 2920 2020  Regular(f1,E)   
│ │ │ │ +0002ca50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca90: 2020 2020 7c0a 7c6f 3820 3d20 6661 6c73      |.|o8 = fals
│ │ │ │ -0002caa0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0002cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cac0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002ca80: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +0002ca90: 2066 616c 7365 2020 2020 2020 2020 2020   false          
│ │ │ │ +0002caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cab0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002caf0: 3920 3a20 6973 5175 6173 6952 6567 756c  9 : isQuasiRegul
│ │ │ │ -0002cb00: 6172 2866 322c 4529 2020 2020 2020 2020  ar(f2,E)        
│ │ │ │ -0002cb10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002cae0: 2d2b 0a7c 6939 203a 2069 7351 7561 7369  -+.|i9 : isQuasi
│ │ │ │ +0002caf0: 5265 6775 6c61 7228 6632 2c45 2920 2020  Regular(f2,E)   
│ │ │ │ +0002cb00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb40: 2020 2020 7c0a 7c6f 3920 3d20 7472 7565      |.|o9 = true
│ │ │ │ +0002cb30: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +0002cb40: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  0002cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002cb60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002cba0: 3130 203a 2069 7351 7561 7369 5265 6775  10 : isQuasiRegu
│ │ │ │ -0002cbb0: 6c61 7228 6633 2c45 2920 2020 2020 2020  lar(f3,E)       
│ │ │ │ -0002cbc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002cb90: 2d2b 0a7c 6931 3020 3a20 6973 5175 6173  -+.|i10 : isQuas
│ │ │ │ +0002cba0: 6952 6567 756c 6172 2866 332c 4529 2020  iRegular(f3,E)  
│ │ │ │ +0002cbb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbf0: 2020 2020 7c0a 7c6f 3130 203d 2074 7275      |.|o10 = tru
│ │ │ │ -0002cc00: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002cbe0: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ +0002cbf0: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │ +0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002cc50: 3131 203a 2069 7351 7561 7369 5265 6775  11 : isQuasiRegu
│ │ │ │ -0002cc60: 6c61 7228 6634 2c45 2920 2020 2020 2020  lar(f4,E)       
│ │ │ │ -0002cc70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002cc40: 2d2b 0a7c 6931 3120 3a20 6973 5175 6173  -+.|i11 : isQuas
│ │ │ │ +0002cc50: 6952 6567 756c 6172 2866 342c 4529 2020  iRegular(f4,E)  
│ │ │ │ +0002cc60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cca0: 2020 2020 7c0a 7c6f 3131 203d 2066 616c      |.|o11 = fal
│ │ │ │ -0002ccb0: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │ -0002ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccd0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002cc90: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ +0002cca0: 3d20 6661 6c73 6520 2020 2020 2020 2020  = false         
│ │ │ │ +0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ccc0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ -0002cd00: 6179 7320 746f 2075 7365 2069 7351 7561  ays to use isQua
│ │ │ │ -0002cd10: 7369 5265 6775 6c61 723a 0a3d 3d3d 3d3d  siRegular:.=====
│ │ │ │ -0002cd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002cd30: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 7351  ======..  * "isQ
│ │ │ │ -0002cd40: 7561 7369 5265 6775 6c61 7228 4c69 7374  uasiRegular(List
│ │ │ │ -0002cd50: 2c4d 6f64 756c 6529 220a 2020 2a20 2269  ,Module)".  * "i
│ │ │ │ -0002cd60: 7351 7561 7369 5265 6775 6c61 7228 4d61  sQuasiRegular(Ma
│ │ │ │ -0002cd70: 7472 6978 2c4d 6f64 756c 6529 220a 2020  trix,Module)".  
│ │ │ │ -0002cd80: 2a20 2269 7351 7561 7369 5265 6775 6c61  * "isQuasiRegula
│ │ │ │ -0002cd90: 7228 5365 7175 656e 6365 2c4d 6f64 756c  r(Sequence,Modul
│ │ │ │ -0002cda0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ -0002cdb0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -0002cdc0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -0002cdd0: 6f62 6a65 6374 202a 6e6f 7465 2069 7351  object *note isQ
│ │ │ │ -0002cde0: 7561 7369 5265 6775 6c61 723a 2069 7351  uasiRegular: isQ
│ │ │ │ -0002cdf0: 7561 7369 5265 6775 6c61 722c 2069 7320  uasiRegular, is 
│ │ │ │ -0002ce00: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -0002ce10: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ -0002ce20: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -0002ce30: 6374 696f 6e2c 2e0a 1f0a 4669 6c65 3a20  ction,....File: 
│ │ │ │ -0002ce40: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -0002ce50: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -0002ce60: 696e 666f 2c20 4e6f 6465 3a20 6973 5374  info, Node: isSt
│ │ │ │ -0002ce70: 6162 6c79 5472 6976 6961 6c2c 204e 6578  ablyTrivial, Nex
│ │ │ │ -0002ce80: 743a 206b 6f73 7a75 6c45 7874 656e 7369  t: koszulExtensi
│ │ │ │ -0002ce90: 6f6e 2c20 5072 6576 3a20 6973 5175 6173  on, Prev: isQuas
│ │ │ │ -0002cea0: 6952 6567 756c 6172 2c20 5570 3a20 546f  iRegular, Up: To
│ │ │ │ -0002ceb0: 700a 0a69 7353 7461 626c 7954 7269 7669  p..isStablyTrivi
│ │ │ │ -0002cec0: 616c 202d 2d20 7265 7475 726e 7320 7472  al -- returns tr
│ │ │ │ -0002ced0: 7565 2069 6620 7468 6520 6d61 7020 676f  ue if the map go
│ │ │ │ -0002cee0: 6573 2074 6f20 3020 756e 6465 7220 7374  es to 0 under st
│ │ │ │ -0002cef0: 6162 6c65 486f 6d0a 2a2a 2a2a 2a2a 2a2a  ableHom.********
│ │ │ │ +0002ccf0: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520  -+..Ways to use 
│ │ │ │ +0002cd00: 6973 5175 6173 6952 6567 756c 6172 3a0a  isQuasiRegular:.
│ │ │ │ +0002cd10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002cd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0002cd30: 2022 6973 5175 6173 6952 6567 756c 6172   "isQuasiRegular
│ │ │ │ +0002cd40: 284c 6973 742c 4d6f 6475 6c65 2922 0a20  (List,Module)". 
│ │ │ │ +0002cd50: 202a 2022 6973 5175 6173 6952 6567 756c   * "isQuasiRegul
│ │ │ │ +0002cd60: 6172 284d 6174 7269 782c 4d6f 6475 6c65  ar(Matrix,Module
│ │ │ │ +0002cd70: 2922 0a20 202a 2022 6973 5175 6173 6952  )".  * "isQuasiR
│ │ │ │ +0002cd80: 6567 756c 6172 2853 6571 7565 6e63 652c  egular(Sequence,
│ │ │ │ +0002cd90: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ +0002cda0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +0002cdb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002cdc0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0002cdd0: 6520 6973 5175 6173 6952 6567 756c 6172  e isQuasiRegular
│ │ │ │ +0002cde0: 3a20 6973 5175 6173 6952 6567 756c 6172  : isQuasiRegular
│ │ │ │ +0002cdf0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +0002ce00: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +0002ce10: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +0002ce20: 6f64 4675 6e63 7469 6f6e 2c2e 0a1f 0a46  odFunction,....F
│ │ │ │ +0002ce30: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +0002ce40: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +0002ce50: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +0002ce60: 2069 7353 7461 626c 7954 7269 7669 616c   isStablyTrivial
│ │ │ │ +0002ce70: 2c20 4e65 7874 3a20 6b6f 737a 756c 4578  , Next: koszulEx
│ │ │ │ +0002ce80: 7465 6e73 696f 6e2c 2050 7265 763a 2069  tension, Prev: i
│ │ │ │ +0002ce90: 7351 7561 7369 5265 6775 6c61 722c 2055  sQuasiRegular, U
│ │ │ │ +0002cea0: 703a 2054 6f70 0a0a 6973 5374 6162 6c79  p: Top..isStably
│ │ │ │ +0002ceb0: 5472 6976 6961 6c20 2d2d 2072 6574 7572  Trivial -- retur
│ │ │ │ +0002cec0: 6e73 2074 7275 6520 6966 2074 6865 206d  ns true if the m
│ │ │ │ +0002ced0: 6170 2067 6f65 7320 746f 2030 2075 6e64  ap goes to 0 und
│ │ │ │ +0002cee0: 6572 2073 7461 626c 6548 6f6d 0a2a 2a2a  er stableHom.***
│ │ │ │ +0002cef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002cf00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002cf10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002cf20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002cf30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -0002cf40: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -0002cf50: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -0002cf60: 2020 2020 2062 203d 2069 7353 7461 626c       b = isStabl
│ │ │ │ -0002cf70: 7954 7269 7669 616c 2066 0a20 202a 2049  yTrivial f.  * I
│ │ │ │ -0002cf80: 6e70 7574 733a 0a20 2020 2020 202a 2066  nputs:.      * f
│ │ │ │ -0002cf90: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ -0002cfa0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0002cfb0: 4d61 7472 6978 2c2c 206d 6170 204d 2074  Matrix,, map M t
│ │ │ │ -0002cfc0: 6f20 4e0a 2020 2a20 4f75 7470 7574 733a  o N.  * Outputs:
│ │ │ │ -0002cfd0: 0a20 2020 2020 202a 2062 2c20 6120 2a6e  .      * b, a *n
│ │ │ │ -0002cfe0: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75  ote Boolean valu
│ │ │ │ -0002cff0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0002d000: 2942 6f6f 6c65 616e 2c2c 2074 7275 6520  )Boolean,, true 
│ │ │ │ -0002d010: 6966 6620 6620 6661 6374 6f72 730a 2020  iff f factors.  
│ │ │ │ -0002d020: 2020 2020 2020 7468 726f 7567 6820 6120        through a 
│ │ │ │ -0002d030: 7072 6f6a 6563 7469 7665 0a0a 4465 7363  projective..Desc
│ │ │ │ -0002d040: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0002d050: 3d3d 3d0a 0a41 2070 6f73 7369 626c 6520  ===..A possible 
│ │ │ │ -0002d060: 6f62 7374 7275 6374 696f 6e20 746f 2074  obstruction to t
│ │ │ │ -0002d070: 6865 2063 6f6d 6d75 7461 7469 7669 7479  he commutativity
│ │ │ │ -0002d080: 206f 6620 7468 6520 4349 206f 7065 7261   of the CI opera
│ │ │ │ -0002d090: 746f 7273 2069 6e20 636f 6469 6d20 632c  tors in codim c,
│ │ │ │ -0002d0a0: 0a65 7665 6e20 6173 796d 7074 6f74 6963  .even asymptotic
│ │ │ │ -0002d0b0: 616c 6c79 2c20 776f 756c 6420 6265 2074  ally, would be t
│ │ │ │ -0002d0c0: 6865 206e 6f6e 2d74 7269 7669 616c 6974  he non-trivialit
│ │ │ │ -0002d0d0: 7920 6f66 2074 6865 206d 6170 204d 5f7b  y of the map M_{
│ │ │ │ -0002d0e0: 286b 2b34 297d 202d 2d3e 204d 5f6b 0a5c  (k+4)} --> M_k.\
│ │ │ │ -0002d0f0: 6f74 696d 6573 205c 7765 6467 655e 3228  otimes \wedge^2(
│ │ │ │ -0002d100: 535e 6329 2069 6e20 7468 6520 7374 6162  S^c) in the stab
│ │ │ │ -0002d110: 6c65 2063 6174 6567 6f72 7920 6f66 206d  le category of m
│ │ │ │ -0002d120: 6178 696d 616c 2043 6f68 656e 2d4d 6163  aximal Cohen-Mac
│ │ │ │ -0002d130: 6175 6c61 7920 6d6f 6475 6c65 732e 0a0a  aulay modules...
│ │ │ │ -0002d140: 496e 2074 6865 2066 6f6c 6c6f 7769 6e67  In the following
│ │ │ │ -0002d150: 2065 7861 6d70 6c65 2c20 7374 7564 6965   example, studie
│ │ │ │ -0002d160: 6420 696e 2074 6865 2070 6170 6572 2022  d in the paper "
│ │ │ │ -0002d170: 546f 7220 6173 2061 206d 6f64 756c 6520  Tor as a module 
│ │ │ │ -0002d180: 6f76 6572 2061 6e0a 6578 7465 7269 6f72  over an.exterior
│ │ │ │ -0002d190: 2061 6c67 6562 7261 2220 6f66 2045 6973   algebra" of Eis
│ │ │ │ -0002d1a0: 656e 6275 642c 2050 6565 7661 2061 6e64  enbud, Peeva and
│ │ │ │ -0002d1b0: 2053 6368 7265 7965 722c 2074 6865 206d   Schreyer, the m
│ │ │ │ -0002d1c0: 6170 2069 7320 6e6f 6e2d 7472 6976 6961  ap is non-trivia
│ │ │ │ -0002d1d0: 6c2e 2e2e 6275 740a 6974 2069 7320 7374  l...but.it is st
│ │ │ │ -0002d1e0: 6162 6c79 2074 7269 7669 616c 2e20 5468  ably trivial. Th
│ │ │ │ -0002d1f0: 6520 7361 6d65 2067 6f65 7320 666f 7220  e same goes for 
│ │ │ │ -0002d200: 6869 6768 6572 2076 616c 7565 7320 6f66  higher values of
│ │ │ │ -0002d210: 206b 2028 7768 6963 6820 7461 6b65 206c   k (which take l
│ │ │ │ -0002d220: 6f6e 6765 720a 746f 2063 6f6d 7075 7465  onger.to compute
│ │ │ │ -0002d230: 292e 2028 6e6f 7465 2074 6861 7420 696e  ). (note that in
│ │ │ │ -0002d240: 2074 6869 7320 6361 7365 2c20 7769 7468   this case, with
│ │ │ │ -0002d250: 2063 203d 2033 2c20 7477 6f20 6f66 2074   c = 3, two of t
│ │ │ │ -0002d260: 6865 2074 6872 6565 2061 6c74 6572 6e61  he three alterna
│ │ │ │ -0002d270: 7469 6e67 0a70 726f 6475 6374 7320 6172  ting.products ar
│ │ │ │ -0002d280: 6520 6163 7475 616c 6c79 2065 7175 616c  e actually equal
│ │ │ │ -0002d290: 2074 6f20 302c 2073 6f20 7765 2074 6573   to 0, so we tes
│ │ │ │ -0002d2a0: 7420 6f6e 6c79 2074 6865 2074 6869 7264  t only the third
│ │ │ │ -0002d2b0: 2e29 0a0a 4e6f 7465 2074 6861 7420 5420  .)..Note that T 
│ │ │ │ -0002d2c0: 6973 2077 656c 6c2d 6465 6669 6e65 6420  is well-defined 
│ │ │ │ -0002d2d0: 7570 2074 6f20 686f 6d6f 746f 7079 3b20  up to homotopy; 
│ │ │ │ -0002d2e0: 736f 2054 5e32 2069 7320 7765 6c6c 2d64  so T^2 is well-d
│ │ │ │ -0002d2f0: 6566 696e 6564 206d 6f64 206d 6d5e 322e  efined mod mm^2.
│ │ │ │ -0002d300: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0002cf30: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ +0002cf40: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ +0002cf50: 200a 2020 2020 2020 2020 6220 3d20 6973   .        b = is
│ │ │ │ +0002cf60: 5374 6162 6c79 5472 6976 6961 6c20 660a  StablyTrivial f.
│ │ │ │ +0002cf70: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0002cf80: 2020 2a20 662c 2061 202a 6e6f 7465 206d    * f, a *note m
│ │ │ │ +0002cf90: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ +0002cfa0: 3244 6f63 294d 6174 7269 782c 2c20 6d61  2Doc)Matrix,, ma
│ │ │ │ +0002cfb0: 7020 4d20 746f 204e 0a20 202a 204f 7574  p M to N.  * Out
│ │ │ │ +0002cfc0: 7075 7473 3a0a 2020 2020 2020 2a20 622c  puts:.      * b,
│ │ │ │ +0002cfd0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
│ │ │ │ +0002cfe0: 2076 616c 7565 3a20 284d 6163 6175 6c61   value: (Macaula
│ │ │ │ +0002cff0: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20  y2Doc)Boolean,, 
│ │ │ │ +0002d000: 7472 7565 2069 6666 2066 2066 6163 746f  true iff f facto
│ │ │ │ +0002d010: 7273 0a20 2020 2020 2020 2074 6872 6f75  rs.        throu
│ │ │ │ +0002d020: 6768 2061 2070 726f 6a65 6374 6976 650a  gh a projective.
│ │ │ │ +0002d030: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0002d040: 3d3d 3d3d 3d3d 3d3d 0a0a 4120 706f 7373  ========..A poss
│ │ │ │ +0002d050: 6962 6c65 206f 6273 7472 7563 7469 6f6e  ible obstruction
│ │ │ │ +0002d060: 2074 6f20 7468 6520 636f 6d6d 7574 6174   to the commutat
│ │ │ │ +0002d070: 6976 6974 7920 6f66 2074 6865 2043 4920  ivity of the CI 
│ │ │ │ +0002d080: 6f70 6572 6174 6f72 7320 696e 2063 6f64  operators in cod
│ │ │ │ +0002d090: 696d 2063 2c0a 6576 656e 2061 7379 6d70  im c,.even asymp
│ │ │ │ +0002d0a0: 746f 7469 6361 6c6c 792c 2077 6f75 6c64  totically, would
│ │ │ │ +0002d0b0: 2062 6520 7468 6520 6e6f 6e2d 7472 6976   be the non-triv
│ │ │ │ +0002d0c0: 6961 6c69 7479 206f 6620 7468 6520 6d61  iality of the ma
│ │ │ │ +0002d0d0: 7020 4d5f 7b28 6b2b 3429 7d20 2d2d 3e20  p M_{(k+4)} --> 
│ │ │ │ +0002d0e0: 4d5f 6b0a 5c6f 7469 6d65 7320 5c77 6564  M_k.\otimes \wed
│ │ │ │ +0002d0f0: 6765 5e32 2853 5e63 2920 696e 2074 6865  ge^2(S^c) in the
│ │ │ │ +0002d100: 2073 7461 626c 6520 6361 7465 676f 7279   stable category
│ │ │ │ +0002d110: 206f 6620 6d61 7869 6d61 6c20 436f 6865   of maximal Cohe
│ │ │ │ +0002d120: 6e2d 4d61 6361 756c 6179 206d 6f64 756c  n-Macaulay modul
│ │ │ │ +0002d130: 6573 2e0a 0a49 6e20 7468 6520 666f 6c6c  es...In the foll
│ │ │ │ +0002d140: 6f77 696e 6720 6578 616d 706c 652c 2073  owing example, s
│ │ │ │ +0002d150: 7475 6469 6564 2069 6e20 7468 6520 7061  tudied in the pa
│ │ │ │ +0002d160: 7065 7220 2254 6f72 2061 7320 6120 6d6f  per "Tor as a mo
│ │ │ │ +0002d170: 6475 6c65 206f 7665 7220 616e 0a65 7874  dule over an.ext
│ │ │ │ +0002d180: 6572 696f 7220 616c 6765 6272 6122 206f  erior algebra" o
│ │ │ │ +0002d190: 6620 4569 7365 6e62 7564 2c20 5065 6576  f Eisenbud, Peev
│ │ │ │ +0002d1a0: 6120 616e 6420 5363 6872 6579 6572 2c20  a and Schreyer, 
│ │ │ │ +0002d1b0: 7468 6520 6d61 7020 6973 206e 6f6e 2d74  the map is non-t
│ │ │ │ +0002d1c0: 7269 7669 616c 2e2e 2e62 7574 0a69 7420  rivial...but.it 
│ │ │ │ +0002d1d0: 6973 2073 7461 626c 7920 7472 6976 6961  is stably trivia
│ │ │ │ +0002d1e0: 6c2e 2054 6865 2073 616d 6520 676f 6573  l. The same goes
│ │ │ │ +0002d1f0: 2066 6f72 2068 6967 6865 7220 7661 6c75   for higher valu
│ │ │ │ +0002d200: 6573 206f 6620 6b20 2877 6869 6368 2074  es of k (which t
│ │ │ │ +0002d210: 616b 6520 6c6f 6e67 6572 0a74 6f20 636f  ake longer.to co
│ │ │ │ +0002d220: 6d70 7574 6529 2e20 286e 6f74 6520 7468  mpute). (note th
│ │ │ │ +0002d230: 6174 2069 6e20 7468 6973 2063 6173 652c  at in this case,
│ │ │ │ +0002d240: 2077 6974 6820 6320 3d20 332c 2074 776f   with c = 3, two
│ │ │ │ +0002d250: 206f 6620 7468 6520 7468 7265 6520 616c   of the three al
│ │ │ │ +0002d260: 7465 726e 6174 696e 670a 7072 6f64 7563  ternating.produc
│ │ │ │ +0002d270: 7473 2061 7265 2061 6374 7561 6c6c 7920  ts are actually 
│ │ │ │ +0002d280: 6571 7561 6c20 746f 2030 2c20 736f 2077  equal to 0, so w
│ │ │ │ +0002d290: 6520 7465 7374 206f 6e6c 7920 7468 6520  e test only the 
│ │ │ │ +0002d2a0: 7468 6972 642e 290a 0a4e 6f74 6520 7468  third.)..Note th
│ │ │ │ +0002d2b0: 6174 2054 2069 7320 7765 6c6c 2d64 6566  at T is well-def
│ │ │ │ +0002d2c0: 696e 6564 2075 7020 746f 2068 6f6d 6f74  ined up to homot
│ │ │ │ +0002d2d0: 6f70 793b 2073 6f20 545e 3220 6973 2077  opy; so T^2 is w
│ │ │ │ +0002d2e0: 656c 6c2d 6465 6669 6e65 6420 6d6f 6420  ell-defined mod 
│ │ │ │ +0002d2f0: 6d6d 5e32 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  mm^2...+--------
│ │ │ │ +0002d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d350: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f  +.|i1 : kk = ZZ/
│ │ │ │ -0002d360: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +0002d340: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20  -----+.|i1 : kk 
│ │ │ │ +0002d350: 3d20 5a5a 2f31 3031 2020 2020 2020 2020  = ZZ/101        
│ │ │ │ +0002d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d3a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d3f0: 7c0a 7c6f 3120 3d20 6b6b 2020 2020 2020  |.|o1 = kk      
│ │ │ │ +0002d3e0: 2020 2020 207c 0a7c 6f31 203d 206b 6b20       |.|o1 = kk 
│ │ │ │ +0002d3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d440: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d490: 7c0a 7c6f 3120 3a20 5175 6f74 6965 6e74  |.|o1 : Quotient
│ │ │ │ -0002d4a0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0002d480: 2020 2020 207c 0a7c 6f31 203a 2051 756f       |.|o1 : Quo
│ │ │ │ +0002d490: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +0002d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002d4d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d530: 2b0a 7c69 3220 3a20 5320 3d20 6b6b 5b61  +.|i2 : S = kk[a
│ │ │ │ -0002d540: 2c62 2c63 5d20 2020 2020 2020 2020 2020  ,b,c]           
│ │ │ │ +0002d520: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d  -----+.|i2 : S =
│ │ │ │ +0002d530: 206b 6b5b 612c 622c 635d 2020 2020 2020   kk[a,b,c]      
│ │ │ │ +0002d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d580: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5d0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020  |.|o2 = S       
│ │ │ │ +0002d5c0: 2020 2020 207c 0a7c 6f32 203d 2053 2020       |.|o2 = S  
│ │ │ │ +0002d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d670: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69  |.|o2 : Polynomi
│ │ │ │ -0002d680: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +0002d660: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ +0002d670: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0002d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d6c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002d6b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d710: 2b0a 7c69 3320 3a20 6666 203d 206d 6174  +.|i3 : ff = mat
│ │ │ │ -0002d720: 7269 7822 6132 2c62 322c 6332 2220 2020  rix"a2,b2,c2"   
│ │ │ │ +0002d700: 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 6620  -----+.|i3 : ff 
│ │ │ │ +0002d710: 3d20 6d61 7472 6978 2261 322c 6232 2c63  = matrix"a2,b2,c
│ │ │ │ +0002d720: 3222 2020 2020 2020 2020 2020 2020 2020  2"              
│ │ │ │  0002d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d760: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d7b0: 7c0a 7c6f 3320 3d20 7c20 6132 2062 3220  |.|o3 = | a2 b2 
│ │ │ │ -0002d7c0: 6332 207c 2020 2020 2020 2020 2020 2020  c2 |            
│ │ │ │ +0002d7a0: 2020 2020 207c 0a7c 6f33 203d 207c 2061       |.|o3 = | a
│ │ │ │ +0002d7b0: 3220 6232 2063 3220 7c20 2020 2020 2020  2 b2 c2 |       
│ │ │ │ +0002d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d800: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d7f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d850: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002d860: 3120 2020 2020 2033 2020 2020 2020 2020  1      3        
│ │ │ │ +0002d840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d850: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8a0: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -0002d8b0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +0002d890: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ +0002d8a0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0002d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002d8e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d940: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964  +.|i4 : R = S/id
│ │ │ │ -0002d950: 6561 6c20 6666 2020 2020 2020 2020 2020  eal ff          
│ │ │ │ +0002d930: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d  -----+.|i4 : R =
│ │ │ │ +0002d940: 2053 2f69 6465 616c 2066 6620 2020 2020   S/ideal ff     
│ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d990: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002d980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9e0: 7c0a 7c6f 3420 3d20 5220 2020 2020 2020  |.|o4 = R       
│ │ │ │ +0002d9d0: 2020 2020 207c 0a7c 6f34 203d 2052 2020       |.|o4 = R  
│ │ │ │ +0002d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002da20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da80: 7c0a 7c6f 3420 3a20 5175 6f74 6965 6e74  |.|o4 : Quotient
│ │ │ │ -0002da90: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0002da70: 2020 2020 207c 0a7c 6f34 203a 2051 756f       |.|o4 : Quo
│ │ │ │ +0002da80: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +0002da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dad0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002dac0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002dad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002daf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002db20: 2b0a 7c69 3520 3a20 4d20 3d20 525e 312f  +.|i5 : M = R^1/
│ │ │ │ -0002db30: 6964 6561 6c22 612c 6263 2220 2020 2020  ideal"a,bc"     
│ │ │ │ +0002db10: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 203d  -----+.|i5 : M =
│ │ │ │ +0002db20: 2052 5e31 2f69 6465 616c 2261 2c62 6322   R^1/ideal"a,bc"
│ │ │ │ +0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002db60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbc0: 7c0a 7c6f 3520 3d20 636f 6b65 726e 656c  |.|o5 = cokernel
│ │ │ │ -0002dbd0: 207c 2061 2062 6320 7c20 2020 2020 2020   | a bc |       
│ │ │ │ +0002dbb0: 2020 2020 207c 0a7c 6f35 203d 2063 6f6b       |.|o5 = cok
│ │ │ │ +0002dbc0: 6572 6e65 6c20 7c20 6120 6263 207c 2020  ernel | a bc |  
│ │ │ │ +0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002dc00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002dc70: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0002dc50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc70: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  0002dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcb0: 7c0a 7c6f 3520 3a20 522d 6d6f 6475 6c65  |.|o5 : R-module
│ │ │ │ -0002dcc0: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +0002dca0: 2020 2020 207c 0a7c 6f35 203a 2052 2d6d       |.|o5 : R-m
│ │ │ │ +0002dcb0: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +0002dcc0: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0002dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002dcf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dd50: 2b0a 7c69 3620 3a20 6b20 3d20 3120 2020  +.|i6 : k = 1   
│ │ │ │ +0002dd40: 2d2d 2d2d 2d2b 0a7c 6936 203a 206b 203d  -----+.|i6 : k =
│ │ │ │ +0002dd50: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dda0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002dd90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddf0: 7c0a 7c6f 3620 3d20 3120 2020 2020 2020  |.|o6 = 1       
│ │ │ │ +0002dde0: 2020 2020 207c 0a7c 6f36 203d 2031 2020       |.|o6 = 1  
│ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002de30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de90: 2b0a 7c69 3720 3a20 6d20 3d20 6b2b 3520  +.|i7 : m = k+5 
│ │ │ │ +0002de80: 2d2d 2d2d 2d2b 0a7c 6937 203a 206d 203d  -----+.|i7 : m =
│ │ │ │ +0002de90: 206b 2b35 2020 2020 2020 2020 2020 2020   k+5            
│ │ │ │  0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dee0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002ded0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df30: 7c0a 7c6f 3720 3d20 3620 2020 2020 2020  |.|o7 = 6       
│ │ │ │ +0002df20: 2020 2020 207c 0a7c 6f37 203d 2036 2020       |.|o7 = 6  
│ │ │ │ +0002df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002df70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dfd0: 2b0a 7c69 3820 3a20 4620 3d20 7265 7328  +.|i8 : F = res(
│ │ │ │ -0002dfe0: 4d2c 204c 656e 6774 684c 696d 6974 203d  M, LengthLimit =
│ │ │ │ -0002dff0: 3e20 6d29 2020 2020 2020 2020 2020 2020  > m)            
│ │ │ │ +0002dfc0: 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 203d  -----+.|i8 : F =
│ │ │ │ +0002dfd0: 2072 6573 284d 2c20 4c65 6e67 7468 4c69   res(M, LengthLi
│ │ │ │ +0002dfe0: 6d69 7420 3d3e 206d 2920 2020 2020 2020  mit => m)       
│ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e020: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002e010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e070: 7c0a 7c20 2020 2020 2031 2020 2020 2020  |.|      1      
│ │ │ │ -0002e080: 3220 2020 2020 2034 2020 2020 2020 3720  2      4      7 
│ │ │ │ -0002e090: 2020 2020 2031 3120 2020 2020 2031 3620       11      16 
│ │ │ │ -0002e0a0: 2020 2020 2032 3220 2020 2020 2020 2020       22         
│ │ │ │ -0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0c0: 7c0a 7c6f 3820 3d20 5220 203c 2d2d 2052  |.|o8 = R  <-- R
│ │ │ │ -0002e0d0: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020    <-- R  <-- R  
│ │ │ │ -0002e0e0: 3c2d 2d20 5220 2020 3c2d 2d20 5220 2020  <-- R   <-- R   
│ │ │ │ -0002e0f0: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ -0002e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e110: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002e060: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ +0002e070: 2020 2020 2032 2020 2020 2020 3420 2020       2      4   
│ │ │ │ +0002e080: 2020 2037 2020 2020 2020 3131 2020 2020     7      11    
│ │ │ │ +0002e090: 2020 3136 2020 2020 2020 3232 2020 2020    16      22    
│ │ │ │ +0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e0b0: 2020 2020 207c 0a7c 6f38 203d 2052 2020       |.|o8 = R  
│ │ │ │ +0002e0c0: 3c2d 2d20 5220 203c 2d2d 2052 2020 3c2d  <-- R  <-- R  <-
│ │ │ │ +0002e0d0: 2d20 5220 203c 2d2d 2052 2020 203c 2d2d  - R  <-- R   <--
│ │ │ │ +0002e0e0: 2052 2020 203c 2d2d 2052 2020 2020 2020   R   <-- R      
│ │ │ │ +0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e160: 7c0a 7c20 2020 2020 3020 2020 2020 2031  |.|     0      1
│ │ │ │ -0002e170: 2020 2020 2020 3220 2020 2020 2033 2020        2      3  
│ │ │ │ -0002e180: 2020 2020 3420 2020 2020 2020 3520 2020      4       5   
│ │ │ │ -0002e190: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ -0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002e150: 2020 2020 207c 0a7c 2020 2020 2030 2020       |.|     0  
│ │ │ │ +0002e160: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ +0002e170: 2020 3320 2020 2020 2034 2020 2020 2020    3      4      
│ │ │ │ +0002e180: 2035 2020 2020 2020 2036 2020 2020 2020   5       6      
│ │ │ │ +0002e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e200: 7c0a 7c6f 3820 3a20 4368 6169 6e43 6f6d  |.|o8 : ChainCom
│ │ │ │ -0002e210: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │ +0002e1f0: 2020 2020 207c 0a7c 6f38 203a 2043 6861       |.|o8 : Cha
│ │ │ │ +0002e200: 696e 436f 6d70 6c65 7820 2020 2020 2020  inComplex       
│ │ │ │ +0002e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e250: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002e240: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e2a0: 2b0a 7c69 3920 3a20 7379 7a79 6769 6573  +.|i9 : syzygies
│ │ │ │ -0002e2b0: 203d 2061 7070 6c79 2831 2e2e 6d2c 2069   = apply(1..m, i
│ │ │ │ -0002e2c0: 2d3e 636f 6b65 7220 462e 6464 5f69 293b  ->coker F.dd_i);
│ │ │ │ +0002e290: 2d2d 2d2d 2d2b 0a7c 6939 203a 2073 797a  -----+.|i9 : syz
│ │ │ │ +0002e2a0: 7967 6965 7320 3d20 6170 706c 7928 312e  ygies = apply(1.
│ │ │ │ +0002e2b0: 2e6d 2c20 692d 3e63 6f6b 6572 2046 2e64  .m, i->coker F.d
│ │ │ │ +0002e2c0: 645f 6929 3b20 2020 2020 2020 2020 2020  d_i);           
│ │ │ │  0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002e2e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002e2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e340: 2b0a 7c69 3130 203a 2074 3120 3d20 6d61  +.|i10 : t1 = ma
│ │ │ │ -0002e350: 6b65 5428 6666 2c46 2c6b 2b34 293b 2020  keT(ff,F,k+4);  
│ │ │ │ +0002e330: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 7431  -----+.|i10 : t1
│ │ │ │ +0002e340: 203d 206d 616b 6554 2866 662c 462c 6b2b   = makeT(ff,F,k+
│ │ │ │ +0002e350: 3429 3b20 2020 2020 2020 2020 2020 2020  4);             
│ │ │ │  0002e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e390: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002e380: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e3e0: 2b0a 7c69 3131 203a 2074 3220 3d20 6d61  +.|i11 : t2 = ma
│ │ │ │ -0002e3f0: 6b65 5428 6666 2c46 2c6b 2b32 293b 2020  keT(ff,F,k+2);  
│ │ │ │ +0002e3d0: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7432  -----+.|i11 : t2
│ │ │ │ +0002e3e0: 203d 206d 616b 6554 2866 662c 462c 6b2b   = makeT(ff,F,k+
│ │ │ │ +0002e3f0: 3229 3b20 2020 2020 2020 2020 2020 2020  2);             
│ │ │ │  0002e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e430: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002e420: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e480: 2b0a 7c69 3132 203a 2054 3243 6f6d 706f  +.|i12 : T2Compo
│ │ │ │ -0002e490: 6e65 6e74 7320 3d20 666c 6174 7465 6e20  nents = flatten 
│ │ │ │ -0002e4a0: 666f 7220 6920 6672 6f6d 2030 2074 6f20  for i from 0 to 
│ │ │ │ -0002e4b0: 3120 6c69 7374 2866 6f72 206a 2066 726f  1 list(for j fro
│ │ │ │ -0002e4c0: 6d20 692b 3120 746f 2032 206c 6973 7420  m i+1 to 2 list 
│ │ │ │ -0002e4d0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +0002e470: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 5432  -----+.|i12 : T2
│ │ │ │ +0002e480: 436f 6d70 6f6e 656e 7473 203d 2066 6c61  Components = fla
│ │ │ │ +0002e490: 7474 656e 2066 6f72 2069 2066 726f 6d20  tten for i from 
│ │ │ │ +0002e4a0: 3020 746f 2031 206c 6973 7428 666f 7220  0 to 1 list(for 
│ │ │ │ +0002e4b0: 6a20 6672 6f6d 2069 2b31 2074 6f20 3220  j from i+1 to 2 
│ │ │ │ +0002e4c0: 6c69 7374 207c 0a7c 2d2d 2d2d 2d2d 2d2d  list |.|--------
│ │ │ │ +0002e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e520: 7c0a 7c6d 6170 2846 5f6b 2c20 465f 286b  |.|map(F_k, F_(k
│ │ │ │ -0002e530: 2b34 292c 2074 325f 692a 7431 5f6a 2d74  +4), t2_i*t1_j-t
│ │ │ │ -0002e540: 325f 6a2a 7431 5f69 2929 3b20 2020 2020  2_j*t1_i));     
│ │ │ │ +0002e510: 2d2d 2d2d 2d7c 0a7c 6d61 7028 465f 6b2c  -----|.|map(F_k,
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│ │ │ │ +0002e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002e570: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
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│ │ │ │  0002e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002e5e0: 7967 6965 735f 286b 2b34 292c 2054 3243  ygies_(k+4), T2C
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│ │ │ │ -0002e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e700: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ +0002e680: 2030 2030 207c 2020 2020 2020 2020 2020   0 0 |          
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│ │ │ │ +0002e6f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -0002e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002e740: 2020 2020 207c 0a7c 6f31 3320 3a20 4d61       |.|o13 : Ma
│ │ │ │ +0002e750: 7472 6978 2020 2020 2020 2020 2020 2020  trix            
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│ │ │ │  0002e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
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│ │ │ │ +0002e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002e800: 7954 7269 7669 616c 2067 2020 2020 2020  yTrivial g      
│ │ │ │ +0002e7e0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 6973  -----+.|i14 : is
│ │ │ │ +0002e7f0: 5374 6162 6c79 5472 6976 6961 6c20 6720  StablyTrivial g 
│ │ │ │ +0002e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002e830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -0002e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e890: 7c0a 7c6f 3134 203d 2074 7275 6520 2020  |.|o14 = true   
│ │ │ │ +0002e880: 2020 2020 207c 0a7c 6f31 3420 3d20 7472       |.|o14 = tr
│ │ │ │ +0002e890: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
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│ │ │ │  0002e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e930: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ -0002e940: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -0002e950: 7374 6162 6c65 486f 6d3a 2073 7461 626c  stableHom: stabl
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│ │ │ │ -0002e970: 6d20 486f 6d28 4d2c 4e29 2074 6f20 7468  m Hom(M,N) to th
│ │ │ │ -0002e980: 6520 7374 6162 6c65 2048 6f6d 206d 6f64  e stable Hom mod
│ │ │ │ -0002e990: 756c 650a 0a57 6179 7320 746f 2075 7365  ule..Ways to use
│ │ │ │ -0002e9a0: 2069 7353 7461 626c 7954 7269 7669 616c   isStablyTrivial
│ │ │ │ -0002e9b0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -0002e9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -0002e9d0: 2020 2a20 2269 7353 7461 626c 7954 7269    * "isStablyTri
│ │ │ │ -0002e9e0: 7669 616c 284d 6174 7269 7829 220a 0a46  vial(Matrix)"..F
│ │ │ │ -0002e9f0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -0002ea00: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -0002ea10: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -0002ea20: 202a 6e6f 7465 2069 7353 7461 626c 7954   *note isStablyT
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│ │ │ │ -0002ea40: 5472 6976 6961 6c2c 2069 7320 6120 2a6e  Trivial, is a *n
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│ │ │ │ -0002eac0: 7465 6e73 696f 6e2c 204e 6578 743a 204c  tension, Next: L
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│ │ │ │ -0002eae0: 5374 6162 6c79 5472 6976 6961 6c2c 2055  StablyTrivial, U
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│ │ │ │ +0002e940: 6e6f 7465 2073 7461 626c 6548 6f6d 3a20  note stableHom: 
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│ │ │ │ +0002e960: 7020 6672 6f6d 2048 6f6d 284d 2c4e 2920  p from Hom(M,N) 
│ │ │ │ +0002e970: 746f 2074 6865 2073 7461 626c 6520 486f  to the stable Ho
│ │ │ │ +0002e980: 6d20 6d6f 6475 6c65 0a0a 5761 7973 2074  m module..Ways t
│ │ │ │ +0002e990: 6f20 7573 6520 6973 5374 6162 6c79 5472  o use isStablyTr
│ │ │ │ +0002e9a0: 6976 6961 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d  ivial:.=========
│ │ │ │ +0002e9b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002e9c0: 3d3d 3d0a 0a20 202a 2022 6973 5374 6162  ===..  * "isStab
│ │ │ │ +0002e9d0: 6c79 5472 6976 6961 6c28 4d61 7472 6978  lyTrivial(Matrix
│ │ │ │ +0002e9e0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +0002e9f0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +0002ea00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
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│ │ │ │ +0002ea20: 6162 6c79 5472 6976 6961 6c3a 2069 7353  ablyTrivial: isS
│ │ │ │ +0002ea30: 7461 626c 7954 7269 7669 616c 2c20 6973  tablyTrivial, is
│ │ │ │ +0002ea40: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +0002ea50: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
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│ │ │ │ +0002eb10: 756c 2065 7874 656e 7369 6f6e 2063 6f6d  ul extension com
│ │ │ │ +0002eb20: 706c 6578 206f 6620 6120 6d61 700a 2a2a  plex of a map.**
│ │ │ │ +0002eb30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002eb40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002eb50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002eb60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002eb70: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ -0002eb80: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ -0002eb90: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0002eba0: 2020 4d4d 203d 206b 6f73 7a75 6c45 7874    MM = koszulExt
│ │ │ │ -0002ebb0: 656e 7369 6f6e 2846 462c 4242 2c70 7369  ension(FF,BB,psi
│ │ │ │ -0002ebc0: 312c 6666 290a 2020 2a20 496e 7075 7473  1,ff).  * Inputs
│ │ │ │ -0002ebd0: 3a0a 2020 2020 2020 2a20 4646 2c20 6120  :.      * FF, a 
│ │ │ │ -0002ebe0: 2a6e 6f74 6520 6368 6169 6e20 636f 6d70  *note chain comp
│ │ │ │ -0002ebf0: 6c65 783a 2028 4d61 6361 756c 6179 3244  lex: (Macaulay2D
│ │ │ │ -0002ec00: 6f63 2943 6861 696e 436f 6d70 6c65 782c  oc)ChainComplex,
│ │ │ │ -0002ec10: 2c20 7265 736f 6c75 7469 6f6e 206f 7665  , resolution ove
│ │ │ │ -0002ec20: 720a 2020 2020 2020 2020 530a 2020 2020  r.        S.    
│ │ │ │ -0002ec30: 2020 2a20 4242 2c20 6120 2a6e 6f74 6520    * BB, a *note 
│ │ │ │ -0002ec40: 6368 6169 6e20 636f 6d70 6c65 783a 2028  chain complex: (
│ │ │ │ -0002ec50: 4d61 6361 756c 6179 3244 6f63 2943 6861  Macaulay2Doc)Cha
│ │ │ │ -0002ec60: 696e 436f 6d70 6c65 782c 2c20 7477 6f2d  inComplex,, two-
│ │ │ │ -0002ec70: 7465 726d 0a20 2020 2020 2020 2063 6f6d  term.        com
│ │ │ │ -0002ec80: 706c 6578 2042 425f 312d 2d3e 4242 5f30  plex BB_1-->BB_0
│ │ │ │ -0002ec90: 0a20 2020 2020 202a 2070 7369 312c 2061  .      * psi1, a
│ │ │ │ -0002eca0: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ -0002ecb0: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ -0002ecc0: 7269 782c 2c20 6672 6f6d 2042 425f 3120  rix,, from BB_1 
│ │ │ │ -0002ecd0: 746f 2046 465f 300a 2020 2020 2020 2a20  to FF_0.      * 
│ │ │ │ -0002ece0: 6666 2c20 6120 2a6e 6f74 6520 6d61 7472  ff, a *note matr
│ │ │ │ -0002ecf0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -0002ed00: 6329 4d61 7472 6978 2c2c 2072 6567 756c  c)Matrix,, regul
│ │ │ │ -0002ed10: 6172 2073 6571 7565 6e63 650a 2020 2020  ar sequence.    
│ │ │ │ -0002ed20: 2020 2020 616e 6e69 6869 6c61 7469 6e67      annihilating
│ │ │ │ -0002ed30: 2074 6865 206d 6f64 756c 6520 7265 736f   the module reso
│ │ │ │ -0002ed40: 6c76 6564 2062 7920 4646 0a20 202a 204f  lved by FF.  * O
│ │ │ │ -0002ed50: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0002ed60: 4d4d 2c20 6120 2a6e 6f74 6520 6368 6169  MM, a *note chai
│ │ │ │ -0002ed70: 6e20 636f 6d70 6c65 783a 2028 4d61 6361  n complex: (Maca
│ │ │ │ -0002ed80: 756c 6179 3244 6f63 2943 6861 696e 436f  ulay2Doc)ChainCo
│ │ │ │ -0002ed90: 6d70 6c65 782c 2c20 7468 6520 6d61 7070  mplex,, the mapp
│ │ │ │ -0002eda0: 696e 670a 2020 2020 2020 2020 636f 6e65  ing.        cone
│ │ │ │ -0002edb0: 206f 6620 7468 6520 696e 6475 6365 6420   of the induced 
│ │ │ │ -0002edc0: 6d61 7020 425b 2d31 5d5c 6f74 696d 6573  map B[-1]\otimes
│ │ │ │ -0002edd0: 204b 4b28 6666 2920 746f 2057 2065 7874   KK(ff) to W ext
│ │ │ │ -0002ede0: 656e 6469 6e67 2070 7369 0a0a 4465 7363  ending psi..Desc
│ │ │ │ -0002edf0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0002ee00: 3d3d 3d0a 0a49 6d70 6c65 6d65 6e74 7320  ===..Implements 
│ │ │ │ -0002ee10: 7468 6520 636f 6e73 7472 7563 7469 6f6e  the construction
│ │ │ │ -0002ee20: 2069 6e20 7468 6520 7061 7065 7220 224d   in the paper "M
│ │ │ │ -0002ee30: 6174 7269 7820 4661 6374 6f72 697a 6174  atrix Factorizat
│ │ │ │ -0002ee40: 696f 6e73 2069 6e20 4869 6768 6572 0a43  ions in Higher.C
│ │ │ │ -0002ee50: 6f64 696d 656e 7369 6f6e 2220 6279 2045  odimension" by E
│ │ │ │ -0002ee60: 6973 656e 6275 6420 616e 6420 5065 6576  isenbud and Peev
│ │ │ │ -0002ee70: 612e 0a0a 5365 6520 616c 736f 0a3d 3d3d  a...See also.===
│ │ │ │ -0002ee80: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0002ee90: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ -0002eea0: 7574 696f 6e3a 206d 616b 6546 696e 6974  ution: makeFinit
│ │ │ │ -0002eeb0: 6552 6573 6f6c 7574 696f 6e2c 202d 2d20  eResolution, -- 
│ │ │ │ -0002eec0: 6669 6e69 7465 2072 6573 6f6c 7574 696f  finite resolutio
│ │ │ │ -0002eed0: 6e20 6f66 2061 0a20 2020 206d 6174 7269  n of a.    matri
│ │ │ │ -0002eee0: 7820 6661 6374 6f72 697a 6174 696f 6e20  x factorization 
│ │ │ │ -0002eef0: 6d6f 6475 6c65 204d 0a0a 5761 7973 2074  module M..Ways t
│ │ │ │ -0002ef00: 6f20 7573 6520 6b6f 737a 756c 4578 7465  o use koszulExte
│ │ │ │ -0002ef10: 6e73 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  nsion:.=========
│ │ │ │ -0002ef20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002ef30: 3d3d 3d0a 0a20 202a 2022 6b6f 737a 756c  ===..  * "koszul
│ │ │ │ -0002ef40: 4578 7465 6e73 696f 6e28 4368 6169 6e43  Extension(ChainC
│ │ │ │ -0002ef50: 6f6d 706c 6578 2c43 6861 696e 436f 6d70  omplex,ChainComp
│ │ │ │ -0002ef60: 6c65 782c 4d61 7472 6978 2c4d 6174 7269  lex,Matrix,Matri
│ │ │ │ -0002ef70: 7829 220a 0a46 6f72 2074 6865 2070 726f  x)"..For the pro
│ │ │ │ -0002ef80: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -0002ef90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -0002efa0: 6f62 6a65 6374 202a 6e6f 7465 206b 6f73  object *note kos
│ │ │ │ -0002efb0: 7a75 6c45 7874 656e 7369 6f6e 3a20 6b6f  zulExtension: ko
│ │ │ │ -0002efc0: 737a 756c 4578 7465 6e73 696f 6e2c 2069  szulExtension, i
│ │ │ │ -0002efd0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ -0002efe0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ -0002eff0: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -0002f000: 756e 6374 696f 6e2c 2e0a 1f0a 4669 6c65  unction,....File
│ │ │ │ -0002f010: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -0002f020: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0002f030: 732e 696e 666f 2c20 4e6f 6465 3a20 4c61  s.info, Node: La
│ │ │ │ -0002f040: 7965 7265 642c 204e 6578 743a 206c 6179  yered, Next: lay
│ │ │ │ -0002f050: 6572 6564 5265 736f 6c75 7469 6f6e 2c20  eredResolution, 
│ │ │ │ -0002f060: 5072 6576 3a20 6b6f 737a 756c 4578 7465  Prev: koszulExte
│ │ │ │ -0002f070: 6e73 696f 6e2c 2055 703a 2054 6f70 0a0a  nsion, Up: Top..
│ │ │ │ -0002f080: 4c61 7965 7265 6420 2d2d 204f 7074 696f  Layered -- Optio
│ │ │ │ -0002f090: 6e20 666f 7220 6d61 7472 6978 4661 6374  n for matrixFact
│ │ │ │ -0002f0a0: 6f72 697a 6174 696f 6e0a 2a2a 2a2a 2a2a  orization.******
│ │ │ │ +0002eb60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0002eb70: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ +0002eb80: 3d0a 0a20 202a 2055 7361 6765 3a20 0a20  =..  * Usage: . 
│ │ │ │ +0002eb90: 2020 2020 2020 204d 4d20 3d20 6b6f 737a         MM = kosz
│ │ │ │ +0002eba0: 756c 4578 7465 6e73 696f 6e28 4646 2c42  ulExtension(FF,B
│ │ │ │ +0002ebb0: 422c 7073 6931 2c66 6629 0a20 202a 2049  B,psi1,ff).  * I
│ │ │ │ +0002ebc0: 6e70 7574 733a 0a20 2020 2020 202a 2046  nputs:.      * F
│ │ │ │ +0002ebd0: 462c 2061 202a 6e6f 7465 2063 6861 696e  F, a *note chain
│ │ │ │ +0002ebe0: 2063 6f6d 706c 6578 3a20 284d 6163 6175   complex: (Macau
│ │ │ │ +0002ebf0: 6c61 7932 446f 6329 4368 6169 6e43 6f6d  lay2Doc)ChainCom
│ │ │ │ +0002ec00: 706c 6578 2c2c 2072 6573 6f6c 7574 696f  plex,, resolutio
│ │ │ │ +0002ec10: 6e20 6f76 6572 0a20 2020 2020 2020 2053  n over.        S
│ │ │ │ +0002ec20: 0a20 2020 2020 202a 2042 422c 2061 202a  .      * BB, a *
│ │ │ │ +0002ec30: 6e6f 7465 2063 6861 696e 2063 6f6d 706c  note chain compl
│ │ │ │ +0002ec40: 6578 3a20 284d 6163 6175 6c61 7932 446f  ex: (Macaulay2Do
│ │ │ │ +0002ec50: 6329 4368 6169 6e43 6f6d 706c 6578 2c2c  c)ChainComplex,,
│ │ │ │ +0002ec60: 2074 776f 2d74 6572 6d0a 2020 2020 2020   two-term.      
│ │ │ │ +0002ec70: 2020 636f 6d70 6c65 7820 4242 5f31 2d2d    complex BB_1--
│ │ │ │ +0002ec80: 3e42 425f 300a 2020 2020 2020 2a20 7073  >BB_0.      * ps
│ │ │ │ +0002ec90: 6931 2c20 6120 2a6e 6f74 6520 6d61 7472  i1, a *note matr
│ │ │ │ +0002eca0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +0002ecb0: 6329 4d61 7472 6978 2c2c 2066 726f 6d20  c)Matrix,, from 
│ │ │ │ +0002ecc0: 4242 5f31 2074 6f20 4646 5f30 0a20 2020  BB_1 to FF_0.   
│ │ │ │ +0002ecd0: 2020 202a 2066 662c 2061 202a 6e6f 7465     * ff, a *note
│ │ │ │ +0002ece0: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +0002ecf0: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +0002ed00: 7265 6775 6c61 7220 7365 7175 656e 6365  regular sequence
│ │ │ │ +0002ed10: 0a20 2020 2020 2020 2061 6e6e 6968 696c  .        annihil
│ │ │ │ +0002ed20: 6174 696e 6720 7468 6520 6d6f 6475 6c65  ating the module
│ │ │ │ +0002ed30: 2072 6573 6f6c 7665 6420 6279 2046 460a   resolved by FF.
│ │ │ │ +0002ed40: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0002ed50: 2020 202a 204d 4d2c 2061 202a 6e6f 7465     * MM, a *note
│ │ │ │ +0002ed60: 2063 6861 696e 2063 6f6d 706c 6578 3a20   chain complex: 
│ │ │ │ +0002ed70: 284d 6163 6175 6c61 7932 446f 6329 4368  (Macaulay2Doc)Ch
│ │ │ │ +0002ed80: 6169 6e43 6f6d 706c 6578 2c2c 2074 6865  ainComplex,, the
│ │ │ │ +0002ed90: 206d 6170 7069 6e67 0a20 2020 2020 2020   mapping.       
│ │ │ │ +0002eda0: 2063 6f6e 6520 6f66 2074 6865 2069 6e64   cone of the ind
│ │ │ │ +0002edb0: 7563 6564 206d 6170 2042 5b2d 315d 5c6f  uced map B[-1]\o
│ │ │ │ +0002edc0: 7469 6d65 7320 4b4b 2866 6629 2074 6f20  times KK(ff) to 
│ │ │ │ +0002edd0: 5720 6578 7465 6e64 696e 6720 7073 690a  W extending psi.
│ │ │ │ +0002ede0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0002edf0: 3d3d 3d3d 3d3d 3d3d 0a0a 496d 706c 656d  ========..Implem
│ │ │ │ +0002ee00: 656e 7473 2074 6865 2063 6f6e 7374 7275  ents the constru
│ │ │ │ +0002ee10: 6374 696f 6e20 696e 2074 6865 2070 6170  ction in the pap
│ │ │ │ +0002ee20: 6572 2022 4d61 7472 6978 2046 6163 746f  er "Matrix Facto
│ │ │ │ +0002ee30: 7269 7a61 7469 6f6e 7320 696e 2048 6967  rizations in Hig
│ │ │ │ +0002ee40: 6865 720a 436f 6469 6d65 6e73 696f 6e22  her.Codimension"
│ │ │ │ +0002ee50: 2062 7920 4569 7365 6e62 7564 2061 6e64   by Eisenbud and
│ │ │ │ +0002ee60: 2050 6565 7661 2e0a 0a53 6565 2061 6c73   Peeva...See als
│ │ │ │ +0002ee70: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +0002ee80: 2a6e 6f74 6520 6d61 6b65 4669 6e69 7465  *note makeFinite
│ │ │ │ +0002ee90: 5265 736f 6c75 7469 6f6e 3a20 6d61 6b65  Resolution: make
│ │ │ │ +0002eea0: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ +0002eeb0: 2c20 2d2d 2066 696e 6974 6520 7265 736f  , -- finite reso
│ │ │ │ +0002eec0: 6c75 7469 6f6e 206f 6620 610a 2020 2020  lution of a.    
│ │ │ │ +0002eed0: 6d61 7472 6978 2066 6163 746f 7269 7a61  matrix factoriza
│ │ │ │ +0002eee0: 7469 6f6e 206d 6f64 756c 6520 4d0a 0a57  tion module M..W
│ │ │ │ +0002eef0: 6179 7320 746f 2075 7365 206b 6f73 7a75  ays to use koszu
│ │ │ │ +0002ef00: 6c45 7874 656e 7369 6f6e 3a0a 3d3d 3d3d  lExtension:.====
│ │ │ │ +0002ef10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002ef20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226b  ========..  * "k
│ │ │ │ +0002ef30: 6f73 7a75 6c45 7874 656e 7369 6f6e 2843  oszulExtension(C
│ │ │ │ +0002ef40: 6861 696e 436f 6d70 6c65 782c 4368 6169  hainComplex,Chai
│ │ │ │ +0002ef50: 6e43 6f6d 706c 6578 2c4d 6174 7269 782c  nComplex,Matrix,
│ │ │ │ +0002ef60: 4d61 7472 6978 2922 0a0a 466f 7220 7468  Matrix)"..For th
│ │ │ │ +0002ef70: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +0002ef80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002ef90: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0002efa0: 6520 6b6f 737a 756c 4578 7465 6e73 696f  e koszulExtensio
│ │ │ │ +0002efb0: 6e3a 206b 6f73 7a75 6c45 7874 656e 7369  n: koszulExtensi
│ │ │ │ +0002efc0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d  on, is a *note m
│ │ │ │ +0002efd0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ +0002efe0: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ +0002eff0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a1f  thodFunction,...
│ │ │ │ +0002f000: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ +0002f010: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0002f020: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ +0002f030: 653a 204c 6179 6572 6564 2c20 4e65 7874  e: Layered, Next
│ │ │ │ +0002f040: 3a20 6c61 7965 7265 6452 6573 6f6c 7574  : layeredResolut
│ │ │ │ +0002f050: 696f 6e2c 2050 7265 763a 206b 6f73 7a75  ion, Prev: koszu
│ │ │ │ +0002f060: 6c45 7874 656e 7369 6f6e 2c20 5570 3a20  lExtension, Up: 
│ │ │ │ +0002f070: 546f 700a 0a4c 6179 6572 6564 202d 2d20  Top..Layered -- 
│ │ │ │ +0002f080: 4f70 7469 6f6e 2066 6f72 206d 6174 7269  Option for matri
│ │ │ │ +0002f090: 7846 6163 746f 7269 7a61 7469 6f6e 0a2a  xFactorization.*
│ │ │ │ +0002f0a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f0b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002f0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002f0d0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -0002f0e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ -0002f0f0: 653a 200a 2020 2020 2020 2020 6d61 7472  e: .        matr
│ │ │ │ -0002f100: 6978 4661 6374 6f72 697a 6174 696f 6e28  ixFactorization(
│ │ │ │ -0002f110: 6666 2c6d 2c4c 6179 6572 6564 203d 3e20  ff,m,Layered => 
│ │ │ │ -0002f120: 7472 7565 290a 2020 2a20 496e 7075 7473  true).  * Inputs
│ │ │ │ -0002f130: 3a0a 2020 2020 2020 2a20 4368 6563 6b2c  :.      * Check,
│ │ │ │ -0002f140: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
│ │ │ │ -0002f150: 2076 616c 7565 3a20 284d 6163 6175 6c61   value: (Macaula
│ │ │ │ -0002f160: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20  y2Doc)Boolean,, 
│ │ │ │ -0002f170: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -0002f180: 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d 616b 6573  =========..Makes
│ │ │ │ -0002f190: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0002f1a0: 7469 6f6e 2075 7365 2074 6865 2022 6c61  tion use the "la
│ │ │ │ -0002f1b0: 7965 7265 6422 2061 6c67 6f72 6974 686d  yered" algorithm
│ │ │ │ -0002f1c0: 2c20 7768 6963 6820 776f 726b 7320 666f  , which works fo
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│ │ │ │ -0002f1e0: 2c20 6275 7420 7265 7475 726e 7320 736f  , but returns so
│ │ │ │ -0002f1f0: 6d65 7468 696e 6720 6e6f 6e2d 6d69 6e69  mething non-mini
│ │ │ │ -0002f200: 6d61 6c20 6966 2074 6865 206d 6f64 756c  mal if the modul
│ │ │ │ -0002f210: 6520 6973 206e 6f74 2061 2022 6869 6768  e is not a "high
│ │ │ │ -0002f220: 2073 797a 7967 7922 0a69 6e20 6120 7375   syzygy".in a su
│ │ │ │ -0002f230: 6974 6162 6c65 2073 656e 7365 2e20 4465  itable sense. De
│ │ │ │ -0002f240: 6661 756c 7420 6973 2022 7472 7565 222e  fault is "true".
│ │ │ │ -0002f250: 204e 6f74 6520 7468 6174 2077 6865 6e20   Note that when 
│ │ │ │ -0002f260: 7468 6520 6d6f 6475 6c65 2069 7320 6120  the module is a 
│ │ │ │ -0002f270: 6869 6768 0a73 797a 7967 792c 204c 6179  high.syzygy, Lay
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│ │ │ │ -0002f290: 6d75 6368 2066 6173 7465 722e 0a0a 5365  much faster...Se
│ │ │ │ -0002f2a0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -0002f2b0: 0a20 202a 202a 6e6f 7465 206d 6174 7269  .  * *note matri
│ │ │ │ -0002f2c0: 7846 6163 746f 7269 7a61 7469 6f6e 3a20  xFactorization: 
│ │ │ │ -0002f2d0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
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│ │ │ │ -0002f2f0: 6120 6869 6768 6572 0a20 2020 2063 6f64  a higher.    cod
│ │ │ │ -0002f300: 696d 656e 7369 6f6e 206d 6174 7269 7820  imension matrix 
│ │ │ │ -0002f310: 6661 6374 6f72 697a 6174 696f 6e0a 0a46  factorization..F
│ │ │ │ -0002f320: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
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│ │ │ │ -0002f340: 6e61 6d65 6420 4c61 7965 7265 643a 0a3d  named Layered:.=
│ │ │ │ +0002f0c0: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ +0002f0d0: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ +0002f0e0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0002f0f0: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0002f100: 7469 6f6e 2866 662c 6d2c 4c61 7965 7265  tion(ff,m,Layere
│ │ │ │ +0002f110: 6420 3d3e 2074 7275 6529 0a20 202a 2049  d => true).  * I
│ │ │ │ +0002f120: 6e70 7574 733a 0a20 2020 2020 202a 2043  nputs:.      * C
│ │ │ │ +0002f130: 6865 636b 2c20 6120 2a6e 6f74 6520 426f  heck, a *note Bo
│ │ │ │ +0002f140: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61  olean value: (Ma
│ │ │ │ +0002f150: 6361 756c 6179 3244 6f63 2942 6f6f 6c65  caulay2Doc)Boole
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│ │ │ │ +0002f180: 4d61 6b65 7320 6d61 7472 6978 4661 6374  Makes matrixFact
│ │ │ │ +0002f190: 6f72 697a 6174 696f 6e20 7573 6520 7468  orization use th
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│ │ │ │ +0002f1b0: 7269 7468 6d2c 2077 6869 6368 2077 6f72  rithm, which wor
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│ │ │ │ +0002f200: 6d6f 6475 6c65 2069 7320 6e6f 7420 6120  module is not a 
│ │ │ │ +0002f210: 2268 6967 6820 7379 7a79 6779 220a 696e  "high syzygy".in
│ │ │ │ +0002f220: 2061 2073 7569 7461 626c 6520 7365 6e73   a suitable sens
│ │ │ │ +0002f230: 652e 2044 6566 6175 6c74 2069 7320 2274  e. Default is "t
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│ │ │ │ +0002f260: 6973 2061 2068 6967 680a 7379 7a79 6779  is a high.syzygy
│ │ │ │ +0002f270: 2c20 4c61 7965 7265 643d 3e20 6661 6c73  , Layered=> fals
│ │ │ │ +0002f280: 6520 6973 206d 7563 6820 6661 7374 6572  e is much faster
│ │ │ │ +0002f290: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ +0002f2a0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +0002f2b0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0002f2c0: 696f 6e3a 206d 6174 7269 7846 6163 746f  ion: matrixFacto
│ │ │ │ +0002f2d0: 7269 7a61 7469 6f6e 2c20 2d2d 204d 6170  rization, -- Map
│ │ │ │ +0002f2e0: 7320 696e 2061 2068 6967 6865 720a 2020  s in a higher.  
│ │ │ │ +0002f2f0: 2020 636f 6469 6d65 6e73 696f 6e20 6d61    codimension ma
│ │ │ │ +0002f300: 7472 6978 2066 6163 746f 7269 7a61 7469  trix factorizati
│ │ │ │ +0002f310: 6f6e 0a0a 4675 6e63 7469 6f6e 7320 7769  on..Functions wi
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│ │ │ │ +0002f340: 6564 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ed:.============
│ │ │ │  0002f350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0002f360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -0002f380: 2020 2a20 226d 6174 7269 7846 6163 746f    * "matrixFacto
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│ │ │ │ +0002f940: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0002f950: 3d3d 3d3d 3d0a 0a54 6865 2072 6573 6f6c  =====..The resol
│ │ │ │ +0002f960: 7574 696f 6e73 2063 6f6d 7075 7465 6420  utions computed 
│ │ │ │ +0002f970: 6172 6520 7468 6f73 6520 6465 7363 7269  are those descri
│ │ │ │ +0002f980: 6265 6420 696e 2074 6865 2070 6170 6572  bed in the paper
│ │ │ │ +0002f990: 2022 4c61 7965 7265 6420 5265 736f 6c75   "Layered Resolu
│ │ │ │ +0002f9a0: 7469 6f6e 730a 6f66 2043 6f68 656e 2d4d  tions.of Cohen-M
│ │ │ │ +0002f9b0: 6163 6175 6c61 7920 6d6f 6475 6c65 7322  acaulay modules"
│ │ │ │ +0002f9c0: 2062 7920 4569 7365 6e62 7564 2061 6e64   by Eisenbud and
│ │ │ │ +0002f9d0: 2050 6565 7661 2e20 5468 6579 2061 7265   Peeva. They are
│ │ │ │ +0002f9e0: 2062 6f74 6820 6d69 6e69 6d61 6c20 7768   both minimal wh
│ │ │ │ +0002f9f0: 656e 204d 0a69 7320 6120 7375 6666 6963  en M.is a suffic
│ │ │ │ +0002fa00: 6965 6e74 6c79 2068 6967 6820 7379 7a79  iently high syzy
│ │ │ │ +0002fa10: 6779 206f 6620 6120 6d6f 6475 6c65 204e  gy of a module N
│ │ │ │ +0002fa20: 2e20 4966 2074 6865 206f 7074 696f 6e20  . If the option 
│ │ │ │ +0002fa30: 5665 7262 6f73 653d 3e74 7275 6520 6973  Verbose=>true is
│ │ │ │ +0002fa40: 0a73 6574 2c20 7468 656e 2028 696e 2074  .set, then (in t
│ │ │ │ +0002fa50: 6865 2063 6173 6520 6f66 2074 6865 2072  he case of the r
│ │ │ │ +0002fa60: 6573 6f6c 7574 696f 6e20 6f76 6572 2053  esolution over S
│ │ │ │ +0002fa70: 2920 7468 6520 7261 6e6b 7320 6f66 2074  ) the ranks of t
│ │ │ │ +0002fa80: 6865 206d 6f64 756c 6573 2042 5f73 0a69  he modules B_s.i
│ │ │ │ +0002fa90: 6e20 7468 6520 7265 736f 6c75 7469 6f6e  n the resolution
│ │ │ │ +0002faa0: 2061 7265 206f 7574 7075 742e 0a0a 4865   are output...He
│ │ │ │ +0002fab0: 7265 2069 7320 616e 2065 7861 6d70 6c65  re is an example
│ │ │ │ +0002fac0: 2063 6f6d 7075 7469 6e67 2035 2074 6572   computing 5 ter
│ │ │ │ +0002fad0: 6d73 206f 6620 616e 2069 6e66 696e 6974  ms of an infinit
│ │ │ │ +0002fae0: 6520 7265 736f 6c75 7469 6f6e 3a0a 0a2b  e resolution:..+
│ │ │ │ +0002faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fb40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -0002fb50: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63  S = ZZ/101[a,b,c
│ │ │ │ -0002fb60: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0002fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002fb40: 6931 203a 2053 203d 205a 5a2f 3130 315b  i1 : S = ZZ/101[
│ │ │ │ +0002fb50: 612c 622c 635d 2020 2020 2020 2020 2020  a,b,c]          
│ │ │ │ +0002fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002fb80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fbe0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -0002fbf0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +0002fbd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fbe0: 6f31 203d 2053 2020 2020 2020 2020 2020  o1 = S          
│ │ │ │ +0002fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002fc20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc80: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -0002fc90: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0002fc70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fc80: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ +0002fc90: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  0002fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fcd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002fcc0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fd20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -0002fd30: 6666 203d 206d 6174 7269 7822 6133 2c20  ff = matrix"a3, 
│ │ │ │ -0002fd40: 6233 2c20 6333 2220 2020 2020 2020 2020  b3, c3"         
│ │ │ │ +0002fd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002fd20: 6932 203a 2066 6620 3d20 6d61 7472 6978  i2 : ff = matrix
│ │ │ │ +0002fd30: 2261 332c 2062 332c 2063 3322 2020 2020  "a3, b3, c3"    
│ │ │ │ +0002fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002fd60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fdc0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -0002fdd0: 7c20 6133 2062 3320 6333 207c 2020 2020  | a3 b3 c3 |    
│ │ │ │ +0002fdb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fdc0: 6f32 203d 207c 2061 3320 6233 2063 3320  o2 = | a3 b3 c3 
│ │ │ │ +0002fdd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fe10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002fe00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fe60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002fe70: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +0002fe50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fe60: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0002fe70: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  0002fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002feb0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -0002fec0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +0002fea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002feb0: 6f32 203a 204d 6174 7269 7820 5320 203c  o2 : Matrix S  <
│ │ │ │ +0002fec0: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  0002fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ff00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002fef0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002ff00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ff10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ff20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ff30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ff50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -0002ff60: 5220 3d20 532f 6964 6561 6c20 6666 2020  R = S/ideal ff  
│ │ │ │ +0002ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002ff50: 6933 203a 2052 203d 2053 2f69 6465 616c  i3 : R = S/ideal
│ │ │ │ +0002ff60: 2066 6620 2020 2020 2020 2020 2020 2020   ff             
│ │ │ │  0002ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ffa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002ff90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fff0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00030000: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0002ffe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002fff0: 6f33 203d 2052 2020 2020 2020 2020 2020  o3 = R          
│ │ │ │ +00030000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030040: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00030030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030090: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -000300a0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00030080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030090: 6f33 203a 2051 756f 7469 656e 7452 696e  o3 : QuotientRin
│ │ │ │ +000300a0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  000300b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000300c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000300d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000300e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000300d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000300e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030130: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00030140: 4d20 3d20 7379 7a79 6779 4d6f 6475 6c65  M = syzygyModule
│ │ │ │ -00030150: 2832 2c63 6f6b 6572 2076 6172 7320 5229  (2,coker vars R)
│ │ │ │ +00030120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00030130: 6934 203a 204d 203d 2073 797a 7967 794d  i4 : M = syzygyM
│ │ │ │ +00030140: 6f64 756c 6528 322c 636f 6b65 7220 7661  odule(2,coker va
│ │ │ │ +00030150: 7273 2052 2920 2020 2020 2020 2020 2020  rs R)           
│ │ │ │  00030160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030180: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00030170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301d0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -000301e0: 636f 6b65 726e 656c 207b 327d 207c 2061  cokernel {2} | a
│ │ │ │ -000301f0: 2020 3020 2d63 3220 3020 2020 6232 2030    0 -c2 0   b2 0
│ │ │ │ -00030200: 2030 2020 2030 2020 3020 2030 207c 2020   0   0  0  0 |  
│ │ │ │ -00030210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030220: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030230: 2020 2020 2020 2020 207b 327d 207c 202d           {2} | -
│ │ │ │ -00030240: 6220 3020 3020 2020 2d63 3220 3020 2030  b 0 0   -c2 0  0
│ │ │ │ -00030250: 2030 2020 2061 3220 3020 2030 207c 2020   0   a2 0  0 |  
│ │ │ │ -00030260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030270: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030280: 2020 2020 2020 2020 207b 327d 207c 2063           {2} | c
│ │ │ │ -00030290: 2020 3020 3020 2020 3020 2020 3020 2030    0 0   0   0  0
│ │ │ │ -000302a0: 202d 6232 2030 2020 6132 2030 207c 2020   -b2 0  a2 0 |  
│ │ │ │ -000302b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000302c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000302d0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -000302e0: 2020 6320 6220 2020 6120 2020 3020 2030    c b   a   0  0
│ │ │ │ -000302f0: 2030 2020 2030 2020 3020 2030 207c 2020   0   0  0  0 |  
│ │ │ │ -00030300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030310: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030320: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -00030330: 2020 3020 3020 2020 3020 2020 6320 2062    0 0   0   c  b
│ │ │ │ -00030340: 2061 2020 2030 2020 3020 2030 207c 2020   a   0  0  0 |  
│ │ │ │ -00030350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030360: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030370: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -00030380: 2020 3020 3020 2020 3020 2020 3020 2030    0 0   0   0  0
│ │ │ │ -00030390: 2030 2020 2063 2020 6220 2061 207c 2020   0   c  b  a |  
│ │ │ │ -000303a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000303b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000301c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000301d0: 6f34 203d 2063 6f6b 6572 6e65 6c20 7b32  o4 = cokernel {2
│ │ │ │ +000301e0: 7d20 7c20 6120 2030 202d 6332 2030 2020  } | a  0 -c2 0  
│ │ │ │ +000301f0: 2062 3220 3020 3020 2020 3020 2030 2020   b2 0 0   0  0  
│ │ │ │ +00030200: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00030210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030220: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00030230: 7d20 7c20 2d62 2030 2030 2020 202d 6332  } | -b 0 0   -c2
│ │ │ │ +00030240: 2030 2020 3020 3020 2020 6132 2030 2020   0  0 0   a2 0  
│ │ │ │ +00030250: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00030260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030270: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00030280: 7d20 7c20 6320 2030 2030 2020 2030 2020  } | c  0 0   0  
│ │ │ │ +00030290: 2030 2020 3020 2d62 3220 3020 2061 3220   0  0 -b2 0  a2 
│ │ │ │ +000302a0: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +000302b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000302c0: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ +000302d0: 7d20 7c20 3020 2063 2062 2020 2061 2020  } | 0  c b   a  
│ │ │ │ +000302e0: 2030 2020 3020 3020 2020 3020 2030 2020   0  0 0   0  0  
│ │ │ │ +000302f0: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00030300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030310: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ +00030320: 7d20 7c20 3020 2030 2030 2020 2030 2020  } | 0  0 0   0  
│ │ │ │ +00030330: 2063 2020 6220 6120 2020 3020 2030 2020   c  b a   0  0  
│ │ │ │ +00030340: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00030350: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030360: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ +00030370: 7d20 7c20 3020 2030 2030 2020 2030 2020  } | 0  0 0   0  
│ │ │ │ +00030380: 2030 2020 3020 3020 2020 6320 2062 2020   0  0 0   c  b  
│ │ │ │ +00030390: 6120 7c20 2020 2020 2020 2020 2020 2020  a |             
│ │ │ │ +000303a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000303b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030400: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030420: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ +000303f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030410: 2020 2020 2020 2020 2020 2020 3620 2020              6   
│ │ │ │ +00030420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030450: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -00030460: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965  R-module, quotie
│ │ │ │ -00030470: 6e74 206f 6620 5220 2020 2020 2020 2020  nt of R         
│ │ │ │ +00030440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030450: 6f34 203a 2052 2d6d 6f64 756c 652c 2071  o4 : R-module, q
│ │ │ │ +00030460: 756f 7469 656e 7420 6f66 2052 2020 2020  uotient of R    
│ │ │ │ +00030470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00030490: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000304a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000304c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000304d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -00030500: 2846 462c 2061 7567 2920 3d20 6c61 7965  (FF, aug) = laye
│ │ │ │ -00030510: 7265 6452 6573 6f6c 7574 696f 6e28 6666  redResolution(ff
│ │ │ │ -00030520: 2c4d 2c35 2920 2020 2020 2020 2020 2020  ,M,5)           
│ │ │ │ -00030530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030550: 2020 3020 3120 3220 3320 3420 3520 2020    0 1 2 3 4 5   
│ │ │ │ +000304e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000304f0: 6935 203a 2028 4646 2c20 6175 6729 203d  i5 : (FF, aug) =
│ │ │ │ +00030500: 206c 6179 6572 6564 5265 736f 6c75 7469   layeredResoluti
│ │ │ │ +00030510: 6f6e 2866 662c 4d2c 3529 2020 2020 2020  on(ff,M,5)      
│ │ │ │ +00030520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030540: 2020 2020 2020 2030 2031 2032 2033 2034         0 1 2 3 4
│ │ │ │ +00030550: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │  00030560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030590: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -000305a0: 3a20 3420 3420 3420 3420 3420 3420 2020  : 4 4 4 4 4 4   
│ │ │ │ +00030580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030590: 746f 7461 6c3a 2034 2034 2034 2034 2034  total: 4 4 4 4 4
│ │ │ │ +000305a0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │  000305b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305e0: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000305f0: 3a20 3320 3120 2e20 2e20 2e20 2e20 2020  : 3 1 . . . .   
│ │ │ │ +000305d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000305e0: 2020 2020 323a 2033 2031 202e 202e 202e      2: 3 1 . . .
│ │ │ │ +000305f0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  00030600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030630: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -00030640: 3a20 3120 3320 3320 3120 2e20 2e20 2020  : 1 3 3 1 . .   
│ │ │ │ +00030620: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030630: 2020 2020 333a 2031 2033 2033 2031 202e      3: 1 3 3 1 .
│ │ │ │ +00030640: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  00030650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030680: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00030690: 3a20 2e20 2e20 3120 3320 3320 3120 2020  : . . 1 3 3 1   
│ │ │ │ +00030670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030680: 2020 2020 343a 202e 202e 2031 2033 2033      4: . . 1 3 3
│ │ │ │ +00030690: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  000306a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000306c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000306d0: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -000306e0: 3a20 2e20 2e20 2e20 2e20 3120 3320 2020  : . . . . 1 3   
│ │ │ │ +000306c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000306d0: 2020 2020 353a 202e 202e 202e 202e 2031      5: . . . . 1
│ │ │ │ +000306e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  000306f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030720: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030730: 2020 3020 3120 3220 2033 2020 3420 2035    0 1 2  3  4  5
│ │ │ │ +00030710: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030720: 2020 2020 2020 2030 2031 2032 2020 3320         0 1 2  3 
│ │ │ │ +00030730: 2034 2020 3520 2020 2020 2020 2020 2020   4  5           
│ │ │ │  00030740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030770: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00030780: 3a20 3520 3720 3920 3131 2031 3320 3135  : 5 7 9 11 13 15
│ │ │ │ +00030760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030770: 746f 7461 6c3a 2035 2037 2039 2031 3120  total: 5 7 9 11 
│ │ │ │ +00030780: 3133 2031 3520 2020 2020 2020 2020 2020  13 15           
│ │ │ │  00030790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000307b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000307c0: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000307d0: 3a20 3320 3120 2e20 202e 2020 2e20 202e  : 3 1 .  .  .  .
│ │ │ │ +000307b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000307c0: 2020 2020 323a 2033 2031 202e 2020 2e20      2: 3 1 .  . 
│ │ │ │ +000307d0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │  000307e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030810: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -00030820: 3a20 3220 3620 3620 2032 2020 2e20 202e  : 2 6 6  2  .  .
│ │ │ │ +00030800: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030810: 2020 2020 333a 2032 2036 2036 2020 3220      3: 2 6 6  2 
│ │ │ │ +00030820: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │  00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030860: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00030870: 3a20 2e20 2e20 3320 2039 2020 3920 2033  : . . 3  9  9  3
│ │ │ │ +00030850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030860: 2020 2020 343a 202e 202e 2033 2020 3920      4: . . 3  9 
│ │ │ │ +00030870: 2039 2020 3320 2020 2020 2020 2020 2020   9  3           
│ │ │ │  00030880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000308a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000308b0: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -000308c0: 3a20 2e20 2e20 2e20 202e 2020 3420 3132  : . . .  .  4 12
│ │ │ │ +000308a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000308b0: 2020 2020 353a 202e 202e 202e 2020 2e20      5: . . .  . 
│ │ │ │ +000308c0: 2034 2031 3220 2020 2020 2020 2020 2020   4 12           
│ │ │ │  000308d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000308f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030900: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030910: 2020 3020 2031 2020 3220 2033 2020 3420    0  1  2  3  4 
│ │ │ │ -00030920: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +000308f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030900: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ +00030910: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030950: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00030960: 3a20 3620 3130 2031 3520 3231 2032 3820  : 6 10 15 21 28 
│ │ │ │ -00030970: 3336 2020 2020 2020 2020 2020 2020 2020  36              
│ │ │ │ +00030940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030950: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ +00030960: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ +00030970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309a0: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000309b0: 3a20 3320 2031 2020 2e20 202e 2020 2e20  : 3  1  .  .  . 
│ │ │ │ -000309c0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00030990: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000309a0: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ +000309b0: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309f0: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -00030a00: 3a20 3320 2039 2020 3920 2033 2020 2e20  : 3  9  9  3  . 
│ │ │ │ -00030a10: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +000309e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000309f0: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ +00030a00: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ +00030a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a40: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00030a50: 3a20 2e20 202e 2020 3620 3138 2031 3820  : .  .  6 18 18 
│ │ │ │ -00030a60: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00030a30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030a40: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ +00030a50: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ +00030a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a90: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -00030aa0: 3a20 2e20 202e 2020 2e20 202e 2031 3020  : .  .  .  . 10 
│ │ │ │ -00030ab0: 3330 2020 2020 2020 2020 2020 2020 2020  30              
│ │ │ │ +00030a80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030a90: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ +00030aa0: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ +00030ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00030ad0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030b40: 2020 3620 2020 2020 2031 3020 2020 2020    6      10     
│ │ │ │ -00030b50: 2031 3520 2020 2020 2032 3120 2020 2020   15      21     
│ │ │ │ -00030b60: 2032 3820 2020 2020 2033 3620 2020 2020   28      36     
│ │ │ │ -00030b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b80: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -00030b90: 2852 2020 3c2d 2d20 5220 2020 3c2d 2d20  (R  <-- R   <-- 
│ │ │ │ -00030ba0: 5220 2020 3c2d 2d20 5220 2020 3c2d 2d20  R   <-- R   <-- 
│ │ │ │ -00030bb0: 5220 2020 3c2d 2d20 5220 202c 207b 327d  R   <-- R  , {2}
│ │ │ │ -00030bc0: 207c 2030 2030 2020 3020 2031 2030 2030   | 0 0  0  1 0 0
│ │ │ │ -00030bd0: 207c 2920 2020 2020 7c0a 7c20 2020 2020   |)     |.|     
│ │ │ │ +00030b20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030b30: 2020 2020 2020 2036 2020 2020 2020 3130         6      10
│ │ │ │ +00030b40: 2020 2020 2020 3135 2020 2020 2020 3231        15      21
│ │ │ │ +00030b50: 2020 2020 2020 3238 2020 2020 2020 3336        28      36
│ │ │ │ +00030b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030b70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030b80: 6f35 203d 2028 5220 203c 2d2d 2052 2020  o5 = (R  <-- R  
│ │ │ │ +00030b90: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
│ │ │ │ +00030ba0: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
│ │ │ │ +00030bb0: 2c20 7b32 7d20 7c20 3020 3020 2030 2020  , {2} | 0 0  0  
│ │ │ │ +00030bc0: 3120 3020 3020 7c29 2020 2020 207c 0a7c  1 0 0 |)     |.|
│ │ │ │ +00030bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030c00: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -00030c10: 207c 2030 202d 3120 3020 2030 2030 2030   | 0 -1 0  0 0 0
│ │ │ │ -00030c20: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ -00030c30: 2030 2020 2020 2020 3120 2020 2020 2020   0      1       
│ │ │ │ -00030c40: 3220 2020 2020 2020 3320 2020 2020 2020  2       3       
│ │ │ │ -00030c50: 3420 2020 2020 2020 3520 2020 207b 327d  4       5    {2}
│ │ │ │ -00030c60: 207c 2030 2030 2020 2d31 2030 2030 2030   | 0 0  -1 0 0 0
│ │ │ │ -00030c70: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00030c00: 2020 7b32 7d20 7c20 3020 2d31 2030 2020    {2} | 0 -1 0  
│ │ │ │ +00030c10: 3020 3020 3020 7c20 2020 2020 207c 0a7c  0 0 0 |      |.|
│ │ │ │ +00030c20: 2020 2020 2020 3020 2020 2020 2031 2020        0      1  
│ │ │ │ +00030c30: 2020 2020 2032 2020 2020 2020 2033 2020       2       3  
│ │ │ │ +00030c40: 2020 2020 2034 2020 2020 2020 2035 2020       4       5  
│ │ │ │ +00030c50: 2020 7b32 7d20 7c20 3020 3020 202d 3120    {2} | 0 0  -1 
│ │ │ │ +00030c60: 3020 3020 3020 7c20 2020 2020 207c 0a7c  0 0 0 |      |.|
│ │ │ │ +00030c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ca0: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00030cb0: 207c 2030 2030 2020 3020 2030 2030 2031   | 0 0  0  0 0 1
│ │ │ │ -00030cc0: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00030ca0: 2020 7b33 7d20 7c20 3020 3020 2030 2020    {3} | 0 0  0  
│ │ │ │ +00030cb0: 3020 3020 3120 7c20 2020 2020 207c 0a7c  0 0 1 |      |.|
│ │ │ │ +00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030cf0: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00030d00: 207c 2030 2030 2020 3020 2030 2031 2030   | 0 0  0  0 1 0
│ │ │ │ -00030d10: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00030cf0: 2020 7b33 7d20 7c20 3020 3020 2030 2020    {3} | 0 0  0  
│ │ │ │ +00030d00: 3020 3120 3020 7c20 2020 2020 207c 0a7c  0 1 0 |      |.|
│ │ │ │ +00030d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d40: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00030d50: 207c 2031 2030 2020 3020 2030 2030 2030   | 1 0  0  0 0 0
│ │ │ │ -00030d60: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00030d40: 2020 7b33 7d20 7c20 3120 3020 2030 2020    {3} | 1 0  0  
│ │ │ │ +00030d50: 3020 3020 3020 7c20 2020 2020 207c 0a7c  0 0 0 |      |.|
│ │ │ │ +00030d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030db0: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -00030dc0: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +00030da0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030db0: 6f35 203a 2053 6571 7565 6e63 6520 2020  o5 : Sequence   
│ │ │ │ +00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00030df0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00030e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030e50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00030e60: 6265 7474 6920 4646 2020 2020 2020 2020  betti FF        
│ │ │ │ +00030e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00030e50: 6936 203a 2062 6574 7469 2046 4620 2020  i6 : betti FF   
│ │ │ │ +00030e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00030e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030f00: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -00030f10: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ +00030ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030ef0: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ +00030f00: 2020 3220 2033 2020 3420 2035 2020 2020    2  3  4  5    
│ │ │ │ +00030f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f40: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -00030f50: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ -00030f60: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ +00030f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030f40: 6f36 203d 2074 6f74 616c 3a20 3620 3130  o6 = total: 6 10
│ │ │ │ +00030f50: 2031 3520 3231 2032 3820 3336 2020 2020   15 21 28 36    
│ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030fa0: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ -00030fb0: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +00030f80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030f90: 2020 2020 2020 2020 2032 3a20 3320 2031           2: 3  1
│ │ │ │ +00030fa0: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +00030fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00030ff0: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ -00031000: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ +00030fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00030fe0: 2020 2020 2020 2020 2033 3a20 3320 2039           3: 3  9
│ │ │ │ +00030ff0: 2020 3920 2033 2020 2e20 202e 2020 2020    9  3  .  .    
│ │ │ │ +00031000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00031040: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ -00031050: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ +00031020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031030: 2020 2020 2020 2020 2034 3a20 2e20 202e           4: .  .
│ │ │ │ +00031040: 2020 3620 3138 2031 3820 2036 2020 2020    6 18 18  6    
│ │ │ │ +00031050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00031090: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ -000310a0: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ +00031070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031080: 2020 2020 2020 2020 2035 3a20 2e20 202e           5: .  .
│ │ │ │ +00031090: 2020 2e20 202e 2031 3020 3330 2020 2020    .  . 10 30    
│ │ │ │ +000310a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000310b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000310c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000310d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000310e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000310f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031120: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -00031130: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00031110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031120: 6f36 203a 2042 6574 7469 5461 6c6c 7920  o6 : BettiTally 
│ │ │ │ +00031130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031170: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00031160: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000311a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000311b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000311c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ -000311d0: 6265 7474 6920 7265 7328 4d2c 204c 656e  betti res(M, Len
│ │ │ │ -000311e0: 6774 684c 696d 6974 3d3e 3529 2020 2020  gthLimit=>5)    
│ │ │ │ +000311b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000311c0: 6937 203a 2062 6574 7469 2072 6573 284d  i7 : betti res(M
│ │ │ │ +000311d0: 2c20 4c65 6e67 7468 4c69 6d69 743d 3e35  , LengthLimit=>5
│ │ │ │ +000311e0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000311f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00031270: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -00031280: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ +00031250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031260: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ +00031270: 2020 3220 2033 2020 3420 2035 2020 2020    2  3  4  5    
│ │ │ │ +00031280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312b0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -000312c0: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ -000312d0: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ +000312a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000312b0: 6f37 203d 2074 6f74 616c 3a20 3620 3130  o7 = total: 6 10
│ │ │ │ +000312c0: 2031 3520 3231 2032 3820 3336 2020 2020   15 21 28 36    
│ │ │ │ +000312d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00031310: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ -00031320: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +000312f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031300: 2020 2020 2020 2020 2032 3a20 3320 2031           2: 3  1
│ │ │ │ +00031310: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +00031320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00031360: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ -00031370: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ +00031340: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031350: 2020 2020 2020 2020 2033 3a20 3320 2039           3: 3  9
│ │ │ │ +00031360: 2020 3920 2033 2020 2e20 202e 2020 2020    9  3  .  .    
│ │ │ │ +00031370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000313b0: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ -000313c0: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ +00031390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000313a0: 2020 2020 2020 2020 2034 3a20 2e20 202e           4: .  .
│ │ │ │ +000313b0: 2020 3620 3138 2031 3820 2036 2020 2020    6 18 18  6    
│ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00031400: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ -00031410: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ +000313e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000313f0: 2020 2020 2020 2020 2035 3a20 2e20 202e           5: .  .
│ │ │ │ +00031400: 2020 2e20 202e 2031 3020 3330 2020 2020    .  . 10 30    
│ │ │ │ +00031410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031440: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031490: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -000314a0: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00031480: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031490: 6f37 203a 2042 6574 7469 5461 6c6c 7920  o7 : BettiTally 
│ │ │ │ +000314a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000314d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000314e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000314f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031530: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -00031540: 4320 3d20 6368 6169 6e43 6f6d 706c 6578  C = chainComplex
│ │ │ │ -00031550: 2066 6c61 7474 656e 207b 7b61 7567 7d20   flatten {{aug} 
│ │ │ │ -00031560: 7c61 7070 6c79 2834 2c20 692d 3e20 4646  |apply(4, i-> FF
│ │ │ │ -00031570: 2e64 645f 2869 2b31 2929 7d20 2020 2020  .dd_(i+1))}     
│ │ │ │ -00031580: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00031530: 6938 203a 2043 203d 2063 6861 696e 436f  i8 : C = chainCo
│ │ │ │ +00031540: 6d70 6c65 7820 666c 6174 7465 6e20 7b7b  mplex flatten {{
│ │ │ │ +00031550: 6175 677d 207c 6170 706c 7928 342c 2069  aug} |apply(4, i
│ │ │ │ +00031560: 2d3e 2046 462e 6464 5f28 692b 3129 297d  -> FF.dd_(i+1))}
│ │ │ │ +00031570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000315c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000315d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000315e0: 2020 2020 2020 2036 2020 2020 2020 3130         6      10
│ │ │ │ -000315f0: 2020 2020 2020 3135 2020 2020 2020 3231        15      21
│ │ │ │ -00031600: 2020 2020 2020 3238 2020 2020 2020 2020        28        
│ │ │ │ -00031610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031620: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ -00031630: 4d20 3c2d 2d20 5220 203c 2d2d 2052 2020  M <-- R  <-- R  
│ │ │ │ -00031640: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
│ │ │ │ -00031650: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ -00031660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000315c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000315d0: 2020 2020 2020 2020 2020 2020 3620 2020              6   
│ │ │ │ +000315e0: 2020 2031 3020 2020 2020 2031 3520 2020     10      15   
│ │ │ │ +000315f0: 2020 2032 3120 2020 2020 2032 3820 2020     21      28   
│ │ │ │ +00031600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031620: 6f38 203d 204d 203c 2d2d 2052 2020 3c2d  o8 = M <-- R  <-
│ │ │ │ +00031630: 2d20 5220 2020 3c2d 2d20 5220 2020 3c2d  - R   <-- R   <-
│ │ │ │ +00031640: 2d20 5220 2020 3c2d 2d20 5220 2020 2020  - R   <-- R     
│ │ │ │ +00031650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000316a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000316d0: 3020 2020 2020 3120 2020 2020 2032 2020  0     1      2  
│ │ │ │ -000316e0: 2020 2020 2033 2020 2020 2020 2034 2020       3       4  
│ │ │ │ -000316f0: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ -00031700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031710: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000316b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000316c0: 2020 2020 2030 2020 2020 2031 2020 2020       0     1    
│ │ │ │ +000316d0: 2020 3220 2020 2020 2020 3320 2020 2020    2       3     
│ │ │ │ +000316e0: 2020 3420 2020 2020 2020 3520 2020 2020    4       5     
│ │ │ │ +000316f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031760: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ -00031770: 4368 6169 6e43 6f6d 706c 6578 2020 2020  ChainComplex    
│ │ │ │ +00031750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031760: 6f38 203a 2043 6861 696e 436f 6d70 6c65  o8 : ChainComple
│ │ │ │ +00031770: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │  00031780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000317a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000317b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000317c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000317d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000317e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000317f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031800: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ -00031810: 6170 706c 7928 342c 2069 202d 3e46 462e  apply(4, i ->FF.
│ │ │ │ -00031820: 6464 5f28 692b 3129 2920 2020 2020 2020  dd_(i+1))       
│ │ │ │ +000317f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00031800: 6939 203a 2061 7070 6c79 2834 2c20 6920  i9 : apply(4, i 
│ │ │ │ +00031810: 2d3e 4646 2e64 645f 2869 2b31 2929 2020  ->FF.dd_(i+1))  
│ │ │ │ +00031820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031850: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000318a0: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -000318b0: 7b7b 337d 207c 2030 2020 6120 2d63 202d  {{3} | 0  a -c -
│ │ │ │ -000318c0: 6220 3020 2030 2030 2020 3020 3020 2030  b 0  0 0  0 0  0
│ │ │ │ -000318d0: 2020 7c2c 207b 337d 207c 2030 2020 2030    |, {3} | 0   0
│ │ │ │ -000318e0: 2020 2d63 3220 3020 3020 2020 3020 2030    -c2 0 0   0  0
│ │ │ │ -000318f0: 2020 2062 3220 3020 7c0a 7c20 2020 2020     b2 0 |.|     
│ │ │ │ -00031900: 207b 327d 207c 2062 2020 3020 6132 2030   {2} | b  0 a2 0
│ │ │ │ -00031910: 2020 3020 2030 2030 2020 3020 3020 2063    0  0 0  0 0  c
│ │ │ │ -00031920: 3220 7c20 207b 347d 207c 2030 2020 2030  2 |  {4} | 0   0
│ │ │ │ -00031930: 2020 3020 2020 3020 3020 2020 3020 2030    0   0 0   0  0
│ │ │ │ -00031940: 2020 2030 2020 3020 7c0a 7c20 2020 2020     0  0 |.|     
│ │ │ │ -00031950: 207b 327d 207c 202d 6320 3020 3020 2061   {2} | -c 0 0  a
│ │ │ │ -00031960: 3220 3020 2030 2062 3220 3020 3020 2030  2 0  0 b2 0 0  0
│ │ │ │ -00031970: 2020 7c20 207b 347d 207c 202d 6332 2030    |  {4} | -c2 0
│ │ │ │ -00031980: 2020 3020 2020 3020 3020 2020 3020 2030    0   0 0   0  0
│ │ │ │ -00031990: 2020 2030 2020 3020 7c0a 7c20 2020 2020     0  0 |.|     
│ │ │ │ -000319a0: 207b 327d 207c 2061 2020 3020 3020 2030   {2} | a  0 0  0
│ │ │ │ -000319b0: 2020 6232 2030 2030 2020 3020 6332 2030    b2 0 0  0 c2 0
│ │ │ │ -000319c0: 2020 7c20 207b 347d 207c 2030 2020 2030    |  {4} | 0   0
│ │ │ │ -000319d0: 2020 3020 2020 3020 3020 2020 3020 202d    0   0 0   0  -
│ │ │ │ -000319e0: 6232 2030 2020 3020 7c0a 7c20 2020 2020  b2 0  0 |.|     
│ │ │ │ -000319f0: 207b 337d 207c 2030 2020 3020 3020 2030   {3} | 0  0 0  0
│ │ │ │ -00031a00: 2020 6320 2062 2061 2020 3020 3020 2030    c  b a  0 0  0
│ │ │ │ -00031a10: 2020 7c20 207b 347d 207c 2030 2020 2030    |  {4} | 0   0
│ │ │ │ -00031a20: 2020 3020 2020 3020 6332 2020 3020 2030    0   0 c2  0  0
│ │ │ │ -00031a30: 2020 202d 6120 3020 7c0a 7c20 2020 2020     -a 0 |.|     
│ │ │ │ -00031a40: 207b 337d 207c 2030 2020 3020 3020 2030   {3} | 0  0 0  0
│ │ │ │ -00031a50: 2020 3020 2030 2030 2020 6320 2d62 2061    0  0 0  c -b a
│ │ │ │ -00031a60: 2020 7c20 207b 347d 207c 2030 2020 2030    |  {4} | 0   0
│ │ │ │ -00031a70: 2020 3020 2020 3020 3020 2020 3020 2030    0   0 0   0  0
│ │ │ │ -00031a80: 2020 2030 2020 2d61 7c0a 7c20 2020 2020     0  -a|.|     
│ │ │ │ +00031890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000318a0: 6f39 203d 207b 7b33 7d20 7c20 3020 2061  o9 = {{3} | 0  a
│ │ │ │ +000318b0: 202d 6320 2d62 2030 2020 3020 3020 2030   -c -b 0  0 0  0
│ │ │ │ +000318c0: 2030 2020 3020 207c 2c20 7b33 7d20 7c20   0  0  |, {3} | 
│ │ │ │ +000318d0: 3020 2020 3020 202d 6332 2030 2030 2020  0   0  -c2 0 0  
│ │ │ │ +000318e0: 2030 2020 3020 2020 6232 2030 207c 0a7c   0  0   b2 0 |.|
│ │ │ │ +000318f0: 2020 2020 2020 7b32 7d20 7c20 6220 2030        {2} | b  0
│ │ │ │ +00031900: 2061 3220 3020 2030 2020 3020 3020 2030   a2 0  0  0 0  0
│ │ │ │ +00031910: 2030 2020 6332 207c 2020 7b34 7d20 7c20   0  c2 |  {4} | 
│ │ │ │ +00031920: 3020 2020 3020 2030 2020 2030 2030 2020  0   0  0   0 0  
│ │ │ │ +00031930: 2030 2020 3020 2020 3020 2030 207c 0a7c   0  0   0  0 |.|
│ │ │ │ +00031940: 2020 2020 2020 7b32 7d20 7c20 2d63 2030        {2} | -c 0
│ │ │ │ +00031950: 2030 2020 6132 2030 2020 3020 6232 2030   0  a2 0  0 b2 0
│ │ │ │ +00031960: 2030 2020 3020 207c 2020 7b34 7d20 7c20   0  0  |  {4} | 
│ │ │ │ +00031970: 2d63 3220 3020 2030 2020 2030 2030 2020  -c2 0  0   0 0  
│ │ │ │ +00031980: 2030 2020 3020 2020 3020 2030 207c 0a7c   0  0   0  0 |.|
│ │ │ │ +00031990: 2020 2020 2020 7b32 7d20 7c20 6120 2030        {2} | a  0
│ │ │ │ +000319a0: 2030 2020 3020 2062 3220 3020 3020 2030   0  0  b2 0 0  0
│ │ │ │ +000319b0: 2063 3220 3020 207c 2020 7b34 7d20 7c20   c2 0  |  {4} | 
│ │ │ │ +000319c0: 3020 2020 3020 2030 2020 2030 2030 2020  0   0  0   0 0  
│ │ │ │ +000319d0: 2030 2020 2d62 3220 3020 2030 207c 0a7c   0  -b2 0  0 |.|
│ │ │ │ +000319e0: 2020 2020 2020 7b33 7d20 7c20 3020 2030        {3} | 0  0
│ │ │ │ +000319f0: 2030 2020 3020 2063 2020 6220 6120 2030   0  0  c  b a  0
│ │ │ │ +00031a00: 2030 2020 3020 207c 2020 7b34 7d20 7c20   0  0  |  {4} | 
│ │ │ │ +00031a10: 3020 2020 3020 2030 2020 2030 2063 3220  0   0  0   0 c2 
│ │ │ │ +00031a20: 2030 2020 3020 2020 2d61 2030 207c 0a7c   0  0   -a 0 |.|
│ │ │ │ +00031a30: 2020 2020 2020 7b33 7d20 7c20 3020 2030        {3} | 0  0
│ │ │ │ +00031a40: 2030 2020 3020 2030 2020 3020 3020 2063   0  0  0  0 0  c
│ │ │ │ +00031a50: 202d 6220 6120 207c 2020 7b34 7d20 7c20   -b a  |  {4} | 
│ │ │ │ +00031a60: 3020 2020 3020 2030 2020 2030 2030 2020  0   0  0   0 0  
│ │ │ │ +00031a70: 2030 2020 3020 2020 3020 202d 617c 0a7c   0  0   0  -a|.|
│ │ │ │ +00031a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ab0: 2020 2020 207b 347d 207c 2030 2020 2030       {4} | 0   0
│ │ │ │ -00031ac0: 2020 3020 2020 3020 3020 2020 3020 2061    0   0 0   0  a
│ │ │ │ -00031ad0: 3220 2063 2020 6220 7c0a 7c20 2020 2020  2  c  b |.|     
│ │ │ │ +00031aa0: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00031ab0: 3020 2020 3020 2030 2020 2030 2030 2020  0   0  0   0 0  
│ │ │ │ +00031ac0: 2030 2020 6132 2020 6320 2062 207c 0a7c   0  a2  c  b |.|
│ │ │ │ +00031ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b00: 2020 2020 207b 347d 207c 2030 2020 202d       {4} | 0   -
│ │ │ │ -00031b10: 6120 3020 2020 6220 3020 2020 6332 2030  a 0   b 0   c2 0
│ │ │ │ -00031b20: 2020 2030 2020 3020 7c0a 7c20 2020 2020     0  0 |.|     
│ │ │ │ +00031af0: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00031b00: 3020 2020 2d61 2030 2020 2062 2030 2020  0   -a 0   b 0  
│ │ │ │ +00031b10: 2063 3220 3020 2020 3020 2030 207c 0a7c   c2 0   0  0 |.|
│ │ │ │ +00031b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b50: 2020 2020 207b 347d 207c 2030 2020 2030       {4} | 0   0
│ │ │ │ -00031b60: 2020 6120 2020 6320 2d62 3220 3020 2030    a   c -b2 0  0
│ │ │ │ -00031b70: 2020 2030 2020 3020 7c0a 7c20 2020 2020     0  0 |.|     
│ │ │ │ +00031b40: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00031b50: 3020 2020 3020 2061 2020 2063 202d 6232  0   0  a   c -b2
│ │ │ │ +00031b60: 2030 2020 3020 2020 3020 2030 207c 0a7c   0  0   0  0 |.|
│ │ │ │ +00031b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ba0: 2020 2020 207b 347d 207c 2061 3220 2063       {4} | a2  c
│ │ │ │ -00031bb0: 2020 6220 2020 3020 3020 2020 3020 2030    b   0 0   0  0
│ │ │ │ -00031bc0: 2020 2030 2020 3020 7c0a 7c20 2020 2020     0  0 |.|     
│ │ │ │ +00031b90: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +00031ba0: 6132 2020 6320 2062 2020 2030 2030 2020  a2  c  b   0 0  
│ │ │ │ +00031bb0: 2030 2020 3020 2020 3020 2030 207c 0a7c   0  0   0  0 |.|
│ │ │ │ +00031bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031c50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031cb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031df0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031de0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031e30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ee0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00031f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031f30: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │  00031f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031f80: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00031f90: 3020 2030 2020 3020 2030 2030 2061 3220  0  0  0  0 0 a2 
│ │ │ │ -00031fa0: 7c2c 207b 367d 207c 2030 2020 6120 202d  |, {6} | 0  a  -
│ │ │ │ -00031fb0: 6320 2d62 2030 2020 3020 2020 3020 2030  c -b 0  0   0  0
│ │ │ │ -00031fc0: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00031fd0: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00031fe0: 3020 2030 2020 6132 2063 2062 2030 2020  0  0  a2 c b 0  
│ │ │ │ -00031ff0: 7c20 207b 357d 207c 2062 2020 3020 2061  |  {5} | b  0  a
│ │ │ │ -00032000: 3220 3020 2030 2020 3020 2020 3020 2030  2 0  0  0   0  0
│ │ │ │ -00032010: 2030 2020 6332 2030 2020 3020 2030 2020   0  c2 0  0  0  
│ │ │ │ -00032020: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00032030: 3020 2030 2020 3020 2061 2030 202d 6220  0  0  0  a 0 -b 
│ │ │ │ -00032040: 7c20 207b 357d 207c 202d 6320 3020 2030  |  {5} | -c 0  0
│ │ │ │ -00032050: 2020 6132 2030 2020 3020 2020 6232 2030    a2 0  0   b2 0
│ │ │ │ -00032060: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00032070: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00032080: 3020 2030 2020 3020 2030 2061 2063 2020  0  0  0  0 a c  
│ │ │ │ -00032090: 7c20 207b 357d 207c 2061 2020 3020 2030  |  {5} | a  0  0
│ │ │ │ -000320a0: 2020 3020 2062 3220 3020 2020 3020 2030    0  b2 0   0  0
│ │ │ │ -000320b0: 2063 3220 3020 2030 2020 3020 2030 2020   c2 0  0  0  0  
│ │ │ │ -000320c0: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -000320d0: 6220 2030 2020 3020 2030 2030 2030 2020  b  0  0  0 0 0  
│ │ │ │ -000320e0: 7c20 207b 367d 207c 2030 2020 3020 2030  |  {6} | 0  0  0
│ │ │ │ -000320f0: 2020 3020 2063 2020 6220 2020 6120 2030    0  c  b   a  0
│ │ │ │ -00032100: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00032110: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00032120: 2d63 2062 3220 3020 2030 2030 2030 2020  -c b2 0  0 0 0  
│ │ │ │ -00032130: 7c20 207b 367d 207c 2030 2020 3020 2030  |  {6} | 0  0  0
│ │ │ │ -00032140: 2020 3020 2030 2020 3020 2020 3020 2063    0  0  0   0  c
│ │ │ │ -00032150: 202d 6220 6120 2030 2020 3020 2030 2020   -b a  0  0  0  
│ │ │ │ -00032160: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00032170: 3020 2030 2020 3020 2030 2030 2030 2020  0  0  0  0 0 0  
│ │ │ │ -00032180: 7c20 207b 367d 207c 2030 2020 3020 2030  |  {6} | 0  0  0
│ │ │ │ -00032190: 2020 3020 2030 2020 3020 2020 3020 2030    0  0  0   0  0
│ │ │ │ -000321a0: 2030 2020 3020 2030 2020 6120 202d 6320   0  0  0  a  -c 
│ │ │ │ -000321b0: 202d 6220 3020 2030 7c0a 7c20 2020 2020   -b 0  0|.|     
│ │ │ │ -000321c0: 3020 2030 2020 3020 2030 2030 2030 2020  0  0  0  0 0 0  
│ │ │ │ -000321d0: 7c20 207b 357d 207c 2030 2020 3020 2030  |  {5} | 0  0  0
│ │ │ │ -000321e0: 2020 3020 2030 2020 3020 2020 6332 2030    0  0  0   c2 0
│ │ │ │ -000321f0: 2030 2020 3020 2062 2020 3020 2061 3220   0  0  b  0  a2 
│ │ │ │ -00032200: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00032210: 3020 2030 2020 3020 2030 2030 2030 2020  0  0  0  0 0 0  
│ │ │ │ -00032220: 7c20 207b 357d 207c 2030 2020 3020 2030  |  {5} | 0  0  0
│ │ │ │ -00032230: 2020 3020 2030 2020 3020 2020 3020 2030    0  0  0   0  0
│ │ │ │ -00032240: 2030 2020 3020 202d 6320 3020 2030 2020   0  0  -c 0  0  
│ │ │ │ -00032250: 2061 3220 3020 2030 7c0a 7c20 2020 2020   a2 0  0|.|     
│ │ │ │ -00032260: 3020 2030 2020 3020 2030 2030 2030 2020  0  0  0  0 0 0  
│ │ │ │ -00032270: 7c20 207b 357d 207c 2030 2020 3020 2030  |  {5} | 0  0  0
│ │ │ │ -00032280: 2020 3020 2030 2020 2d63 3220 3020 2030    0  0  -c2 0  0
│ │ │ │ -00032290: 2030 2020 3020 2061 2020 3020 2030 2020   0  0  a  0  0  
│ │ │ │ -000322a0: 2030 2020 6232 2030 7c0a 7c20 2020 2020   0  b2 0|.|     
│ │ │ │ -000322b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000322c0: 2020 207b 367d 207c 2030 2020 3020 2030     {6} | 0  0  0
│ │ │ │ -000322d0: 2020 3020 2030 2020 3020 2020 3020 2030    0  0  0   0  0
│ │ │ │ -000322e0: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -000322f0: 2030 2020 6320 2062 7c0a 7c20 2020 2020   0  c  b|.|     
│ │ │ │ -00032300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032310: 2020 207b 367d 207c 2030 2020 3020 2030     {6} | 0  0  0
│ │ │ │ -00032320: 2020 3020 2030 2020 3020 2020 3020 2030    0  0  0   0  0
│ │ │ │ -00032330: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00032340: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -00032350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032360: 2020 207b 357d 207c 2030 2020 6332 2030     {5} | 0  c2 0
│ │ │ │ -00032370: 2020 3020 2030 2020 3020 2020 3020 2030    0  0  0   0  0
│ │ │ │ -00032380: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00032390: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -000323a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000323b0: 2020 207b 357d 207c 2030 2020 3020 2030     {5} | 0  0  0
│ │ │ │ -000323c0: 2020 3020 2030 2020 3020 2020 3020 2030    0  0  0   0  0
│ │ │ │ -000323d0: 2030 2020 3020 2030 2020 6232 2030 2020   0  0  0  b2 0  
│ │ │ │ -000323e0: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ -000323f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032400: 2020 207b 357d 207c 2030 2020 3020 2030     {5} | 0  0  0
│ │ │ │ -00032410: 2020 6332 2030 2020 3020 2020 3020 2030    c2 0  0   0  0
│ │ │ │ -00032420: 2030 2020 3020 2030 2020 3020 202d 6232   0  0  0  0  -b2
│ │ │ │ -00032430: 2030 2020 3020 2030 7c0a 7c20 2020 2020   0  0  0|.|     
│ │ │ │ +00031f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00031f80: 2020 2020 2030 2020 3020 2030 2020 3020       0  0  0  0 
│ │ │ │ +00031f90: 3020 6132 207c 2c20 7b36 7d20 7c20 3020  0 a2 |, {6} | 0 
│ │ │ │ +00031fa0: 2061 2020 2d63 202d 6220 3020 2030 2020   a  -c -b 0  0  
│ │ │ │ +00031fb0: 2030 2020 3020 3020 2030 2020 3020 2030   0  0 0  0  0  0
│ │ │ │ +00031fc0: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00031fd0: 2020 2020 2030 2020 3020 2061 3220 6320       0  0  a2 c 
│ │ │ │ +00031fe0: 6220 3020 207c 2020 7b35 7d20 7c20 6220  b 0  |  {5} | b 
│ │ │ │ +00031ff0: 2030 2020 6132 2030 2020 3020 2030 2020   0  a2 0  0  0  
│ │ │ │ +00032000: 2030 2020 3020 3020 2063 3220 3020 2030   0  0 0  c2 0  0
│ │ │ │ +00032010: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00032020: 2020 2020 2030 2020 3020 2030 2020 6120       0  0  0  a 
│ │ │ │ +00032030: 3020 2d62 207c 2020 7b35 7d20 7c20 2d63  0 -b |  {5} | -c
│ │ │ │ +00032040: 2030 2020 3020 2061 3220 3020 2030 2020   0  0  a2 0  0  
│ │ │ │ +00032050: 2062 3220 3020 3020 2030 2020 3020 2030   b2 0 0  0  0  0
│ │ │ │ +00032060: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00032070: 2020 2020 2030 2020 3020 2030 2020 3020       0  0  0  0 
│ │ │ │ +00032080: 6120 6320 207c 2020 7b35 7d20 7c20 6120  a c  |  {5} | a 
│ │ │ │ +00032090: 2030 2020 3020 2030 2020 6232 2030 2020   0  0  0  b2 0  
│ │ │ │ +000320a0: 2030 2020 3020 6332 2030 2020 3020 2030   0  0 c2 0  0  0
│ │ │ │ +000320b0: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +000320c0: 2020 2020 2062 2020 3020 2030 2020 3020       b  0  0  0 
│ │ │ │ +000320d0: 3020 3020 207c 2020 7b36 7d20 7c20 3020  0 0  |  {6} | 0 
│ │ │ │ +000320e0: 2030 2020 3020 2030 2020 6320 2062 2020   0  0  0  c  b  
│ │ │ │ +000320f0: 2061 2020 3020 3020 2030 2020 3020 2030   a  0 0  0  0  0
│ │ │ │ +00032100: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00032110: 2020 2020 202d 6320 6232 2030 2020 3020       -c b2 0  0 
│ │ │ │ +00032120: 3020 3020 207c 2020 7b36 7d20 7c20 3020  0 0  |  {6} | 0 
│ │ │ │ +00032130: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +00032140: 2030 2020 6320 2d62 2061 2020 3020 2030   0  c -b a  0  0
│ │ │ │ +00032150: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00032160: 2020 2020 2030 2020 3020 2030 2020 3020       0  0  0  0 
│ │ │ │ +00032170: 3020 3020 207c 2020 7b36 7d20 7c20 3020  0 0  |  {6} | 0 
│ │ │ │ +00032180: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +00032190: 2030 2020 3020 3020 2030 2020 3020 2061   0  0 0  0  0  a
│ │ │ │ +000321a0: 2020 2d63 2020 2d62 2030 2020 307c 0a7c    -c  -b 0  0|.|
│ │ │ │ +000321b0: 2020 2020 2030 2020 3020 2030 2020 3020       0  0  0  0 
│ │ │ │ +000321c0: 3020 3020 207c 2020 7b35 7d20 7c20 3020  0 0  |  {5} | 0 
│ │ │ │ +000321d0: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +000321e0: 2063 3220 3020 3020 2030 2020 6220 2030   c2 0 0  0  b  0
│ │ │ │ +000321f0: 2020 6132 2020 3020 2030 2020 307c 0a7c    a2  0  0  0|.|
│ │ │ │ +00032200: 2020 2020 2030 2020 3020 2030 2020 3020       0  0  0  0 
│ │ │ │ +00032210: 3020 3020 207c 2020 7b35 7d20 7c20 3020  0 0  |  {5} | 0 
│ │ │ │ +00032220: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +00032230: 2030 2020 3020 3020 2030 2020 2d63 2030   0  0 0  0  -c 0
│ │ │ │ +00032240: 2020 3020 2020 6132 2030 2020 307c 0a7c    0   a2 0  0|.|
│ │ │ │ +00032250: 2020 2020 2030 2020 3020 2030 2020 3020       0  0  0  0 
│ │ │ │ +00032260: 3020 3020 207c 2020 7b35 7d20 7c20 3020  0 0  |  {5} | 0 
│ │ │ │ +00032270: 2030 2020 3020 2030 2020 3020 202d 6332   0  0  0  0  -c2
│ │ │ │ +00032280: 2030 2020 3020 3020 2030 2020 6120 2030   0  0 0  0  a  0
│ │ │ │ +00032290: 2020 3020 2020 3020 2062 3220 307c 0a7c    0   0  b2 0|.|
│ │ │ │ +000322a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000322b0: 2020 2020 2020 2020 7b36 7d20 7c20 3020          {6} | 0 
│ │ │ │ +000322c0: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +000322d0: 2030 2020 3020 3020 2030 2020 3020 2030   0  0 0  0  0  0
│ │ │ │ +000322e0: 2020 3020 2020 3020 2063 2020 627c 0a7c    0   0  c  b|.|
│ │ │ │ +000322f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032300: 2020 2020 2020 2020 7b36 7d20 7c20 3020          {6} | 0 
│ │ │ │ +00032310: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +00032320: 2030 2020 3020 3020 2030 2020 3020 2030   0  0 0  0  0  0
│ │ │ │ +00032330: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00032340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032350: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ +00032360: 2063 3220 3020 2030 2020 3020 2030 2020   c2 0  0  0  0  
│ │ │ │ +00032370: 2030 2020 3020 3020 2030 2020 3020 2030   0  0 0  0  0  0
│ │ │ │ +00032380: 2020 3020 2020 3020 2030 2020 307c 0a7c    0   0  0  0|.|
│ │ │ │ +00032390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000323a0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ +000323b0: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ +000323c0: 2030 2020 3020 3020 2030 2020 3020 2062   0  0 0  0  0  b
│ │ │ │ +000323d0: 3220 3020 2020 3020 2030 2020 307c 0a7c  2 0   0  0  0|.|
│ │ │ │ +000323e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000323f0: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ +00032400: 2030 2020 3020 2063 3220 3020 2030 2020   0  0  c2 0  0  
│ │ │ │ +00032410: 2030 2020 3020 3020 2030 2020 3020 2030   0  0 0  0  0  0
│ │ │ │ +00032420: 2020 2d62 3220 3020 2030 2020 307c 0a7c    -b2 0  0  0|.|
│ │ │ │ +00032430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000324c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000324d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000325b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000325c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000325b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000325c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032610: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │  00032620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032660: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00032670: 3020 2030 2020 3020 3020 2030 2020 7c2c  0  0  0 0  0  |,
│ │ │ │ -00032680: 207b 367d 207c 2030 2020 2030 2020 2d63   {6} | 0   0  -c
│ │ │ │ -00032690: 3220 3020 3020 2020 3020 2030 2020 2062  2 0 0   0  0   b
│ │ │ │ -000326a0: 3220 3020 2020 3020 2030 2020 3020 2020  2 0   0  0  0   
│ │ │ │ -000326b0: 3020 3020 2020 6132 7c0a 7c20 2020 2020  0 0   a2|.|     
│ │ │ │ -000326c0: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -000326d0: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -000326e0: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -000326f0: 2020 3020 2020 3020 2030 2020 6132 2020    0   0  0  a2  
│ │ │ │ -00032700: 6320 6220 2020 3020 7c0a 7c20 2020 2020  c b   0 |.|     
│ │ │ │ -00032710: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -00032720: 207b 377d 207c 202d 6332 2030 2020 3020   {7} | -c2 0  0 
│ │ │ │ -00032730: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032740: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032750: 6120 3020 2020 2d62 7c0a 7c20 2020 2020  a 0   -b|.|     
│ │ │ │ -00032760: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -00032770: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032780: 2020 3020 3020 2020 3020 202d 6232 2030    0 0   0  -b2 0
│ │ │ │ -00032790: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -000327a0: 3020 6120 2020 6320 7c0a 7c20 2020 2020  0 a   c |.|     
│ │ │ │ -000327b0: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -000327c0: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -000327d0: 2020 3020 6332 2020 3020 2030 2020 202d    0 c2  0  0   -
│ │ │ │ -000327e0: 6120 3020 2020 6220 2030 2020 3020 2020  a 0   b  0  0   
│ │ │ │ -000327f0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032800: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -00032810: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032820: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032830: 2020 2d61 2020 2d63 2062 3220 3020 2020    -a  -c b2 0   
│ │ │ │ -00032840: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032850: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -00032860: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032870: 2020 3020 3020 2020 3020 2061 3220 2063    0 0   0  a2  c
│ │ │ │ -00032880: 2020 6220 2020 3020 2030 2020 3020 2020    b   0  0  0   
│ │ │ │ -00032890: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -000328a0: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -000328b0: 207b 377d 207c 2030 2020 202d 6120 3020   {7} | 0   -a 0 
│ │ │ │ -000328c0: 2020 6220 3020 2020 6332 2030 2020 2030    b 0   c2 0   0
│ │ │ │ -000328d0: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -000328e0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -000328f0: 6232 2030 2020 3020 3020 2030 2020 7c20  b2 0  0 0  0  | 
│ │ │ │ -00032900: 207b 377d 207c 2030 2020 2030 2020 6120   {7} | 0   0  a 
│ │ │ │ -00032910: 2020 6320 2d62 3220 3020 2030 2020 2030    c -b2 0  0   0
│ │ │ │ -00032920: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032930: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032940: 3020 2030 2020 3020 3020 2030 2020 7c20  0  0  0 0  0  | 
│ │ │ │ -00032950: 207b 377d 207c 2061 3220 2063 2020 6220   {7} | a2  c  b 
│ │ │ │ -00032960: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032970: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032980: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032990: 6120 2030 2020 3020 3020 2030 2020 7c20  a  0  0 0  0  | 
│ │ │ │ -000329a0: 207b 367d 207c 2030 2020 2030 2020 3020   {6} | 0   0  0 
│ │ │ │ -000329b0: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -000329c0: 2020 2d63 3220 3020 2030 2020 3020 2020    -c2 0  0  0   
│ │ │ │ -000329d0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -000329e0: 3020 2030 2020 6120 2d63 202d 6220 7c20  0  0  a -c -b | 
│ │ │ │ -000329f0: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032a00: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032a10: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032a20: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032a30: 3020 2062 2020 3020 6132 2030 2020 7c20  0  b  0 a2 0  | 
│ │ │ │ -00032a40: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032a50: 2020 3020 3020 2020 3020 202d 6332 2030    0 0   0  -c2 0
│ │ │ │ -00032a60: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032a70: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032a80: 3020 202d 6320 3020 3020 2061 3220 7c20  0  -c 0 0  a2 | 
│ │ │ │ -00032a90: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032aa0: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032ab0: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032ac0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032ad0: 3020 2061 2020 3020 3020 2030 2020 7c20  0  a  0 0  0  | 
│ │ │ │ -00032ae0: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032af0: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032b00: 2020 3020 2020 3020 2063 3220 3020 2020    0   0  c2 0   
│ │ │ │ -00032b10: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032b30: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032b40: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032b50: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032b60: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032b80: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032b90: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032ba0: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032bb0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032bd0: 207b 367d 207c 2030 2020 2030 2020 3020   {6} | 0   0  0 
│ │ │ │ -00032be0: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032bf0: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032c00: 3020 2d63 3220 3020 7c0a 7c20 2020 2020  0 -c2 0 |.|     
│ │ │ │ -00032c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c20: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032c30: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032c40: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032c50: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c70: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032c80: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032c90: 2020 3020 2020 3020 2030 2020 2d63 3220    0   0  0  -c2 
│ │ │ │ -00032ca0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ -00032cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032cc0: 207b 377d 207c 2030 2020 2030 2020 3020   {7} | 0   0  0 
│ │ │ │ -00032cd0: 2020 3020 3020 2020 3020 2030 2020 2030    0 0   0  0   0
│ │ │ │ -00032ce0: 2020 3020 2020 3020 2030 2020 3020 2020    0   0  0  0   
│ │ │ │ -00032cf0: 3020 3020 2020 3020 7c0a 7c20 2020 2020  0 0   0 |.|     
│ │ │ │ +00032650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00032660: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +00032670: 3020 207c 2c20 7b36 7d20 7c20 3020 2020  0  |, {6} | 0   
│ │ │ │ +00032680: 3020 202d 6332 2030 2030 2020 2030 2020  0  -c2 0 0   0  
│ │ │ │ +00032690: 3020 2020 6232 2030 2020 2030 2020 3020  0   b2 0   0  0 
│ │ │ │ +000326a0: 2030 2020 2030 2030 2020 2061 327c 0a7c   0   0 0   a2|.|
│ │ │ │ +000326b0: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +000326c0: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +000326d0: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +000326e0: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +000326f0: 2061 3220 2063 2062 2020 2030 207c 0a7c   a2  c b   0 |.|
│ │ │ │ +00032700: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +00032710: 3020 207c 2020 7b37 7d20 7c20 2d63 3220  0  |  {7} | -c2 
│ │ │ │ +00032720: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032730: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032740: 2030 2020 2061 2030 2020 202d 627c 0a7c   0   a 0   -b|.|
│ │ │ │ +00032750: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +00032760: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +00032770: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032780: 2d62 3220 3020 2030 2020 2030 2020 3020  -b2 0  0   0  0 
│ │ │ │ +00032790: 2030 2020 2030 2061 2020 2063 207c 0a7c   0   0 a   c |.|
│ │ │ │ +000327a0: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +000327b0: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +000327c0: 3020 2030 2020 2030 2063 3220 2030 2020  0  0   0 c2  0  
│ │ │ │ +000327d0: 3020 2020 2d61 2030 2020 2062 2020 3020  0   -a 0   b  0 
│ │ │ │ +000327e0: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +000327f0: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +00032800: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +00032810: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032820: 3020 2020 3020 202d 6120 202d 6320 6232  0   0  -a  -c b2
│ │ │ │ +00032830: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032840: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +00032850: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +00032860: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032870: 6132 2020 6320 2062 2020 2030 2020 3020  a2  c  b   0  0 
│ │ │ │ +00032880: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032890: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +000328a0: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +000328b0: 2d61 2030 2020 2062 2030 2020 2063 3220  -a 0   b 0   c2 
│ │ │ │ +000328c0: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +000328d0: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +000328e0: 2020 2020 2062 3220 3020 2030 2030 2020       b2 0  0 0  
│ │ │ │ +000328f0: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +00032900: 3020 2061 2020 2063 202d 6232 2030 2020  0  a   c -b2 0  
│ │ │ │ +00032910: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032920: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032930: 2020 2020 2030 2020 3020 2030 2030 2020       0  0  0 0  
│ │ │ │ +00032940: 3020 207c 2020 7b37 7d20 7c20 6132 2020  0  |  {7} | a2  
│ │ │ │ +00032950: 6320 2062 2020 2030 2030 2020 2030 2020  c  b   0 0   0  
│ │ │ │ +00032960: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032970: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032980: 2020 2020 2061 2020 3020 2030 2030 2020       a  0  0 0  
│ │ │ │ +00032990: 3020 207c 2020 7b36 7d20 7c20 3020 2020  0  |  {6} | 0   
│ │ │ │ +000329a0: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +000329b0: 3020 2020 3020 202d 6332 2030 2020 3020  0   0  -c2 0  0 
│ │ │ │ +000329c0: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +000329d0: 2020 2020 2030 2020 3020 2061 202d 6320       0  0  a -c 
│ │ │ │ +000329e0: 2d62 207c 2020 7b37 7d20 7c20 3020 2020  -b |  {7} | 0   
│ │ │ │ +000329f0: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032a00: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032a10: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032a20: 2020 2020 2030 2020 6220 2030 2061 3220       0  b  0 a2 
│ │ │ │ +00032a30: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +00032a40: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032a50: 2d63 3220 3020 2030 2020 2030 2020 3020  -c2 0  0   0  0 
│ │ │ │ +00032a60: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032a70: 2020 2020 2030 2020 2d63 2030 2030 2020       0  -c 0 0  
│ │ │ │ +00032a80: 6132 207c 2020 7b37 7d20 7c20 3020 2020  a2 |  {7} | 0   
│ │ │ │ +00032a90: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032aa0: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032ab0: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032ac0: 2020 2020 2030 2020 6120 2030 2030 2020       0  a  0 0  
│ │ │ │ +00032ad0: 3020 207c 2020 7b37 7d20 7c20 3020 2020  0  |  {7} | 0   
│ │ │ │ +00032ae0: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032af0: 3020 2020 3020 2030 2020 2030 2020 6332  0   0  0   0  c2
│ │ │ │ +00032b00: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032b20: 2020 2020 2020 7b37 7d20 7c20 3020 2020        {7} | 0   
│ │ │ │ +00032b30: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032b40: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032b50: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032b70: 2020 2020 2020 7b37 7d20 7c20 3020 2020        {7} | 0   
│ │ │ │ +00032b80: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032b90: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032ba0: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032bc0: 2020 2020 2020 7b36 7d20 7c20 3020 2020        {6} | 0   
│ │ │ │ +00032bd0: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032be0: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032bf0: 2030 2020 2030 202d 6332 2030 207c 0a7c   0   0 -c2 0 |.|
│ │ │ │ +00032c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032c10: 2020 2020 2020 7b37 7d20 7c20 3020 2020        {7} | 0   
│ │ │ │ +00032c20: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032c30: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032c40: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032c60: 2020 2020 2020 7b37 7d20 7c20 3020 2020        {7} | 0   
│ │ │ │ +00032c70: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032c80: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032c90: 202d 6332 2030 2030 2020 2030 207c 0a7c   -c2 0 0   0 |.|
│ │ │ │ +00032ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cb0: 2020 2020 2020 7b37 7d20 7c20 3020 2020        {7} | 0   
│ │ │ │ +00032cc0: 3020 2030 2020 2030 2030 2020 2030 2020  0  0   0 0   0  
│ │ │ │ +00032cd0: 3020 2020 3020 2030 2020 2030 2020 3020  0   0  0   0  0 
│ │ │ │ +00032ce0: 2030 2020 2030 2030 2020 2030 207c 0a7c   0   0 0   0 |.|
│ │ │ │ +00032cf0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │  00032d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d40: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00032d50: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032d60: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032d70: 2030 2030 2020 7c7d 2020 2020 2020 2020   0 0  |}        
│ │ │ │ -00032d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032da0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032db0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032dc0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032df0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032e00: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032e10: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032e40: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032e50: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032e60: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032e90: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032ea0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032eb0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032ee0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032ef0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032f00: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032f20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032f30: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032f40: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032f50: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032f80: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032f90: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032fa0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00032fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032fc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032fd0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00032fe0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00032ff0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00033000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033020: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00033030: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00033040: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00033050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033060: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033070: 3020 2020 6232 2030 2020 3020 2030 2020  0   b2 0  0  0  
│ │ │ │ -00033080: 3020 2020 3020 2030 2061 3220 3020 2030  0   0  0 a2 0  0
│ │ │ │ -00033090: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -000330a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000330b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000330c0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -000330d0: 6132 2020 6320 2062 2030 2020 3020 2030  a2  c  b 0  0  0
│ │ │ │ -000330e0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -000330f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033110: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00033120: 3020 2020 6120 2030 202d 6220 3020 2030  0   a  0 -b 0  0
│ │ │ │ -00033130: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00033140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033160: 2d62 3220 3020 2030 2020 3020 2030 2020  -b2 0  0  0  0  
│ │ │ │ -00033170: 3020 2020 3020 2061 2063 2020 3020 2030  0   0  a c  0  0
│ │ │ │ -00033180: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000331b0: 3020 2020 2d61 2030 2020 6220 2030 2020  0   -a 0  b  0  
│ │ │ │ -000331c0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -000331d0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -000331e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033200: 3020 2020 3020 202d 6120 2d63 2062 3220  0   0  -a -c b2 
│ │ │ │ -00033210: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00033220: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00033230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033250: 6132 2020 6320 2062 2020 3020 2030 2020  a2  c  b  0  0  
│ │ │ │ -00033260: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00033270: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00033280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000332a0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -000332b0: 3020 2020 6232 2030 2030 2020 3020 2030  0   b2 0 0  0  0
│ │ │ │ -000332c0: 2030 2061 3220 7c20 2020 2020 2020 2020   0 a2 |         
│ │ │ │ -000332d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000332e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000332f0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00033300: 3020 2020 3020 2030 2030 2020 6132 2063  0   0  0 0  a2 c
│ │ │ │ -00033310: 2062 2030 2020 7c20 2020 2020 2020 2020   b 0  |         
│ │ │ │ -00033320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033340: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00033350: 3020 2020 3020 2030 2030 2020 3020 2061  0   0  0 0  0  a
│ │ │ │ -00033360: 2030 202d 6220 7c20 2020 2020 2020 2020   0 -b |         
│ │ │ │ -00033370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033390: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -000333a0: 2d62 3220 3020 2030 2030 2020 3020 2030  -b2 0  0 0  0  0
│ │ │ │ -000333b0: 2061 2063 2020 7c20 2020 2020 2020 2020   a c  |         
│ │ │ │ -000333c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00032d40: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032d50: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032d60: 2030 2020 3020 3020 3020 207c 7d20 2020   0  0 0 0  |}   
│ │ │ │ +00032d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032d90: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032da0: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032db0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032de0: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032df0: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032e00: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032e30: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032e40: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032e50: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032e80: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032e90: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032ea0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032ed0: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032ee0: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032ef0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032f10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032f20: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032f30: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032f40: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032f60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032f70: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032f80: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032f90: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032fc0: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00032fd0: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00032fe0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00032ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033010: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00033020: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00033030: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00033040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033060: 2020 2020 2030 2020 2062 3220 3020 2030       0   b2 0  0
│ │ │ │ +00033070: 2020 3020 2030 2020 2030 2020 3020 6132    0  0   0  0 a2
│ │ │ │ +00033080: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00033090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000330a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000330b0: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +000330c0: 2020 3020 2061 3220 2063 2020 6220 3020    0  a2  c  b 0 
│ │ │ │ +000330d0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +000330e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000330f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033100: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00033110: 2020 3020 2030 2020 2061 2020 3020 2d62    0  0   a  0 -b
│ │ │ │ +00033120: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00033130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033150: 2020 2020 202d 6232 2030 2020 3020 2030       -b2 0  0  0
│ │ │ │ +00033160: 2020 3020 2030 2020 2030 2020 6120 6320    0  0   0  a c 
│ │ │ │ +00033170: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00033180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000331a0: 2020 2020 2030 2020 202d 6120 3020 2062       0   -a 0  b
│ │ │ │ +000331b0: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +000331c0: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +000331d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000331e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000331f0: 2020 2020 2030 2020 2030 2020 2d61 202d       0   0  -a -
│ │ │ │ +00033200: 6320 6232 2030 2020 2030 2020 3020 3020  c b2 0   0  0 0 
│ │ │ │ +00033210: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00033220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033240: 2020 2020 2061 3220 2063 2020 6220 2030       a2  c  b  0
│ │ │ │ +00033250: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00033260: 2030 2020 3020 3020 3020 207c 2020 2020   0  0 0 0  |    
│ │ │ │ +00033270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033290: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +000332a0: 2020 3020 2030 2020 2062 3220 3020 3020    0  0   b2 0 0 
│ │ │ │ +000332b0: 2030 2020 3020 3020 6132 207c 2020 2020   0  0 0 a2 |    
│ │ │ │ +000332c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000332d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000332e0: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +000332f0: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00033300: 2061 3220 6320 6220 3020 207c 2020 2020   a2 c b 0  |    
│ │ │ │ +00033310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033330: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00033340: 2020 3020 2030 2020 2030 2020 3020 3020    0  0   0  0 0 
│ │ │ │ +00033350: 2030 2020 6120 3020 2d62 207c 2020 2020   0  a 0 -b |    
│ │ │ │ +00033360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033370: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033380: 2020 2020 2030 2020 2030 2020 3020 2030       0   0  0  0
│ │ │ │ +00033390: 2020 3020 202d 6232 2030 2020 3020 3020    0  -b2 0  0 0 
│ │ │ │ +000333a0: 2030 2020 3020 6120 6320 207c 2020 2020   0  0 a c  |    
│ │ │ │ +000333b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000333d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000333e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000333f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033420: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ -00033430: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +00033410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033420: 6f39 203a 204c 6973 7420 2020 2020 2020  o9 : List       
│ │ │ │ +00033430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033470: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033460: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000334a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ -000334d0: 2061 7070 6c79 2835 2c20 6a2d 3e20 7072   apply(5, j-> pr
│ │ │ │ -000334e0: 756e 6520 4848 5f6a 2043 203d 3d20 3029  une HH_j C == 0)
│ │ │ │ +000334b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000334c0: 6931 3020 3a20 6170 706c 7928 352c 206a  i10 : apply(5, j
│ │ │ │ +000334d0: 2d3e 2070 7275 6e65 2048 485f 6a20 4320  -> prune HH_j C 
│ │ │ │ +000334e0: 3d3d 2030 2920 2020 2020 2020 2020 2020  == 0)           
│ │ │ │  000334f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033560: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -00033570: 207b 7472 7565 2c20 7472 7565 2c20 7472   {true, true, tr
│ │ │ │ -00033580: 7565 2c20 7472 7565 2c20 7472 7565 7d20  ue, true, true} 
│ │ │ │ +00033550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033560: 6f31 3020 3d20 7b74 7275 652c 2074 7275  o10 = {true, tru
│ │ │ │ +00033570: 652c 2074 7275 652c 2074 7275 652c 2074  e, true, true, t
│ │ │ │ +00033580: 7275 657d 2020 2020 2020 2020 2020 2020  rue}            
│ │ │ │  00033590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000335a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000335b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000335c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000335d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000335e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033600: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ -00033610: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +000335f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033600: 6f31 3020 3a20 4c69 7374 2020 2020 2020  o10 : List      
│ │ │ │ +00033610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033650: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033640: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000336a0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6e64 206f  --------+..And o
│ │ │ │ -000336b0: 6e65 2063 6f6d 7075 7469 6e67 2074 6865  ne computing the
│ │ │ │ -000336c0: 2077 686f 6c65 2066 696e 6974 6520 7265   whole finite re
│ │ │ │ -000336d0: 736f 6c75 7469 6f6e 3a0a 0a2b 2d2d 2d2d  solution:..+----
│ │ │ │ +00033690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +000336a0: 416e 6420 6f6e 6520 636f 6d70 7574 696e  And one computin
│ │ │ │ +000336b0: 6720 7468 6520 7768 6f6c 6520 6669 6e69  g the whole fini
│ │ │ │ +000336c0: 7465 2072 6573 6f6c 7574 696f 6e3a 0a0a  te resolution:..
│ │ │ │ +000336d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000336e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000336f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033720: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ -00033730: 204d 5320 3d20 7075 7368 466f 7277 6172   MS = pushForwar
│ │ │ │ -00033740: 6428 6d61 7028 522c 5329 2c20 4d29 3b20  d(map(R,S), M); 
│ │ │ │ +00033710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033720: 6931 3120 3a20 4d53 203d 2070 7573 6846  i11 : MS = pushF
│ │ │ │ +00033730: 6f72 7761 7264 286d 6170 2852 2c53 292c  orward(map(R,S),
│ │ │ │ +00033740: 204d 293b 2020 2020 2020 2020 2020 2020   M);            
│ │ │ │  00033750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033770: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00033760: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00033770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000337a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000337b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000337c0: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2028  ------+.|i12 : (
│ │ │ │ -000337d0: 4747 2c20 6175 6729 203d 206c 6179 6572  GG, aug) = layer
│ │ │ │ -000337e0: 6564 5265 736f 6c75 7469 6f6e 2866 662c  edResolution(ff,
│ │ │ │ -000337f0: 4d53 2920 2020 2020 2020 2020 2020 2020  MS)             
│ │ │ │ -00033800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000337b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000337c0: 3220 3a20 2847 472c 2061 7567 2920 3d20  2 : (GG, aug) = 
│ │ │ │ +000337d0: 6c61 7965 7265 6452 6573 6f6c 7574 696f  layeredResolutio
│ │ │ │ +000337e0: 6e28 6666 2c4d 5329 2020 2020 2020 2020  n(ff,MS)        
│ │ │ │ +000337f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033800: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00033810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033860: 2020 2020 7c0a 7c20 2020 2020 2020 2036      |.|        6
│ │ │ │ -00033870: 2020 2020 2020 3133 2020 2020 2020 3130        13      10
│ │ │ │ -00033880: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00033850: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00033860: 2020 2020 3620 2020 2020 2031 3320 2020      6      13   
│ │ │ │ +00033870: 2020 2031 3020 2020 2020 2033 2020 2020     10      3    
│ │ │ │ +00033880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338b0: 2020 207c 0a7c 6f31 3220 3d20 2853 2020     |.|o12 = (S  
│ │ │ │ -000338c0: 3c2d 2d20 5320 2020 3c2d 2d20 5320 2020  <-- S   <-- S   
│ │ │ │ -000338d0: 3c2d 2d20 5320 2c20 7b32 7d20 7c20 3020  <-- S , {2} | 0 
│ │ │ │ -000338e0: 3020 2030 2020 3120 3020 3020 7c29 2020  0  0  1 0 0 |)  
│ │ │ │ -000338f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033900: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00033910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033920: 2020 2020 2020 207b 327d 207c 2030 202d         {2} | 0 -
│ │ │ │ -00033930: 3120 3020 2030 2030 2030 207c 2020 2020  1 0  0 0 0 |    
│ │ │ │ -00033940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033950: 207c 0a7c 2020 2020 2020 2030 2020 2020   |.|       0    
│ │ │ │ -00033960: 2020 3120 2020 2020 2020 3220 2020 2020    1       2     
│ │ │ │ -00033970: 2020 3320 2020 7b32 7d20 7c20 3020 3020    3   {2} | 0 0 
│ │ │ │ -00033980: 202d 3120 3020 3020 3020 7c20 2020 2020   -1 0 0 0 |     
│ │ │ │ -00033990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000339b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339c0: 2020 2020 207b 337d 207c 2030 2030 2020       {3} | 0 0  
│ │ │ │ -000339d0: 3020 2030 2030 2031 207c 2020 2020 2020  0  0 0 1 |      
│ │ │ │ -000339e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000339f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00033a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a10: 2020 2020 7b33 7d20 7c20 3020 3020 2030      {3} | 0 0  0
│ │ │ │ -00033a20: 2020 3020 3120 3020 7c20 2020 2020 2020    0 1 0 |       
│ │ │ │ -00033a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00033a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00033a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a60: 2020 207b 337d 207c 2031 2030 2020 3020     {3} | 1 0  0 
│ │ │ │ -00033a70: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00033a80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000338a0: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ +000338b0: 2028 5320 203c 2d2d 2053 2020 203c 2d2d   (S  <-- S   <--
│ │ │ │ +000338c0: 2053 2020 203c 2d2d 2053 202c 207b 327d   S   <-- S , {2}
│ │ │ │ +000338d0: 207c 2030 2030 2020 3020 2031 2030 2030   | 0 0  0  1 0 0
│ │ │ │ +000338e0: 207c 2920 2020 2020 2020 2020 2020 2020   |)             
│ │ │ │ +000338f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033910: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ +00033920: 7c20 3020 2d31 2030 2020 3020 3020 3020  | 0 -1 0  0 0 0 
│ │ │ │ +00033930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00033940: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00033950: 3020 2020 2020 2031 2020 2020 2020 2032  0      1       2
│ │ │ │ +00033960: 2020 2020 2020 2033 2020 207b 327d 207c         3   {2} |
│ │ │ │ +00033970: 2030 2030 2020 2d31 2030 2030 2030 207c   0 0  -1 0 0 0 |
│ │ │ │ +00033980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000339a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000339b0: 2020 2020 2020 2020 2020 7b33 7d20 7c20            {3} | 
│ │ │ │ +000339c0: 3020 3020 2030 2020 3020 3020 3120 7c20  0 0  0  0 0 1 | 
│ │ │ │ +000339d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000339e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000339f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a00: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ +00033a10: 2030 2020 3020 2030 2031 2030 207c 2020   0  0  0 1 0 |  
│ │ │ │ +00033a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00033a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a50: 2020 2020 2020 2020 7b33 7d20 7c20 3120          {3} | 1 
│ │ │ │ +00033a60: 3020 2030 2020 3020 3020 3020 7c20 2020  0  0  0 0 0 |   
│ │ │ │ +00033a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00033a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00033ae0: 3132 203a 2053 6571 7565 6e63 6520 2020  12 : Sequence   
│ │ │ │ +00033ad0: 207c 0a7c 6f31 3220 3a20 5365 7175 656e   |.|o12 : Sequen
│ │ │ │ +00033ae0: 6365 2020 2020 2020 2020 2020 2020 2020  ce              
│ │ │ │  00033af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00033b20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00033b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -00033b80: 203a 2028 4747 2c20 6175 6729 203d 206c   : (GG, aug) = l
│ │ │ │ -00033b90: 6179 6572 6564 5265 736f 6c75 7469 6f6e  ayeredResolution
│ │ │ │ -00033ba0: 2866 662c 4d53 2c20 5665 7262 6f73 6520  (ff,MS, Verbose 
│ │ │ │ -00033bb0: 3d3e 7472 7565 2920 2020 2020 2020 2020  =>true)         
│ │ │ │ -00033bc0: 2020 2020 2020 2020 207c 0a7c 7b33 2c20           |.|{3, 
│ │ │ │ -00033bd0: 317d 2069 6e20 636f 6469 6d65 6e73 696f  1} in codimensio
│ │ │ │ -00033be0: 6e20 3320 2020 2020 2020 2020 2020 2020  n 3             
│ │ │ │ +00033b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00033b70: 0a7c 6931 3320 3a20 2847 472c 2061 7567  .|i13 : (GG, aug
│ │ │ │ +00033b80: 2920 3d20 6c61 7965 7265 6452 6573 6f6c  ) = layeredResol
│ │ │ │ +00033b90: 7574 696f 6e28 6666 2c4d 532c 2056 6572  ution(ff,MS, Ver
│ │ │ │ +00033ba0: 626f 7365 203d 3e74 7275 6529 2020 2020  bose =>true)    
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│ │ │ │ +00033be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c10: 2020 2020 2020 2020 7c0a 7c7b 332c 2031          |.|{3, 1
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│ │ │ │ +00033c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033c10: 7b33 2c20 317d 2069 6e20 636f 6469 6d65  {3, 1} in codime
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│ │ │ │  00033c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00033c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00033c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00033cb0: 2020 2020 2020 3620 2020 2020 2031 3320        6      13 
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│ │ │ │ +00033cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d00: 2020 2020 207c 0a7c 6f31 3320 3d20 2853       |.|o13 = (S
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│ │ │ │ -00033d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00033d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d70: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
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│ │ │ │ -00033df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -00033e10: 2020 2020 2020 207b 337d 207c 2030 2030         {3} | 0 0
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│ │ │ │ -00033e60: 2020 2020 2020 7b33 7d20 7c20 3020 3020        {3} | 0 0 
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│ │ │ │ +00033e00: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
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│ │ │ │ +00033e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e50: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +00033e60: 2030 2030 2020 3020 2030 2031 2030 207c   0 0  0  0 1 0 |
│ │ │ │ +00033e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00033e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ea0: 2020 2020 2020 2020 2020 7b33 7d20 7c20            {3} | 
│ │ │ │ +00033eb0: 3120 3020 2030 2020 3020 3020 3020 7c20  1 0  0  0 0 0 | 
│ │ │ │ +00033ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00033ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00033f30: 7c6f 3133 203a 2053 6571 7565 6e63 6520  |o13 : Sequence 
│ │ │ │ +00033f20: 2020 207c 0a7c 6f31 3320 3a20 5365 7175     |.|o13 : Sequ
│ │ │ │ +00033f30: 656e 6365 2020 2020 2020 2020 2020 2020  ence            
│ │ │ │  00033f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f70: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033f70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00033f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00033fd0: 3134 203a 2062 6574 7469 2047 4720 2020  14 : betti GG   
│ │ │ │ +00033fc0: 2d2b 0a7c 6931 3420 3a20 6265 7474 6920  -+.|i14 : betti 
│ │ │ │ +00033fd0: 4747 2020 2020 2020 2020 2020 2020 2020  GG              
│ │ │ │  00033fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034010: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034010: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00034020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034060: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00034070: 2020 2020 2020 2020 2020 3020 2031 2020            0  1  
│ │ │ │ -00034080: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
│ │ │ │ +00034050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00034060: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ +00034070: 2020 3120 2032 2033 2020 2020 2020 2020    1  2 3        
│ │ │ │ +00034080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340b0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -000340c0: 3d20 746f 7461 6c3a 2036 2031 3320 3130  = total: 6 13 10
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│ │ │ │ +000340a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000340b0: 7c6f 3134 203d 2074 6f74 616c 3a20 3620  |o14 = total: 6 
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│ │ │ │ +000340d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034110: 2020 2020 2032 3a20 3320 2031 2020 2e20       2: 3  1  . 
│ │ │ │ -00034120: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +000340f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00034110: 3120 202e 202e 2020 2020 2020 2020 2020  1  . .          
│ │ │ │ +00034120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00034160: 2020 2020 333a 2033 2020 3920 202e 202e      3: 3  9  . .
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│ │ │ │ +00034150: 2020 2020 2020 2020 2033 3a20 3320 2039           3: 3  9
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│ │ │ │  00034170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000341b0: 2020 2034 3a20 2e20 202e 2020 2e20 2e20     4: .  .  . . 
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│ │ │ │ +000341b0: 202e 202e 2020 2020 2020 2020 2020 2020   . .            
│ │ │ │  000341c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000341d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00034200: 2020 353a 202e 2020 3320 2039 202e 2020    5: .  3  9 .  
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│ │ │ │ +000341f0: 2020 2020 2020 2035 3a20 2e20 2033 2020         5: .  3  
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│ │ │ │  00034210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034240: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00034250: 2036 3a20 2e20 202e 2020 2e20 2e20 2020   6: .  .  . .   
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│ │ │ │ +00034240: 2020 2020 2020 363a 202e 2020 2e20 202e        6: .  .  .
│ │ │ │ +00034250: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │  00034260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000342a0: 373a 202e 2020 2e20 2031 2033 2020 2020  7: .  .  1 3    
│ │ │ │ +00034280: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034290: 2020 2020 2037 3a20 2e20 202e 2020 3120       7: .  .  1 
│ │ │ │ +000342a0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  000342b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000342c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000342d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000342e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000342f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034330: 207c 0a7c 6f31 3420 3a20 4265 7474 6954   |.|o14 : BettiT
│ │ │ │ -00034340: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │ +00034320: 2020 2020 2020 7c0a 7c6f 3134 203a 2042        |.|o14 : B
│ │ │ │ +00034330: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00034340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034380: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00034370: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000343a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000343b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000343c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000343d0: 0a7c 6931 3520 3a20 6265 7474 6920 7265  .|i15 : betti re
│ │ │ │ -000343e0: 7320 4d53 2020 2020 2020 2020 2020 2020  s MS            
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│ │ │ │  000343f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034420: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034410: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00034420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00034470: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00034480: 3120 2032 2033 2020 2020 2020 2020 2020  1  2 3          
│ │ │ │ +00034460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00034470: 2020 3020 2031 2020 3220 3320 2020 2020    0  1  2 3     
│ │ │ │ +00034480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000344c0: 3135 203d 2074 6f74 616c 3a20 3620 3133  15 = total: 6 13
│ │ │ │ -000344d0: 2031 3020 3320 2020 2020 2020 2020 2020   10 3           
│ │ │ │ +000344b0: 207c 0a7c 6f31 3520 3d20 746f 7461 6c3a   |.|o15 = total:
│ │ │ │ +000344c0: 2036 2031 3320 3130 2033 2020 2020 2020   6 13 10 3      
│ │ │ │ +000344d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034500: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00034510: 2020 2020 2020 2020 323a 2033 2020 3120          2: 3  1 
│ │ │ │ -00034520: 202e 202e 2020 2020 2020 2020 2020 2020   . .            
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│ │ │ │ +00034510: 3320 2031 2020 2e20 2e20 2020 2020 2020  3  1  . .       
│ │ │ │ +00034520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034550: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00034560: 2020 2020 2020 2033 3a20 3320 2039 2020         3: 3  9  
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│ │ │ │ +00034570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034590: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000345b0: 2020 2020 2020 343a 202e 2020 2e20 202e        4: .  .  .
│ │ │ │ -000345c0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00034590: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000345a0: 7c20 2020 2020 2020 2020 2034 3a20 2e20  |          4: . 
│ │ │ │ +000345b0: 202e 2020 2e20 2e20 2020 2020 2020 2020   .  . .         
│ │ │ │ +000345c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000345d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034600: 2020 2020 2035 3a20 2e20 2033 2020 3920       5: .  3  9 
│ │ │ │ -00034610: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +000345e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000345f0: 2020 2020 2020 2020 2020 353a 202e 2020            5: .  
│ │ │ │ +00034600: 3320 2039 202e 2020 2020 2020 2020 2020  3  9 .          
│ │ │ │ +00034610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034620: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00034640: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00034650: 2020 2020 363a 202e 2020 2e20 202e 202e      6: .  .  . .
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│ │ │ │ +00034640: 2020 2020 2020 2020 2036 3a20 2e20 202e           6: .  .
│ │ │ │ +00034650: 2020 2e20 2e20 2020 2020 2020 2020 2020    . .           
│ │ │ │  00034660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00034690: 2020 2020 2020 2020 373a 202e 2020 2e20          7: .  . 
│ │ │ │ +000346a0: 2031 2033 2020 2020 2020 2020 2020 2020   1 3            
│ │ │ │  000346b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000348b0: 2020 2020 7c0a 7c6f 3136 203d 204d 5320      |.|o16 = MS 
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│ │ │ │ +000348d0: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │  000348e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000348f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034900: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │  000349c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000349e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000349f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00034a00: 3620 3a20 4368 6169 6e43 6f6d 706c 6578  6 : ChainComplex
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│ │ │ │ +00034a00: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
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│ │ │ │  00034a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -00034aa0: 3a20 6170 706c 7928 6c65 6e67 7468 2047  : apply(length G
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│ │ │ │ -00034ac0: 2048 485f 6a20 4320 3d3d 2030 2920 2020   HH_j C == 0)   
│ │ │ │ -00034ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00034a90: 7c69 3137 203a 2061 7070 6c79 286c 656e  |i17 : apply(len
│ │ │ │ +00034aa0: 6774 6820 4747 202b 3120 2c20 6a2d 3e20  gth GG +1 , j-> 
│ │ │ │ +00034ab0: 7072 756e 6520 4848 5f6a 2043 203d 3d20  prune HH_j C == 
│ │ │ │ +00034ac0: 3029 2020 2020 2020 2020 2020 2020 2020  0)              
│ │ │ │ +00034ad0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00034b40: 7b74 7275 652c 2074 7275 652c 2074 7275  {true, true, tru
│ │ │ │ -00034b50: 652c 2066 616c 7365 7d20 2020 2020 2020  e, false}       
│ │ │ │ +00034b20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00034b30: 3137 203d 207b 7472 7565 2c20 7472 7565  17 = {true, true
│ │ │ │ +00034b40: 2c20 7472 7565 2c20 6661 6c73 657d 2020  , true, false}  
│ │ │ │ +00034b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00034bd0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
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│ │ │ │  00034bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00034c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00034e40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00034e60: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ +00034e70: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00034e80: 206e 6577 4578 7428 4d2c 4e2c 4368 6563   newExt(M,N,Chec
│ │ │ │ +00034e90: 6b20 3d3e 7472 7565 290a 2020 2a20 496e  k =>true).  * In
│ │ │ │ +00034ea0: 7075 7473 3a0a 2020 2020 2020 2a20 4368  puts:.      * Ch
│ │ │ │ +00034eb0: 6563 6b2c 2061 202a 6e6f 7465 2042 6f6f  eck, a *note Boo
│ │ │ │ +00034ec0: 6c65 616e 2076 616c 7565 3a20 284d 6163  lean value: (Mac
│ │ │ │ +00034ed0: 6175 6c61 7932 446f 6329 426f 6f6c 6561  aulay2Doc)Boolea
│ │ │ │ +00034ee0: 6e2c 2c20 0a0a 4465 7363 7269 7074 696f  n,, ..Descriptio
│ │ │ │ +00034ef0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d  n.===========..M
│ │ │ │ +00034f00: 616b 6573 206e 6577 4578 7420 7065 7266  akes newExt perf
│ │ │ │ +00034f10: 6f72 6d20 7661 7269 6f75 7320 6368 6563  orm various chec
│ │ │ │ +00034f20: 6b73 2061 7320 6974 2063 6f6d 7075 7465  ks as it compute
│ │ │ │ +00034f30: 732e 0a0a 5365 6520 616c 736f 0a3d 3d3d  s...See also.===
│ │ │ │ +00034f40: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00034f50: 206e 6577 4578 743a 206e 6577 4578 742c   newExt: newExt,
│ │ │ │ +00034f60: 202d 2d20 476c 6f62 616c 2045 7874 2066   -- Global Ext f
│ │ │ │ +00034f70: 6f72 206d 6f64 756c 6573 206f 7665 7220  or modules over 
│ │ │ │ +00034f80: 6120 636f 6d70 6c65 7465 0a20 2020 2049  a complete.    I
│ │ │ │ +00034f90: 6e74 6572 7365 6374 696f 6e0a 0a46 756e  ntersection..Fun
│ │ │ │ +00034fa0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +00034fb0: 6f6e 616c 2061 7267 756d 656e 7420 6e61  onal argument na
│ │ │ │ +00034fc0: 6d65 6420 4c69 6674 3a0a 3d3d 3d3d 3d3d  med Lift:.======
│ │ │ │ +00034fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00034fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00034ff0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00035000: 3d0a 0a20 202a 2022 6e65 7745 7874 282e  =..  * "newExt(.
│ │ │ │ -00035010: 2e2e 2c4c 6966 743d 3e2e 2e2e 2922 202d  ..,Lift=>...)" -
│ │ │ │ -00035020: 2d20 7365 6520 2a6e 6f74 6520 6e65 7745  - see *note newE
│ │ │ │ -00035030: 7874 3a20 6e65 7745 7874 2c20 2d2d 2047  xt: newExt, -- G
│ │ │ │ -00035040: 6c6f 6261 6c20 4578 7420 666f 720a 2020  lobal Ext for.  
│ │ │ │ -00035050: 2020 6d6f 6475 6c65 7320 6f76 6572 2061    modules over a
│ │ │ │ -00035060: 2063 6f6d 706c 6574 6520 496e 7465 7273   complete Inters
│ │ │ │ -00035070: 6563 7469 6f6e 0a0a 466f 7220 7468 6520  ection..For the 
│ │ │ │ -00035080: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00035090: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -000350a0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -000350b0: 4c69 6674 3a20 4c69 6674 2c20 6973 2061  Lift: Lift, is a
│ │ │ │ -000350c0: 202a 6e6f 7465 2073 796d 626f 6c3a 2028   *note symbol: (
│ │ │ │ -000350d0: 4d61 6361 756c 6179 3244 6f63 2953 796d  Macaulay2Doc)Sym
│ │ │ │ -000350e0: 626f 6c2c 2e0a 1f0a 4669 6c65 3a20 436f  bol,....File: Co
│ │ │ │ -000350f0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -00035100: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ -00035110: 666f 2c20 4e6f 6465 3a20 6d61 6b65 4669  fo, Node: makeFi
│ │ │ │ -00035120: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20  niteResolution, 
│ │ │ │ -00035130: 4e65 7874 3a20 6d61 6b65 4669 6e69 7465  Next: makeFinite
│ │ │ │ -00035140: 5265 736f 6c75 7469 6f6e 436f 6469 6d32  ResolutionCodim2
│ │ │ │ -00035150: 2c20 5072 6576 3a20 4c69 6674 2c20 5570  , Prev: Lift, Up
│ │ │ │ -00035160: 3a20 546f 700a 0a6d 616b 6546 696e 6974  : Top..makeFinit
│ │ │ │ -00035170: 6552 6573 6f6c 7574 696f 6e20 2d2d 2066  eResolution -- f
│ │ │ │ -00035180: 696e 6974 6520 7265 736f 6c75 7469 6f6e  inite resolution
│ │ │ │ -00035190: 206f 6620 6120 6d61 7472 6978 2066 6163   of a matrix fac
│ │ │ │ -000351a0: 746f 7269 7a61 7469 6f6e 206d 6f64 756c  torization modul
│ │ │ │ -000351b0: 6520 4d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  e M.************
│ │ │ │ +00034ff0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226e 6577  ======..  * "new
│ │ │ │ +00035000: 4578 7428 2e2e 2e2c 4c69 6674 3d3e 2e2e  Ext(...,Lift=>..
│ │ │ │ +00035010: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ +00035020: 206e 6577 4578 743a 206e 6577 4578 742c   newExt: newExt,
│ │ │ │ +00035030: 202d 2d20 476c 6f62 616c 2045 7874 2066   -- Global Ext f
│ │ │ │ +00035040: 6f72 0a20 2020 206d 6f64 756c 6573 206f  or.    modules o
│ │ │ │ +00035050: 7665 7220 6120 636f 6d70 6c65 7465 2049  ver a complete I
│ │ │ │ +00035060: 6e74 6572 7365 6374 696f 6e0a 0a46 6f72  ntersection..For
│ │ │ │ +00035070: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00035080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00035090: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +000350a0: 6e6f 7465 204c 6966 743a 204c 6966 742c  note Lift: Lift,
│ │ │ │ +000350b0: 2069 7320 6120 2a6e 6f74 6520 7379 6d62   is a *note symb
│ │ │ │ +000350c0: 6f6c 3a20 284d 6163 6175 6c61 7932 446f  ol: (Macaulay2Do
│ │ │ │ +000350d0: 6329 5379 6d62 6f6c 2c2e 0a1f 0a46 696c  c)Symbol,....Fil
│ │ │ │ +000350e0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ +000350f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +00035100: 6e73 2e69 6e66 6f2c 204e 6f64 653a 206d  ns.info, Node: m
│ │ │ │ +00035110: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ +00035120: 696f 6e2c 204e 6578 743a 206d 616b 6546  ion, Next: makeF
│ │ │ │ +00035130: 696e 6974 6552 6573 6f6c 7574 696f 6e43  initeResolutionC
│ │ │ │ +00035140: 6f64 696d 322c 2050 7265 763a 204c 6966  odim2, Prev: Lif
│ │ │ │ +00035150: 742c 2055 703a 2054 6f70 0a0a 6d61 6b65  t, Up: Top..make
│ │ │ │ +00035160: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ +00035170: 202d 2d20 6669 6e69 7465 2072 6573 6f6c   -- finite resol
│ │ │ │ +00035180: 7574 696f 6e20 6f66 2061 206d 6174 7269  ution of a matri
│ │ │ │ +00035190: 7820 6661 6374 6f72 697a 6174 696f 6e20  x factorization 
│ │ │ │ +000351a0: 6d6f 6475 6c65 204d 0a2a 2a2a 2a2a 2a2a  module M.*******
│ │ │ │ +000351b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000351c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000351d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000351e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000351f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00035200: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ -00035210: 3d3d 3d0a 0a20 202a 2055 7361 6765 3a20  ===..  * Usage: 
│ │ │ │ -00035220: 0a20 2020 2020 2020 2041 203d 206d 616b  .        A = mak
│ │ │ │ -00035230: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ -00035240: 6e28 6666 2c6d 6629 0a20 202a 2049 6e70  n(ff,mf).  * Inp
│ │ │ │ -00035250: 7574 733a 0a20 2020 2020 202a 206d 662c  uts:.      * mf,
│ │ │ │ -00035260: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ -00035270: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ -00035280: 742c 2c20 6f75 7470 7574 206f 6620 6d61  t,, output of ma
│ │ │ │ -00035290: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -000352a0: 6e0a 2020 2020 2020 2a20 6666 2c20 6120  n.      * ff, a 
│ │ │ │ -000352b0: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d  *note matrix: (M
│ │ │ │ -000352c0: 6163 6175 6c61 7932 446f 6329 4d61 7472  acaulay2Doc)Matr
│ │ │ │ -000352d0: 6978 2c2c 2074 6865 2072 6567 756c 6172  ix,, the regular
│ │ │ │ -000352e0: 2073 6571 7565 6e63 6520 7573 6564 0a20   sequence used. 
│ │ │ │ -000352f0: 2020 2020 2020 2066 6f72 2074 6865 206d         for the m
│ │ │ │ -00035300: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00035310: 6f6e 2063 6f6d 7075 7461 7469 6f6e 0a20  on computation. 
│ │ │ │ -00035320: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00035330: 2020 2a20 412c 2061 202a 6e6f 7465 2063    * A, a *note c
│ │ │ │ -00035340: 6861 696e 2063 6f6d 706c 6578 3a20 284d  hain complex: (M
│ │ │ │ -00035350: 6163 6175 6c61 7932 446f 6329 4368 6169  acaulay2Doc)Chai
│ │ │ │ -00035360: 6e43 6f6d 706c 6578 2c2c 2041 2069 7320  nComplex,, A is 
│ │ │ │ -00035370: 7468 6520 6d69 6e69 6d61 6c0a 2020 2020  the minimal.    
│ │ │ │ -00035380: 2020 2020 6669 6e69 7465 2072 6573 6f6c      finite resol
│ │ │ │ -00035390: 7574 696f 6e20 6f66 204d 206f 7665 7220  ution of M over 
│ │ │ │ -000353a0: 522e 0a0a 4465 7363 7269 7074 696f 6e0a  R...Description.
│ │ │ │ -000353b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 7570  ===========..Sup
│ │ │ │ -000353c0: 706f 7365 2074 6861 7420 665f 312e 2e66  pose that f_1..f
│ │ │ │ -000353d0: 5f63 2069 7320 6120 686f 6d6f 6765 6e65  _c is a homogene
│ │ │ │ -000353e0: 6f75 7320 7265 6775 6c61 7220 7365 7175  ous regular sequ
│ │ │ │ -000353f0: 656e 6365 206f 6620 666f 726d 7320 6f66  ence of forms of
│ │ │ │ -00035400: 2074 6865 2073 616d 650a 6465 6772 6565   the same.degree
│ │ │ │ -00035410: 2069 6e20 6120 706f 6c79 6e6f 6d69 616c   in a polynomial
│ │ │ │ -00035420: 2072 696e 6720 5320 616e 6420 4d20 6973   ring S and M is
│ │ │ │ -00035430: 2061 2068 6967 6820 7379 7a79 6779 206d   a high syzygy m
│ │ │ │ -00035440: 6f64 756c 6520 6f76 6572 2053 2f28 665f  odule over S/(f_
│ │ │ │ -00035450: 312c 2e2e 2c66 5f63 290a 3d20 5228 6329  1,..,f_c).= R(c)
│ │ │ │ -00035460: 2c20 616e 6420 6d66 203d 2028 642c 6829  , and mf = (d,h)
│ │ │ │ -00035470: 2069 7320 7468 6520 6f75 7470 7574 206f   is the output o
│ │ │ │ -00035480: 6620 6d61 7472 6978 4661 6374 6f72 697a  f matrixFactoriz
│ │ │ │ -00035490: 6174 696f 6e28 4d2c 6666 292e 2049 6620  ation(M,ff). If 
│ │ │ │ -000354a0: 7468 650a 636f 6d70 6c65 7869 7479 206f  the.complexity o
│ │ │ │ -000354b0: 6620 4d20 6973 2063 272c 2074 6865 6e20  f M is c', then 
│ │ │ │ -000354c0: 4d20 6861 7320 6120 6669 6e69 7465 2066  M has a finite f
│ │ │ │ -000354d0: 7265 6520 7265 736f 6c75 7469 6f6e 206f  ree resolution o
│ │ │ │ -000354e0: 7665 7220 5220 3d0a 532f 2866 5f31 2c2e  ver R =.S/(f_1,.
│ │ │ │ -000354f0: 2e2c 665f 7b28 632d 6327 297d 2920 2861  .,f_{(c-c')}) (a
│ │ │ │ -00035500: 6e64 2c20 6d6f 7265 2067 656e 6572 616c  nd, more general
│ │ │ │ -00035510: 6c79 2c20 6861 7320 636f 6d70 6c65 7869  ly, has complexi
│ │ │ │ -00035520: 7479 2063 2d64 206f 7665 720a 532f 2866  ty c-d over.S/(f
│ │ │ │ -00035530: 5f31 2c2e 2e2c 665f 7b28 632d 6429 7d29  _1,..,f_{(c-d)})
│ │ │ │ -00035540: 2066 6f72 2064 3e3d 6327 292e 0a0a 5468   for d>=c')...Th
│ │ │ │ -00035550: 6520 636f 6d70 6c65 7820 4120 6973 2074  e complex A is t
│ │ │ │ -00035560: 6865 206d 696e 696d 616c 2066 696e 6974  he minimal finit
│ │ │ │ -00035570: 6520 6672 6565 2072 6573 6f6c 7574 696f  e free resolutio
│ │ │ │ -00035580: 6e20 6f66 204d 206f 7665 7220 412c 2063  n of M over A, c
│ │ │ │ -00035590: 6f6e 7374 7275 6374 6564 2061 730a 616e  onstructed as.an
│ │ │ │ -000355a0: 2069 7465 7261 7465 6420 4b6f 737a 756c   iterated Koszul
│ │ │ │ -000355b0: 2065 7874 656e 7369 6f6e 2c20 6d61 6465   extension, made
│ │ │ │ -000355c0: 2066 726f 6d20 7468 6520 6d61 7073 2069   from the maps i
│ │ │ │ -000355d0: 6e20 624d 6170 7320 6d66 2061 6e64 2070  n bMaps mf and p
│ │ │ │ -000355e0: 7369 4d61 7073 206d 662c 2061 730a 6465  siMaps mf, as.de
│ │ │ │ -000355f0: 7363 7269 6265 6420 696e 2045 6973 656e  scribed in Eisen
│ │ │ │ -00035600: 6275 642d 5065 6576 612e 0a0a 2b2d 2d2d  bud-Peeva...+---
│ │ │ │ +000351f0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +00035200: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +00035210: 6167 653a 200a 2020 2020 2020 2020 4120  age: .        A 
│ │ │ │ +00035220: 3d20 6d61 6b65 4669 6e69 7465 5265 736f  = makeFiniteReso
│ │ │ │ +00035230: 6c75 7469 6f6e 2866 662c 6d66 290a 2020  lution(ff,mf).  
│ │ │ │ +00035240: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +00035250: 2a20 6d66 2c20 6120 2a6e 6f74 6520 6c69  * mf, a *note li
│ │ │ │ +00035260: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +00035270: 6329 4c69 7374 2c2c 206f 7574 7075 7420  c)List,, output 
│ │ │ │ +00035280: 6f66 206d 6174 7269 7846 6163 746f 7269  of matrixFactori
│ │ │ │ +00035290: 7a61 7469 6f6e 0a20 2020 2020 202a 2066  zation.      * f
│ │ │ │ +000352a0: 662c 2061 202a 6e6f 7465 206d 6174 7269  f, a *note matri
│ │ │ │ +000352b0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +000352c0: 294d 6174 7269 782c 2c20 7468 6520 7265  )Matrix,, the re
│ │ │ │ +000352d0: 6775 6c61 7220 7365 7175 656e 6365 2075  gular sequence u
│ │ │ │ +000352e0: 7365 640a 2020 2020 2020 2020 666f 7220  sed.        for 
│ │ │ │ +000352f0: 7468 6520 6d61 7472 6978 4661 6374 6f72  the matrixFactor
│ │ │ │ +00035300: 697a 6174 696f 6e20 636f 6d70 7574 6174  ization computat
│ │ │ │ +00035310: 696f 6e0a 2020 2a20 4f75 7470 7574 733a  ion.  * Outputs:
│ │ │ │ +00035320: 0a20 2020 2020 202a 2041 2c20 6120 2a6e  .      * A, a *n
│ │ │ │ +00035330: 6f74 6520 6368 6169 6e20 636f 6d70 6c65  ote chain comple
│ │ │ │ +00035340: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +00035350: 2943 6861 696e 436f 6d70 6c65 782c 2c20  )ChainComplex,, 
│ │ │ │ +00035360: 4120 6973 2074 6865 206d 696e 696d 616c  A is the minimal
│ │ │ │ +00035370: 0a20 2020 2020 2020 2066 696e 6974 6520  .        finite 
│ │ │ │ +00035380: 7265 736f 6c75 7469 6f6e 206f 6620 4d20  resolution of M 
│ │ │ │ +00035390: 6f76 6572 2052 2e0a 0a44 6573 6372 6970  over R...Descrip
│ │ │ │ +000353a0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +000353b0: 0a0a 5375 7070 6f73 6520 7468 6174 2066  ..Suppose that f
│ │ │ │ +000353c0: 5f31 2e2e 665f 6320 6973 2061 2068 6f6d  _1..f_c is a hom
│ │ │ │ +000353d0: 6f67 656e 656f 7573 2072 6567 756c 6172  ogeneous regular
│ │ │ │ +000353e0: 2073 6571 7565 6e63 6520 6f66 2066 6f72   sequence of for
│ │ │ │ +000353f0: 6d73 206f 6620 7468 6520 7361 6d65 0a64  ms of the same.d
│ │ │ │ +00035400: 6567 7265 6520 696e 2061 2070 6f6c 796e  egree in a polyn
│ │ │ │ +00035410: 6f6d 6961 6c20 7269 6e67 2053 2061 6e64  omial ring S and
│ │ │ │ +00035420: 204d 2069 7320 6120 6869 6768 2073 797a   M is a high syz
│ │ │ │ +00035430: 7967 7920 6d6f 6475 6c65 206f 7665 7220  ygy module over 
│ │ │ │ +00035440: 532f 2866 5f31 2c2e 2e2c 665f 6329 0a3d  S/(f_1,..,f_c).=
│ │ │ │ +00035450: 2052 2863 292c 2061 6e64 206d 6620 3d20   R(c), and mf = 
│ │ │ │ +00035460: 2864 2c68 2920 6973 2074 6865 206f 7574  (d,h) is the out
│ │ │ │ +00035470: 7075 7420 6f66 206d 6174 7269 7846 6163  put of matrixFac
│ │ │ │ +00035480: 746f 7269 7a61 7469 6f6e 284d 2c66 6629  torization(M,ff)
│ │ │ │ +00035490: 2e20 4966 2074 6865 0a63 6f6d 706c 6578  . If the.complex
│ │ │ │ +000354a0: 6974 7920 6f66 204d 2069 7320 6327 2c20  ity of M is c', 
│ │ │ │ +000354b0: 7468 656e 204d 2068 6173 2061 2066 696e  then M has a fin
│ │ │ │ +000354c0: 6974 6520 6672 6565 2072 6573 6f6c 7574  ite free resolut
│ │ │ │ +000354d0: 696f 6e20 6f76 6572 2052 203d 0a53 2f28  ion over R =.S/(
│ │ │ │ +000354e0: 665f 312c 2e2e 2c66 5f7b 2863 2d63 2729  f_1,..,f_{(c-c')
│ │ │ │ +000354f0: 7d29 2028 616e 642c 206d 6f72 6520 6765  }) (and, more ge
│ │ │ │ +00035500: 6e65 7261 6c6c 792c 2068 6173 2063 6f6d  nerally, has com
│ │ │ │ +00035510: 706c 6578 6974 7920 632d 6420 6f76 6572  plexity c-d over
│ │ │ │ +00035520: 0a53 2f28 665f 312c 2e2e 2c66 5f7b 2863  .S/(f_1,..,f_{(c
│ │ │ │ +00035530: 2d64 297d 2920 666f 7220 643e 3d63 2729  -d)}) for d>=c')
│ │ │ │ +00035540: 2e0a 0a54 6865 2063 6f6d 706c 6578 2041  ...The complex A
│ │ │ │ +00035550: 2069 7320 7468 6520 6d69 6e69 6d61 6c20   is the minimal 
│ │ │ │ +00035560: 6669 6e69 7465 2066 7265 6520 7265 736f  finite free reso
│ │ │ │ +00035570: 6c75 7469 6f6e 206f 6620 4d20 6f76 6572  lution of M over
│ │ │ │ +00035580: 2041 2c20 636f 6e73 7472 7563 7465 6420   A, constructed 
│ │ │ │ +00035590: 6173 0a61 6e20 6974 6572 6174 6564 204b  as.an iterated K
│ │ │ │ +000355a0: 6f73 7a75 6c20 6578 7465 6e73 696f 6e2c  oszul extension,
│ │ │ │ +000355b0: 206d 6164 6520 6672 6f6d 2074 6865 206d   made from the m
│ │ │ │ +000355c0: 6170 7320 696e 2062 4d61 7073 206d 6620  aps in bMaps mf 
│ │ │ │ +000355d0: 616e 6420 7073 694d 6170 7320 6d66 2c20  and psiMaps mf, 
│ │ │ │ +000355e0: 6173 0a64 6573 6372 6962 6564 2069 6e20  as.described in 
│ │ │ │ +000355f0: 4569 7365 6e62 7564 2d50 6565 7661 2e0a  Eisenbud-Peeva..
│ │ │ │ +00035600: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00035610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035650: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -00035660: 3a20 7365 7452 616e 646f 6d53 6565 6420  : setRandomSeed 
│ │ │ │ -00035670: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00035640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00035650: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d  .|i1 : setRandom
│ │ │ │ +00035660: 5365 6564 2030 2020 2020 2020 2020 2020  Seed 0          
│ │ │ │ +00035670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000356a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00035690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000356a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000356b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000356c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000356d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000356e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000356f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -00035700: 3d20 3020 2020 2020 2020 2020 2020 2020  = 0             
│ │ │ │ +000356e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000356f0: 0a7c 6f31 203d 2030 2020 2020 2020 2020  .|o1 = 0        
│ │ │ │ +00035700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035740: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00035730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035740: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00035750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035790: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -000357a0: 3a20 5320 3d20 5a5a 2f31 3031 5b61 2c62  : S = ZZ/101[a,b
│ │ │ │ -000357b0: 2c63 5d3b 2020 2020 2020 2020 2020 2020  ,c];            
│ │ │ │ +00035780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00035790: 0a7c 6932 203a 2053 203d 205a 5a2f 3130  .|i2 : S = ZZ/10
│ │ │ │ +000357a0: 315b 612c 622c 635d 3b20 2020 2020 2020  1[a,b,c];       
│ │ │ │ +000357b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000357d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000357e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000357d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000357e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000357f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035830: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -00035840: 3a20 6666 203d 206d 6174 7269 7822 6133  : ff = matrix"a3
│ │ │ │ -00035850: 2c62 3322 3b20 2020 2020 2020 2020 2020  ,b3";           
│ │ │ │ +00035820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00035830: 0a7c 6933 203a 2066 6620 3d20 6d61 7472  .|i3 : ff = matr
│ │ │ │ +00035840: 6978 2261 332c 6233 223b 2020 2020 2020  ix"a3,b3";      
│ │ │ │ +00035850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035880: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00035870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00035890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000358c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000358d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000358e0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -000358f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000358c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000358d0: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +000358e0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000358f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035920: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -00035930: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -00035940: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00035910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035920: 0a7c 6f33 203a 204d 6174 7269 7820 5320  .|o3 : Matrix S 
│ │ │ │ +00035930: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00035940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035970: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00035960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035970: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00035980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000359a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000359b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000359c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -000359d0: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ -000359e0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +000359b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000359c0: 0a7c 6934 203a 2052 203d 2053 2f69 6465  .|i4 : R = S/ide
│ │ │ │ +000359d0: 616c 2066 663b 2020 2020 2020 2020 2020  al ff;          
│ │ │ │ +000359e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035a10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00035a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035a10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00035a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -00035a70: 3a20 4d20 3d20 6869 6768 5379 7a79 6779  : M = highSyzygy
│ │ │ │ -00035a80: 2028 525e 312f 6964 6561 6c20 7661 7273   (R^1/ideal vars
│ │ │ │ -00035a90: 2052 293b 2020 2020 2020 2020 2020 2020   R);            
│ │ │ │ -00035aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ab0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00035a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00035a60: 0a7c 6935 203a 204d 203d 2068 6967 6853  .|i5 : M = highS
│ │ │ │ +00035a70: 797a 7967 7920 2852 5e31 2f69 6465 616c  yzygy (R^1/ideal
│ │ │ │ +00035a80: 2076 6172 7320 5229 3b20 2020 2020 2020   vars R);       
│ │ │ │ +00035a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035ab0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00035ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -00035b10: 3a20 6d66 203d 206d 6174 7269 7846 6163  : mf = matrixFac
│ │ │ │ -00035b20: 746f 7269 7a61 7469 6f6e 2028 6666 2c20  torization (ff, 
│ │ │ │ -00035b30: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ -00035b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00035af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00035b00: 0a7c 6936 203a 206d 6620 3d20 6d61 7472  .|i6 : mf = matr
│ │ │ │ +00035b10: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ +00035b20: 2866 662c 204d 2920 2020 2020 2020 2020  (ff, M)         
│ │ │ │ +00035b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035b50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00035b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ba0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00035bb0: 3d20 7b7b 347d 207c 202d 6320 6220 3020  = {{4} | -c b 0 
│ │ │ │ -00035bc0: 2061 3220 3020 2030 2020 3020 2030 2020   a2 0  0  0  0  
│ │ │ │ -00035bd0: 3020 2020 7c2c 207b 357d 207c 2030 2061  0   |, {5} | 0 a
│ │ │ │ -00035be0: 3220 3020 202d 6220 3020 2030 2020 2030  2 0  -b 0  0   0
│ │ │ │ -00035bf0: 2020 2030 2030 2020 3020 7c0a 7c20 2020     0 0  0 |.|   
│ │ │ │ -00035c00: 2020 207b 347d 207c 2061 2020 3020 6220     {4} | a  0 b 
│ │ │ │ -00035c10: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00035c20: 3020 2020 7c20 207b 357d 207c 2030 2030  0   |  {5} | 0 0
│ │ │ │ -00035c30: 2020 6132 202d 6320 6232 2030 2020 2030    a2 -c b2 0   0
│ │ │ │ -00035c40: 2020 2030 2030 2020 3020 7c0a 7c20 2020     0 0  0 |.|   
│ │ │ │ -00035c50: 2020 207b 347d 207c 2030 2020 6120 6320     {4} | 0  a c 
│ │ │ │ -00035c60: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00035c70: 2d62 3220 7c20 207b 357d 207c 2030 2030  -b2 |  {5} | 0 0
│ │ │ │ -00035c80: 2020 3020 2061 2020 3020 2062 3220 2030    0  a  0  b2  0
│ │ │ │ -00035c90: 2020 2030 2030 2020 3020 7c0a 7c20 2020     0 0  0 |.|   
│ │ │ │ -00035ca0: 2020 207b 337d 207c 2030 2020 3020 6132     {3} | 0  0 a2
│ │ │ │ -00035cb0: 2030 2020 3020 2062 3220 3020 2030 2020   0  0  b2 0  0  
│ │ │ │ -00035cc0: 3020 2020 7c20 207b 367d 207c 2061 2063  0   |  {6} | a c
│ │ │ │ -00035cd0: 2020 2d62 2030 2020 3020 2030 2020 2030    -b 0  0  0   0
│ │ │ │ -00035ce0: 2020 2030 2030 2020 3020 7c0a 7c20 2020     0 0  0 |.|   
│ │ │ │ -00035cf0: 2020 207b 347d 207c 2030 2020 3020 3020     {4} | 0  0 0 
│ │ │ │ -00035d00: 2030 2020 6220 202d 6120 3020 2030 2020   0  b  -a 0  0  
│ │ │ │ -00035d10: 3020 2020 7c20 207b 357d 207c 2030 2030  0   |  {5} | 0 0
│ │ │ │ -00035d20: 2020 3020 2030 2020 3020 2030 2020 2030    0  0  0  0   0
│ │ │ │ -00035d30: 2020 2061 2062 3220 3020 7c0a 7c20 2020     a b2 0 |.|   
│ │ │ │ -00035d40: 2020 207b 347d 207c 2030 2020 3020 3020     {4} | 0  0 0 
│ │ │ │ -00035d50: 2030 2020 2d63 2030 2020 6120 2062 3220   0  -c 0  a  b2 
│ │ │ │ -00035d60: 3020 2020 7c20 207b 357d 207c 2030 2030  0   |  {5} | 0 0
│ │ │ │ -00035d70: 2020 3020 2030 2020 3020 202d 6132 2030    0  0  0  -a2 0
│ │ │ │ -00035d80: 2020 2062 2030 2020 3020 7c0a 7c20 2020     b 0  0 |.|   
│ │ │ │ -00035d90: 2020 207b 347d 207c 2030 2020 3020 3020     {4} | 0  0 0 
│ │ │ │ -00035da0: 2030 2020 3020 2063 2020 2d62 2030 2020   0  0  c  -b 0  
│ │ │ │ -00035db0: 6132 2020 7c20 207b 357d 207c 2030 2030  a2  |  {5} | 0 0
│ │ │ │ -00035dc0: 2020 3020 2030 2020 3020 2030 2020 202d    0  0  0  0   -
│ │ │ │ -00035dd0: 6132 2063 2030 2020 3020 7c0a 7c20 2020  a2 c 0  0 |.|   
│ │ │ │ +00035b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035ba0: 0a7c 6f36 203d 207b 7b34 7d20 7c20 2d63  .|o6 = {{4} | -c
│ │ │ │ +00035bb0: 2062 2030 2020 6132 2030 2020 3020 2030   b 0  a2 0  0  0
│ │ │ │ +00035bc0: 2020 3020 2030 2020 207c 2c20 7b35 7d20    0  0   |, {5} 
│ │ │ │ +00035bd0: 7c20 3020 6132 2030 2020 2d62 2030 2020  | 0 a2 0  -b 0  
│ │ │ │ +00035be0: 3020 2020 3020 2020 3020 3020 2030 207c  0   0   0 0  0 |
│ │ │ │ +00035bf0: 0a7c 2020 2020 2020 7b34 7d20 7c20 6120  .|      {4} | a 
│ │ │ │ +00035c00: 2030 2062 2020 3020 2030 2020 3020 2030   0 b  0  0  0  0
│ │ │ │ +00035c10: 2020 3020 2030 2020 207c 2020 7b35 7d20    0  0   |  {5} 
│ │ │ │ +00035c20: 7c20 3020 3020 2061 3220 2d63 2062 3220  | 0 0  a2 -c b2 
│ │ │ │ +00035c30: 3020 2020 3020 2020 3020 3020 2030 207c  0   0   0 0  0 |
│ │ │ │ +00035c40: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00035c50: 2061 2063 2020 3020 2030 2020 3020 2030   a c  0  0  0  0
│ │ │ │ +00035c60: 2020 3020 202d 6232 207c 2020 7b35 7d20    0  -b2 |  {5} 
│ │ │ │ +00035c70: 7c20 3020 3020 2030 2020 6120 2030 2020  | 0 0  0  a  0  
│ │ │ │ +00035c80: 6232 2020 3020 2020 3020 3020 2030 207c  b2  0   0 0  0 |
│ │ │ │ +00035c90: 0a7c 2020 2020 2020 7b33 7d20 7c20 3020  .|      {3} | 0 
│ │ │ │ +00035ca0: 2030 2061 3220 3020 2030 2020 6232 2030   0 a2 0  0  b2 0
│ │ │ │ +00035cb0: 2020 3020 2030 2020 207c 2020 7b36 7d20    0  0   |  {6} 
│ │ │ │ +00035cc0: 7c20 6120 6320 202d 6220 3020 2030 2020  | a c  -b 0  0  
│ │ │ │ +00035cd0: 3020 2020 3020 2020 3020 3020 2030 207c  0   0   0 0  0 |
│ │ │ │ +00035ce0: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00035cf0: 2030 2030 2020 3020 2062 2020 2d61 2030   0 0  0  b  -a 0
│ │ │ │ +00035d00: 2020 3020 2030 2020 207c 2020 7b35 7d20    0  0   |  {5} 
│ │ │ │ +00035d10: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00035d20: 3020 2020 3020 2020 6120 6232 2030 207c  0   0   a b2 0 |
│ │ │ │ +00035d30: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00035d40: 2030 2030 2020 3020 202d 6320 3020 2061   0 0  0  -c 0  a
│ │ │ │ +00035d50: 2020 6232 2030 2020 207c 2020 7b35 7d20    b2 0   |  {5} 
│ │ │ │ +00035d60: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00035d70: 2d61 3220 3020 2020 6220 3020 2030 207c  -a2 0   b 0  0 |
│ │ │ │ +00035d80: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00035d90: 2030 2030 2020 3020 2030 2020 6320 202d   0 0  0  0  c  -
│ │ │ │ +00035da0: 6220 3020 2061 3220 207c 2020 7b35 7d20  b 0  a2  |  {5} 
│ │ │ │ +00035db0: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00035dc0: 3020 2020 2d61 3220 6320 3020 2030 207c  0   -a2 c 0  0 |
│ │ │ │ +00035dd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00035de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e00: 2020 2020 2020 207b 367d 207c 2030 2030         {6} | 0 0
│ │ │ │ -00035e10: 2020 3020 2030 2020 3020 2030 2020 2030    0  0  0  0   0
│ │ │ │ -00035e20: 2020 2030 2063 2020 6220 7c0a 7c20 2020     0 c  b |.|   
│ │ │ │ +00035df0: 2020 2020 2020 2020 2020 2020 7b36 7d20              {6} 
│ │ │ │ +00035e00: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00035e10: 3020 2020 3020 2020 3020 6320 2062 207c  0   0   0 c  b |
│ │ │ │ +00035e20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00035e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e50: 2020 2020 2020 207b 367d 207c 2030 2030         {6} | 0 0
│ │ │ │ -00035e60: 2020 3020 2030 2020 6120 2063 2020 202d    0  0  a  c   -
│ │ │ │ -00035e70: 6220 2030 2030 2020 3020 7c0a 7c20 2020  b  0 0  0 |.|   
│ │ │ │ -00035e80: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │ +00035e40: 2020 2020 2020 2020 2020 2020 7b36 7d20              {6} 
│ │ │ │ +00035e50: 7c20 3020 3020 2030 2020 3020 2061 2020  | 0 0  0  0  a  
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│ │ │ │ +00035e70: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
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│ │ │ │ -00035ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ -00035ed0: 2020 3020 2020 7c2c 207b 337d 207c 2030    0   |, {3} | 0
│ │ │ │ -00035ee0: 2030 2030 2020 3120 3020 3020 3020 7c7d   0 0  1 0 0 0 |}
│ │ │ │ +00035eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00035ec0: 0a7c 2020 2020 2030 2020 207c 2c20 7b33  .|     0   |, {3
│ │ │ │ +00035ed0: 7d20 7c20 3020 3020 3020 2031 2030 2030  } | 0 0 0  1 0 0
│ │ │ │ +00035ee0: 2030 207c 7d20 2020 2020 2020 2020 2020   0 |}           
│ │ │ │  00035ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00035f20: 2020 3020 2020 7c20 207b 347d 207c 2030    0   |  {4} | 0
│ │ │ │ -00035f30: 2030 2030 2020 3020 3120 3020 3020 7c20   0 0  0 1 0 0 | 
│ │ │ │ +00035f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035f10: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00035f20: 7d20 7c20 3020 3020 3020 2030 2031 2030  } | 0 0 0  0 1 0
│ │ │ │ +00035f30: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  00035f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00035f70: 2020 3020 2020 7c20 207b 347d 207c 2030    0   |  {4} | 0
│ │ │ │ -00035f80: 2030 2030 2020 3020 3020 3120 3020 7c20   0 0  0 0 1 0 | 
│ │ │ │ +00035f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035f60: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
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│ │ │ │ +00035f80: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  00035f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035fb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00035fc0: 2020 3020 2020 7c20 207b 347d 207c 2030    0   |  {4} | 0
│ │ │ │ -00035fd0: 2030 2030 2020 3020 3020 3020 3120 7c20   0 0  0 0 0 1 | 
│ │ │ │ +00035fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00035fb0: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00035fc0: 7d20 7c20 3020 3020 3020 2030 2030 2030  } | 0 0 0  0 0 0
│ │ │ │ +00035fd0: 2031 207c 2020 2020 2020 2020 2020 2020   1 |            
│ │ │ │  00035fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036000: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036010: 2020 3020 2020 7c20 207b 347d 207c 2030    0   |  {4} | 0
│ │ │ │ -00036020: 2031 2030 2020 3020 3020 3020 3020 7c20   1 0  0 0 0 0 | 
│ │ │ │ +00035ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036000: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00036010: 7d20 7c20 3020 3120 3020 2030 2030 2030  } | 0 1 0  0 0 0
│ │ │ │ +00036020: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  00036030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036050: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036060: 2020 3020 2020 7c20 207b 347d 207c 2030    0   |  {4} | 0
│ │ │ │ -00036070: 2030 202d 3120 3020 3020 3020 3020 7c20   0 -1 0 0 0 0 | 
│ │ │ │ +00036040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036050: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00036060: 7d20 7c20 3020 3020 2d31 2030 2030 2030  } | 0 0 -1 0 0 0
│ │ │ │ +00036070: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  00036080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000360b0: 2020 2d62 3220 7c20 207b 347d 207c 2031    -b2 |  {4} | 1
│ │ │ │ -000360c0: 2030 2030 2020 3020 3020 3020 3020 7c20   0 0  0 0 0 0 | 
│ │ │ │ +00036090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000360a0: 0a7c 2020 2020 202d 6232 207c 2020 7b34  .|     -b2 |  {4
│ │ │ │ +000360b0: 7d20 7c20 3120 3020 3020 2030 2030 2030  } | 1 0 0  0 0 0
│ │ │ │ +000360c0: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │  000360d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036100: 2020 6120 2020 7c20 2020 2020 2020 2020    a   |         
│ │ │ │ +000360e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000360f0: 0a7c 2020 2020 2061 2020 207c 2020 2020  .|     a   |    
│ │ │ │ +00036100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00036130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036140: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036150: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ +00036130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036140: 0a7c 2020 2020 2030 2020 207c 2020 2020  .|     0   |    
│ │ │ │ +00036150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036160: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00036180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000361a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000361b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000361c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -000361f0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +000361d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +000361f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036230: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00036220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036230: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00036240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036280: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -00036290: 3a20 4720 3d20 6d61 6b65 4669 6e69 7465  : G = makeFinite
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│ │ │ │ -000362c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00036280: 0a7c 6937 203a 2047 203d 206d 616b 6546  .|i7 : G = makeF
│ │ │ │ +00036290: 696e 6974 6552 6573 6f6c 7574 696f 6e28  initeResolution(
│ │ │ │ +000362a0: 6666 2c6d 6629 2020 2020 2020 2020 2020  ff,mf)          
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│ │ │ │ +000362c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  000362e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000362f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036330: 2020 2037 2020 2020 2020 3132 2020 2020     7      12    
│ │ │ │ -00036340: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +00036310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036320: 0a7c 2020 2020 2020 3720 2020 2020 2031  .|      7      1
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│ │ │ │ +00036340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036370: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -00036380: 3d20 5320 203c 2d2d 2053 2020 203c 2d2d  = S  <-- S   <--
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│ │ │ │ +00036360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00036390: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000363b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000363b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000363c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000363d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036410: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036420: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ -00036430: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00036400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036410: 0a7c 2020 2020 2030 2020 2020 2020 3120  .|     0      1 
│ │ │ │ +00036420: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00036430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036460: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00036470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -000364c0: 3a20 4368 6169 6e43 6f6d 706c 6578 2020  : ChainComplex  
│ │ │ │ +000364a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000364b0: 0a7c 6f37 203a 2043 6861 696e 436f 6d70  .|o7 : ChainComp
│ │ │ │ +000364c0: 6c65 7820 2020 2020 2020 2020 2020 2020  lex             
│ │ │ │  000364d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000364e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036500: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000364f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00036510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036550: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -00036560: 3a20 4620 3d20 7265 7320 7075 7368 466f  : F = res pushFo
│ │ │ │ -00036570: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -00036580: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00036590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00036550: 0a7c 6938 203a 2046 203d 2072 6573 2070  .|i8 : F = res p
│ │ │ │ +00036560: 7573 6846 6f72 7761 7264 286d 6170 2852  ushForward(map(R
│ │ │ │ +00036570: 2c53 292c 4d29 2020 2020 2020 2020 2020  ,S),M)          
│ │ │ │ +00036580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000365a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000365b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036600: 2020 2037 2020 2020 2020 3132 2020 2020     7      12    
│ │ │ │ -00036610: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +000365e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000365f0: 0a7c 2020 2020 2020 3720 2020 2020 2031  .|      7      1
│ │ │ │ +00036600: 3220 2020 2020 2035 2020 2020 2020 2020  2      5        
│ │ │ │ +00036610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036640: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -00036650: 3d20 5320 203c 2d2d 2053 2020 203c 2d2d  = S  <-- S   <--
│ │ │ │ -00036660: 2053 2020 3c2d 2d20 3020 2020 2020 2020   S  <-- 0       
│ │ │ │ +00036630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036640: 0a7c 6f38 203d 2053 2020 3c2d 2d20 5320  .|o8 = S  <-- S 
│ │ │ │ +00036650: 2020 3c2d 2d20 5320 203c 2d2d 2030 2020    <-- S  <-- 0  
│ │ │ │ +00036660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036690: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036690: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000366a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000366b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000366c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000366d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000366e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000366f0: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ -00036700: 2032 2020 2020 2020 3320 2020 2020 2020   2      3       
│ │ │ │ +000366d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000366e0: 0a7c 2020 2020 2030 2020 2020 2020 3120  .|     0      1 
│ │ │ │ +000366f0: 2020 2020 2020 3220 2020 2020 2033 2020        2      3  
│ │ │ │ +00036700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036730: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00036740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036780: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -00036790: 3a20 4368 6169 6e43 6f6d 706c 6578 2020  : ChainComplex  
│ │ │ │ +00036770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036780: 0a7c 6f38 203a 2043 6861 696e 436f 6d70  .|o8 : ChainComp
│ │ │ │ +00036790: 6c65 7820 2020 2020 2020 2020 2020 2020  lex             
│ │ │ │  000367a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000367b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000367c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000367d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000367e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000367f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036820: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920  ----------+.|i9 
│ │ │ │ -00036830: 3a20 472e 6464 5f31 2020 2020 2020 2020  : G.dd_1        
│ │ │ │ +00036810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00036820: 0a7c 6939 203a 2047 2e64 645f 3120 2020  .|i9 : G.dd_1   
│ │ │ │ +00036830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036870: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036870: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00036880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000368a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -000368d0: 3d20 7b34 7d20 7c20 2d63 2062 2030 2020  = {4} | -c b 0  
│ │ │ │ -000368e0: 6132 2030 2020 3020 2030 2020 3020 2030  a2 0  0  0  0  0
│ │ │ │ -000368f0: 2020 2030 2020 2030 2020 2030 2020 207c     0   0   0   |
│ │ │ │ -00036900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036910: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036920: 2020 7b34 7d20 7c20 6120 2030 2062 2020    {4} | a  0 b  
│ │ │ │ -00036930: 3020 2030 2020 3020 2030 2020 3020 2030  0  0  0  0  0  0
│ │ │ │ -00036940: 2020 2030 2020 2030 2020 2030 2020 207c     0   0   0   |
│ │ │ │ -00036950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036960: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036970: 2020 7b34 7d20 7c20 3020 2061 2063 2020    {4} | 0  a c  
│ │ │ │ -00036980: 3020 2030 2020 3020 2030 2020 3020 202d  0  0  0  0  0  -
│ │ │ │ -00036990: 6232 2030 2020 2030 2020 2030 2020 207c  b2 0   0   0   |
│ │ │ │ -000369a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000369c0: 2020 7b33 7d20 7c20 3020 2030 2061 3220    {3} | 0  0 a2 
│ │ │ │ -000369d0: 3020 2030 2020 6232 2030 2020 3020 2030  0  0  b2 0  0  0
│ │ │ │ -000369e0: 2020 2030 2020 2030 2020 2030 2020 207c     0   0   0   |
│ │ │ │ -000369f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036a10: 2020 7b34 7d20 7c20 3020 2030 2030 2020    {4} | 0  0 0  
│ │ │ │ -00036a20: 3020 2062 2020 2d61 2030 2020 3020 2030  0  b  -a 0  0  0
│ │ │ │ -00036a30: 2020 202d 6133 2030 2020 2030 2020 207c     -a3 0   0   |
│ │ │ │ -00036a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036a60: 2020 7b34 7d20 7c20 3020 2030 2030 2020    {4} | 0  0 0  
│ │ │ │ -00036a70: 3020 202d 6320 3020 2061 2020 6232 2030  0  -c 0  a  b2 0
│ │ │ │ -00036a80: 2020 2030 2020 202d 6133 2030 2020 207c     0   -a3 0   |
│ │ │ │ -00036a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036aa0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036ab0: 2020 7b34 7d20 7c20 3020 2030 2030 2020    {4} | 0  0 0  
│ │ │ │ -00036ac0: 3020 2030 2020 6320 202d 6220 3020 2061  0  0  c  -b 0  a
│ │ │ │ -00036ad0: 3220 2030 2020 2030 2020 202d 6133 207c  2  0   0   -a3 |
│ │ │ │ -00036ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036af0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000368b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000368c0: 0a7c 6f39 203d 207b 347d 207c 202d 6320  .|o9 = {4} | -c 
│ │ │ │ +000368d0: 6220 3020 2061 3220 3020 2030 2020 3020  b 0  a2 0  0  0 
│ │ │ │ +000368e0: 2030 2020 3020 2020 3020 2020 3020 2020   0  0   0   0   
│ │ │ │ +000368f0: 3020 2020 7c20 2020 2020 2020 2020 2020  0   |           
│ │ │ │ +00036900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036910: 0a7c 2020 2020 207b 347d 207c 2061 2020  .|     {4} | a  
│ │ │ │ +00036920: 3020 6220 2030 2020 3020 2030 2020 3020  0 b  0  0  0  0 
│ │ │ │ +00036930: 2030 2020 3020 2020 3020 2020 3020 2020   0  0   0   0   
│ │ │ │ +00036940: 3020 2020 7c20 2020 2020 2020 2020 2020  0   |           
│ │ │ │ +00036950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036960: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00036970: 6120 6320 2030 2020 3020 2030 2020 3020  a c  0  0  0  0 
│ │ │ │ +00036980: 2030 2020 2d62 3220 3020 2020 3020 2020   0  -b2 0   0   
│ │ │ │ +00036990: 3020 2020 7c20 2020 2020 2020 2020 2020  0   |           
│ │ │ │ +000369a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000369b0: 0a7c 2020 2020 207b 337d 207c 2030 2020  .|     {3} | 0  
│ │ │ │ +000369c0: 3020 6132 2030 2020 3020 2062 3220 3020  0 a2 0  0  b2 0 
│ │ │ │ +000369d0: 2030 2020 3020 2020 3020 2020 3020 2020   0  0   0   0   
│ │ │ │ +000369e0: 3020 2020 7c20 2020 2020 2020 2020 2020  0   |           
│ │ │ │ +000369f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036a00: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00036a10: 3020 3020 2030 2020 6220 202d 6120 3020  0 0  0  b  -a 0 
│ │ │ │ +00036a20: 2030 2020 3020 2020 2d61 3320 3020 2020   0  0   -a3 0   
│ │ │ │ +00036a30: 3020 2020 7c20 2020 2020 2020 2020 2020  0   |           
│ │ │ │ +00036a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036a50: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00036a60: 3020 3020 2030 2020 2d63 2030 2020 6120  0 0  0  -c 0  a 
│ │ │ │ +00036a70: 2062 3220 3020 2020 3020 2020 2d61 3320   b2 0   0   -a3 
│ │ │ │ +00036a80: 3020 2020 7c20 2020 2020 2020 2020 2020  0   |           
│ │ │ │ +00036a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036aa0: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00036ab0: 3020 3020 2030 2020 3020 2063 2020 2d62  0 0  0  0  c  -b
│ │ │ │ +00036ac0: 2030 2020 6132 2020 3020 2020 3020 2020   0  a2  0   0   
│ │ │ │ +00036ad0: 2d61 3320 7c20 2020 2020 2020 2020 2020  -a3 |           
│ │ │ │ +00036ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00036b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036b50: 2020 2020 2020 2020 2020 3720 2020 2020            7     
│ │ │ │ -00036b60: 2031 3220 2020 2020 2020 2020 2020 2020   12             
│ │ │ │ +00036b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036b40: 0a7c 2020 2020 2020 2020 2020 2020 2037  .|             7
│ │ │ │ +00036b50: 2020 2020 2020 3132 2020 2020 2020 2020        12        
│ │ │ │ +00036b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b90: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -00036ba0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -00036bb0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00036b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036b90: 0a7c 6f39 203a 204d 6174 7269 7820 5320  .|o9 : Matrix S 
│ │ │ │ +00036ba0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00036bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036be0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00036bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036be0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00036bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ -00036c40: 203a 2046 2e64 645f 3120 2020 2020 2020   : F.dd_1       
│ │ │ │ +00036c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00036c30: 0a7c 6931 3020 3a20 462e 6464 5f31 2020  .|i10 : F.dd_1  
│ │ │ │ +00036c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036c80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00036c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036cd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -00036ce0: 203d 207b 337d 207c 2061 3220 6232 2030   = {3} | a2 b2 0
│ │ │ │ -00036cf0: 2020 3020 2030 2020 3020 2030 2020 3020    0  0  0  0  0 
│ │ │ │ -00036d00: 2030 2020 3020 2030 2020 3020 207c 2020   0  0  0  0  |  
│ │ │ │ -00036d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036d30: 2020 207b 347d 207c 2030 2020 2d61 2030     {4} | 0  -a 0
│ │ │ │ -00036d40: 2020 3020 2030 2020 6220 2030 2020 3020    0  0  b  0  0 
│ │ │ │ -00036d50: 2030 2020 3020 2030 2020 3020 207c 2020   0  0  0  0  |  
│ │ │ │ -00036d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036d80: 2020 207b 347d 207c 2030 2020 3020 2061     {4} | 0  0  a
│ │ │ │ -00036d90: 2020 3020 2030 2020 2d63 2030 2020 3020    0  0  -c 0  0 
│ │ │ │ -00036da0: 2062 3220 3020 2030 2020 3020 207c 2020   b2 0  0  0  |  
│ │ │ │ -00036db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036dd0: 2020 207b 347d 207c 2030 2020 6320 202d     {4} | 0  c  -
│ │ │ │ -00036de0: 6220 3020 2030 2020 3020 2061 3220 3020  b 0  0  0  a2 0 
│ │ │ │ -00036df0: 2030 2020 3020 2030 2020 3020 207c 2020   0  0  0  0  |  
│ │ │ │ -00036e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036e20: 2020 207b 347d 207c 2062 2020 3020 2030     {4} | b  0  0
│ │ │ │ -00036e30: 2020 6120 2030 2020 3020 2030 2020 3020    a  0  0  0  0 
│ │ │ │ -00036e40: 2030 2020 6233 2030 2020 3020 207c 2020   0  b3 0  0  |  
│ │ │ │ -00036e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036e70: 2020 207b 347d 207c 202d 6320 3020 2030     {4} | -c 0  0
│ │ │ │ -00036e80: 2020 3020 2061 2020 3020 2062 3220 3020    0  a  0  b2 0 
│ │ │ │ -00036e90: 2030 2020 3020 2062 3320 3020 207c 2020   0  0  b3 0  |  
│ │ │ │ -00036ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036ec0: 2020 207b 347d 207c 2030 2020 3020 2030     {4} | 0  0  0
│ │ │ │ -00036ed0: 2020 2d63 202d 6220 3020 2030 2020 6132    -c -b 0  0  a2
│ │ │ │ -00036ee0: 2030 2020 3020 2030 2020 6233 207c 2020   0  0  0  b3 |  
│ │ │ │ -00036ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036cd0: 0a7c 6f31 3020 3d20 7b33 7d20 7c20 6132  .|o10 = {3} | a2
│ │ │ │ +00036ce0: 2062 3220 3020 2030 2020 3020 2030 2020   b2 0  0  0  0  
│ │ │ │ +00036cf0: 3020 2030 2020 3020 2030 2020 3020 2030  0  0  0  0  0  0
│ │ │ │ +00036d00: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00036d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036d20: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00036d30: 202d 6120 3020 2030 2020 3020 2062 2020   -a 0  0  0  b  
│ │ │ │ +00036d40: 3020 2030 2020 3020 2030 2020 3020 2030  0  0  0  0  0  0
│ │ │ │ +00036d50: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00036d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036d70: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00036d80: 2030 2020 6120 2030 2020 3020 202d 6320   0  a  0  0  -c 
│ │ │ │ +00036d90: 3020 2030 2020 6232 2030 2020 3020 2030  0  0  b2 0  0  0
│ │ │ │ +00036da0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00036db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036dc0: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00036dd0: 2063 2020 2d62 2030 2020 3020 2030 2020   c  -b 0  0  0  
│ │ │ │ +00036de0: 6132 2030 2020 3020 2030 2020 3020 2030  a2 0  0  0  0  0
│ │ │ │ +00036df0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00036e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036e10: 0a7c 2020 2020 2020 7b34 7d20 7c20 6220  .|      {4} | b 
│ │ │ │ +00036e20: 2030 2020 3020 2061 2020 3020 2030 2020   0  0  a  0  0  
│ │ │ │ +00036e30: 3020 2030 2020 3020 2062 3320 3020 2030  0  0  0  b3 0  0
│ │ │ │ +00036e40: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00036e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036e60: 0a7c 2020 2020 2020 7b34 7d20 7c20 2d63  .|      {4} | -c
│ │ │ │ +00036e70: 2030 2020 3020 2030 2020 6120 2030 2020   0  0  0  a  0  
│ │ │ │ +00036e80: 6232 2030 2020 3020 2030 2020 6233 2030  b2 0  0  0  b3 0
│ │ │ │ +00036e90: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00036ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036eb0: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00036ec0: 2030 2020 3020 202d 6320 2d62 2030 2020   0  0  -c -b 0  
│ │ │ │ +00036ed0: 3020 2061 3220 3020 2030 2020 3020 2062  0  a2 0  0  0  b
│ │ │ │ +00036ee0: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │ +00036ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036f00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00036f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00036f60: 2020 2020 2020 2020 2020 2037 2020 2020             7    
│ │ │ │ -00036f70: 2020 3132 2020 2020 2020 2020 2020 2020    12            
│ │ │ │ +00036f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036f50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00036f60: 3720 2020 2020 2031 3220 2020 2020 2020  7      12       
│ │ │ │ +00036f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -00036fb0: 203a 204d 6174 7269 7820 5320 203c 2d2d   : Matrix S  <--
│ │ │ │ -00036fc0: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ +00036f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036fa0: 0a7c 6f31 3020 3a20 4d61 7472 6978 2053  .|o10 : Matrix S
│ │ │ │ +00036fb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +00036fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ff0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00036fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00036ff0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00037000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037040: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -00037050: 203a 2047 2e64 645f 3220 2020 2020 2020   : G.dd_2       
│ │ │ │ +00037030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00037040: 0a7c 6931 3120 3a20 472e 6464 5f32 2020  .|i11 : G.dd_2  
│ │ │ │ +00037050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00037080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00037090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000370a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -000370f0: 203d 207b 357d 207c 2030 2020 202d 6233   = {5} | 0   -b3
│ │ │ │ -00037100: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ -00037110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00037120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00037140: 2020 207b 357d 207c 2030 2020 202d 6232     {5} | 0   -b2
│ │ │ │ -00037150: 6320 3020 2020 3020 2020 2d61 3262 3220  c 0   0   -a2b2 
│ │ │ │ -00037160: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00037170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00037190: 2020 207b 357d 207c 2030 2020 2061 6232     {5} | 0   ab2
│ │ │ │ -000371a0: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ -000371b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000371c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000371e0: 2020 207b 367d 207c 2030 2020 2030 2020     {6} | 0   0  
│ │ │ │ -000371f0: 2020 3020 2020 3020 2020 6233 2020 2020    0   0   b3    
│ │ │ │ -00037200: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037220: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00037230: 2020 207b 357d 207c 202d 6133 2030 2020     {5} | -a3 0  
│ │ │ │ -00037240: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ -00037250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00037260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00037280: 2020 207b 357d 207c 2030 2020 202d 6133     {5} | 0   -a3
│ │ │ │ -00037290: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ -000372a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000372b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00037820: 2020 207b 357d 207c 202d 6133 2020 3020     {5} | -a3  0 
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│ │ │ │ -00037870: 2020 207b 367d 207c 2030 2020 2020 3020     {6} | 0    0 
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│ │ │ │ -00037a00: 2020 207b 377d 207c 2030 2020 2020 2d63     {7} | 0    -c
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│ │ │ │ +00037720: 0a7c 2020 2020 2020 7b35 7d20 7c20 2d61  .|      {5} | -a
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│ │ │ │ +00037770: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
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│ │ │ │ +000377b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000377c0: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +000377d0: 2020 2030 2020 202d 6233 2030 2020 2020     0   -b3 0    
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│ │ │ │ +00037810: 0a7c 2020 2020 2020 7b35 7d20 7c20 2d61  .|      {5} | -a
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│ │ │ │ +00037840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00037860: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
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│ │ │ │ +00037890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000378a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000378b0: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
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│ │ │ │ +000378d0: 202d 6233 207c 2020 2020 2020 2020 2020   -b3 |          
│ │ │ │ +000378e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000378f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00037900: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
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│ │ │ │ +00037950: 0a7c 2020 2020 2020 7b37 7d20 7c20 2d62  .|      {7} | -b
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│ │ │ │ +00037980: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000379a0: 0a7c 2020 2020 2020 7b37 7d20 7c20 6320  .|      {7} | c 
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│ │ │ │ +000379e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000379f0: 0a7c 2020 2020 2020 7b37 7d20 7c20 3020  .|      {7} | 0 
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│ │ │ │ +00037a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00037aa0: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
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│ │ │ │ -00037ae0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
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│ │ │ │ -00037b30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00037b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00037b30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00037b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6620  ----------+..If 
│ │ │ │ -00037b90: 7468 6520 636f 6d70 6c65 7869 7479 206f  the complexity o
│ │ │ │ -00037ba0: 6620 4d20 6973 206e 6f74 206d 6178 696d  f M is not maxim
│ │ │ │ -00037bb0: 616c 2c20 7468 656e 2074 6865 2066 696e  al, then the fin
│ │ │ │ -00037bc0: 6974 6520 7265 736f 6c75 7469 6f6e 2074  ite resolution t
│ │ │ │ -00037bd0: 616b 6573 2070 6c61 6365 0a6f 7665 7220  akes place.over 
│ │ │ │ -00037be0: 616e 2069 6e74 6572 6d65 6469 6174 6520  an intermediate 
│ │ │ │ -00037bf0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ -00037c00: 6374 696f 6e3a 0a0a 2b2d 2d2d 2d2d 2d2d  ction:..+-------
│ │ │ │ +00037b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00037b80: 0a0a 4966 2074 6865 2063 6f6d 706c 6578  ..If the complex
│ │ │ │ +00037b90: 6974 7920 6f66 204d 2069 7320 6e6f 7420  ity of M is not 
│ │ │ │ +00037ba0: 6d61 7869 6d61 6c2c 2074 6865 6e20 7468  maximal, then th
│ │ │ │ +00037bb0: 6520 6669 6e69 7465 2072 6573 6f6c 7574  e finite resolut
│ │ │ │ +00037bc0: 696f 6e20 7461 6b65 7320 706c 6163 650a  ion takes place.
│ │ │ │ +00037bd0: 6f76 6572 2061 6e20 696e 7465 726d 6564  over an intermed
│ │ │ │ +00037be0: 6961 7465 2063 6f6d 706c 6574 6520 696e  iate complete in
│ │ │ │ +00037bf0: 7465 7273 6563 7469 6f6e 3a0a 0a2b 2d2d  tersection:..+--
│ │ │ │ +00037c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037c50: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2053  ------+.|i13 : S
│ │ │ │ -00037c60: 203d 205a 5a2f 3130 315b 612c 622c 632c   = ZZ/101[a,b,c,
│ │ │ │ -00037c70: 645d 2020 2020 2020 2020 2020 2020 2020  d]              
│ │ │ │ +00037c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00037c50: 3320 3a20 5320 3d20 5a5a 2f31 3031 5b61  3 : S = ZZ/101[a
│ │ │ │ +00037c60: 2c62 2c63 2c64 5d20 2020 2020 2020 2020  ,b,c,d]         
│ │ │ │ +00037c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ca0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037c90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037cf0: 2020 2020 2020 7c0a 7c6f 3133 203d 2053        |.|o13 = S
│ │ │ │ +00037ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00037cf0: 3320 3d20 5320 2020 2020 2020 2020 2020  3 = S           
│ │ │ │  00037d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037d30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d90: 2020 2020 2020 7c0a 7c6f 3133 203a 2050        |.|o13 : P
│ │ │ │ -00037da0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00037d80: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00037d90: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +00037da0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00037db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037de0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00037dd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00037de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037e30: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2066  ------+.|i14 : f
│ │ │ │ -00037e40: 6631 203d 206d 6174 7269 7822 6133 2c62  f1 = matrix"a3,b
│ │ │ │ -00037e50: 332c 6333 2c64 3322 2020 2020 2020 2020  3,c3,d3"        
│ │ │ │ +00037e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00037e30: 3420 3a20 6666 3120 3d20 6d61 7472 6978  4 : ff1 = matrix
│ │ │ │ +00037e40: 2261 332c 6233 2c63 332c 6433 2220 2020  "a3,b3,c3,d3"   
│ │ │ │ +00037e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037e70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ed0: 2020 2020 2020 7c0a 7c6f 3134 203d 207c        |.|o14 = |
│ │ │ │ -00037ee0: 2061 3320 6233 2063 3320 6433 207c 2020   a3 b3 c3 d3 |  
│ │ │ │ +00037ec0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00037ed0: 3420 3d20 7c20 6133 2062 3320 6333 2064  4 = | a3 b3 c3 d
│ │ │ │ +00037ee0: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │  00037ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037f10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00037f80: 2020 2020 2020 2031 2020 2020 2020 3420         1      4 
│ │ │ │ +00037f60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037f70: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00037f80: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │  00037f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fc0: 2020 2020 2020 7c0a 7c6f 3134 203a 204d        |.|o14 : M
│ │ │ │ -00037fd0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +00037fb0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00037fc0: 3420 3a20 4d61 7472 6978 2053 2020 3c2d  4 : Matrix S  <-
│ │ │ │ +00037fd0: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │  00037fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038010: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00038000: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00038010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038060: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2066  ------+.|i15 : f
│ │ │ │ -00038070: 6620 3d66 6631 2a72 616e 646f 6d28 736f  f =ff1*random(so
│ │ │ │ -00038080: 7572 6365 2066 6631 2c20 736f 7572 6365  urce ff1, source
│ │ │ │ -00038090: 2066 6631 2920 2020 2020 2020 2020 2020   ff1)           
│ │ │ │ -000380a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00038060: 3520 3a20 6666 203d 6666 312a 7261 6e64  5 : ff =ff1*rand
│ │ │ │ +00038070: 6f6d 2873 6f75 7263 6520 6666 312c 2073  om(source ff1, s
│ │ │ │ +00038080: 6f75 7263 6520 6666 3129 2020 2020 2020  ource ff1)      
│ │ │ │ +00038090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000380a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000380b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038100: 2020 2020 2020 7c0a 7c6f 3135 203d 207c        |.|o15 = |
│ │ │ │ -00038110: 2032 3461 332d 3336 6233 2d33 3063 332d   24a3-36b3-30c3-
│ │ │ │ -00038120: 3239 6433 2031 3961 332b 3139 6233 2d31  29d3 19a3+19b3-1
│ │ │ │ -00038130: 3063 332d 3239 6433 202d 3861 332d 3232  0c3-29d3 -8a3-22
│ │ │ │ -00038140: 6233 2d32 3963 332d 3234 6433 2020 2020  b3-29c3-24d3    
│ │ │ │ -00038150: 2020 2020 2020 7c0a 7c20 2020 2020 202d        |.|      -
│ │ │ │ +000380f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00038100: 3520 3d20 7c20 3234 6133 2d33 3662 332d  5 = | 24a3-36b3-
│ │ │ │ +00038110: 3330 6333 2d32 3964 3320 3139 6133 2b31  30c3-29d3 19a3+1
│ │ │ │ +00038120: 3962 332d 3130 6333 2d32 3964 3320 2d38  9b3-10c3-29d3 -8
│ │ │ │ +00038130: 6133 2d32 3262 332d 3239 6333 2d32 3464  a3-22b3-29c3-24d
│ │ │ │ +00038140: 3320 2020 2020 2020 2020 207c 0a7c 2020  3          |.|  
│ │ │ │ +00038150: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  00038160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000381a0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 202d  ------|.|      -
│ │ │ │ -000381b0: 3338 6133 2d31 3662 332b 3339 6333 2b32  38a3-16b3+39c3+2
│ │ │ │ -000381c0: 3164 3320 7c20 2020 2020 2020 2020 2020  1d3 |           
│ │ │ │ +00038190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +000381a0: 2020 2020 2d33 3861 332d 3136 6233 2b33      -38a3-16b3+3
│ │ │ │ +000381b0: 3963 332b 3231 6433 207c 2020 2020 2020  9c3+21d3 |      
│ │ │ │ +000381c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000381e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000381f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038240: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00038250: 2020 2020 2020 2031 2020 2020 2020 3420         1      4 
│ │ │ │ +00038230: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038240: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00038250: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │  00038260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038290: 2020 2020 2020 7c0a 7c6f 3135 203a 204d        |.|o15 : M
│ │ │ │ -000382a0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +00038280: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00038290: 3520 3a20 4d61 7472 6978 2053 2020 3c2d  5 : Matrix S  <-
│ │ │ │ +000382a0: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │  000382b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000382c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000382d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000382e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000382d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000382e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000382f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038330: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2052  ------+.|i16 : R
│ │ │ │ -00038340: 203d 2053 2f69 6465 616c 2066 6620 2020   = S/ideal ff   
│ │ │ │ +00038320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00038330: 3620 3a20 5220 3d20 532f 6964 6561 6c20  6 : R = S/ideal 
│ │ │ │ +00038340: 6666 2020 2020 2020 2020 2020 2020 2020  ff              
│ │ │ │  00038350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038370: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000383a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000383b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000383c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000383d0: 2020 2020 2020 7c0a 7c6f 3136 203d 2052        |.|o16 = R
│ │ │ │ +000383c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000383d0: 3620 3d20 5220 2020 2020 2020 2020 2020  6 = R           
│ │ │ │  000383e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000383f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038410: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038470: 2020 2020 2020 7c0a 7c6f 3136 203a 2051        |.|o16 : Q
│ │ │ │ -00038480: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00038460: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00038470: 3620 3a20 5175 6f74 6965 6e74 5269 6e67  6 : QuotientRing
│ │ │ │ +00038480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000384a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000384b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000384c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000384b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000384c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000384d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000384e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000384f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038510: 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a 204d  ------+.|i17 : M
│ │ │ │ -00038520: 203d 2068 6967 6853 797a 7967 7920 2852   = highSyzygy (R
│ │ │ │ -00038530: 5e31 2f69 6465 616c 2261 3262 3222 2920  ^1/ideal"a2b2") 
│ │ │ │ +00038500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00038510: 3720 3a20 4d20 3d20 6869 6768 5379 7a79  7 : M = highSyzy
│ │ │ │ +00038520: 6779 2028 525e 312f 6964 6561 6c22 6132  gy (R^1/ideal"a2
│ │ │ │ +00038530: 6232 2229 2020 2020 2020 2020 2020 2020  b2")            
│ │ │ │  00038540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038550: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000385a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000385b0: 2020 2020 2020 7c0a 7c6f 3137 203d 2063        |.|o17 = c
│ │ │ │ -000385c0: 6f6b 6572 6e65 6c20 7b36 7d20 7c20 6232  okernel {6} | b2
│ │ │ │ -000385d0: 2030 202d 6132 2030 207c 2020 2020 2020   0 -a2 0 |      
│ │ │ │ +000385a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000385b0: 3720 3d20 636f 6b65 726e 656c 207b 367d  7 = cokernel {6}
│ │ │ │ +000385c0: 207c 2062 3220 3020 2d61 3220 3020 7c20   | b2 0 -a2 0 | 
│ │ │ │ +000385d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000385e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000385f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038600: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00038610: 2020 2020 2020 2020 7b37 7d20 7c20 6120          {7} | a 
│ │ │ │ -00038620: 2062 2030 2020 2030 207c 2020 2020 2020   b 0   0 |      
│ │ │ │ +000385f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038600: 2020 2020 2020 2020 2020 2020 207b 377d               {7}
│ │ │ │ +00038610: 207c 2061 2020 6220 3020 2020 3020 7c20   | a  b 0   0 | 
│ │ │ │ +00038620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00038660: 2020 2020 2020 2020 7b37 7d20 7c20 3020          {7} | 0 
│ │ │ │ -00038670: 2030 2062 2020 2061 207c 2020 2020 2020   0 b   a |      
│ │ │ │ +00038640: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038650: 2020 2020 2020 2020 2020 2020 207b 377d               {7}
│ │ │ │ +00038660: 207c 2030 2020 3020 6220 2020 6120 7c20   | 0  0 b   a | 
│ │ │ │ +00038670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038690: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000386a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000386b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000386c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000386d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00038700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038710: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +000386e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000386f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038700: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00038710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038740: 2020 2020 2020 7c0a 7c6f 3137 203a 2052        |.|o17 : R
│ │ │ │ -00038750: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00038760: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ +00038730: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00038740: 3720 3a20 522d 6d6f 6475 6c65 2c20 7175  7 : R-module, qu
│ │ │ │ +00038750: 6f74 6965 6e74 206f 6620 5220 2020 2020  otient of R     
│ │ │ │ +00038760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038790: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00038780: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00038790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387e0: 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a 2063  ------+.|i18 : c
│ │ │ │ -000387f0: 6f6d 706c 6578 6974 7920 4d20 2020 2020  omplexity M     
│ │ │ │ +000387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000387e0: 3820 3a20 636f 6d70 6c65 7869 7479 204d  8 : complexity M
│ │ │ │ +000387f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038830: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038880: 2020 2020 2020 7c0a 7c6f 3138 203d 2032        |.|o18 = 2
│ │ │ │ +00038870: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00038880: 3820 3d20 3220 2020 2020 2020 2020 2020  8 = 2           
│ │ │ │  00038890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000388c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000388d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000388e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000388f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038920: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 206d  ------+.|i19 : m
│ │ │ │ -00038930: 6620 3d20 6d61 7472 6978 4661 6374 6f72  f = matrixFactor
│ │ │ │ -00038940: 697a 6174 696f 6e20 2866 662c 204d 2920  ization (ff, M) 
│ │ │ │ +00038910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00038920: 3920 3a20 6d66 203d 206d 6174 7269 7846  9 : mf = matrixF
│ │ │ │ +00038930: 6163 746f 7269 7a61 7469 6f6e 2028 6666  actorization (ff
│ │ │ │ +00038940: 2c20 4d29 2020 2020 2020 2020 2020 2020  , M)            
│ │ │ │  00038950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389c0: 2020 2020 2020 7c0a 7c6f 3139 203d 207b        |.|o19 = {
│ │ │ │ -000389d0: 7b37 7d20 7c20 2d61 202d 3336 6220 3020  {7} | -a -36b 0 
│ │ │ │ -000389e0: 6120 7c2c 207b 387d 207c 2033 3561 3220  a |, {8} | 35a2 
│ │ │ │ -000389f0: 2034 3862 2020 3020 2020 2020 2d33 3362   48b  0     -33b
│ │ │ │ -00038a00: 2030 2020 2020 207c 2c20 7b36 7d20 7c20   0     |, {6} | 
│ │ │ │ -00038a10: 3020 2020 3336 7c0a 7c20 2020 2020 2020  0   36|.|       
│ │ │ │ -00038a20: 7b36 7d20 7c20 6232 2061 3220 2020 3020  {6} | b2 a2   0 
│ │ │ │ -00038a30: 3020 7c20 207b 387d 207c 202d 3335 6232  0 |  {8} | -35b2
│ │ │ │ -00038a40: 202d 3335 6120 3020 2020 2020 3020 2020   -35a 0     0   
│ │ │ │ -00038a50: 2030 2020 2020 207c 2020 7b37 7d20 7c20   0     |  {7} | 
│ │ │ │ -00038a60: 2d33 3620 3020 7c0a 7c20 2020 2020 2020  -36 0 |.|       
│ │ │ │ -00038a70: 7b37 7d20 7c20 3020 2030 2020 2020 6220  {7} | 0  0    b 
│ │ │ │ -00038a80: 6120 7c20 207b 387d 207c 2030 2020 2020  a |  {8} | 0    
│ │ │ │ -00038a90: 2030 2020 2020 3333 6232 2020 3333 6120   0    33b2  33a 
│ │ │ │ -00038aa0: 202d 3333 6232 207c 2020 7b37 7d20 7c20   -33b2 |  {7} | 
│ │ │ │ -00038ab0: 3120 2020 3020 7c0a 7c20 2020 2020 2020  1   0 |.|       
│ │ │ │ -00038ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ad0: 2020 2020 207b 387d 207c 2030 2020 2020       {8} | 0    
│ │ │ │ -00038ae0: 2030 2020 2020 2d34 3361 3220 2d33 3362   0    -43a2 -33b
│ │ │ │ -00038af0: 2030 2020 2020 207c 2020 2020 2020 2020   0     |        
│ │ │ │ -00038b00: 2020 2020 2020 7c0a 7c20 2020 2020 202d        |.|      -
│ │ │ │ +000389b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000389c0: 3920 3d20 7b7b 377d 207c 202d 6120 2d33  9 = {{7} | -a -3
│ │ │ │ +000389d0: 3662 2030 2061 207c 2c20 7b38 7d20 7c20  6b 0 a |, {8} | 
│ │ │ │ +000389e0: 3335 6132 2020 3438 6220 2030 2020 2020  35a2  48b  0    
│ │ │ │ +000389f0: 202d 3333 6220 3020 2020 2020 7c2c 207b   -33b 0     |, {
│ │ │ │ +00038a00: 367d 207c 2030 2020 2033 367c 0a7c 2020  6} | 0   36|.|  
│ │ │ │ +00038a10: 2020 2020 207b 367d 207c 2062 3220 6132       {6} | b2 a2
│ │ │ │ +00038a20: 2020 2030 2030 207c 2020 7b38 7d20 7c20     0 0 |  {8} | 
│ │ │ │ +00038a30: 2d33 3562 3220 2d33 3561 2030 2020 2020  -35b2 -35a 0    
│ │ │ │ +00038a40: 2030 2020 2020 3020 2020 2020 7c20 207b   0    0     |  {
│ │ │ │ +00038a50: 377d 207c 202d 3336 2030 207c 0a7c 2020  7} | -36 0 |.|  
│ │ │ │ +00038a60: 2020 2020 207b 377d 207c 2030 2020 3020       {7} | 0  0 
│ │ │ │ +00038a70: 2020 2062 2061 207c 2020 7b38 7d20 7c20     b a |  {8} | 
│ │ │ │ +00038a80: 3020 2020 2020 3020 2020 2033 3362 3220  0     0    33b2 
│ │ │ │ +00038a90: 2033 3361 2020 2d33 3362 3220 7c20 207b   33a  -33b2 |  {
│ │ │ │ +00038aa0: 377d 207c 2031 2020 2030 207c 0a7c 2020  7} | 1   0 |.|  
│ │ │ │ +00038ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ac0: 2020 2020 2020 2020 2020 7b38 7d20 7c20            {8} | 
│ │ │ │ +00038ad0: 3020 2020 2020 3020 2020 202d 3433 6132  0     0    -43a2
│ │ │ │ +00038ae0: 202d 3333 6220 3020 2020 2020 7c20 2020   -33b 0     |   
│ │ │ │ +00038af0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038b00: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  00038b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b50: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2030  ------|.|      0
│ │ │ │ -00038b60: 2020 7c7d 2020 2020 2020 2020 2020 2020    |}            
│ │ │ │ +00038b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00038b50: 2020 2020 3020 207c 7d20 2020 2020 2020      0  |}       
│ │ │ │ +00038b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ba0: 2020 2020 2020 7c0a 7c20 2020 2020 2033        |.|      3
│ │ │ │ -00038bb0: 3620 7c20 2020 2020 2020 2020 2020 2020  6 |             
│ │ │ │ +00038b90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038ba0: 2020 2020 3336 207c 2020 2020 2020 2020      36 |        
│ │ │ │ +00038bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2030        |.|      0
│ │ │ │ -00038c00: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00038be0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038bf0: 2020 2020 3020 207c 2020 2020 2020 2020      0  |        
│ │ │ │ +00038c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038c30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c90: 2020 2020 2020 7c0a 7c6f 3139 203a 204c        |.|o19 : L
│ │ │ │ -00038ca0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00038c80: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00038c90: 3920 3a20 4c69 7374 2020 2020 2020 2020  9 : List        
│ │ │ │ +00038ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ce0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00038cd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00038ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038d30: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 2063  ------+.|i20 : c
│ │ │ │ -00038d40: 6f6d 706c 6578 6974 7920 6d66 2020 2020  omplexity mf    
│ │ │ │ +00038d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00038d30: 3020 3a20 636f 6d70 6c65 7869 7479 206d  0 : complexity m
│ │ │ │ +00038d40: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │  00038d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038d80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038d70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038dd0: 2020 2020 2020 7c0a 7c6f 3230 203d 2032        |.|o20 = 2
│ │ │ │ +00038dc0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00038dd0: 3020 3d20 3220 2020 2020 2020 2020 2020  0 = 2           
│ │ │ │  00038de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038e20: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00038e10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00038e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038e70: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2042  ------+.|i21 : B
│ │ │ │ -00038e80: 5261 6e6b 7320 6d66 2020 2020 2020 2020  Ranks mf        
│ │ │ │ +00038e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00038e70: 3120 3a20 4252 616e 6b73 206d 6620 2020  1 : BRanks mf   
│ │ │ │ +00038e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038eb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f10: 2020 2020 2020 7c0a 7c6f 3231 203d 207b        |.|o21 = {
│ │ │ │ -00038f20: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ +00038f00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00038f10: 3120 3d20 7b7b 322c 2032 7d2c 207b 312c  1 = {{2, 2}, {1,
│ │ │ │ +00038f20: 2032 7d7d 2020 2020 2020 2020 2020 2020   2}}            
│ │ │ │  00038f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00038f50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038fb0: 2020 2020 2020 7c0a 7c6f 3231 203a 204c        |.|o21 : L
│ │ │ │ -00038fc0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00038fa0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00038fb0: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │ +00038fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039000: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00038ff0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00039000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039050: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2047  ------+.|i22 : G
│ │ │ │ -00039060: 203d 206d 616b 6546 696e 6974 6552 6573   = makeFiniteRes
│ │ │ │ -00039070: 6f6c 7574 696f 6e28 6666 2c6d 6629 3b20  olution(ff,mf); 
│ │ │ │ +00039040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00039050: 3220 3a20 4720 3d20 6d61 6b65 4669 6e69  2 : G = makeFini
│ │ │ │ +00039060: 7465 5265 736f 6c75 7469 6f6e 2866 662c  teResolution(ff,
│ │ │ │ +00039070: 6d66 293b 2020 2020 2020 2020 2020 2020  mf);            
│ │ │ │  00039080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000390a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00039090: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000390a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000390b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000390c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000390d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000390e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000390f0: 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a 2063  ------+.|i23 : c
│ │ │ │ -00039100: 6f64 696d 2072 696e 6720 4720 2020 2020  odim ring G     
│ │ │ │ +000390e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +000390f0: 3320 3a20 636f 6469 6d20 7269 6e67 2047  3 : codim ring G
│ │ │ │ +00039100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039190: 2020 2020 2020 7c0a 7c6f 3233 203d 2032        |.|o23 = 2
│ │ │ │ +00039180: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00039190: 3320 3d20 3220 2020 2020 2020 2020 2020  3 = 2           
│ │ │ │  000391a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000391b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000391c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000391d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000391e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000391f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039230: 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a 2052  ------+.|i24 : R
│ │ │ │ -00039240: 3120 3d20 7269 6e67 2047 2020 2020 2020  1 = ring G      
│ │ │ │ +00039220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00039230: 3420 3a20 5231 203d 2072 696e 6720 4720  4 : R1 = ring G 
│ │ │ │ +00039240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039280: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039270: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000392a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000392b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000392c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000392d0: 2020 2020 2020 7c0a 7c6f 3234 203d 2052        |.|o24 = R
│ │ │ │ -000392e0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000392c0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000392d0: 3420 3d20 5231 2020 2020 2020 2020 2020  4 = R1          
│ │ │ │ +000392e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000392f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039310: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039370: 2020 2020 2020 7c0a 7c6f 3234 203a 2051        |.|o24 : Q
│ │ │ │ -00039380: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00039360: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00039370: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +00039380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000393a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000393b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000393c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000393d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000393e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000393f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039410: 2d2d 2d2d 2d2d 2b0a 7c69 3235 203a 2046  ------+.|i25 : F
│ │ │ │ -00039420: 203d 2072 6573 2070 7275 6e65 2070 7573   = res prune pus
│ │ │ │ -00039430: 6846 6f72 7761 7264 286d 6170 2852 2c52  hForward(map(R,R
│ │ │ │ -00039440: 3129 2c4d 2920 2020 2020 2020 2020 2020  1),M)           
│ │ │ │ -00039450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039460: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00039410: 3520 3a20 4620 3d20 7265 7320 7072 756e  5 : F = res prun
│ │ │ │ +00039420: 6520 7075 7368 466f 7277 6172 6428 6d61  e pushForward(ma
│ │ │ │ +00039430: 7028 522c 5231 292c 4d29 2020 2020 2020  p(R,R1),M)      
│ │ │ │ +00039440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039450: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000394c0: 2033 2020 2020 2020 2035 2020 2020 2020   3       5      
│ │ │ │ -000394d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000394a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000394b0: 2020 2020 2020 3320 2020 2020 2020 3520        3       5 
│ │ │ │ +000394c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000394d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000394e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039500: 2020 2020 2020 7c0a 7c6f 3235 203d 2052        |.|o25 = R
│ │ │ │ -00039510: 3120 203c 2d2d 2052 3120 203c 2d2d 2052  1  <-- R1  <-- R
│ │ │ │ -00039520: 3120 203c 2d2d 2030 2020 2020 2020 2020  1  <-- 0        
│ │ │ │ +000394f0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00039500: 3520 3d20 5231 2020 3c2d 2d20 5231 2020  5 = R1  <-- R1  
│ │ │ │ +00039510: 3c2d 2d20 5231 2020 3c2d 2d20 3020 2020  <-- R1  <-- 0   
│ │ │ │ +00039520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000395a0: 2020 2020 2020 7c0a 7c20 2020 2020 2030        |.|      0
│ │ │ │ -000395b0: 2020 2020 2020 2031 2020 2020 2020 2032         1       2
│ │ │ │ -000395c0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00039590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000395a0: 2020 2020 3020 2020 2020 2020 3120 2020      0       1   
│ │ │ │ +000395b0: 2020 2020 3220 2020 2020 2020 3320 2020      2       3   
│ │ │ │ +000395c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000395d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000395e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000395f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000395e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000395f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039640: 2020 2020 2020 7c0a 7c6f 3235 203a 2043        |.|o25 : C
│ │ │ │ -00039650: 6861 696e 436f 6d70 6c65 7820 2020 2020  hainComplex     
│ │ │ │ +00039630: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00039640: 3520 3a20 4368 6169 6e43 6f6d 706c 6578  5 : ChainComplex
│ │ │ │ +00039650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039690: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00039680: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00039690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000396a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000396b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000396c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000396d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000396e0: 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a 2062  ------+.|i26 : b
│ │ │ │ -000396f0: 6574 7469 2046 2020 2020 2020 2020 2020  etti F          
│ │ │ │ +000396d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +000396e0: 3620 3a20 6265 7474 6920 4620 2020 2020  6 : betti F     
│ │ │ │ +000396f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039730: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039720: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039780: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039790: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +00039770: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039780: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +00039790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397d0: 2020 2020 2020 7c0a 7c6f 3236 203d 2074        |.|o26 = t
│ │ │ │ -000397e0: 6f74 616c 3a20 3320 3520 3220 2020 2020  otal: 3 5 2     
│ │ │ │ +000397c0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000397d0: 3620 3d20 746f 7461 6c3a 2033 2035 2032  6 = total: 3 5 2
│ │ │ │ +000397e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039820: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039830: 2020 2036 3a20 3120 2e20 2e20 2020 2020     6: 1 . .     
│ │ │ │ +00039810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039820: 2020 2020 2020 2020 363a 2031 202e 202e          6: 1 . .
│ │ │ │ +00039830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039880: 2020 2037 3a20 3220 3420 2e20 2020 2020     7: 2 4 .     
│ │ │ │ +00039860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039870: 2020 2020 2020 2020 373a 2032 2034 202e          7: 2 4 .
│ │ │ │ +00039880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000398d0: 2020 2038 3a20 2e20 2e20 2e20 2020 2020     8: . . .     
│ │ │ │ +000398b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000398c0: 2020 2020 2020 2020 383a 202e 202e 202e          8: . . .
│ │ │ │ +000398d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039910: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039920: 2020 2039 3a20 2e20 3120 3220 2020 2020     9: . 1 2     
│ │ │ │ +00039900: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039910: 2020 2020 2020 2020 393a 202e 2031 2032          9: . 1 2
│ │ │ │ +00039920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039960: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00039950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000399a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000399b0: 2020 2020 2020 7c0a 7c6f 3236 203a 2042        |.|o26 : B
│ │ │ │ -000399c0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +000399a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000399b0: 3620 3a20 4265 7474 6954 616c 6c79 2020  6 : BettiTally  
│ │ │ │ +000399c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000399d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000399e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000399f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039a00: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000399f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00039a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039a50: 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a 2062  ------+.|i27 : b
│ │ │ │ -00039a60: 6574 7469 2047 2020 2020 2020 2020 2020  etti G          
│ │ │ │ +00039a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
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│ │ │ │ -00039b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │  00039bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039bf0: 2020 2037 3a20 3220 3420 2e20 2020 2020     7: 2 4 .     
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│ │ │ │ +00039be0: 2020 2020 2020 2020 373a 2032 2034 202e          7: 2 4 .
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│ │ │ │  00039c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039c40: 2020 2038 3a20 2e20 2e20 2e20 2020 2020     8: . . .     
│ │ │ │ +00039c20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039c30: 2020 2020 2020 2020 383a 202e 202e 202e          8: . . .
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│ │ │ │  00039c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00039c90: 2020 2039 3a20 2e20 3120 3220 2020 2020     9: . 1 2     
│ │ │ │ +00039c70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00039c80: 2020 2020 2020 2020 393a 202e 2031 2032          9: . 1 2
│ │ │ │ +00039c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │  00039ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00039d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00039e80: 425f 3028 7029 2069 6e20 610a 2020 2020  B_0(p) in a.    
│ │ │ │ -00039e90: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
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│ │ │ │ -00039ed0: 2020 7073 6928 7029 3a20 425f 3128 7029    psi(p): B_1(p)
│ │ │ │ -00039ee0: 202d 2d3e 2041 5f30 2870 2d31 2920 696e   --> A_0(p-1) in
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│ │ │ │ -00039fd0: 7365 206d 616b 6546 696e 6974 6552 6573  se makeFiniteRes
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│ │ │ │ -00039ff0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -0003a010: 226d 616b 6546 696e 6974 6552 6573 6f6c  "makeFiniteResol
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│ │ │ │ -0003a070: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
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│ │ │ │ +00039dd0: 0a20 202a 202a 6e6f 7465 206d 6174 7269  .  * *note matri
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│ │ │ │ +00039df0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +00039e00: 696f 6e2c 202d 2d20 4d61 7073 2069 6e20  ion, -- Maps in 
│ │ │ │ +00039e10: 6120 6869 6768 6572 0a20 2020 2063 6f64  a higher.    cod
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│ │ │ │ +00039e30: 6661 6374 6f72 697a 6174 696f 6e0a 2020  factorization.  
│ │ │ │ +00039e40: 2a20 2a6e 6f74 6520 624d 6170 733a 2062  * *note bMaps: b
│ │ │ │ +00039e50: 4d61 7073 2c20 2d2d 206c 6973 7420 7468  Maps, -- list th
│ │ │ │ +00039e60: 6520 6d61 7073 2020 645f 703a 425f 3128  e maps  d_p:B_1(
│ │ │ │ +00039e70: 7029 2d2d 3e42 5f30 2870 2920 696e 2061  p)-->B_0(p) in a
│ │ │ │ +00039e80: 0a20 2020 206d 6174 7269 7846 6163 746f  .    matrixFacto
│ │ │ │ +00039e90: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f  rization.  * *no
│ │ │ │ +00039ea0: 7465 2070 7369 4d61 7073 3a20 7073 694d  te psiMaps: psiM
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│ │ │ │ +00039ec0: 206d 6170 7320 2070 7369 2870 293a 2042   maps  psi(p): B
│ │ │ │ +00039ed0: 5f31 2870 2920 2d2d 3e20 415f 3028 702d  _1(p) --> A_0(p-
│ │ │ │ +00039ee0: 3129 2069 6e20 610a 2020 2020 6d61 7472  1) in a.    matr
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│ │ │ │ +00039f00: 2020 2a20 2a6e 6f74 6520 684d 6170 733a    * *note hMaps:
│ │ │ │ +00039f10: 2068 4d61 7073 2c20 2d2d 206c 6973 7420   hMaps, -- list 
│ │ │ │ +00039f20: 7468 6520 6d61 7073 2020 6828 7029 3a20  the maps  h(p): 
│ │ │ │ +00039f30: 415f 3028 7029 2d2d 3e20 415f 3128 7029  A_0(p)--> A_1(p)
│ │ │ │ +00039f40: 2069 6e20 610a 2020 2020 6d61 7472 6978   in a.    matrix
│ │ │ │ +00039f50: 4661 6374 6f72 697a 6174 696f 6e0a 2020  Factorization.  
│ │ │ │ +00039f60: 2a20 2a6e 6f74 6520 636f 6d70 6c65 7869  * *note complexi
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│ │ │ │ +00039f80: 2d2d 2063 6f6d 706c 6578 6974 7920 6f66  -- complexity of
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│ │ │ │ +00039fb0: 7465 7273 6563 7469 6f6e 0a0a 5761 7973  tersection..Ways
│ │ │ │ +00039fc0: 2074 6f20 7573 6520 6d61 6b65 4669 6e69   to use makeFini
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│ │ │ │ +00039fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00039ff0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0003a000: 0a20 202a 2022 6d61 6b65 4669 6e69 7465  .  * "makeFinite
│ │ │ │ +0003a010: 5265 736f 6c75 7469 6f6e 284d 6174 7269  Resolution(Matri
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│ │ │ │ +0003a040: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
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│ │ │ │ +0003a070: 6c75 7469 6f6e 3a20 6d61 6b65 4669 6e69  lution: makeFini
│ │ │ │ +0003a080: 7465 5265 736f 6c75 7469 6f6e 2c20 6973  teResolution, is
│ │ │ │ +0003a090: 2061 202a 6e6f 7465 206d 6574 686f 640a   a *note method.
│ │ │ │ +0003a0a0: 6675 6e63 7469 6f6e 3a20 284d 6163 6175  function: (Macau
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│ │ │ │ +0003a0d0: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
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│ │ │ │ +0003a110: 6e43 6f64 696d 322c 204e 6578 743a 206d  nCodim2, Next: m
│ │ │ │ +0003a120: 616b 6548 6f6d 6f74 6f70 6965 732c 2050  akeHomotopies, P
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│ │ │ │ +0003a1a0: 6869 6768 2073 797a 7967 7920 6d6f 6475  high syzygy modu
│ │ │ │ +0003a1b0: 6c65 2069 6e20 636f 6469 6d20 320a 2a2a  le in codim 2.**
│ │ │ │ +0003a1c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003a1d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003a1e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003a1f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003a200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003a210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003a220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003a230: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ -0003a240: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ -0003a250: 3a20 0a20 2020 2020 2020 206d 6170 7320  : .        maps 
│ │ │ │ -0003a260: 3d20 6d61 6b65 4669 6e69 7465 5265 736f  = makeFiniteReso
│ │ │ │ -0003a270: 6c75 7469 6f6e 436f 6469 6d32 2866 662c  lutionCodim2(ff,
│ │ │ │ -0003a280: 6d66 290a 2020 2a20 496e 7075 7473 3a0a  mf).  * Inputs:.
│ │ │ │ -0003a290: 2020 2020 2020 2a20 6d66 2c20 6120 2a6e        * mf, a *n
│ │ │ │ -0003a2a0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ -0003a2b0: 6c61 7932 446f 6329 4c69 7374 2c2c 206d  lay2Doc)List,, m
│ │ │ │ -0003a2c0: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ -0003a2d0: 696f 6e0a 2020 2020 2020 2a20 6666 2c20  ion.      * ff, 
│ │ │ │ -0003a2e0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -0003a2f0: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -0003a300: 7472 6978 2c2c 2072 6567 756c 6172 2073  trix,, regular s
│ │ │ │ -0003a310: 6571 7565 6e63 650a 2020 2a20 2a6e 6f74  equence.  * *not
│ │ │ │ -0003a320: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ -0003a330: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ -0003a340: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ -0003a350: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ -0003a360: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ -0003a370: 4368 6563 6b20 3d3e 202e 2e2e 2c20 6465  Check => ..., de
│ │ │ │ -0003a380: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -0003a390: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ -0003a3a0: 2020 2020 202a 206d 6170 732c 2061 202a       * maps, a *
│ │ │ │ -0003a3b0: 6e6f 7465 2068 6173 6820 7461 626c 653a  note hash table:
│ │ │ │ -0003a3c0: 2028 4d61 6361 756c 6179 3244 6f63 2948   (Macaulay2Doc)H
│ │ │ │ -0003a3d0: 6173 6854 6162 6c65 2c2c 206d 616e 7920  ashTable,, many 
│ │ │ │ -0003a3e0: 6d61 7073 0a0a 4465 7363 7269 7074 696f  maps..Descriptio
│ │ │ │ -0003a3f0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a47  n.===========..G
│ │ │ │ -0003a400: 6976 656e 2061 2063 6f64 696d 2032 206d  iven a codim 2 m
│ │ │ │ -0003a410: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ -0003a420: 696f 6e2c 206d 616b 6573 2061 6c6c 2074  ion, makes all t
│ │ │ │ -0003a430: 6865 2063 6f6d 706f 6e65 6e74 7320 6f66  he components of
│ │ │ │ -0003a440: 2074 6865 0a64 6966 6665 7265 6e74 6961   the.differentia
│ │ │ │ -0003a450: 6c20 616e 6420 6f66 2074 6865 2068 6f6d  l and of the hom
│ │ │ │ -0003a460: 6f74 6f70 6965 7320 7468 6174 2061 7265  otopies that are
│ │ │ │ -0003a470: 2072 656c 6576 616e 7420 746f 2074 6865   relevant to the
│ │ │ │ -0003a480: 2066 696e 6974 6520 7265 736f 6c75 7469   finite resoluti
│ │ │ │ -0003a490: 6f6e 2c0a 6173 2069 6e20 342e 322e 3320  on,.as in 4.2.3 
│ │ │ │ -0003a4a0: 6f66 2045 6973 656e 6275 642d 5065 6576  of Eisenbud-Peev
│ │ │ │ -0003a4b0: 6120 224d 696e 696d 616c 2046 7265 6520  a "Minimal Free 
│ │ │ │ -0003a4c0: 5265 736f 6c75 7469 6f6e 7320 616e 6420  Resolutions and 
│ │ │ │ -0003a4d0: 4869 6768 6572 204d 6174 7269 780a 4661  Higher Matrix.Fa
│ │ │ │ -0003a4e0: 6374 6f72 697a 6174 696f 6e73 220a 0a2b  ctorizations"..+
│ │ │ │ +0003a220: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +0003a230: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +0003a240: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0003a250: 6d61 7073 203d 206d 616b 6546 696e 6974  maps = makeFinit
│ │ │ │ +0003a260: 6552 6573 6f6c 7574 696f 6e43 6f64 696d  eResolutionCodim
│ │ │ │ +0003a270: 3228 6666 2c6d 6629 0a20 202a 2049 6e70  2(ff,mf).  * Inp
│ │ │ │ +0003a280: 7574 733a 0a20 2020 2020 202a 206d 662c  uts:.      * mf,
│ │ │ │ +0003a290: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +0003a2a0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +0003a2b0: 742c 2c20 6d61 7472 6978 2066 6163 746f  t,, matrix facto
│ │ │ │ +0003a2c0: 7269 7a61 7469 6f6e 0a20 2020 2020 202a  rization.      *
│ │ │ │ +0003a2d0: 2066 662c 2061 202a 6e6f 7465 206d 6174   ff, a *note mat
│ │ │ │ +0003a2e0: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ +0003a2f0: 6f63 294d 6174 7269 782c 2c20 7265 6775  oc)Matrix,, regu
│ │ │ │ +0003a300: 6c61 7220 7365 7175 656e 6365 0a20 202a  lar sequence.  *
│ │ │ │ +0003a310: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +0003a320: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +0003a330: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +0003a340: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +0003a350: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +0003a360: 2020 202a 2043 6865 636b 203d 3e20 2e2e     * Check => ..
│ │ │ │ +0003a370: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +0003a380: 2066 616c 7365 0a20 202a 204f 7574 7075   false.  * Outpu
│ │ │ │ +0003a390: 7473 3a0a 2020 2020 2020 2a20 6d61 7073  ts:.      * maps
│ │ │ │ +0003a3a0: 2c20 6120 2a6e 6f74 6520 6861 7368 2074  , a *note hash t
│ │ │ │ +0003a3b0: 6162 6c65 3a20 284d 6163 6175 6c61 7932  able: (Macaulay2
│ │ │ │ +0003a3c0: 446f 6329 4861 7368 5461 626c 652c 2c20  Doc)HashTable,, 
│ │ │ │ +0003a3d0: 6d61 6e79 206d 6170 730a 0a44 6573 6372  many maps..Descr
│ │ │ │ +0003a3e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0003a3f0: 3d3d 0a0a 4769 7665 6e20 6120 636f 6469  ==..Given a codi
│ │ │ │ +0003a400: 6d20 3220 6d61 7472 6978 2066 6163 746f  m 2 matrix facto
│ │ │ │ +0003a410: 7269 7a61 7469 6f6e 2c20 6d61 6b65 7320  rization, makes 
│ │ │ │ +0003a420: 616c 6c20 7468 6520 636f 6d70 6f6e 656e  all the componen
│ │ │ │ +0003a430: 7473 206f 6620 7468 650a 6469 6666 6572  ts of the.differ
│ │ │ │ +0003a440: 656e 7469 616c 2061 6e64 206f 6620 7468  ential and of th
│ │ │ │ +0003a450: 6520 686f 6d6f 746f 7069 6573 2074 6861  e homotopies tha
│ │ │ │ +0003a460: 7420 6172 6520 7265 6c65 7661 6e74 2074  t are relevant t
│ │ │ │ +0003a470: 6f20 7468 6520 6669 6e69 7465 2072 6573  o the finite res
│ │ │ │ +0003a480: 6f6c 7574 696f 6e2c 0a61 7320 696e 2034  olution,.as in 4
│ │ │ │ +0003a490: 2e32 2e33 206f 6620 4569 7365 6e62 7564  .2.3 of Eisenbud
│ │ │ │ +0003a4a0: 2d50 6565 7661 2022 4d69 6e69 6d61 6c20  -Peeva "Minimal 
│ │ │ │ +0003a4b0: 4672 6565 2052 6573 6f6c 7574 696f 6e73  Free Resolutions
│ │ │ │ +0003a4c0: 2061 6e64 2048 6967 6865 7220 4d61 7472   and Higher Matr
│ │ │ │ +0003a4d0: 6978 0a46 6163 746f 7269 7a61 7469 6f6e  ix.Factorization
│ │ │ │ +0003a4e0: 7322 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  s"..+-----------
│ │ │ │  0003a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a530: 2b0a 7c69 3120 3a20 6b6b 3d5a 5a2f 3130  +.|i1 : kk=ZZ/10
│ │ │ │ -0003a540: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0003a520: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b3d  -----+.|i1 : kk=
│ │ │ │ +0003a530: 5a5a 2f31 3031 2020 2020 2020 2020 2020  ZZ/101          
│ │ │ │ +0003a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a570: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003a560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5b0: 2020 2020 2020 7c0a 7c6f 3120 3d20 6b6b        |.|o1 = kk
│ │ │ │ +0003a5a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003a5b0: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │  0003a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003a5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003a5f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0003a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a630: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003a640: 3120 3a20 5175 6f74 6965 6e74 5269 6e67  1 : QuotientRing
│ │ │ │ +0003a630: 207c 0a7c 6f31 203a 2051 756f 7469 656e   |.|o1 : Quotien
│ │ │ │ +0003a640: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │  0003a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003a680: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003a670: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a6c0: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b  --+.|i2 : S = kk
│ │ │ │ -0003a6d0: 5b61 2c62 5d20 2020 2020 2020 2020 2020  [a,b]           
│ │ │ │ +0003a6b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ +0003a6c0: 203d 206b 6b5b 612c 625d 2020 2020 2020   = kk[a,b]      
│ │ │ │ +0003a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003a6f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a740: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -0003a750: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +0003a730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003a740: 6f32 203d 2053 2020 2020 2020 2020 2020  o2 = S          
│ │ │ │ +0003a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a780: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003a780: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0003a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003a7d0: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ -0003a7e0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0003a7c0: 2020 207c 0a7c 6f32 203a 2050 6f6c 796e     |.|o2 : Polyn
│ │ │ │ +0003a7d0: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +0003a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a810: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0003a800: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0003a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a850: 2d2d 2d2d 2b0a 7c69 3320 3a20 6666 203d  ----+.|i3 : ff =
│ │ │ │ -0003a860: 206d 6174 7269 7822 6134 2c62 3422 2020   matrix"a4,b4"  
│ │ │ │ +0003a840: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0003a850: 2066 6620 3d20 6d61 7472 6978 2261 342c   ff = matrix"a4,
│ │ │ │ +0003a860: 6234 2220 2020 2020 2020 2020 2020 2020  b4"             
│ │ │ │  0003a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a890: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003a880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0003a8e0: 3d20 7c20 6134 2062 3420 7c20 2020 2020  = | a4 b4 |     
│ │ │ │ +0003a8c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a8d0: 0a7c 6f33 203d 207c 2061 3420 6234 207c  .|o3 = | a4 b4 |
│ │ │ │ +0003a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003a910: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0003a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a960: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003a970: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ +0003a950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003a960: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +0003a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9a0: 2020 207c 0a7c 6f33 203a 204d 6174 7269     |.|o3 : Matri
│ │ │ │ -0003a9b0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +0003a990: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0003a9a0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +0003a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0003a9d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003aa20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0003aa30: 2052 203d 2053 2f69 6465 616c 2066 6620   R = S/ideal ff 
│ │ │ │ +0003aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0003aa20: 7c69 3420 3a20 5220 3d20 532f 6964 6561  |i4 : R = S/idea
│ │ │ │ +0003aa30: 6c20 6666 2020 2020 2020 2020 2020 2020  l ff            
│ │ │ │  0003aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003aa60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aaa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003aab0: 0a7c 6f34 203d 2052 2020 2020 2020 2020  .|o4 = R        
│ │ │ │ +0003aaa0: 2020 2020 7c0a 7c6f 3420 3d20 5220 2020      |.|o4 = R   
│ │ │ │ +0003aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aaf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003aae0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab30: 2020 2020 207c 0a7c 6f34 203a 2051 756f       |.|o4 : Quo
│ │ │ │ -0003ab40: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +0003ab20: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +0003ab30: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +0003ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003ab60: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -0003abc0: 203a 204e 203d 2052 5e31 2f69 6465 616c   : N = R^1/ideal
│ │ │ │ -0003abd0: 2261 322c 2061 622c 2062 3322 2020 2020  "a2, ab, b3"    
│ │ │ │ +0003abb0: 2b0a 7c69 3520 3a20 4e20 3d20 525e 312f  +.|i5 : N = R^1/
│ │ │ │ +0003abc0: 6964 6561 6c22 6132 2c20 6162 2c20 6233  ideal"a2, ab, b3
│ │ │ │ +0003abd0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  0003abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003abf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003ac00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003abf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac40: 207c 0a7c 6f35 203d 2063 6f6b 6572 6e65   |.|o5 = cokerne
│ │ │ │ -0003ac50: 6c20 7c20 6132 2061 6220 6233 207c 2020  l | a2 ab b3 |  
│ │ │ │ +0003ac30: 2020 2020 2020 7c0a 7c6f 3520 3d20 636f        |.|o5 = co
│ │ │ │ +0003ac40: 6b65 726e 656c 207c 2061 3220 6162 2062  kernel | a2 ab b
│ │ │ │ +0003ac50: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │  0003ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003ac70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003acc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ace0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad00: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -0003ad10: 3a20 522d 6d6f 6475 6c65 2c20 7175 6f74  : R-module, quot
│ │ │ │ -0003ad20: 6965 6e74 206f 6620 5220 2020 2020 2020  ient of R       
│ │ │ │ +0003acb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003acd0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0003ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003acf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003ad00: 0a7c 6f35 203a 2052 2d6d 6f64 756c 652c  .|o5 : R-module,
│ │ │ │ +0003ad10: 2071 756f 7469 656e 7420 6f66 2052 2020   quotient of R  
│ │ │ │ +0003ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad40: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003ad40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0003ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ad90: 2b0a 7c69 3620 3a20 4e20 3d20 636f 6b65  +.|i6 : N = coke
│ │ │ │ -0003ada0: 7220 7661 7273 2052 2020 2020 2020 2020  r vars R        
│ │ │ │ +0003ad80: 2d2d 2d2d 2d2b 0a7c 6936 203a 204e 203d  -----+.|i6 : N =
│ │ │ │ +0003ad90: 2063 6f6b 6572 2076 6172 7320 5220 2020   coker vars R   
│ │ │ │ +0003ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003add0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003adc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae10: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ -0003ae20: 6b65 726e 656c 207c 2061 2062 207c 2020  kernel | a b |  
│ │ │ │ +0003ae00: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +0003ae10: 203d 2063 6f6b 6572 6e65 6c20 7c20 6120   = cokernel | a 
│ │ │ │ +0003ae20: 6220 7c20 2020 2020 2020 2020 2020 2020  b |             
│ │ │ │  0003ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003ae40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003ae50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0003ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003ae90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aeb0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0003aeb0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  0003aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003aee0: 0a7c 6f36 203a 2052 2d6d 6f64 756c 652c  .|o6 : R-module,
│ │ │ │ -0003aef0: 2071 756f 7469 656e 7420 6f66 2052 2020   quotient of R  
│ │ │ │ +0003aed0: 2020 2020 7c0a 7c6f 3620 3a20 522d 6d6f      |.|o6 : R-mo
│ │ │ │ +0003aee0: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +0003aef0: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │  0003af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af20: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0003af10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0003af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003af30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003af40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003af60: 2d2d 2d2d 2d2b 0a7c 6937 203a 204d 203d  -----+.|i7 : M =
│ │ │ │ -0003af70: 2068 6967 6853 797a 7967 7920 4e20 2020   highSyzygy N   
│ │ │ │ +0003af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0003af60: 3a20 4d20 3d20 6869 6768 5379 7a79 6779  : M = highSyzygy
│ │ │ │ +0003af70: 204e 2020 2020 2020 2020 2020 2020 2020   N              
│ │ │ │  0003af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003af90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afe0: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0003aff0: 203d 2063 6f6b 6572 6e65 6c20 7b32 7d20   = cokernel {2} 
│ │ │ │ -0003b000: 7c20 3020 2d62 3320 6133 2030 207c 2020  | 0 -b3 a3 0 |  
│ │ │ │ +0003afe0: 7c0a 7c6f 3720 3d20 636f 6b65 726e 656c  |.|o7 = cokernel
│ │ │ │ +0003aff0: 207b 327d 207c 2030 202d 6233 2061 3320   {2} | 0 -b3 a3 
│ │ │ │ +0003b000: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │  0003b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003b030: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ -0003b040: 347d 207c 2062 2061 2020 2030 2020 3020  4} | b a   0  0 
│ │ │ │ -0003b050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003b080: 2020 7b34 7d20 7c20 3020 3020 2020 6220    {4} | 0 0   b 
│ │ │ │ -0003b090: 2061 207c 2020 2020 2020 2020 2020 2020   a |            
│ │ │ │ -0003b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b0b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003b020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b030: 2020 2020 7b34 7d20 7c20 6220 6120 2020      {4} | b a   
│ │ │ │ +0003b040: 3020 2030 207c 2020 2020 2020 2020 2020  0  0 |          
│ │ │ │ +0003b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b060: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003b070: 2020 2020 2020 207b 347d 207c 2030 2030         {4} | 0 0
│ │ │ │ +0003b080: 2020 2062 2020 6120 7c20 2020 2020 2020     b  a |       
│ │ │ │ +0003b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b0a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b0f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b110: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -0003b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b130: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0003b140: 3a20 522d 6d6f 6475 6c65 2c20 7175 6f74  : R-module, quot
│ │ │ │ -0003b150: 6965 6e74 206f 6620 5220 2020 2020 2020  ient of R       
│ │ │ │ +0003b0e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b100: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +0003b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003b130: 0a7c 6f37 203a 2052 2d6d 6f64 756c 652c  .|o7 : R-module,
│ │ │ │ +0003b140: 2071 756f 7469 656e 7420 6f66 2052 2020   quotient of R  
│ │ │ │ +0003b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b170: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003b170: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0003b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1c0: 2b0a 7c69 3820 3a20 4d53 203d 2070 7573  +.|i8 : MS = pus
│ │ │ │ -0003b1d0: 6846 6f72 7761 7264 286d 6170 2852 2c53  hForward(map(R,S
│ │ │ │ -0003b1e0: 292c 4d29 2020 2020 2020 2020 2020 2020  ),M)            
│ │ │ │ -0003b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b200: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b1b0: 2d2d 2d2d 2d2b 0a7c 6938 203a 204d 5320  -----+.|i8 : MS 
│ │ │ │ +0003b1c0: 3d20 7075 7368 466f 7277 6172 6428 6d61  = pushForward(ma
│ │ │ │ +0003b1d0: 7028 522c 5329 2c4d 2920 2020 2020 2020  p(R,S),M)       
│ │ │ │ +0003b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b1f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b240: 2020 2020 2020 7c0a 7c6f 3820 3d20 636f        |.|o8 = co
│ │ │ │ -0003b250: 6b65 726e 656c 207b 327d 207c 2030 2062  kernel {2} | 0 b
│ │ │ │ -0003b260: 3320 6133 2030 2030 2020 7c20 2020 2020  3 a3 0 0  |     
│ │ │ │ -0003b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b280: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003b290: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ -0003b2a0: 6220 2d61 2030 2020 3020 3020 207c 2020  b -a 0  0 0  |  
│ │ │ │ +0003b230: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +0003b240: 203d 2063 6f6b 6572 6e65 6c20 7b32 7d20   = cokernel {2} 
│ │ │ │ +0003b250: 7c20 3020 6233 2061 3320 3020 3020 207c  | 0 b3 a3 0 0  |
│ │ │ │ +0003b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003b280: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0003b290: 347d 207c 2062 202d 6120 3020 2030 2030  4} | b -a 0  0 0
│ │ │ │ +0003b2a0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b2d0: 2020 2020 2020 2020 2020 2020 207b 347d               {4}
│ │ │ │ -0003b2e0: 207c 2030 2030 2020 6220 2061 2062 3420   | 0 0  b  a b4 
│ │ │ │ -0003b2f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003b300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003b310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003b2c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003b2d0: 2020 7b34 7d20 7c20 3020 3020 2062 2020    {4} | 0 0  b  
│ │ │ │ +0003b2e0: 6120 6234 207c 2020 2020 2020 2020 2020  a b4 |          
│ │ │ │ +0003b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b300: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b350: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b370: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -0003b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b390: 2020 2020 207c 0a7c 6f38 203a 2053 2d6d       |.|o8 : S-m
│ │ │ │ -0003b3a0: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ -0003b3b0: 6f66 2053 2020 2020 2020 2020 2020 2020  of S            
│ │ │ │ -0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b3d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003b340: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b360: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0003b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b380: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +0003b390: 3a20 532d 6d6f 6475 6c65 2c20 7175 6f74  : S-module, quot
│ │ │ │ +0003b3a0: 6965 6e74 206f 6620 5320 2020 2020 2020  ient of S       
│ │ │ │ +0003b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b3c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ -0003b420: 203a 206d 6620 3d20 6d61 7472 6978 4661   : mf = matrixFa
│ │ │ │ -0003b430: 6374 6f72 697a 6174 696f 6e28 6666 2c20  ctorization(ff, 
│ │ │ │ -0003b440: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ -0003b450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003b460: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003b410: 2b0a 7c69 3920 3a20 6d66 203d 206d 6174  +.|i9 : mf = mat
│ │ │ │ +0003b420: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +0003b430: 2866 662c 204d 2920 2020 2020 2020 2020  (ff, M)         
│ │ │ │ +0003b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b450: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4a0: 207c 0a7c 6f39 203d 207b 7b34 7d20 7c20   |.|o9 = {{4} | 
│ │ │ │ -0003b4b0: 6120 2d62 2030 2030 2020 7c2c 207b 357d  a -b 0 0  |, {5}
│ │ │ │ -0003b4c0: 207c 2061 3320 6220 3020 2020 3020 2030   | a3 b 0   0  0
│ │ │ │ -0003b4d0: 2020 7c2c 207b 327d 207c 2030 202d 3120    |, {2} | 0 -1 
│ │ │ │ -0003b4e0: 3020 7c7d 7c0a 7c20 2020 2020 207b 327d  0 |}|.|      {2}
│ │ │ │ -0003b4f0: 207c 2030 2061 3320 3020 6233 207c 2020   | 0 a3 0 b3 |  
│ │ │ │ -0003b500: 7b35 7d20 7c20 3020 2061 202d 6233 2030  {5} | 0  a -b3 0
│ │ │ │ -0003b510: 2020 3020 207c 2020 7b34 7d20 7c20 3020    0  |  {4} | 0 
│ │ │ │ -0003b520: 3020 2031 207c 207c 0a7c 2020 2020 2020  0  1 | |.|      
│ │ │ │ -0003b530: 7b34 7d20 7c20 3020 3020 2062 2061 2020  {4} | 0 0  b a  
│ │ │ │ -0003b540: 7c20 207b 357d 207c 2030 2020 3020 3020  |  {5} | 0  0 0 
│ │ │ │ -0003b550: 2020 2d61 2062 3320 7c20 207b 347d 207c    -a b3 |  {4} |
│ │ │ │ -0003b560: 2031 2030 2020 3020 7c20 7c0a 7c20 2020   1 0  0 | |.|   
│ │ │ │ -0003b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b580: 2020 2020 2020 7b35 7d20 7c20 3020 2030        {5} | 0  0
│ │ │ │ -0003b590: 2061 3320 2062 2020 3020 207c 2020 2020   a3  b  0  |    
│ │ │ │ -0003b5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003b490: 2020 2020 2020 7c0a 7c6f 3920 3d20 7b7b        |.|o9 = {{
│ │ │ │ +0003b4a0: 347d 207c 2061 202d 6220 3020 3020 207c  4} | a -b 0 0  |
│ │ │ │ +0003b4b0: 2c20 7b35 7d20 7c20 6133 2062 2030 2020  , {5} | a3 b 0  
│ │ │ │ +0003b4c0: 2030 2020 3020 207c 2c20 7b32 7d20 7c20   0  0  |, {2} | 
│ │ │ │ +0003b4d0: 3020 2d31 2030 207c 7d7c 0a7c 2020 2020  0 -1 0 |}|.|    
│ │ │ │ +0003b4e0: 2020 7b32 7d20 7c20 3020 6133 2030 2062    {2} | 0 a3 0 b
│ │ │ │ +0003b4f0: 3320 7c20 207b 357d 207c 2030 2020 6120  3 |  {5} | 0  a 
│ │ │ │ +0003b500: 2d62 3320 3020 2030 2020 7c20 207b 347d  -b3 0  0  |  {4}
│ │ │ │ +0003b510: 207c 2030 2030 2020 3120 7c20 7c0a 7c20   | 0 0  1 | |.| 
│ │ │ │ +0003b520: 2020 2020 207b 347d 207c 2030 2030 2020       {4} | 0 0  
│ │ │ │ +0003b530: 6220 6120 207c 2020 7b35 7d20 7c20 3020  b a  |  {5} | 0 
│ │ │ │ +0003b540: 2030 2030 2020 202d 6120 6233 207c 2020   0 0   -a b3 |  
│ │ │ │ +0003b550: 7b34 7d20 7c20 3120 3020 2030 207c 207c  {4} | 1 0  0 | |
│ │ │ │ +0003b560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003b570: 2020 2020 2020 2020 2020 207b 357d 207c             {5} |
│ │ │ │ +0003b580: 2030 2020 3020 6133 2020 6220 2030 2020   0  0 a3  b  0  
│ │ │ │ +0003b590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003b5a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0003b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5f0: 7c0a 7c6f 3920 3a20 4c69 7374 2020 2020  |.|o9 : List    
│ │ │ │ +0003b5e0: 2020 2020 207c 0a7c 6f39 203a 204c 6973       |.|o9 : Lis
│ │ │ │ +0003b5f0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0003b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b630: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003b620: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b670: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2047  ------+.|i10 : G
│ │ │ │ -0003b680: 203d 206d 616b 6546 696e 6974 6552 6573   = makeFiniteRes
│ │ │ │ -0003b690: 6f6c 7574 696f 6e43 6f64 696d 3228 6666  olutionCodim2(ff
│ │ │ │ -0003b6a0: 2c6d 6629 2020 2020 2020 2020 2020 2020  ,mf)            
│ │ │ │ -0003b6b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003b670: 3020 3a20 4720 3d20 6d61 6b65 4669 6e69  0 : G = makeFini
│ │ │ │ +0003b680: 7465 5265 736f 6c75 7469 6f6e 436f 6469  teResolutionCodi
│ │ │ │ +0003b690: 6d32 2866 662c 6d66 2920 2020 2020 2020  m2(ff,mf)       
│ │ │ │ +0003b6a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003b6b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0003b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b700: 3130 203d 2048 6173 6854 6162 6c65 7b22  10 = HashTable{"
│ │ │ │ -0003b710: 616c 7068 6122 203d 3e20 7b35 7d20 7c20  alpha" => {5} | 
│ │ │ │ -0003b720: 3020 2020 3020 7c20 2020 2020 2020 2020  0   0 |         
│ │ │ │ -0003b730: 2020 2020 207d 2020 2020 2020 2020 207c       }         |
│ │ │ │ -0003b740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003b750: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -0003b760: 207c 202d 6233 2030 207c 2020 2020 2020   | -b3 0 |      
│ │ │ │ -0003b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b780: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003b790: 2020 2020 2022 6222 203d 3e20 7b34 7d20       "b" => {4} 
│ │ │ │ -0003b7a0: 7c20 6220 6120 7c20 2020 2020 2020 2020  | b a |         
│ │ │ │ -0003b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b7c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0003b7d0: 2020 2020 2020 2020 2268 3127 2220 3d3e          "h1'" =>
│ │ │ │ -0003b7e0: 207b 357d 207c 2030 2020 2030 2020 3020   {5} | 0   0  0 
│ │ │ │ -0003b7f0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003b800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b820: 2020 2020 7b35 7d20 7c20 2d62 3320 3020      {5} | -b3 0 
│ │ │ │ -0003b830: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ -0003b840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b860: 2020 2020 2020 207b 357d 207c 2030 2020         {5} | 0  
│ │ │ │ -0003b870: 202d 6120 6233 207c 2020 2020 2020 2020   -a b3 |        
│ │ │ │ -0003b880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003b890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003b8a0: 2020 2020 2020 2020 2020 7b35 7d20 7c20            {5} | 
│ │ │ │ -0003b8b0: 6133 2020 6220 2030 2020 7c20 2020 2020  a3  b  0  |     
│ │ │ │ -0003b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003b8e0: 2020 2020 2268 3122 203d 3e20 7b35 7d20      "h1" => {5} 
│ │ │ │ -0003b8f0: 7c20 3020 2020 3020 2030 2020 7c20 2020  | 0   0  0  |   
│ │ │ │ -0003b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b910: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003b920: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -0003b930: 357d 207c 202d 6233 2030 2020 3020 207c  5} | -b3 0  0  |
│ │ │ │ -0003b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b970: 2020 7b35 7d20 7c20 3020 2020 2d61 2062    {5} | 0   -a b
│ │ │ │ -0003b980: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │ -0003b990: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b9b0: 2020 2020 207b 357d 207c 2061 3320 2062       {5} | a3  b
│ │ │ │ -0003b9c0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -0003b9d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b9f0: 226d 7522 203d 3e20 7b35 7d20 7c20 6133  "mu" => {5} | a3
│ │ │ │ -0003ba00: 2062 207c 2020 2020 2020 2020 2020 2020   b |            
│ │ │ │ -0003ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ba20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003ba30: 2020 2020 2020 2020 2020 207b 357d 207c             {5} |
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│ │ │ │ -0003ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ba60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ba70: 2020 2020 2020 2270 6172 7469 616c 2220        "partial" 
│ │ │ │ -0003ba80: 3d3e 207b 347d 207c 2061 202d 6220 7c20  => {4} | a -b | 
│ │ │ │ -0003ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003baa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bac0: 2020 2020 2020 7b32 7d20 7c20 3020 6133        {2} | 0 a3
│ │ │ │ -0003bad0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003bae0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003baf0: 2020 2020 2020 2020 2020 2020 2270 7369              "psi
│ │ │ │ -0003bb00: 2220 3d3e 207b 347d 207c 2030 2030 2020  " => {4} | 0 0  
│ │ │ │ -0003bb10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003bb20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bb40: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ -0003bb50: 6233 207c 2020 2020 2020 2020 2020 2020  b3 |            
│ │ │ │ -0003bb60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003bb70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bb90: 2020 2033 2020 2020 2020 3520 2020 2020     3      5     
│ │ │ │ -0003bba0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0003bbb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003bbc0: 2020 2020 2022 7265 736f 6c75 7469 6f6e       "resolution
│ │ │ │ -0003bbd0: 2220 3d3e 2053 2020 3c2d 2d20 5320 203c  " => S  <-- S  <
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│ │ │ │ -0003bbf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003b6f0: 207c 0a7c 6f31 3020 3d20 4861 7368 5461   |.|o10 = HashTa
│ │ │ │ +0003b700: 626c 657b 2261 6c70 6861 2220 3d3e 207b  ble{"alpha" => {
│ │ │ │ +0003b710: 357d 207c 2030 2020 2030 207c 2020 2020  5} | 0   0 |    
│ │ │ │ +0003b720: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │ +0003b730: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b750: 2020 7b35 7d20 7c20 2d62 3320 3020 7c20    {5} | -b3 0 | 
│ │ │ │ +0003b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003b780: 2020 2020 2020 2020 2020 2262 2220 3d3e            "b" =>
│ │ │ │ +0003b790: 207b 347d 207c 2062 2061 207c 2020 2020   {4} | b a |    
│ │ │ │ +0003b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b7b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003b7c0: 2020 2020 2020 2020 2020 2020 2022 6831               "h1
│ │ │ │ +0003b7d0: 2722 203d 3e20 7b35 7d20 7c20 3020 2020  '" => {5} | 0   
│ │ │ │ +0003b7e0: 3020 2030 2020 7c20 2020 2020 2020 2020  0  0  |         
│ │ │ │ +0003b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b810: 2020 2020 2020 2020 207b 357d 207c 202d           {5} | -
│ │ │ │ +0003b820: 6233 2030 2020 3020 207c 2020 2020 2020  b3 0  0  |      
│ │ │ │ +0003b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003b850: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ +0003b860: 7c20 3020 2020 2d61 2062 3320 7c20 2020  | 0   -a b3 |   
│ │ │ │ +0003b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b880: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b890: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003b8a0: 357d 207c 2061 3320 2062 2020 3020 207c  5} | a3  b  0  |
│ │ │ │ +0003b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b8c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003b8d0: 2020 2020 2020 2020 2022 6831 2220 3d3e           "h1" =>
│ │ │ │ +0003b8e0: 207b 357d 207c 2030 2020 2030 2020 3020   {5} | 0   0  0 
│ │ │ │ +0003b8f0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003b900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b920: 2020 2020 7b35 7d20 7c20 2d62 3320 3020      {5} | -b3 0 
│ │ │ │ +0003b930: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +0003b940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b960: 2020 2020 2020 207b 357d 207c 2030 2020         {5} | 0  
│ │ │ │ +0003b970: 202d 6120 6233 207c 2020 2020 2020 2020   -a b3 |        
│ │ │ │ +0003b980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003b990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003b9a0: 2020 2020 2020 2020 2020 7b35 7d20 7c20            {5} | 
│ │ │ │ +0003b9b0: 6133 2020 6220 2030 2020 7c20 2020 2020  a3  b  0  |     
│ │ │ │ +0003b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b9d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003b9e0: 2020 2020 2022 6d75 2220 3d3e 207b 357d       "mu" => {5}
│ │ │ │ +0003b9f0: 207c 2061 3320 6220 7c20 2020 2020 2020   | a3 b |       
│ │ │ │ +0003ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ba10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ba30: 7b35 7d20 7c20 3020 2061 207c 2020 2020  {5} | 0  a |    
│ │ │ │ +0003ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ba50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ba60: 2020 2020 2020 2020 2020 2022 7061 7274             "part
│ │ │ │ +0003ba70: 6961 6c22 203d 3e20 7b34 7d20 7c20 6120  ial" => {4} | a 
│ │ │ │ +0003ba80: 2d62 207c 2020 2020 2020 2020 2020 2020  -b |            
│ │ │ │ +0003ba90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bab0: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +0003bac0: 2030 2061 3320 7c20 2020 2020 2020 2020   0 a3 |         
│ │ │ │ +0003bad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003bae0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003baf0: 2022 7073 6922 203d 3e20 7b34 7d20 7c20   "psi" => {4} | 
│ │ │ │ +0003bb00: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +0003bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003bb30: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ +0003bb40: 207c 2030 2062 3320 7c20 2020 2020 2020   | 0 b3 |       
│ │ │ │ +0003bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb80: 2020 2020 2020 2020 3320 2020 2020 2035          3      5
│ │ │ │ +0003bb90: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003bba0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003bbb0: 2020 2020 2020 2020 2020 2272 6573 6f6c            "resol
│ │ │ │ +0003bbc0: 7574 696f 6e22 203d 3e20 5320 203c 2d2d  ution" => S  <--
│ │ │ │ +0003bbd0: 2053 2020 3c2d 2d20 5320 203c 2d2d 2030   S  <-- S  <-- 0
│ │ │ │ +0003bbe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003bc20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc50: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -0003bc60: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ -0003bc70: 3320 2020 2020 2020 2020 207c 0a7c 2020  3          |.|  
│ │ │ │ -0003bc80: 2020 2020 2020 2020 2020 2020 2020 2273                "s
│ │ │ │ -0003bc90: 6967 6d61 2220 3d3e 207b 357d 207c 2062  igma" => {5} | b
│ │ │ │ -0003bca0: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │ -0003bcb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003bcc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003bcd0: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ -0003bce0: 7c20 3020 207c 2020 2020 2020 2020 2020  | 0  |          
│ │ │ │ -0003bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003bd10: 2020 2020 2274 6175 2220 3d3e 2030 2020      "tau" => 0  
│ │ │ │ +0003bc50: 3020 2020 2020 2031 2020 2020 2020 3220  0      1      2 
│ │ │ │ +0003bc60: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +0003bc70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003bc80: 2020 2022 7369 676d 6122 203d 3e20 7b35     "sigma" => {5
│ │ │ │ +0003bc90: 7d20 7c20 6233 207c 2020 2020 2020 2020  } | b3 |        
│ │ │ │ +0003bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bcb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bcd0: 207b 357d 207c 2030 2020 7c20 2020 2020   {5} | 0  |     
│ │ │ │ +0003bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bcf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003bd00: 2020 2020 2020 2020 2022 7461 7522 203d           "tau" =
│ │ │ │ +0003bd10: 3e20 3020 2020 2020 2020 2020 2020 2020  > 0             
│ │ │ │  0003bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003bd50: 2020 2020 2020 2022 7522 203d 3e20 7b38         "u" => {8
│ │ │ │ -0003bd60: 7d20 7c20 3120 3020 7c20 2020 2020 2020  } | 1 0 |       
│ │ │ │ -0003bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003bd90: 2020 2020 2020 2020 2020 2276 2220 3d3e            "v" =>
│ │ │ │ -0003bda0: 207b 397d 207c 2030 202d 3120 7c20 2020   {9} | 0 -1 |   
│ │ │ │ -0003bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bdc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bde0: 2020 2020 7b39 7d20 7c20 3120 3020 207c      {9} | 1 0  |
│ │ │ │ +0003bd30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003bd40: 2020 2020 2020 2020 2020 2020 2275 2220              "u" 
│ │ │ │ +0003bd50: 3d3e 207b 387d 207c 2031 2030 207c 2020  => {8} | 1 0 |  
│ │ │ │ +0003bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bd70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bd80: 2020 2020 2020 2020 2020 2020 2020 2022                 "
│ │ │ │ +0003bd90: 7622 203d 3e20 7b39 7d20 7c20 3020 2d31  v" => {9} | 0 -1
│ │ │ │ +0003bda0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003bdb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003bdc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003bdd0: 2020 2020 2020 2020 207b 397d 207c 2031           {9} | 1
│ │ │ │ +0003bde0: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │  0003bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -0003be60: 2020 2022 5922 203d 3e20 3020 2020 2020     "Y" => 0     
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│ │ │ │ +0003c020: 203d 2053 2020 3c2d 2d20 5320 203c 2d2d   = S  <-- S  <--
│ │ │ │ +0003c030: 2053 2020 3c2d 2d20 3020 2020 2020 2020   S  <-- 0       
│ │ │ │ +0003c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003c0c0: 2020 3220 2020 2020 2033 2020 2020 2020    2      3      
│ │ │ │ +0003c0a0: 7c0a 7c20 2020 2020 2030 2020 2020 2020  |.|      0      
│ │ │ │ +0003c0b0: 3120 2020 2020 2032 2020 2020 2020 3320  1      2      3 
│ │ │ │ +0003c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003c0f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003c0e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003c140: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
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│ │ │ │ +0003c130: 6861 696e 436f 6d70 6c65 7820 2020 2020  hainComplex     
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│ │ │ │ +0003c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c1b0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -0003c1c0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -0003c1d0: 202a 6e6f 7465 206d 616b 6546 696e 6974   *note makeFinit
│ │ │ │ -0003c1e0: 6552 6573 6f6c 7574 696f 6e3a 206d 616b  eResolution: mak
│ │ │ │ -0003c1f0: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ -0003c200: 6e2c 202d 2d20 6669 6e69 7465 2072 6573  n, -- finite res
│ │ │ │ -0003c210: 6f6c 7574 696f 6e20 6f66 2061 0a20 2020  olution of a.   
│ │ │ │ -0003c220: 206d 6174 7269 7820 6661 6374 6f72 697a   matrix factoriz
│ │ │ │ -0003c230: 6174 696f 6e20 6d6f 6475 6c65 204d 0a0a  ation module M..
│ │ │ │ -0003c240: 5761 7973 2074 6f20 7573 6520 6d61 6b65  Ways to use make
│ │ │ │ -0003c250: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ -0003c260: 436f 6469 6d32 3a0a 3d3d 3d3d 3d3d 3d3d  Codim2:.========
│ │ │ │ +0003c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +0003c1b0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +0003c1c0: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 6b65  ..  * *note make
│ │ │ │ +0003c1d0: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ +0003c1e0: 3a20 6d61 6b65 4669 6e69 7465 5265 736f  : makeFiniteReso
│ │ │ │ +0003c1f0: 6c75 7469 6f6e 2c20 2d2d 2066 696e 6974  lution, -- finit
│ │ │ │ +0003c200: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ +0003c210: 610a 2020 2020 6d61 7472 6978 2066 6163  a.    matrix fac
│ │ │ │ +0003c220: 746f 7269 7a61 7469 6f6e 206d 6f64 756c  torization modul
│ │ │ │ +0003c230: 6520 4d0a 0a57 6179 7320 746f 2075 7365  e M..Ways to use
│ │ │ │ +0003c240: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ +0003c250: 7574 696f 6e43 6f64 696d 323a 0a3d 3d3d  utionCodim2:.===
│ │ │ │ +0003c260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0003c270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003c280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0003c290: 0a20 202a 2022 6d61 6b65 4669 6e69 7465  .  * "makeFinite
│ │ │ │ -0003c2a0: 5265 736f 6c75 7469 6f6e 436f 6469 6d32  ResolutionCodim2
│ │ │ │ -0003c2b0: 284d 6174 7269 782c 4c69 7374 2922 0a0a  (Matrix,List)"..
│ │ │ │ -0003c2c0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -0003c2d0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -0003c2e0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0003c2f0: 7420 2a6e 6f74 6520 6d61 6b65 4669 6e69  t *note makeFini
│ │ │ │ -0003c300: 7465 5265 736f 6c75 7469 6f6e 436f 6469  teResolutionCodi
│ │ │ │ -0003c310: 6d32 3a20 6d61 6b65 4669 6e69 7465 5265  m2: makeFiniteRe
│ │ │ │ -0003c320: 736f 6c75 7469 6f6e 436f 6469 6d32 2c20  solutionCodim2, 
│ │ │ │ -0003c330: 6973 2061 0a2a 6e6f 7465 206d 6574 686f  is a.*note metho
│ │ │ │ -0003c340: 6420 6675 6e63 7469 6f6e 2077 6974 6820  d function with 
│ │ │ │ -0003c350: 6f70 7469 6f6e 733a 2028 4d61 6361 756c  options: (Macaul
│ │ │ │ -0003c360: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -0003c370: 6374 696f 6e57 6974 684f 7074 696f 6e73  ctionWithOptions
│ │ │ │ -0003c380: 2c2e 0a1f 0a46 696c 653a 2043 6f6d 706c  ,....File: Compl
│ │ │ │ -0003c390: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -0003c3a0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -0003c3b0: 204e 6f64 653a 206d 616b 6548 6f6d 6f74   Node: makeHomot
│ │ │ │ -0003c3c0: 6f70 6965 732c 204e 6578 743a 206d 616b  opies, Next: mak
│ │ │ │ -0003c3d0: 6548 6f6d 6f74 6f70 6965 7331 2c20 5072  eHomotopies1, Pr
│ │ │ │ -0003c3e0: 6576 3a20 6d61 6b65 4669 6e69 7465 5265  ev: makeFiniteRe
│ │ │ │ -0003c3f0: 736f 6c75 7469 6f6e 436f 6469 6d32 2c20  solutionCodim2, 
│ │ │ │ -0003c400: 5570 3a20 546f 700a 0a6d 616b 6548 6f6d  Up: Top..makeHom
│ │ │ │ -0003c410: 6f74 6f70 6965 7320 2d2d 2072 6574 7572  otopies -- retur
│ │ │ │ -0003c420: 6e73 2061 2073 7973 7465 6d20 6f66 2068  ns a system of h
│ │ │ │ -0003c430: 6967 6865 7220 686f 6d6f 746f 7069 6573  igher homotopies
│ │ │ │ -0003c440: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0003c280: 3d3d 3d3d 0a0a 2020 2a20 226d 616b 6546  ====..  * "makeF
│ │ │ │ +0003c290: 696e 6974 6552 6573 6f6c 7574 696f 6e43  initeResolutionC
│ │ │ │ +0003c2a0: 6f64 696d 3228 4d61 7472 6978 2c4c 6973  odim2(Matrix,Lis
│ │ │ │ +0003c2b0: 7429 220a 0a46 6f72 2074 6865 2070 726f  t)"..For the pro
│ │ │ │ +0003c2c0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0003c2d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +0003c2e0: 6f62 6a65 6374 202a 6e6f 7465 206d 616b  object *note mak
│ │ │ │ +0003c2f0: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ +0003c300: 6e43 6f64 696d 323a 206d 616b 6546 696e  nCodim2: makeFin
│ │ │ │ +0003c310: 6974 6552 6573 6f6c 7574 696f 6e43 6f64  iteResolutionCod
│ │ │ │ +0003c320: 696d 322c 2069 7320 610a 2a6e 6f74 6520  im2, is a.*note 
│ │ │ │ +0003c330: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ +0003c340: 7769 7468 206f 7074 696f 6e73 3a20 284d  with options: (M
│ │ │ │ +0003c350: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +0003c360: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ +0003c370: 7469 6f6e 732c 2e0a 1f0a 4669 6c65 3a20  tions,....File: 
│ │ │ │ +0003c380: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +0003c390: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +0003c3a0: 696e 666f 2c20 4e6f 6465 3a20 6d61 6b65  info, Node: make
│ │ │ │ +0003c3b0: 486f 6d6f 746f 7069 6573 2c20 4e65 7874  Homotopies, Next
│ │ │ │ +0003c3c0: 3a20 6d61 6b65 486f 6d6f 746f 7069 6573  : makeHomotopies
│ │ │ │ +0003c3d0: 312c 2050 7265 763a 206d 616b 6546 696e  1, Prev: makeFin
│ │ │ │ +0003c3e0: 6974 6552 6573 6f6c 7574 696f 6e43 6f64  iteResolutionCod
│ │ │ │ +0003c3f0: 696d 322c 2055 703a 2054 6f70 0a0a 6d61  im2, Up: Top..ma
│ │ │ │ +0003c400: 6b65 486f 6d6f 746f 7069 6573 202d 2d20  keHomotopies -- 
│ │ │ │ +0003c410: 7265 7475 726e 7320 6120 7379 7374 656d  returns a system
│ │ │ │ +0003c420: 206f 6620 6869 6768 6572 2068 6f6d 6f74   of higher homot
│ │ │ │ +0003c430: 6f70 6965 730a 2a2a 2a2a 2a2a 2a2a 2a2a  opies.**********
│ │ │ │ +0003c440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003c450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003c460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003c470: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ -0003c480: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ -0003c490: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0003c4a0: 2048 203d 206d 616b 6548 6f6d 6f74 6f70   H = makeHomotop
│ │ │ │ -0003c4b0: 6965 7328 662c 462c 6229 0a20 202a 2049  ies(f,F,b).  * I
│ │ │ │ -0003c4c0: 6e70 7574 733a 0a20 2020 2020 202a 2066  nputs:.      * f
│ │ │ │ -0003c4d0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ -0003c4e0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0003c4f0: 4d61 7472 6978 2c2c 2031 786e 206d 6174  Matrix,, 1xn mat
│ │ │ │ -0003c500: 7269 7820 6f66 2065 6c65 6d65 6e74 7320  rix of elements 
│ │ │ │ -0003c510: 6f66 2053 0a20 2020 2020 202a 2046 2c20  of S.      * F, 
│ │ │ │ -0003c520: 6120 2a6e 6f74 6520 6368 6169 6e20 636f  a *note chain co
│ │ │ │ -0003c530: 6d70 6c65 783a 2028 4d61 6361 756c 6179  mplex: (Macaulay
│ │ │ │ -0003c540: 3244 6f63 2943 6861 696e 436f 6d70 6c65  2Doc)ChainComple
│ │ │ │ -0003c550: 782c 2c20 6164 6d69 7474 696e 670a 2020  x,, admitting.  
│ │ │ │ -0003c560: 2020 2020 2020 686f 6d6f 746f 7069 6573        homotopies
│ │ │ │ -0003c570: 2066 6f72 2074 6865 2065 6e74 7269 6573   for the entries
│ │ │ │ -0003c580: 206f 6620 660a 2020 2020 2020 2a20 622c   of f.      * b,
│ │ │ │ -0003c590: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ -0003c5a0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -0003c5b0: 295a 5a2c 2c20 686f 7720 6661 7220 6261  )ZZ,, how far ba
│ │ │ │ -0003c5c0: 636b 2074 6f20 636f 6d70 7574 6520 7468  ck to compute th
│ │ │ │ -0003c5d0: 650a 2020 2020 2020 2020 686f 6d6f 746f  e.        homoto
│ │ │ │ -0003c5e0: 7069 6573 2028 6465 6661 756c 7473 2074  pies (defaults t
│ │ │ │ -0003c5f0: 6f20 6c65 6e67 7468 206f 6620 4629 0a20  o length of F). 
│ │ │ │ -0003c600: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -0003c610: 2020 2a20 482c 2061 202a 6e6f 7465 2068    * H, a *note h
│ │ │ │ -0003c620: 6173 6820 7461 626c 653a 2028 4d61 6361  ash table: (Maca
│ │ │ │ -0003c630: 756c 6179 3244 6f63 2948 6173 6854 6162  ulay2Doc)HashTab
│ │ │ │ -0003c640: 6c65 2c2c 2067 6976 6573 2074 6865 2068  le,, gives the h
│ │ │ │ -0003c650: 6967 6865 720a 2020 2020 2020 2020 686f  igher.        ho
│ │ │ │ -0003c660: 6d6f 746f 7079 2066 726f 6d20 465f 6920  motopy from F_i 
│ │ │ │ -0003c670: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ -0003c680: 2061 206d 6f6e 6f6d 6961 6c20 7769 7468   a monomial with
│ │ │ │ -0003c690: 2065 7870 6f6e 656e 7420 7665 6374 6f72   exponent vector
│ │ │ │ -0003c6a0: 204c 2061 730a 2020 2020 2020 2020 7468   L as.        th
│ │ │ │ -0003c6b0: 6520 7661 6c75 6520 2448 235c 7b4c 2c69  e value $H#\{L,i
│ │ │ │ -0003c6c0: 5c7d 240a 0a44 6573 6372 6970 7469 6f6e  \}$..Description
│ │ │ │ -0003c6d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4769  .===========..Gi
│ │ │ │ -0003c6e0: 7665 6e20 6120 2431 5c74 696d 6573 206e  ven a $1\times n
│ │ │ │ -0003c6f0: 2420 6d61 7472 6978 2066 2c20 616e 6420  $ matrix f, and 
│ │ │ │ -0003c700: 6120 6368 6169 6e20 636f 6d70 6c65 7820  a chain complex 
│ │ │ │ -0003c710: 462c 2074 6865 2073 6372 6970 7420 6174  F, the script at
│ │ │ │ -0003c720: 7465 6d70 7473 2074 6f0a 6d61 6b65 2061  tempts to.make a
│ │ │ │ -0003c730: 2066 616d 696c 7920 6f66 2068 6967 6865   family of highe
│ │ │ │ -0003c740: 7220 686f 6d6f 746f 7069 6573 206f 6e20  r homotopies on 
│ │ │ │ -0003c750: 4620 666f 7220 7468 6520 656c 656d 656e  F for the elemen
│ │ │ │ -0003c760: 7473 206f 6620 662c 2069 6e20 7468 6520  ts of f, in the 
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│ │ │ │ -0003c800: 245c 7b4a 2c69 5c7d 3d3e 7324 2c20 7768  $\{J,i\}=>s$, wh
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│ │ │ │ -0003c8a0: 2024 2420 616e 6420 2424 0a48 235c 7b65   $$ and $$.H#\{e
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│ │ │ │ -0003c8d0: 7b65 302c 695c 7d20 3d20 665f 692c 2024  {e0,i\} = f_i, $
│ │ │ │ -0003c8e0: 2420 7768 6572 6520 2465 3020 3d0a 5c7b  $ where $e0 =.\{
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│ │ │ │ +0003c800: 242c 2077 6865 7265 204a 2069 7320 610a  $, where J is a.
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│ │ │ │ +0003c840: 485c 235c 7b4a 2c69 5c7d 3a20 465f 692d  H\#\{J,i\}: F_i-
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│ │ │ │ +0003c890: 203d 2064 3b20 2424 2061 6e64 2024 240a   = d; $$ and $$.
│ │ │ │ +0003c8a0: 4823 5c7b 6530 2c69 2b31 5c7d 2a48 235c  H#\{e0,i+1\}*H#\
│ │ │ │ +0003c8b0: 7b65 2c69 5c7d 2b48 235c 7b65 2c69 2d31  {e,i\}+H#\{e,i-1
│ │ │ │ +0003c8c0: 5c7d 4823 5c7b 6530 2c69 5c7d 203d 2066  \}H#\{e0,i\} = f
│ │ │ │ +0003c8d0: 5f69 2c20 2424 2077 6865 7265 2024 6530  _i, $$ where $e0
│ │ │ │ +0003c8e0: 203d 0a5c 7b30 2c5c 646f 7473 2c30 5c7d   =.\{0,\dots,0\}
│ │ │ │ +0003c8f0: 2420 616e 6420 2465 2420 6973 2074 6865  $ and $e$ is the
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│ │ │ │ -0003d1c0: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ -0003d1d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003d1e0: 2020 7b7b 302c 2030 2c20 307d 2c20 317d    {{0, 0, 0}, 1}
│ │ │ │ -0003d1f0: 203d 3e20 7c20 6120 6220 6320 6420 7c20   => | a b c d | 
│ │ │ │ -0003d200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003d210: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -0003d240: 202d 6420 3020 2030 2020 7c20 7c0a 7c20   -d 0  0  | |.| 
│ │ │ │ +0003d180: 2020 2020 2020 207c 0a7c 6f35 203d 2048         |.|o5 = H
│ │ │ │ +0003d190: 6173 6854 6162 6c65 7b7b 7b30 2c20 302c  ashTable{{{0, 0,
│ │ │ │ +0003d1a0: 2030 7d2c 2030 7d20 3d3e 2030 2020 2020   0}, 0} => 0    
│ │ │ │ +0003d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d1c0: 2020 2020 7d7c 0a7c 2020 2020 2020 2020      }|.|        
│ │ │ │ +0003d1d0: 2020 2020 2020 207b 7b30 2c20 302c 2030         {{0, 0, 0
│ │ │ │ +0003d1e0: 7d2c 2031 7d20 3d3e 207c 2061 2062 2063  }, 1} => | a b c
│ │ │ │ +0003d1f0: 2064 207c 2020 2020 2020 2020 2020 2020   d |            
│ │ │ │ +0003d200: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003d210: 2020 2020 207b 7b30 2c20 302c 2030 7d2c       {{0, 0, 0},
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│ │ │ │ +0003d230: 2d63 2030 2020 2d64 2030 2020 3020 207c  -c 0  -d 0  0  |
│ │ │ │ +0003d240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d280: 2020 2d64 2030 2020 7c20 7c0a 7c20 2020    -d 0  | |.|   
│ │ │ │ +0003d260: 2020 2020 207b 317d 207c 2061 2020 3020       {1} | a  0 
│ │ │ │ +0003d270: 202d 6320 3020 202d 6420 3020 207c 207c   -c 0  -d 0  | |
│ │ │ │ +0003d280: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d2a0: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -0003d2b0: 7d20 7c20 3020 2061 2020 6220 2030 2020  } | 0  a  b  0  
│ │ │ │ -0003d2c0: 3020 202d 6420 7c20 7c0a 7c20 2020 2020  0  -d | |.|     
│ │ │ │ +0003d2a0: 2020 207b 317d 207c 2030 2020 6120 2062     {1} | 0  a  b
│ │ │ │ +0003d2b0: 2020 3020 2030 2020 2d64 207c 207c 0a7c    0  0  -d | |.|
│ │ │ │ +0003d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d2e0: 2020 2020 2020 2020 2020 2020 7b31 7d20              {1} 
│ │ │ │ -0003d2f0: 7c20 3020 2030 2020 3020 2061 2020 6220  | 0  0  0  a  b 
│ │ │ │ -0003d300: 2063 2020 7c20 7c0a 7c20 2020 2020 2020   c  | |.|       
│ │ │ │ -0003d310: 2020 2020 2020 2020 7b7b 302c 2030 2c20          {{0, 0, 
│ │ │ │ -0003d320: 307d 2c20 337d 203d 3e20 7b32 7d20 7c20  0}, 3} => {2} | 
│ │ │ │ -0003d330: 6320 2064 2020 3020 2030 2020 7c20 2020  c  d  0  0  |   
│ │ │ │ -0003d340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d360: 2020 2020 2020 2020 7b32 7d20 7c20 2d62          {2} | -b
│ │ │ │ -0003d370: 2030 2020 6420 2030 2020 7c20 2020 2020   0  d  0  |     
│ │ │ │ -0003d380: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d3a0: 2020 2020 2020 7b32 7d20 7c20 6120 2030        {2} | a  0
│ │ │ │ -0003d3b0: 2020 3020 2064 2020 7c20 2020 2020 2020    0  d  |       
│ │ │ │ -0003d3c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003d3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d3e0: 2020 2020 7b32 7d20 7c20 3020 202d 6220      {2} | 0  -b 
│ │ │ │ -0003d3f0: 2d63 2030 2020 7c20 2020 2020 2020 7c0a  -c 0  |       |.
│ │ │ │ -0003d400: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d420: 2020 7b32 7d20 7c20 3020 2061 2020 3020    {2} | 0  a  0 
│ │ │ │ -0003d430: 202d 6320 7c20 2020 2020 2020 7c0a 7c20   -c |       |.| 
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│ │ │ │ +0003d2f0: 6120 2062 2020 6320 207c 207c 0a7c 2020  a  b  c  | |.|  
│ │ │ │ +0003d300: 2020 2020 2020 2020 2020 2020 207b 7b30               {{0
│ │ │ │ +0003d310: 2c20 302c 2030 7d2c 2033 7d20 3d3e 207b  , 0, 0}, 3} => {
│ │ │ │ +0003d320: 327d 207c 2063 2020 6420 2030 2020 3020  2} | c  d  0  0 
│ │ │ │ +0003d330: 207c 2020 2020 2020 207c 0a7c 2020 2020   |       |.|    
│ │ │ │ +0003d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d350: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ +0003d360: 207c 202d 6220 3020 2064 2020 3020 207c   | -b 0  d  0  |
│ │ │ │ +0003d370: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d390: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +0003d3a0: 2061 2020 3020 2030 2020 6420 207c 2020   a  0  0  d  |  
│ │ │ │ +0003d3b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003d3e0: 2020 2d62 202d 6320 3020 207c 2020 2020    -b -c 0  |    
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│ │ │ │ +0003d420: 6120 2030 2020 2d63 207c 2020 2020 2020  a  0  -c |      
│ │ │ │ +0003d430: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d460: 7b32 7d20 7c20 3020 2030 2020 6120 2062  {2} | 0  0  a  b
│ │ │ │ -0003d470: 2020 7c20 2020 2020 2020 7c0a 7c20 2020    |       |.|   
│ │ │ │ -0003d480: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
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│ │ │ │ -0003d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d4b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003d4c0: 2020 2020 2020 2020 2020 7b7b 302c 2030            {{0, 0
│ │ │ │ -0003d4d0: 2c20 317d 2c20 307d 203d 3e20 7b31 7d20  , 1}, 0} => {1} 
│ │ │ │ -0003d4e0: 7c20 3020 7c20 2020 2020 2020 2020 2020  | 0 |           
│ │ │ │ -0003d4f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d530: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d560: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003d570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d5d0: 203d 3e20 7b32 7d20 7c20 3020 2030 2020   => {2} | 0  0  
│ │ │ │ -0003d5e0: 3020 3020 7c20 2020 2020 2020 2020 7c0a  0 0 |         |.
│ │ │ │ -0003d5f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d610: 2020 7b32 7d20 7c20 2d31 2030 2020 3020    {2} | -1 0  0 
│ │ │ │ -0003d620: 3020 7c20 2020 2020 2020 2020 7c0a 7c20  0 |         |.| 
│ │ │ │ +0003d450: 2020 2020 207b 327d 207c 2030 2020 3020       {2} | 0  0 
│ │ │ │ +0003d460: 2061 2020 6220 207c 2020 2020 2020 207c   a  b  |       |
│ │ │ │ +0003d470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003d480: 207b 7b30 2c20 302c 2031 7d2c 202d 317d   {{0, 0, 1}, -1}
│ │ │ │ +0003d490: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +0003d4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003d4b0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003d4c0: 7b30 2c20 302c 2031 7d2c 2030 7d20 3d3e  {0, 0, 1}, 0} =>
│ │ │ │ +0003d4d0: 207b 317d 207c 2030 207c 2020 2020 2020   {1} | 0 |      
│ │ │ │ +0003d4e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d500: 2020 2020 2020 2020 2020 2020 2020 207b                 {
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│ │ │ │ +0003d520: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d540: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +0003d550: 207c 2031 207c 2020 2020 2020 2020 2020   | 1 |          
│ │ │ │ +0003d560: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d580: 2020 2020 2020 2020 2020 207b 317d 207c             {1} |
│ │ │ │ +0003d590: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +0003d5a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +0003d5d0: 2020 3020 2030 2030 207c 2020 2020 2020    0  0 0 |      
│ │ │ │ +0003d5e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d600: 2020 2020 2020 207b 327d 207c 202d 3120         {2} | -1 
│ │ │ │ +0003d610: 3020 2030 2030 207c 2020 2020 2020 2020  0  0 0 |        
│ │ │ │ +0003d620: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d650: 7b32 7d20 7c20 3020 202d 3120 3020 3020  {2} | 0  -1 0 0 
│ │ │ │ -0003d660: 7c20 2020 2020 2020 2020 7c0a 7c20 2020  |         |.|   
│ │ │ │ +0003d640: 2020 2020 207b 327d 207c 2030 2020 2d31       {2} | 0  -1
│ │ │ │ +0003d650: 2030 2030 207c 2020 2020 2020 2020 207c   0 0 |         |
│ │ │ │ +0003d660: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d680: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
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│ │ │ │ -0003d6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003d680: 2020 207b 327d 207c 2030 2020 3020 2030     {2} | 0  0  0
│ │ │ │ +0003d690: 2030 207c 2020 2020 2020 2020 207c 0a7c   0 |         |.|
│ │ │ │ +0003d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d6c0: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -0003d6d0: 7c20 3020 2030 2020 3020 3020 7c20 2020  | 0  0  0 0 |   
│ │ │ │ -0003d6e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d700: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -0003d710: 3020 2030 2020 3020 3120 7c20 2020 2020  0  0  0 1 |     
│ │ │ │ -0003d720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003d730: 2020 2020 2020 7b7b 302c 2030 2c20 317d        {{0, 0, 1}
│ │ │ │ -0003d740: 2c20 327d 203d 3e20 7b33 7d20 7c20 3120  , 2} => {3} | 1 
│ │ │ │ -0003d750: 3020 3020 3020 2030 2020 3020 7c20 2020  0 0 0  0  0 |   
│ │ │ │ -0003d760: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d790: 3020 3020 2030 2020 3020 7c20 2020 2020  0 0  0  0 |     
│ │ │ │ -0003d7a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003d7d0: 2d31 2030 2020 3020 7c20 2020 2020 7c0a  -1 0  0 |     |.
│ │ │ │ -0003d7e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d800: 2020 7b33 7d20 7c20 3020 3020 3020 3020    {3} | 0 0 0 0 
│ │ │ │ -0003d810: 202d 3120 3020 7c20 2020 2020 7c0a 7c20   -1 0 |     |.| 
│ │ │ │ -0003d820: 2020 2020 2020 2020 2020 2020 2020 7b7b                {{
│ │ │ │ -0003d830: 302c 2030 2c20 327d 2c20 2d31 7d20 3d3e  0, 0, 2}, -1} =>
│ │ │ │ -0003d840: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -0003d850: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003d860: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
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│ │ │ │ -0003d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003d8a0: 2020 2020 2020 2020 2020 7b7b 302c 2031            {{0, 1
│ │ │ │ -0003d8b0: 2c20 307d 2c20 307d 203d 3e20 7b31 7d20  , 0}, 0} => {1} 
│ │ │ │ -0003d8c0: 7c20 3020 7c20 2020 2020 2020 2020 2020  | 0 |           
│ │ │ │ -0003d8d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8f0: 2020 2020 2020 2020 2020 7b31 7d20 7c20            {1} | 
│ │ │ │ -0003d900: 3120 7c20 2020 2020 2020 2020 2020 2020  1 |             
│ │ │ │ -0003d910: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d930: 2020 2020 2020 2020 7b31 7d20 7c20 3020          {1} | 0 
│ │ │ │ -0003d940: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003d950: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d970: 2020 2020 2020 7b31 7d20 7c20 3020 7c20        {1} | 0 | 
│ │ │ │ -0003d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d990: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003d9a0: 2020 7b7b 302c 2031 2c20 307d 2c20 317d    {{0, 1, 0}, 1}
│ │ │ │ -0003d9b0: 203d 3e20 7b32 7d20 7c20 2d31 2030 2030   => {2} | -1 0 0
│ │ │ │ -0003d9c0: 2030 207c 2020 2020 2020 2020 2020 7c0a   0 |          |.
│ │ │ │ -0003d9d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d9f0: 2020 7b32 7d20 7c20 3020 2030 2030 2030    {2} | 0  0 0 0
│ │ │ │ -0003da00: 207c 2020 2020 2020 2020 2020 7c0a 7c20   |          |.| 
│ │ │ │ +0003d6c0: 207b 327d 207c 2030 2020 3020 2030 2030   {2} | 0  0  0 0
│ │ │ │ +0003d6d0: 207c 2020 2020 2020 2020 207c 0a7c 2020   |         |.|  
│ │ │ │ +0003d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d6f0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003d700: 327d 207c 2030 2020 3020 2030 2031 207c  2} | 0  0  0 1 |
│ │ │ │ +0003d710: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d720: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003d730: 302c 2031 7d2c 2032 7d20 3d3e 207b 337d  0, 1}, 2} => {3}
│ │ │ │ +0003d740: 207c 2031 2030 2030 2030 2020 3020 2030   | 1 0 0 0  0  0
│ │ │ │ +0003d750: 207c 2020 2020 207c 0a7c 2020 2020 2020   |     |.|      
│ │ │ │ +0003d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d770: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +0003d780: 2030 2030 2030 2030 2020 3020 2030 207c   0 0 0 0  0  0 |
│ │ │ │ +0003d790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7b0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ +0003d7c0: 2030 2030 202d 3120 3020 2030 207c 2020   0 0 -1 0  0 |  
│ │ │ │ +0003d7d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7f0: 2020 2020 2020 207b 337d 207c 2030 2030         {3} | 0 0
│ │ │ │ +0003d800: 2030 2030 2020 2d31 2030 207c 2020 2020   0 0  -1 0 |    
│ │ │ │ +0003d810: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003d820: 2020 207b 7b30 2c20 302c 2032 7d2c 202d     {{0, 0, 2}, -
│ │ │ │ +0003d830: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003d840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003d850: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003d860: 207b 7b30 2c20 312c 2030 7d2c 202d 317d   {{0, 1, 0}, -1}
│ │ │ │ +0003d870: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +0003d880: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003d890: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003d8a0: 7b30 2c20 312c 2030 7d2c 2030 7d20 3d3e  {0, 1, 0}, 0} =>
│ │ │ │ +0003d8b0: 207b 317d 207c 2030 207c 2020 2020 2020   {1} | 0 |      
│ │ │ │ +0003d8c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d8e0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003d8f0: 317d 207c 2031 207c 2020 2020 2020 2020  1} | 1 |        
│ │ │ │ +0003d900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d920: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +0003d930: 207c 2030 207c 2020 2020 2020 2020 2020   | 0 |          
│ │ │ │ +0003d940: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d960: 2020 2020 2020 2020 2020 207b 317d 207c             {1} |
│ │ │ │ +0003d970: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +0003d980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003d990: 2020 2020 2020 207b 7b30 2c20 312c 2030         {{0, 1, 0
│ │ │ │ +0003d9a0: 7d2c 2031 7d20 3d3e 207b 327d 207c 202d  }, 1} => {2} | -
│ │ │ │ +0003d9b0: 3120 3020 3020 3020 7c20 2020 2020 2020  1 0 0 0 |       
│ │ │ │ +0003d9c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d9e0: 2020 2020 2020 207b 327d 207c 2030 2020         {2} | 0  
│ │ │ │ +0003d9f0: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +0003da00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da30: 7b32 7d20 7c20 3020 2030 2031 2030 207c  {2} | 0  0 1 0 |
│ │ │ │ -0003da40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003da20: 2020 2020 207b 327d 207c 2030 2020 3020       {2} | 0  0 
│ │ │ │ +0003da30: 3120 3020 7c20 2020 2020 2020 2020 207c  1 0 |          |
│ │ │ │ +0003da40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da60: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -0003da70: 7d20 7c20 3020 2030 2030 2030 207c 2020  } | 0  0 0 0 |  
│ │ │ │ -0003da80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003da60: 2020 207b 327d 207c 2030 2020 3020 3020     {2} | 0  0 0 
│ │ │ │ +0003da70: 3020 7c20 2020 2020 2020 2020 207c 0a7c  0 |          |.|
│ │ │ │ +0003da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003daa0: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -0003dab0: 7c20 3020 2030 2030 2031 207c 2020 2020  | 0  0 0 1 |    
│ │ │ │ -0003dac0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dae0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -0003daf0: 3020 2030 2030 2030 207c 2020 2020 2020  0  0 0 0 |      
│ │ │ │ -0003db00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003db10: 2020 2020 2020 7b7b 302c 2031 2c20 307d        {{0, 1, 0}
│ │ │ │ -0003db20: 2c20 327d 203d 3e20 7b33 7d20 7c20 3020  , 2} => {3} | 0 
│ │ │ │ -0003db30: 2d31 2030 2030 2020 3020 3020 7c20 2020  -1 0 0  0 0 |   
│ │ │ │ -0003db40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db60: 2020 2020 2020 7b33 7d20 7c20 3020 3020        {3} | 0 0 
│ │ │ │ -0003db70: 2030 202d 3120 3020 3020 7c20 2020 2020   0 -1 0 0 |     
│ │ │ │ -0003db80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dba0: 2020 2020 7b33 7d20 7c20 3020 3020 2030      {3} | 0 0  0
│ │ │ │ -0003dbb0: 2030 2020 3020 3020 7c20 2020 2020 7c0a   0  0 0 |     |.
│ │ │ │ -0003dbc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dbe0: 2020 7b33 7d20 7c20 3020 3020 2030 2030    {3} | 0 0  0 0
│ │ │ │ -0003dbf0: 2020 3020 3120 7c20 2020 2020 7c0a 7c20    0 1 |     |.| 
│ │ │ │ -0003dc00: 2020 2020 2020 2020 2020 2020 2020 7b7b                {{
│ │ │ │ -0003dc10: 302c 2031 2c20 317d 2c20 2d31 7d20 3d3e  0, 1, 1}, -1} =>
│ │ │ │ -0003dc20: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -0003dc30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003dc40: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
│ │ │ │ -0003dc50: 2032 2c20 307d 2c20 2d31 7d20 3d3e 2030   2, 0}, -1} => 0
│ │ │ │ -0003dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003dc80: 2020 2020 2020 2020 2020 7b7b 312c 2030            {{1, 0
│ │ │ │ -0003dc90: 2c20 307d 2c20 2d31 7d20 3d3e 2030 2020  , 0}, -1} => 0  
│ │ │ │ -0003dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dcb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003dcc0: 2020 2020 2020 2020 7b7b 312c 2030 2c20          {{1, 0, 
│ │ │ │ -0003dcd0: 307d 2c20 307d 203d 3e20 7b31 7d20 7c20  0}, 0} => {1} | 
│ │ │ │ -0003dce0: 3120 7c20 2020 2020 2020 2020 2020 2020  1 |             
│ │ │ │ -0003dcf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd10: 2020 2020 2020 2020 7b31 7d20 7c20 3020          {1} | 0 
│ │ │ │ -0003dd20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003dd30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd50: 2020 2020 2020 7b31 7d20 7c20 3020 7c20        {1} | 0 | 
│ │ │ │ -0003dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd90: 2020 2020 7b31 7d20 7c20 3020 7c20 2020      {1} | 0 |   
│ │ │ │ -0003dda0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003ddb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003ddc0: 7b7b 312c 2030 2c20 307d 2c20 317d 203d  {{1, 0, 0}, 1} =
│ │ │ │ -0003ddd0: 3e20 7b32 7d20 7c20 3020 3120 3020 3020  > {2} | 0 1 0 0 
│ │ │ │ -0003dde0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0003daa0: 207b 327d 207c 2030 2020 3020 3020 3120   {2} | 0  0 0 1 
│ │ │ │ +0003dab0: 7c20 2020 2020 2020 2020 207c 0a7c 2020  |          |.|  
│ │ │ │ +0003dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dad0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003dae0: 327d 207c 2030 2020 3020 3020 3020 7c20  2} | 0  0 0 0 | 
│ │ │ │ +0003daf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003db00: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003db10: 312c 2030 7d2c 2032 7d20 3d3e 207b 337d  1, 0}, 2} => {3}
│ │ │ │ +0003db20: 207c 2030 202d 3120 3020 3020 2030 2030   | 0 -1 0 0  0 0
│ │ │ │ +0003db30: 207c 2020 2020 207c 0a7c 2020 2020 2020   |     |.|      
│ │ │ │ +0003db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003db50: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +0003db60: 2030 2030 2020 3020 2d31 2030 2030 207c   0 0  0 -1 0 0 |
│ │ │ │ +0003db70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003db90: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ +0003dba0: 2030 2020 3020 3020 2030 2030 207c 2020   0  0 0  0 0 |  
│ │ │ │ +0003dbb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dbd0: 2020 2020 2020 207b 337d 207c 2030 2030         {3} | 0 0
│ │ │ │ +0003dbe0: 2020 3020 3020 2030 2031 207c 2020 2020    0 0  0 1 |    
│ │ │ │ +0003dbf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003dc00: 2020 207b 7b30 2c20 312c 2031 7d2c 202d     {{0, 1, 1}, -
│ │ │ │ +0003dc10: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003dc20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003dc30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003dc40: 207b 7b30 2c20 322c 2030 7d2c 202d 317d   {{0, 2, 0}, -1}
│ │ │ │ +0003dc50: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +0003dc60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003dc70: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003dc80: 7b31 2c20 302c 2030 7d2c 202d 317d 203d  {1, 0, 0}, -1} =
│ │ │ │ +0003dc90: 3e20 3020 2020 2020 2020 2020 2020 2020  > 0             
│ │ │ │ +0003dca0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003dcb0: 2020 2020 2020 2020 2020 2020 207b 7b31               {{1
│ │ │ │ +0003dcc0: 2c20 302c 2030 7d2c 2030 7d20 3d3e 207b  , 0, 0}, 0} => {
│ │ │ │ +0003dcd0: 317d 207c 2031 207c 2020 2020 2020 2020  1} | 1 |        
│ │ │ │ +0003dce0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dd00: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +0003dd10: 207c 2030 207c 2020 2020 2020 2020 2020   | 0 |          
│ │ │ │ +0003dd20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dd40: 2020 2020 2020 2020 2020 207b 317d 207c             {1} |
│ │ │ │ +0003dd50: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +0003dd60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dd80: 2020 2020 2020 2020 207b 317d 207c 2030           {1} | 0
│ │ │ │ +0003dd90: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003dda0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ddb0: 2020 2020 207b 7b31 2c20 302c 2030 7d2c       {{1, 0, 0},
│ │ │ │ +0003ddc0: 2031 7d20 3d3e 207b 327d 207c 2030 2031   1} => {2} | 0 1
│ │ │ │ +0003ddd0: 2030 2030 207c 2020 2020 2020 2020 2020   0 0 |          
│ │ │ │ +0003dde0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de10: 7b32 7d20 7c20 3020 3020 3120 3020 7c20  {2} | 0 0 1 0 | 
│ │ │ │ -0003de20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003de00: 2020 2020 207b 327d 207c 2030 2030 2031       {2} | 0 0 1
│ │ │ │ +0003de10: 2030 207c 2020 2020 2020 2020 2020 207c   0 |           |
│ │ │ │ +0003de20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de40: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -0003de50: 7d20 7c20 3020 3020 3020 3020 7c20 2020  } | 0 0 0 0 |   
│ │ │ │ -0003de60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003de40: 2020 207b 327d 207c 2030 2030 2030 2030     {2} | 0 0 0 0
│ │ │ │ +0003de50: 207c 2020 2020 2020 2020 2020 207c 0a7c   |           |.|
│ │ │ │ +0003de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de80: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -0003de90: 7c20 3020 3020 3020 3120 7c20 2020 2020  | 0 0 0 1 |     
│ │ │ │ -0003dea0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dec0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -0003ded0: 3020 3020 3020 3020 7c20 2020 2020 2020  0 0 0 0 |       
│ │ │ │ -0003dee0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df00: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ -0003df10: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ -0003df20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003df30: 2020 2020 7b7b 312c 2030 2c20 307d 2c20      {{1, 0, 0}, 
│ │ │ │ -0003df40: 327d 203d 3e20 7b33 7d20 7c20 3020 3020  2} => {3} | 0 0 
│ │ │ │ -0003df50: 3120 3020 3020 3020 7c20 2020 2020 2020  1 0 0 0 |       
│ │ │ │ -0003df60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df80: 2020 2020 7b33 7d20 7c20 3020 3020 3020      {3} | 0 0 0 
│ │ │ │ -0003df90: 3020 3120 3020 7c20 2020 2020 2020 7c0a  0 1 0 |       |.
│ │ │ │ -0003dfa0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfc0: 2020 7b33 7d20 7c20 3020 3020 3020 3020    {3} | 0 0 0 0 
│ │ │ │ -0003dfd0: 3020 3120 7c20 2020 2020 2020 7c0a 7c20  0 1 |       |.| 
│ │ │ │ +0003de80: 207b 327d 207c 2030 2030 2030 2031 207c   {2} | 0 0 0 1 |
│ │ │ │ +0003de90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003deb0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003dec0: 327d 207c 2030 2030 2030 2030 207c 2020  2} | 0 0 0 0 |  
│ │ │ │ +0003ded0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003def0: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ +0003df00: 207c 2030 2030 2030 2030 207c 2020 2020   | 0 0 0 0 |    
│ │ │ │ +0003df10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003df20: 2020 2020 2020 2020 207b 7b31 2c20 302c           {{1, 0,
│ │ │ │ +0003df30: 2030 7d2c 2032 7d20 3d3e 207b 337d 207c   0}, 2} => {3} |
│ │ │ │ +0003df40: 2030 2030 2031 2030 2030 2030 207c 2020   0 0 1 0 0 0 |  
│ │ │ │ +0003df50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df70: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ +0003df80: 2030 2030 2030 2031 2030 207c 2020 2020   0 0 0 1 0 |    
│ │ │ │ +0003df90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dfb0: 2020 2020 2020 207b 337d 207c 2030 2030         {3} | 0 0
│ │ │ │ +0003dfc0: 2030 2030 2030 2031 207c 2020 2020 2020   0 0 0 1 |      
│ │ │ │ +0003dfd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e000: 7b33 7d20 7c20 3020 3020 3020 3020 3020  {3} | 0 0 0 0 0 
│ │ │ │ -0003e010: 3020 7c20 2020 2020 2020 7c0a 7c20 2020  0 |       |.|   
│ │ │ │ -0003e020: 2020 2020 2020 2020 2020 2020 7b7b 312c              {{1,
│ │ │ │ -0003e030: 2030 2c20 317d 2c20 2d31 7d20 3d3e 2030   0, 1}, -1} => 0
│ │ │ │ -0003e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003e060: 2020 2020 2020 2020 2020 7b7b 312c 2031            {{1, 1
│ │ │ │ -0003e070: 2c20 307d 2c20 2d31 7d20 3d3e 2030 2020  , 0}, -1} => 0  
│ │ │ │ -0003e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003e0a0: 2020 2020 2020 2020 7b7b 322c 2030 2c20          {{2, 0, 
│ │ │ │ -0003e0b0: 307d 2c20 2d31 7d20 3d3e 2030 2020 2020  0}, -1} => 0    
│ │ │ │ -0003e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e0d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003dff0: 2020 2020 207b 337d 207c 2030 2030 2030       {3} | 0 0 0
│ │ │ │ +0003e000: 2030 2030 2030 207c 2020 2020 2020 207c   0 0 0 |       |
│ │ │ │ +0003e010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003e020: 207b 7b31 2c20 302c 2031 7d2c 202d 317d   {{1, 0, 1}, -1}
│ │ │ │ +0003e030: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +0003e040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003e050: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0003e060: 7b31 2c20 312c 2030 7d2c 202d 317d 203d  {1, 1, 0}, -1} =
│ │ │ │ +0003e070: 3e20 3020 2020 2020 2020 2020 2020 2020  > 0             
│ │ │ │ +0003e080: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003e090: 2020 2020 2020 2020 2020 2020 207b 7b32               {{2
│ │ │ │ +0003e0a0: 2c20 302c 2030 7d2c 202d 317d 203d 3e20  , 0, 0}, -1} => 
│ │ │ │ +0003e0b0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +0003e0c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e110: 2020 7c0a 7c6f 3520 3a20 4861 7368 5461    |.|o5 : HashTa
│ │ │ │ -0003e120: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +0003e100: 2020 2020 2020 207c 0a7c 6f35 203a 2048         |.|o5 : H
│ │ │ │ +0003e110: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │ +0003e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e150: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003e140: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0003e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0003e190: 0a49 6e20 7468 6973 2063 6173 6520 7468  .In this case th
│ │ │ │ -0003e1a0: 6520 6869 6768 6572 2068 6f6d 6f74 6f70  e higher homotop
│ │ │ │ -0003e1b0: 6965 7320 6172 6520 303a 0a0a 2b2d 2d2d  ies are 0:..+---
│ │ │ │ +0003e180: 2d2d 2d2b 0a0a 496e 2074 6869 7320 6361  ---+..In this ca
│ │ │ │ +0003e190: 7365 2074 6865 2068 6967 6865 7220 686f  se the higher ho
│ │ │ │ +0003e1a0: 6d6f 746f 7069 6573 2061 7265 2030 3a0a  motopies are 0:.
│ │ │ │ +0003e1b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0003e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0003e200: 6936 203a 204c 203d 2073 6f72 7420 7365  i6 : L = sort se
│ │ │ │ -0003e210: 6c65 6374 286b 6579 7320 686f 6d6f 742c  lect(keys homot,
│ │ │ │ -0003e220: 206b 2d3e 2868 6f6d 6f74 236b 213d 3020   k->(homot#k!=0 
│ │ │ │ -0003e230: 616e 6420 7375 6d28 6b5f 3029 3e31 2929  and sum(k_0)>1))
│ │ │ │ -0003e240: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003e1f0: 2d2d 2b0a 7c69 3620 3a20 4c20 3d20 736f  --+.|i6 : L = so
│ │ │ │ +0003e200: 7274 2073 656c 6563 7428 6b65 7973 2068  rt select(keys h
│ │ │ │ +0003e210: 6f6d 6f74 2c20 6b2d 3e28 686f 6d6f 7423  omot, k->(homot#
│ │ │ │ +0003e220: 6b21 3d30 2061 6e64 2073 756d 286b 5f30  k!=0 and sum(k_0
│ │ │ │ +0003e230: 293e 3129 297c 0a7c 2020 2020 2020 2020  )>1))|.|        
│ │ │ │ +0003e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e280: 2020 207c 0a7c 6f36 203d 207b 7d20 2020     |.|o6 = {}   
│ │ │ │ +0003e270: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +0003e280: 7b7d 2020 2020 2020 2020 2020 2020 2020  {}              
│ │ │ │  0003e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e2c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003e2b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e300: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ -0003e310: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +0003e2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e300: 7c6f 3620 3a20 4c69 7374 2020 2020 2020  |o6 : List      
│ │ │ │ +0003e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003e340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0003e390: 0a0a 4f6e 2074 6865 206f 7468 6572 2068  ..On the other h
│ │ │ │ -0003e3a0: 616e 642c 2069 6620 7765 2074 616b 6520  and, if we take 
│ │ │ │ -0003e3b0: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -0003e3c0: 7365 6374 696f 6e20 616e 6420 736f 6d65  section and some
│ │ │ │ -0003e3d0: 7468 696e 6720 636f 6e74 6169 6e65 640a  thing contained.
│ │ │ │ -0003e3e0: 696e 2069 7420 696e 2061 206d 6f72 6520  in it in a more 
│ │ │ │ -0003e3f0: 636f 6d70 6c69 6361 7465 6420 7369 7475  complicated situ
│ │ │ │ -0003e400: 6174 696f 6e2c 2074 6865 2070 726f 6772  ation, the progr
│ │ │ │ -0003e410: 616d 2067 6976 6573 206e 6f6e 7a65 726f  am gives nonzero
│ │ │ │ -0003e420: 2068 6967 6865 720a 686f 6d6f 746f 7069   higher.homotopi
│ │ │ │ -0003e430: 6573 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  es:..+----------
│ │ │ │ +0003e380: 2d2d 2d2d 2b0a 0a4f 6e20 7468 6520 6f74  ----+..On the ot
│ │ │ │ +0003e390: 6865 7220 6861 6e64 2c20 6966 2077 6520  her hand, if we 
│ │ │ │ +0003e3a0: 7461 6b65 2061 2063 6f6d 706c 6574 6520  take a complete 
│ │ │ │ +0003e3b0: 696e 7465 7273 6563 7469 6f6e 2061 6e64  intersection and
│ │ │ │ +0003e3c0: 2073 6f6d 6574 6869 6e67 2063 6f6e 7461   something conta
│ │ │ │ +0003e3d0: 696e 6564 0a69 6e20 6974 2069 6e20 6120  ined.in it in a 
│ │ │ │ +0003e3e0: 6d6f 7265 2063 6f6d 706c 6963 6174 6564  more complicated
│ │ │ │ +0003e3f0: 2073 6974 7561 7469 6f6e 2c20 7468 6520   situation, the 
│ │ │ │ +0003e400: 7072 6f67 7261 6d20 6769 7665 7320 6e6f  program gives no
│ │ │ │ +0003e410: 6e7a 6572 6f20 6869 6768 6572 0a68 6f6d  nzero higher.hom
│ │ │ │ +0003e420: 6f74 6f70 6965 733a 0a0a 2b2d 2d2d 2d2d  otopies:..+-----
│ │ │ │ +0003e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e480: 2d2d 2d2b 0a7c 6937 203a 206b 6b3d 205a  ---+.|i7 : kk= Z
│ │ │ │ -0003e490: 5a2f 3332 3030 333b 2020 2020 2020 2020  Z/32003;        
│ │ │ │ +0003e470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +0003e480: 6b6b 3d20 5a5a 2f33 3230 3033 3b20 2020  kk= ZZ/32003;   
│ │ │ │ +0003e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003e4c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e520: 2d2d 2d2b 0a7c 6938 203a 2053 203d 206b  ---+.|i8 : S = k
│ │ │ │ -0003e530: 6b5b 612c 622c 632c 645d 3b20 2020 2020  k[a,b,c,d];     
│ │ │ │ +0003e510: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +0003e520: 5320 3d20 6b6b 5b61 2c62 2c63 2c64 5d3b  S = kk[a,b,c,d];
│ │ │ │ +0003e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e570: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003e560: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e5c0: 2d2d 2d2b 0a7c 6939 203a 204d 203d 2053  ---+.|i9 : M = S
│ │ │ │ -0003e5d0: 5e31 2f28 6964 6561 6c22 6132 2c62 322c  ^1/(ideal"a2,b2,
│ │ │ │ -0003e5e0: 6332 2c64 3222 293b 2020 2020 2020 2020  c2,d2");        
│ │ │ │ +0003e5b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +0003e5c0: 4d20 3d20 535e 312f 2869 6465 616c 2261  M = S^1/(ideal"a
│ │ │ │ +0003e5d0: 322c 6232 2c63 322c 6432 2229 3b20 2020  2,b2,c2,d2");   
│ │ │ │ +0003e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e610: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003e600: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e660: 2d2d 2d2b 0a7c 6931 3020 3a20 4620 3d20  ---+.|i10 : F = 
│ │ │ │ -0003e670: 7265 7320 4d20 2020 2020 2020 2020 2020  res M           
│ │ │ │ +0003e650: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ +0003e660: 2046 203d 2072 6573 204d 2020 2020 2020   F = res M      
│ │ │ │ +0003e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003e6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e700: 2020 207c 0a7c 2020 2020 2020 2031 2020     |.|       1  
│ │ │ │ -0003e710: 2020 2020 3420 2020 2020 2036 2020 2020      4      6    
│ │ │ │ -0003e720: 2020 3420 2020 2020 2031 2020 2020 2020    4      1      
│ │ │ │ +0003e6f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e700: 2020 3120 2020 2020 2034 2020 2020 2020    1      4      
│ │ │ │ +0003e710: 3620 2020 2020 2034 2020 2020 2020 3120  6      4      1 
│ │ │ │ +0003e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e750: 2020 207c 0a7c 6f31 3020 3d20 5320 203c     |.|o10 = S  <
│ │ │ │ -0003e760: 2d2d 2053 2020 3c2d 2d20 5320 203c 2d2d  -- S  <-- S  <--
│ │ │ │ -0003e770: 2053 2020 3c2d 2d20 5320 203c 2d2d 2030   S  <-- S  <-- 0
│ │ │ │ +0003e740: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ +0003e750: 2053 2020 3c2d 2d20 5320 203c 2d2d 2053   S  <-- S  <-- S
│ │ │ │ +0003e760: 2020 3c2d 2d20 5320 203c 2d2d 2053 2020    <-- S  <-- S  
│ │ │ │ +0003e770: 3c2d 2d20 3020 2020 2020 2020 2020 2020  <-- 0           
│ │ │ │  0003e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e7a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003e790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e7f0: 2020 207c 0a7c 2020 2020 2020 3020 2020     |.|      0   
│ │ │ │ -0003e800: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -0003e810: 2033 2020 2020 2020 3420 2020 2020 2035   3      4      5
│ │ │ │ +0003e7e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e7f0: 2030 2020 2020 2020 3120 2020 2020 2032   0      1      2
│ │ │ │ +0003e800: 2020 2020 2020 3320 2020 2020 2034 2020        3      4  
│ │ │ │ +0003e810: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │  0003e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003e830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e890: 2020 207c 0a7c 6f31 3020 3a20 4368 6169     |.|o10 : Chai
│ │ │ │ -0003e8a0: 6e43 6f6d 706c 6578 2020 2020 2020 2020  nComplex        
│ │ │ │ +0003e880: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ +0003e890: 2043 6861 696e 436f 6d70 6c65 7820 2020   ChainComplex   
│ │ │ │ +0003e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003e8d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e930: 2d2d 2d2b 0a7c 6931 3120 3a20 7365 7452  ---+.|i11 : setR
│ │ │ │ -0003e940: 616e 646f 6d53 6565 6420 3020 2020 2020  andomSeed 0     
│ │ │ │ +0003e920: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +0003e930: 2073 6574 5261 6e64 6f6d 5365 6564 2030   setRandomSeed 0
│ │ │ │ +0003e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003e970: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9d0: 2020 207c 0a7c 6f31 3120 3d20 3020 2020     |.|o11 = 0   
│ │ │ │ +0003e9c0: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +0003e9d0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  0003e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003ea10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ea70: 2d2d 2d2b 0a7c 6931 3220 3a20 6620 3d20  ---+.|i12 : f = 
│ │ │ │ -0003ea80: 7261 6e64 6f6d 2853 5e31 2c53 5e7b 323a  random(S^1,S^{2:
│ │ │ │ -0003ea90: 2d35 7d29 3b20 2020 2020 2020 2020 2020  -5});           
│ │ │ │ +0003ea60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │ +0003ea70: 2066 203d 2072 616e 646f 6d28 535e 312c   f = random(S^1,
│ │ │ │ +0003ea80: 535e 7b32 3a2d 357d 293b 2020 2020 2020  S^{2:-5});      
│ │ │ │ +0003ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eac0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003eab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003eb20: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ +0003eb00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003eb10: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +0003eb20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0003eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb60: 2020 207c 0a7c 6f31 3220 3a20 4d61 7472     |.|o12 : Matr
│ │ │ │ -0003eb70: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +0003eb50: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ +0003eb60: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ebb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003eba0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003ebb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ebd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ec00: 2d2d 2d2b 0a7c 6931 3320 3a20 686f 6d6f  ---+.|i13 : homo
│ │ │ │ -0003ec10: 7420 3d20 6d61 6b65 486f 6d6f 746f 7069  t = makeHomotopi
│ │ │ │ -0003ec20: 6573 2866 2c46 2c35 2920 2020 2020 2020  es(f,F,5)       
│ │ │ │ +0003ebf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ +0003ec00: 2068 6f6d 6f74 203d 206d 616b 6548 6f6d   homot = makeHom
│ │ │ │ +0003ec10: 6f74 6f70 6965 7328 662c 462c 3529 2020  otopies(f,F,5)  
│ │ │ │ +0003ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ec40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eca0: 2020 207c 0a7c 6f31 3320 3d20 4861 7368     |.|o13 = Hash
│ │ │ │ -0003ecb0: 5461 626c 657b 7b7b 302c 2030 7d2c 2030  Table{{{0, 0}, 0
│ │ │ │ -0003ecc0: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003ec90: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +0003eca0: 2048 6173 6854 6162 6c65 7b7b 7b30 2c20   HashTable{{{0, 
│ │ │ │ +0003ecb0: 307d 2c20 307d 203d 3e20 3020 2020 2020  0}, 0} => 0     
│ │ │ │ +0003ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ecf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ed00: 2020 2020 2020 7b7b 302c 2030 7d2c 2031        {{0, 0}, 1
│ │ │ │ -0003ed10: 7d20 3d3e 207c 2061 3220 2020 2020 2020  } => | a2       
│ │ │ │ +0003ece0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ecf0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003ed00: 307d 2c20 317d 203d 3e20 7c20 6132 2020  0}, 1} => | a2  
│ │ │ │ +0003ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ed50: 2020 2020 2020 7b7b 302c 2030 7d2c 2032        {{0, 0}, 2
│ │ │ │ -0003ed60: 7d20 3d3e 207b 327d 207c 2020 2020 2020  } => {2} |      
│ │ │ │ +0003ed30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ed40: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003ed50: 307d 2c20 327d 203d 3e20 7b32 7d20 7c20  0}, 2} => {2} | 
│ │ │ │ +0003ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003edb0: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003ed80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eda0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ede0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee00: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003edd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003edf0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee50: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003ee20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ee40: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ee90: 2020 2020 2020 7b7b 302c 2030 7d2c 2033        {{0, 0}, 3
│ │ │ │ -0003eea0: 7d20 3d3e 207b 347d 207c 2020 2020 2020  } => {4} |      
│ │ │ │ +0003ee70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ee80: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003ee90: 307d 2c20 337d 203d 3e20 7b34 7d20 7c20  0}, 3} => {4} | 
│ │ │ │ +0003eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eed0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eef0: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003eec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eee0: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003eef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef40: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003ef10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef30: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef90: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003ef60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef80: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efe0: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003efb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003efd0: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f030: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003f000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f020: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f060: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f070: 2020 2020 2020 7b7b 302c 2030 7d2c 2034        {{0, 0}, 4
│ │ │ │ -0003f080: 7d20 3d3e 207b 367d 207c 2020 2020 2020  } => {6} |      
│ │ │ │ +0003f050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f060: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f070: 307d 2c20 347d 203d 3e20 7b36 7d20 7c20  0}, 4} => {6} | 
│ │ │ │ +0003f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0d0: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f0a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f0c0: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f120: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f110: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f150: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f170: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f140: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f160: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f1b0: 2020 2020 2020 7b7b 302c 2030 7d2c 2035        {{0, 0}, 5
│ │ │ │ -0003f1c0: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003f190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f1a0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f1b0: 307d 2c20 357d 203d 3e20 3020 2020 2020  0}, 5} => 0     
│ │ │ │ +0003f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f200: 2020 2020 2020 7b7b 302c 2030 7d2c 2036        {{0, 0}, 6
│ │ │ │ -0003f210: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003f1e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f1f0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f200: 307d 2c20 367d 203d 3e20 3020 2020 2020  0}, 6} => 0     
│ │ │ │ +0003f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f250: 2020 2020 2020 7b7b 302c 2031 7d2c 202d        {{0, 1}, -
│ │ │ │ -0003f260: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003f230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f240: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f250: 317d 2c20 2d31 7d20 3d3e 2030 2020 2020  1}, -1} => 0    
│ │ │ │ +0003f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f2a0: 2020 2020 2020 7b7b 302c 2031 7d2c 2030        {{0, 1}, 0
│ │ │ │ -0003f2b0: 7d20 3d3e 207b 327d 207c 2020 2020 2020  } => {2} |      
│ │ │ │ +0003f280: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f290: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f2a0: 317d 2c20 307d 203d 3e20 7b32 7d20 7c20  1}, 0} => {2} | 
│ │ │ │ +0003f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f2e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f300: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003f2d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f2f0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f330: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f350: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003f320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f340: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f380: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f3a0: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003f370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f390: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f3d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f3e0: 2020 2020 2020 7b7b 302c 2031 7d2c 2031        {{0, 1}, 1
│ │ │ │ -0003f3f0: 7d20 3d3e 207b 347d 207c 2020 2020 2020  } => {4} |      
│ │ │ │ +0003f3c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f3d0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f3e0: 317d 2c20 317d 203d 3e20 7b34 7d20 7c20  1}, 1} => {4} | 
│ │ │ │ +0003f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f440: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003f410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f430: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f490: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003f460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f480: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f4e0: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
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│ │ │ │ +0003f500: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003f550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f5c0: 2020 2020 2020 7b7b 302c 2031 7d2c 2032        {{0, 1}, 2
│ │ │ │ -0003f5d0: 7d20 3d3e 207b 367d 207c 2020 2020 2020  } => {6} |      
│ │ │ │ +0003f5a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f5b0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f5c0: 317d 2c20 327d 203d 3e20 7b36 7d20 7c20  1}, 2} => {6} | 
│ │ │ │ +0003f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003f5f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f610: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f670: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f640: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f660: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f6c0: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +0003f6b0: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
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│ │ │ │ -0003f700: 2020 2020 2020 7b7b 302c 2031 7d2c 2033        {{0, 1}, 3
│ │ │ │ -0003f710: 7d20 3d3e 207b 387d 207c 2020 2020 2020  } => {8} |      
│ │ │ │ +0003f6e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f6f0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f700: 317d 2c20 337d 203d 3e20 7b38 7d20 7c20  1}, 3} => {8} | 
│ │ │ │ +0003f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f750: 2020 2020 2020 7b7b 302c 2031 7d2c 2034        {{0, 1}, 4
│ │ │ │ -0003f760: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003f730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f740: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f750: 317d 2c20 347d 203d 3e20 3020 2020 2020  1}, 4} => 0     
│ │ │ │ +0003f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f7a0: 2020 2020 2020 7b7b 302c 2031 7d2c 2035        {{0, 1}, 5
│ │ │ │ -0003f7b0: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003f780: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f790: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f7a0: 317d 2c20 357d 203d 3e20 3020 2020 2020  1}, 5} => 0     
│ │ │ │ +0003f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f7f0: 2020 2020 2020 7b7b 302c 2032 7d2c 202d        {{0, 2}, -
│ │ │ │ -0003f800: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003f7d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f7e0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f7f0: 327d 2c20 2d31 7d20 3d3e 2030 2020 2020  2}, -1} => 0    
│ │ │ │ +0003f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f840: 2020 2020 2020 7b7b 302c 2032 7d2c 2030        {{0, 2}, 0
│ │ │ │ -0003f850: 7d20 3d3e 207b 367d 207c 2020 2020 2020  } => {6} |      
│ │ │ │ +0003f820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f830: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f840: 327d 2c20 307d 203d 3e20 7b36 7d20 7c20  2}, 0} => {6} | 
│ │ │ │ +0003f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f8a0: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f870: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f890: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f8f0: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f8c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f8e0: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f940: 2020 2020 207b 367d 207c 2020 2020 2020       {6} |      
│ │ │ │ +0003f910: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f930: 2020 2020 2020 2020 2020 7b36 7d20 7c20            {6} | 
│ │ │ │ +0003f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f970: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f980: 2020 2020 2020 7b7b 302c 2032 7d2c 2031        {{0, 2}, 1
│ │ │ │ -0003f990: 7d20 3d3e 207b 387d 207c 2020 2020 2020  } => {8} |      
│ │ │ │ +0003f960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f970: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f980: 327d 2c20 317d 203d 3e20 7b38 7d20 7c20  2}, 1} => {8} | 
│ │ │ │ +0003f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f9c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003f9d0: 2020 2020 2020 7b7b 302c 2032 7d2c 2032        {{0, 2}, 2
│ │ │ │ -0003f9e0: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003f9b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f9c0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003f9d0: 327d 2c20 327d 203d 3e20 3020 2020 2020  2}, 2} => 0     
│ │ │ │ +0003f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fa20: 2020 2020 2020 7b7b 302c 2033 7d2c 202d        {{0, 3}, -
│ │ │ │ -0003fa30: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003fa00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fa10: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003fa20: 337d 2c20 2d31 7d20 3d3e 2030 2020 2020  3}, -1} => 0    
│ │ │ │ +0003fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fa70: 2020 2020 2020 7b7b 302c 2033 7d2c 2030        {{0, 3}, 0
│ │ │ │ -0003fa80: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ +0003fa50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fa60: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003fa70: 337d 2c20 307d 203d 3e20 3020 2020 2020  3}, 0} => 0     
│ │ │ │ +0003fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fab0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fac0: 2020 2020 2020 7b7b 302c 2034 7d2c 202d        {{0, 4}, -
│ │ │ │ -0003fad0: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003faa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fab0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003fac0: 347d 2c20 2d31 7d20 3d3e 2030 2020 2020  4}, -1} => 0    
│ │ │ │ +0003fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fb10: 2020 2020 2020 7b7b 302c 2035 7d2c 202d        {{0, 5}, -
│ │ │ │ -0003fb20: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003faf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fb00: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +0003fb10: 357d 2c20 2d31 7d20 3d3e 2030 2020 2020  5}, -1} => 0    
│ │ │ │ +0003fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fb60: 2020 2020 2020 7b7b 312c 2030 7d2c 202d        {{1, 0}, -
│ │ │ │ -0003fb70: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +0003fb40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fb50: 2020 2020 2020 2020 2020 207b 7b31 2c20             {{1, 
│ │ │ │ +0003fb60: 307d 2c20 2d31 7d20 3d3e 2030 2020 2020  0}, -1} => 0    
│ │ │ │ +0003fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fba0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fbb0: 2020 2020 2020 7b7b 312c 2030 7d2c 2030        {{1, 0}, 0
│ │ │ │ -0003fbc0: 7d20 3d3e 207b 327d 207c 2020 2020 2020  } => {2} |      
│ │ │ │ +0003fb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fba0: 2020 2020 2020 2020 2020 207b 7b31 2c20             {{1, 
│ │ │ │ +0003fbb0: 307d 2c20 307d 203d 3e20 7b32 7d20 7c20  0}, 0} => {2} | 
│ │ │ │ +0003fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fbf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fc10: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003fbe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fc00: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fc60: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003fc30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fc50: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fc90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fcb0: 2020 2020 207b 327d 207c 2020 2020 2020       {2} |      
│ │ │ │ +0003fc80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fca0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0003fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fce0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fcf0: 2020 2020 2020 7b7b 312c 2030 7d2c 2031        {{1, 0}, 1
│ │ │ │ -0003fd00: 7d20 3d3e 207b 347d 207c 2020 2020 2020  } => {4} |      
│ │ │ │ +0003fcd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fce0: 2020 2020 2020 2020 2020 207b 7b31 2c20             {{1, 
│ │ │ │ +0003fcf0: 307d 2c20 317d 203d 3e20 7b34 7d20 7c20  0}, 1} => {4} | 
│ │ │ │ +0003fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd50: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003fd20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fd40: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fda0: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003fd70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fd90: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fdd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fdf0: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003fdc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fde0: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fe40: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003fe10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fe30: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
│ │ │ │ +0003fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fe70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fe90: 2020 2020 207b 347d 207c 2020 2020 2020       {4} |      
│ │ │ │ +0003fe60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fe80: 2020 2020 2020 2020 2020 7b34 7d20 7c20            {4} | 
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│ │ │ │ +000408e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000408f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00040920: 2020 2020 2020 7b7b 342c 2030 7d2c 202d        {{4, 0}, -
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│ │ │ │ +00040900: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +00040950: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00040960: 2020 2020 2020 2020 2020 207b 7b34 2c20             {{4, 
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│ │ │ │ -000409c0: 2020 2020 2020 7b7b 352c 2030 7d2c 202d        {{5, 0}, -
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│ │ │ │ +000409a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000409b0: 2020 2020 2020 2020 2020 207b 7b35 2c20             {{5, 
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│ │ │ │ -00040a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040a50: 2020 207c 0a7c 6f31 3320 3a20 4861 7368     |.|o13 : Hash
│ │ │ │ -00040a60: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +00040a40: 2020 2020 2020 2020 7c0a 7c6f 3133 203a          |.|o13 :
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│ │ │ │ +00040bd0: 2020 2020 2020 2020 7c0a 7c20 6132 2020          |.| a2  
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│ │ │ │ -00040c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00040c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00040cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040cd0: 2020 207c 0a7c 2063 3220 2064 3220 2030     |.| c2  d2  0
│ │ │ │ -00040ce0: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +00040cc0: 2020 2020 2020 2020 7c0a 7c20 6332 2020          |.| c2  
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│ │ │ │ -00040d70: 2020 207c 0a7c 2061 3220 2030 2020 2030     |.| a2  0   0
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│ │ │ │ +00040d60: 2020 2020 2020 2020 7c0a 7c20 6132 2020          |.| a2  
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│ │ │ │ +00040e00: 2020 2020 2020 2020 7c0a 7c20 3020 2020          |.| 0   
│ │ │ │ +00040e10: 6132 2020 3020 2020 2d63 3220 7c20 2020  a2  0   -c2 |   
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│ │ │ │ -00040e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040e60: 2020 207c 0a7c 2030 2020 2030 2020 2061     |.| 0   0   a
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│ │ │ │ +00040e50: 2020 2020 2020 2020 7c0a 7c20 3020 2020          |.| 0   
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│ │ │ │  00040ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00040ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040f00: 2020 207c 0a7c 2063 3220 207c 2020 2020     |.| c2  |    
│ │ │ │ +00040ef0: 2020 2020 2020 2020 7c0a 7c20 6332 2020          |.| c2  
│ │ │ │ +00040f00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00040f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00040f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040f50: 2020 207c 0a7c 202d 6232 207c 2020 2020     |.| -b2 |    
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│ │ │ │  00040f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00040f90: 2020 2020 2020 2020 7c0a 7c20 6132 2020          |.| a2  
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│ │ │ │  00040fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000410d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000410e0: 2020 207c 0a7c 2031 3233 3661 332b 3839     |.| 1236a3+89
│ │ │ │ -000410f0: 3232 6132 622d 3335 3839 6162 322b 3839  22a2b-3589ab2+89
│ │ │ │ -00041100: 3731 6233 2d35 3030 3661 3263 2d35 3539  71b3-5006a2c-559
│ │ │ │ -00041110: 3961 6263 2d31 3431 3635 6232 632b 3838  9abc-14165b2c+88
│ │ │ │ -00041120: 3830 6132 642b 3432 3539 6162 642d 3330  80a2d+4259abd-30
│ │ │ │ -00041130: 3032 627c 0a7c 2031 3033 3730 6162 322d  02b|.| 10370ab2-
│ │ │ │ -00041140: 3730 3932 6233 2d39 3730 3261 6263 2d36  7092b3-9702abc-6
│ │ │ │ -00041150: 3632 3762 3263 2b38 3838 3661 6264 2b34  627b2c+8886abd+4
│ │ │ │ -00041160: 3730 3062 3264 2d31 3561 6364 2b35 3936  700b2d-15acd+596
│ │ │ │ -00041170: 3962 6364 2020 2020 2020 2020 2020 2020  9bcd            
│ │ │ │ -00041180: 2020 207c 0a7c 2031 3337 3037 6133 2b32     |.| 13707a3+2
│ │ │ │ -00041190: 3137 3761 3262 2d37 3032 3861 6232 2b39  177a2b-7028ab2+9
│ │ │ │ -000411a0: 3739 3762 332d 3730 3231 6132 632b 3633  797b3-7021a2c+63
│ │ │ │ -000411b0: 3737 6162 632b 3538 3734 6232 632d 3736  77abc+5874b2c-76
│ │ │ │ -000411c0: 3030 6163 322d 3131 3732 3662 6332 2b31  00ac2-11726bc2+1
│ │ │ │ -000411d0: 3134 307c 0a7c 202d 3939 3461 332d 3139  140|.| -994a3-19
│ │ │ │ -000411e0: 3436 6132 622d 3637 3233 6162 322d 3534  46a2b-6723ab2-54
│ │ │ │ -000411f0: 3833 6233 2d36 3435 3361 3263 2d31 3139  83b3-6453a2c-119
│ │ │ │ -00041200: 3261 6263 2d31 3532 3530 6232 632d 3331  2abc-15250b2c-31
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│ │ │ │ -00041220: 3063 337c 0a7c 202d 3130 3337 3061 6232  0c3|.| -10370ab2
│ │ │ │ -00041230: 2b37 3039 3262 332b 3937 3032 6162 632b  +7092b3+9702abc+
│ │ │ │ -00041240: 3636 3237 6232 632b 3730 3238 6163 322d  6627b2c+7028ac2-
│ │ │ │ -00041250: 3937 3937 6263 322d 3538 3734 6333 2d38  9797bc2-5874c3-8
│ │ │ │ -00041260: 3838 3661 6264 2d34 3730 3062 3264 2b31  886abd-4700b2d+1
│ │ │ │ -00041270: 3561 637c 0a7c 202d 3133 3730 3761 332d  5ac|.| -13707a3-
│ │ │ │ -00041280: 3231 3737 6132 622b 3730 3231 6132 632d  2177a2b+7021a2c-
│ │ │ │ -00041290: 3633 3737 6162 632b 3736 3030 6163 322b  6377abc+7600ac2+
│ │ │ │ -000412a0: 3131 3732 3662 6332 2d31 3134 3063 332d  11726bc2-1140c3-
│ │ │ │ -000412b0: 3132 3036 6132 642b 3131 3433 3561 6264  1206a2d+11435abd
│ │ │ │ -000412c0: 2b36 307c 0a7c 2037 3032 3861 332d 3937  +60|.| 7028a3-97
│ │ │ │ -000412d0: 3937 6132 622d 3538 3734 6132 632d 3930  97a2b-5874a2c-90
│ │ │ │ -000412e0: 3734 6132 6420 2020 2020 2020 2020 2020  74a2d           
│ │ │ │ +000410d0: 2020 2020 2020 2020 7c0a 7c20 3132 3336          |.| 1236
│ │ │ │ +000410e0: 6133 2b38 3932 3261 3262 2d33 3538 3961  a3+8922a2b-3589a
│ │ │ │ +000410f0: 6232 2b38 3937 3162 332d 3530 3036 6132  b2+8971b3-5006a2
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│ │ │ │ +00041110: 3263 2b38 3838 3061 3264 2b34 3235 3961  2c+8880a2d+4259a
│ │ │ │ +00041120: 6264 2d33 3030 3262 7c0a 7c20 3130 3337  bd-3002b|.| 1037
│ │ │ │ +00041130: 3061 6232 2d37 3039 3262 332d 3937 3032  0ab2-7092b3-9702
│ │ │ │ +00041140: 6162 632d 3636 3237 6232 632b 3838 3836  abc-6627b2c+8886
│ │ │ │ +00041150: 6162 642b 3437 3030 6232 642d 3135 6163  abd+4700b2d-15ac
│ │ │ │ +00041160: 642b 3539 3639 6263 6420 2020 2020 2020  d+5969bcd       
│ │ │ │ +00041170: 2020 2020 2020 2020 7c0a 7c20 3133 3730          |.| 1370
│ │ │ │ +00041180: 3761 332b 3231 3737 6132 622d 3730 3238  7a3+2177a2b-7028
│ │ │ │ +00041190: 6162 322b 3937 3937 6233 2d37 3032 3161  ab2+9797b3-7021a
│ │ │ │ +000411a0: 3263 2b36 3337 3761 6263 2b35 3837 3462  2c+6377abc+5874b
│ │ │ │ +000411b0: 3263 2d37 3630 3061 6332 2d31 3137 3236  2c-7600ac2-11726
│ │ │ │ +000411c0: 6263 322b 3131 3430 7c0a 7c20 2d39 3934  bc2+1140|.| -994
│ │ │ │ +000411d0: 6133 2d31 3934 3661 3262 2d36 3732 3361  a3-1946a2b-6723a
│ │ │ │ +000411e0: 6232 2d35 3438 3362 332d 3634 3533 6132  b2-5483b3-6453a2
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│ │ │ │ +00041200: 3263 2d33 3136 3461 6332 2b32 3935 6263  2c-3164ac2+295bc
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│ │ │ │ +00041230: 3261 6263 2b36 3632 3762 3263 2b37 3032  2abc+6627b2c+702
│ │ │ │ +00041240: 3861 6332 2d39 3739 3762 6332 2d35 3837  8ac2-9797bc2-587
│ │ │ │ +00041250: 3463 332d 3838 3836 6162 642d 3437 3030  4c3-8886abd-4700
│ │ │ │ +00041260: 6232 642b 3135 6163 7c0a 7c20 2d31 3337  b2d+15ac|.| -137
│ │ │ │ +00041270: 3037 6133 2d32 3137 3761 3262 2b37 3032  07a3-2177a2b+702
│ │ │ │ +00041280: 3161 3263 2d36 3337 3761 6263 2b37 3630  1a2c-6377abc+760
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│ │ │ │ +00042390: 3337 6233 632b 3530 7c0a 7c20 3233 3938  37b3c+50|.| 2398
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│ │ │ │ +000423e0: 3530 6132 6364 2d39 7c0a 7c20 3639 3435  50a2cd-9|.| 6945
│ │ │ │ +000423f0: 6134 2b31 3230 3938 6133 622d 3839 3738  a4+12098a3b-8978
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│ │ │ │ -00042ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00042b60: 2020 2020 2020 2020 7c0a 7c63 332b 3132          |.|c3+12
│ │ │ │ +00042b70: 3036 6132 642d 3131 3433 3561 6264 2b39  06a2d-11435abd+9
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│ │ │ │ +00042b90: 3032 3262 6364 2b33 3936 3863 3264 2020  022bcd+3968c2d  
│ │ │ │ +00042ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042bb0: 2020 2020 2020 2020 7c0a 7c2d 3131 3034          |.|-1104
│ │ │ │ +00042bc0: 3561 3264 2d31 3533 3333 6162 642b 3130  5a2d-15333abd+10
│ │ │ │ +00042bd0: 3536 3762 3264 2d33 3536 3061 6364 2b32  567b2d-3560acd+2
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│ │ │ │ +00042c20: 3361 6432 2b35 3438 3362 6432 2b31 3532  3ad2+5483bd2+152
│ │ │ │ +00042c30: 3530 6364 322d 3130 3536 3764 3320 2020  50cd2-10567d3   
│ │ │ │ +00042c40: 3132 3336 6133 2b38 3932 3261 3262 2d33  1236a3+8922a2b-3
│ │ │ │ +00042c50: 3538 3961 6232 2b38 7c0a 7c34 3061 6364  589ab2+8|.|40acd
│ │ │ │ +00042c60: 2d38 3032 3262 6364 2d33 3936 3863 3264  -8022bcd-3968c2d
│ │ │ │ +00042c70: 2b33 3136 3461 6432 2d32 3935 6264 322b  +3164ad2-295bd2+
│ │ │ │ +00042c80: 3736 3530 6364 322d 3134 3338 3864 3320  7650cd2-14388d3 
│ │ │ │ +00042c90: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00042ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00042cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ce0: 2020 2020 2020 2020 2020 202d 3133 3730             -1370
│ │ │ │ -00042cf0: 3761 332d 3231 3737 6132 622b 3730 3238  7a3-2177a2b+7028
│ │ │ │ -00042d00: 6162 327c 0a7c 6264 322b 3836 3339 6364  ab2|.|bd2+8639cd
│ │ │ │ -00042d10: 322d 3934 3236 6433 2020 2020 2020 2020  2-9426d3        
│ │ │ │ +00042ce0: 2d31 3337 3037 6133 2d32 3137 3761 3262  -13707a3-2177a2b
│ │ │ │ +00042cf0: 2b37 3032 3861 6232 7c0a 7c62 6432 2b38  +7028ab2|.|bd2+8
│ │ │ │ +00042d00: 3633 3963 6432 2d39 3432 3664 3320 2020  639cd2-9426d3   
│ │ │ │ +00042d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00042d80: 2020 2020 2020 2020 2020 2039 3934 6133             994a3
│ │ │ │ -00042d90: 2b31 3934 3661 3262 2b36 3732 3361 6232  +1946a2b+6723ab2
│ │ │ │ -00042da0: 2b35 347c 0a7c 2020 2020 2020 2020 2020  +54|.|          
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│ │ │ │ +00042d90: 3233 6162 322b 3534 7c0a 7c20 2020 2020  23ab2+54|.|     
│ │ │ │ +00042da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042dd0: 2020 2020 2020 2020 2020 2033 3136 3461             3164a
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│ │ │ │ -00042df0: 2d31 347c 0a7c 3061 6364 2b38 3032 3262  -14|.|0acd+8022b
│ │ │ │ -00042e00: 6364 2b33 3936 3863 3264 2d33 3136 3461  cd+3968c2d-3164a
│ │ │ │ -00042e10: 6432 2b32 3935 6264 322d 3736 3530 6364  d2+295bd2-7650cd
│ │ │ │ -00042e20: 322b 3134 3338 3864 3320 2d31 3033 3730  2+14388d3 -10370
│ │ │ │ -00042e30: 6162 322b 3730 3932 6233 2b39 3730 3261  ab2+7092b3+9702a
│ │ │ │ -00042e40: 6263 2b7c 0a7c 3562 6432 2d38 3633 3963  bc+|.|5bd2-8639c
│ │ │ │ -00042e50: 6432 2b39 3432 3664 3320 2020 2020 2020  d2+9426d3       
│ │ │ │ -00042e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042e70: 2020 2020 2020 2020 2020 3637 3233 6163            6723ac
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│ │ │ │  00044030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044060: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044080: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000440a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000440b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000440a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000440b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000440c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000440d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000440e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000440f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000440f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000441a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000441a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000441b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000441c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000441d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000441e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000441f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000441e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000441f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044280: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000442a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000442b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000442c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000442d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000442e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000442d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000442e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000442f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044330: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044380: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000443a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000443b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000443c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000443d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000443c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000443d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000443e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000443f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000444a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000444b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000444c0: 2020 207c 0a7c 2020 2020 2020 207c 2020     |.|       |  
│ │ │ │ +000444b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000444c0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  000444d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000444e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000444f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044510: 2020 207c 0a7c 2020 2020 2020 207c 2020     |.|       |  
│ │ │ │ +00044500: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044510: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00044520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044560: 2020 207c 0a7c 2020 2020 2020 207c 2020     |.|       |  
│ │ │ │ +00044550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044560: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │  00044570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000445a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000445b0: 2020 207c 0a7c 3934 3236 6433 207c 2020     |.|9426d3 |  
│ │ │ │ +000445a0: 2020 2020 2020 2020 7c0a 7c39 3432 3664          |.|9426d
│ │ │ │ +000445b0: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │  000445c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000445d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000445e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000445f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044600: 2020 207c 0a7c 3937 3162 332d 3530 3036     |.|971b3-5006
│ │ │ │ -00044610: 6132 632d 3535 3939 6162 632d 3134 3136  a2c-5599abc-1416
│ │ │ │ -00044620: 3562 3263 2b38 3838 3061 3264 2b34 3235  5b2c+8880a2d+425
│ │ │ │ -00044630: 3961 6264 2d33 3030 3262 3264 2d31 3338  9abd-3002b2d-138
│ │ │ │ -00044640: 3932 6163 642d 3130 3532 3162 6364 2020  92acd-10521bcd  
│ │ │ │ -00044650: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000445f0: 2020 2020 2020 2020 7c0a 7c39 3731 6233          |.|971b3
│ │ │ │ +00044600: 2d35 3030 3661 3263 2d35 3539 3961 6263  -5006a2c-5599abc
│ │ │ │ +00044610: 2d31 3431 3635 6232 632b 3838 3830 6132  -14165b2c+8880a2
│ │ │ │ +00044620: 642b 3432 3539 6162 642d 3330 3032 6232  d+4259abd-3002b2
│ │ │ │ +00044630: 642d 3133 3839 3261 6364 2d31 3035 3231  d-13892acd-10521
│ │ │ │ +00044640: 6263 6420 2020 2020 7c0a 7c20 2020 2020  bcd     |.|     
│ │ │ │ +00044650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000446a0: 2020 207c 0a7c 2d39 3739 3762 332b 3730     |.|-9797b3+70
│ │ │ │ -000446b0: 3231 6132 632d 3633 3737 6162 632d 3538  21a2c-6377abc-58
│ │ │ │ -000446c0: 3734 6232 632b 3736 3030 6163 322b 3131  74b2c+7600ac2+11
│ │ │ │ -000446d0: 3732 3662 6332 2d31 3134 3063 332d 3132  726bc2-1140c3-12
│ │ │ │ -000446e0: 3036 6132 642b 3131 3433 3561 6264 2d39  06a2d+11435abd-9
│ │ │ │ -000446f0: 3037 347c 0a7c 2020 2020 2020 2020 2020  074|.|          
│ │ │ │ +00044690: 2020 2020 2020 2020 7c0a 7c2d 3937 3937          |.|-9797
│ │ │ │ +000446a0: 6233 2b37 3032 3161 3263 2d36 3337 3761  b3+7021a2c-6377a
│ │ │ │ +000446b0: 6263 2d35 3837 3462 3263 2b37 3630 3061  bc-5874b2c+7600a
│ │ │ │ +000446c0: 6332 2b31 3137 3236 6263 322d 3131 3430  c2+11726bc2-1140
│ │ │ │ +000446d0: 6333 2d31 3230 3661 3264 2b31 3134 3335  c3-1206a2d+11435
│ │ │ │ +000446e0: 6162 642d 3930 3734 7c0a 7c20 2020 2020  abd-9074|.|     
│ │ │ │ +000446f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044740: 2020 207c 0a7c 3833 6233 2b36 3435 3361     |.|83b3+6453a
│ │ │ │ -00044750: 3263 2b31 3139 3261 6263 2b31 3532 3530  2c+1192abc+15250
│ │ │ │ -00044760: 6232 632b 3131 3034 3561 3264 2b31 3533  b2c+11045a2d+153
│ │ │ │ -00044770: 3333 6162 642d 3130 3536 3762 3264 2b33  33abd-10567b2d+3
│ │ │ │ -00044780: 3536 3061 6364 2d32 3239 3262 6364 2b37  560acd-2292bcd+7
│ │ │ │ -00044790: 3139 347c 0a7c 3338 3862 3264 2020 2020  194|.|388b2d    
│ │ │ │ +00044730: 2020 2020 2020 2020 7c0a 7c38 3362 332b          |.|83b3+
│ │ │ │ +00044740: 3634 3533 6132 632b 3131 3932 6162 632b  6453a2c+1192abc+
│ │ │ │ +00044750: 3135 3235 3062 3263 2b31 3130 3435 6132  15250b2c+11045a2
│ │ │ │ +00044760: 642b 3135 3333 3361 6264 2d31 3035 3637  d+15333abd-10567
│ │ │ │ +00044770: 6232 642b 3335 3630 6163 642d 3232 3932  b2d+3560acd-2292
│ │ │ │ +00044780: 6263 642b 3731 3934 7c0a 7c33 3838 6232  bcd+7194|.|388b2
│ │ │ │ +00044790: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  000447a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000447b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000447c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000447d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000447e0: 2020 207c 0a7c 3636 3237 6232 632b 3730     |.|6627b2c+70
│ │ │ │ -000447f0: 3238 6163 322d 3937 3937 6263 322d 3538  28ac2-9797bc2-58
│ │ │ │ -00044800: 3734 6333 2d38 3838 3661 6264 2d34 3730  74c3-8886abd-470
│ │ │ │ -00044810: 3062 3264 2b31 3561 6364 2d35 3936 3962  0b2d+15acd-5969b
│ │ │ │ -00044820: 6364 2d39 3037 3463 3264 2b35 3438 3362  cd-9074c2d+5483b
│ │ │ │ -00044830: 6432 2b7c 0a7c 2020 2020 2020 2020 2020  d2+|.|          
│ │ │ │ +000447d0: 2020 2020 2020 2020 7c0a 7c36 3632 3762          |.|6627b
│ │ │ │ +000447e0: 3263 2b37 3032 3861 6332 2d39 3739 3762  2c+7028ac2-9797b
│ │ │ │ +000447f0: 6332 2d35 3837 3463 332d 3838 3836 6162  c2-5874c3-8886ab
│ │ │ │ +00044800: 642d 3437 3030 6232 642b 3135 6163 642d  d-4700b2d+15acd-
│ │ │ │ +00044810: 3539 3639 6263 642d 3930 3734 6332 642b  5969bcd-9074c2d+
│ │ │ │ +00044820: 3534 3833 6264 322b 7c0a 7c20 2020 2020  5483bd2+|.|     
│ │ │ │ +00044830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044880: 2020 207c 0a7c 3533 6132 632d 3131 3932     |.|53a2c-1192
│ │ │ │ -00044890: 6162 632d 3131 3034 3561 3264 2d31 3533  abc-11045a2d-153
│ │ │ │ -000448a0: 3333 6162 642d 3335 3630 6163 642b 3232  33abd-3560acd+22
│ │ │ │ -000448b0: 3932 6263 642d 3731 3934 6164 322b 3632  92bcd-7194ad2+62
│ │ │ │ -000448c0: 3435 6264 322d 3836 3339 6364 322b 3934  45bd2-8639cd2+94
│ │ │ │ -000448d0: 3236 647c 0a7c 6420 2020 2020 2020 2020  26d|.|d         
│ │ │ │ +00044870: 2020 2020 2020 2020 7c0a 7c35 3361 3263          |.|53a2c
│ │ │ │ +00044880: 2d31 3139 3261 6263 2d31 3130 3435 6132  -1192abc-11045a2
│ │ │ │ +00044890: 642d 3135 3333 3361 6264 2d33 3536 3061  d-15333abd-3560a
│ │ │ │ +000448a0: 6364 2b32 3239 3262 6364 2d37 3139 3461  cd+2292bcd-7194a
│ │ │ │ +000448b0: 6432 2b36 3234 3562 6432 2d38 3633 3963  d2+6245bd2-8639c
│ │ │ │ +000448c0: 6432 2b39 3432 3664 7c0a 7c64 2020 2020  d2+9426d|.|d    
│ │ │ │ +000448d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000448e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000448f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044920: 2020 207c 0a7c 3732 3662 6332 2b31 3134     |.|726bc2+114
│ │ │ │ -00044930: 3063 332b 3132 3036 6132 642d 3131 3433  0c3+1206a2d-1143
│ │ │ │ -00044940: 3561 6264 2d36 3034 3061 6364 2b38 3032  5abd-6040acd+802
│ │ │ │ -00044950: 3262 6364 2b33 3936 3863 3264 2d33 3136  2bcd+3968c2d-316
│ │ │ │ -00044960: 3461 6432 2b32 3935 6264 322d 3736 3530  4ad2+295bd2-7650
│ │ │ │ -00044970: 6364 327c 0a7c 2020 2020 2020 2020 2020  cd2|.|          
│ │ │ │ +00044910: 2020 2020 2020 2020 7c0a 7c37 3236 6263          |.|726bc
│ │ │ │ +00044920: 322b 3131 3430 6333 2b31 3230 3661 3264  2+1140c3+1206a2d
│ │ │ │ +00044930: 2d31 3134 3335 6162 642d 3630 3430 6163  -11435abd-6040ac
│ │ │ │ +00044940: 642b 3830 3232 6263 642b 3339 3638 6332  d+8022bcd+3968c2
│ │ │ │ +00044950: 642d 3331 3634 6164 322b 3239 3562 6432  d-3164ad2+295bd2
│ │ │ │ +00044960: 2d37 3635 3063 6432 7c0a 7c20 2020 2020  -7650cd2|.|     
│ │ │ │ +00044970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000449a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000449b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000449c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000449d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000449e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000449f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044a10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00044a00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044a60: 2020 207c 0a7c 3532 6162 6364 2d31 3833     |.|52abcd-183
│ │ │ │ -00044a70: 3162 3263 642b 3630 3432 6163 3264 2d32  1b2cd+6042ac2d-2
│ │ │ │ -00044a80: 3536 3162 6332 642d 3837 3039 6132 6432  561bc2d-8709a2d2
│ │ │ │ -00044a90: 2d31 3332 3139 6162 6432 2b34 3230 3962  -13219abd2+4209b
│ │ │ │ -00044aa0: 3264 322b 3132 3232 3561 6364 322d 3236  2d2+12225acd2-26
│ │ │ │ -00044ab0: 3035 627c 0a7c 3135 3838 6132 6364 2b32  05b|.|1588a2cd+2
│ │ │ │ -00044ac0: 3030 3061 6263 642d 3230 3830 6232 6364  000abcd-2080b2cd
│ │ │ │ -00044ad0: 2b39 3137 3561 6332 642d 3634 3962 6332  +9175ac2d-649bc2
│ │ │ │ -00044ae0: 642b 3838 3239 6333 642b 3231 3634 6132  d+8829c3d+2164a2
│ │ │ │ -00044af0: 6432 2b38 3633 3561 6264 322d 3731 3631  d2+8635abd2-7161
│ │ │ │ -00044b00: 6232 647c 0a7c 2020 2020 2020 2020 2020  b2d|.|          
│ │ │ │ +00044a50: 2020 2020 2020 2020 7c0a 7c35 3261 6263          |.|52abc
│ │ │ │ +00044a60: 642d 3138 3331 6232 6364 2b36 3034 3261  d-1831b2cd+6042a
│ │ │ │ +00044a70: 6332 642d 3235 3631 6263 3264 2d38 3730  c2d-2561bc2d-870
│ │ │ │ +00044a80: 3961 3264 322d 3133 3231 3961 6264 322b  9a2d2-13219abd2+
│ │ │ │ +00044a90: 3432 3039 6232 6432 2b31 3232 3235 6163  4209b2d2+12225ac
│ │ │ │ +00044aa0: 6432 2d32 3630 3562 7c0a 7c31 3538 3861  d2-2605b|.|1588a
│ │ │ │ +00044ab0: 3263 642b 3230 3030 6162 6364 2d32 3038  2cd+2000abcd-208
│ │ │ │ +00044ac0: 3062 3263 642b 3931 3735 6163 3264 2d36  0b2cd+9175ac2d-6
│ │ │ │ +00044ad0: 3439 6263 3264 2b38 3832 3963 3364 2b32  49bc2d+8829c3d+2
│ │ │ │ +00044ae0: 3136 3461 3264 322b 3836 3335 6162 6432  164a2d2+8635abd2
│ │ │ │ +00044af0: 2d37 3136 3162 3264 7c0a 7c20 2020 2020  -7161b2d|.|     
│ │ │ │ +00044b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044b50: 2020 207c 0a7c 3732 3362 3263 642d 3133     |.|723b2cd-13
│ │ │ │ -00044b60: 3236 3261 6332 642b 3534 3331 6263 3264  262ac2d+5431bc2d
│ │ │ │ -00044b70: 2b31 3132 3734 6132 6432 2d32 3137 6162  +11274a2d2-217ab
│ │ │ │ -00044b80: 6432 2b31 3236 3162 3264 322b 3832 3031  d2+1261b2d2+8201
│ │ │ │ -00044b90: 6163 6432 2d31 3430 3830 6263 6432 2020  acd2-14080bcd2  
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│ │ │ │ -00044bf0: 3132 377c 0a7c 2020 2020 2020 2020 2020  127|.|          
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│ │ │ │ +00044b90: 6364 3220 2020 2020 7c0a 7c33 6164 3320  cd2     |.|3ad3 
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│ │ │ │ +00044eb0: 2020 2020 2020 2020 7c0a 7c32 2d35 3038          |.|2-508
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│ │ │ │ +00044f00: 2020 2020 2020 2020 7c0a 7c31 3033 3539          |.|10359
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│ │ │ │ +00044f50: 6364 2020 2020 2020 7c0a 7c20 2020 2020  cd      |.|     
│ │ │ │ +00044f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00044fa0: 2020 2020 2020 2020 7c0a 7c62 322b 3133          |.|b2+13
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│ │ │ │ -00045140: 3264 2d7c 0a7c 2020 2020 2020 2020 2020  2d-|.|          
│ │ │ │ +000450e0: 2020 2020 2020 2020 7c0a 7c62 632d 3133          |.|bc-13
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│ │ │ │ -000451e0: 6432 2d7c 0a7c 3264 2020 2020 2020 2020  d2-|.|2d        
│ │ │ │ +00045180: 2020 2020 2020 2020 7c0a 7c62 322d 3934          |.|b2-94
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│ │ │ │ +000451b0: 3939 3661 6364 2d37 3135 3262 6364 2b39  996acd-7152bcd+9
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│ │ │ │ +000451d0: 3639 3232 6364 322d 7c0a 7c32 6420 2020  6922cd2-|.|2d   
│ │ │ │ +000451e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000451f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045200: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045270: 642b 3531 3037 6164 322b 3536 3739 6264  d+5107ad2+5679bd
│ │ │ │ -00045280: 322d 367c 0a7c 2020 2020 2020 2020 2020  2-6|.|          
│ │ │ │ +00045220: 2020 2020 2020 2020 7c0a 7c63 322d 3533          |.|c2-53
│ │ │ │ +00045230: 3236 6263 322b 3130 3331 6333 2d31 3335  26bc2+1031c3-135
│ │ │ │ +00045240: 3038 6132 642d 3130 3132 3561 6264 2b39  08a2d-10125abd+9
│ │ │ │ +00045250: 3033 3361 6364 2b32 3939 3862 6364 2d32  033acd+2998bcd-2
│ │ │ │ +00045260: 3033 3663 3264 2b35 3130 3761 6432 2b35  036c2d+5107ad2+5
│ │ │ │ +00045270: 3637 3962 6432 2d36 7c0a 7c20 2020 2020  679bd2-6|.|     
│ │ │ │ +00045280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000452a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000452b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000452d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045370: 2020 207c 0a7c 3837 6162 6364 2d31 3436     |.|87abcd-146
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│ │ │ │ -000453a0: 3264 322d 3137 3933 6162 6432 2d31 3232  2d2-1793abd2-122
│ │ │ │ -000453b0: 3435 6232 6432 2b31 3232 3833 6163 6432  45b2d2+12283acd2
│ │ │ │ -000453c0: 2b36 377c 0a7c 3264 2b35 3231 3862 3364  +67|.|2d+5218b3d
│ │ │ │ -000453d0: 2d31 3238 3032 6132 6364 2d31 3236 3138  -12802a2cd-12618
│ │ │ │ -000453e0: 6162 6364 2d35 3732 3862 3263 642b 3533  abcd-5728b2cd+53
│ │ │ │ -000453f0: 3437 6163 3264 2b31 3333 3934 6263 3264  47ac2d+13394bc2d
│ │ │ │ -00045400: 2b32 3936 6333 642b 3734 3436 6132 6432  +296c3d+7446a2d2
│ │ │ │ -00045410: 2b31 307c 0a7c 2020 2020 2020 2020 2020  +10|.|          
│ │ │ │ +00045360: 2020 2020 2020 2020 7c0a 7c38 3761 6263          |.|87abc
│ │ │ │ +00045370: 642d 3134 3638 3562 3263 642d 3130 3538  d-14685b2cd-1058
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│ │ │ │ +000453c0: 3138 6233 642d 3132 3830 3261 3263 642d  18b3d-12802a2cd-
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│ │ │ │ +00045400: 3661 3264 322b 3130 7c0a 7c20 2020 2020  6a2d2+10|.|     
│ │ │ │ +00045410: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045460: 2020 207c 0a7c 3433 3961 6263 642d 3132     |.|439abcd-12
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│ │ │ │ -000454b0: 3334 357c 0a7c 3535 3038 6132 6232 2d31  345|.|5508a2b2-1
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│ │ │ │ -00045500: 6232 637c 0a7c 2020 2020 2020 2020 2020  b2c|.|          
│ │ │ │ +00045450: 2020 2020 2020 2020 7c0a 7c34 3339 6162          |.|439ab
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│ │ │ │ +000454f0: 642d 3239 3662 3263 7c0a 7c20 2020 2020  d-296b2c|.|     
│ │ │ │ +00045500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -000456e0: 2020 207c 0a7c 3930 3861 6263 642d 3938     |.|908abcd-98
│ │ │ │ -000456f0: 3934 6232 6364 2b38 3434 3061 6332 642b  94b2cd+8440ac2d+
│ │ │ │ -00045700: 3130 3639 3362 6332 642b 3132 3036 3261  10693bc2d+12062a
│ │ │ │ -00045710: 3264 322d 3133 3632 3661 6264 322b 3432  2d2-13626abd2+42
│ │ │ │ -00045720: 3437 6232 6432 2d31 3237 3033 6163 6432  47b2d2-12703acd2
│ │ │ │ -00045730: 2b32 397c 0a7c 3432 3062 3364 2b35 3233  +29|.|420b3d+523
│ │ │ │ -00045740: 3461 3263 642b 3730 3036 6162 6364 2d31  4a2cd+7006abcd-1
│ │ │ │ -00045750: 3439 3231 6232 6364 2b38 3831 3061 6332  4921b2cd+8810ac2
│ │ │ │ -00045760: 642d 3332 3731 6263 3264 2d31 3036 3633  d-3271bc2d-10663
│ │ │ │ -00045770: 6333 642d 3132 3533 3761 3264 322b 3135  c3d-12537a2d2+15
│ │ │ │ -00045780: 3238 317c 0a7c 2020 2020 2020 2020 2020  281|.|          
│ │ │ │ +000456d0: 2020 2020 2020 2020 7c0a 7c39 3038 6162          |.|908ab
│ │ │ │ +000456e0: 6364 2d39 3839 3462 3263 642b 3834 3430  cd-9894b2cd+8440
│ │ │ │ +000456f0: 6163 3264 2b31 3036 3933 6263 3264 2b31  ac2d+10693bc2d+1
│ │ │ │ +00045700: 3230 3632 6132 6432 2d31 3336 3236 6162  2062a2d2-13626ab
│ │ │ │ +00045710: 6432 2b34 3234 3762 3264 322d 3132 3730  d2+4247b2d2-1270
│ │ │ │ +00045720: 3361 6364 322b 3239 7c0a 7c34 3230 6233  3acd2+29|.|420b3
│ │ │ │ +00045730: 642b 3532 3334 6132 6364 2b37 3030 3661  d+5234a2cd+7006a
│ │ │ │ +00045740: 6263 642d 3134 3932 3162 3263 642b 3838  bcd-14921b2cd+88
│ │ │ │ +00045750: 3130 6163 3264 2d33 3237 3162 6332 642d  10ac2d-3271bc2d-
│ │ │ │ +00045760: 3130 3636 3363 3364 2d31 3235 3337 6132  10663c3d-12537a2
│ │ │ │ +00045770: 6432 2b31 3532 3831 7c0a 7c20 2020 2020  d2+15281|.|     
│ │ │ │ +00045780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045790: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000457d0: 2020 207c 0a7c 3561 6263 642b 3632 3937     |.|5abcd+6297
│ │ │ │ -000457e0: 6232 6364 2b31 3533 3739 6163 3264 2d31  b2cd+15379ac2d-1
│ │ │ │ -000457f0: 3238 3831 6263 3264 2d36 3334 3161 3264  2881bc2d-6341a2d
│ │ │ │ -00045800: 322b 3832 3932 6162 6432 2b31 3138 3632  2+8292abd2+11862
│ │ │ │ -00045810: 6232 6432 2d31 3035 3136 6163 6432 2d31  b2d2-10516acd2-1
│ │ │ │ -00045820: 3234 397c 0a7c 3639 3435 6132 6232 2d31  249|.|6945a2b2-1
│ │ │ │ -00045830: 3230 3938 6162 332d 3732 3437 6234 2b31  2098ab3-7247b4+1
│ │ │ │ -00045840: 3539 3336 6162 3263 2d35 3338 3562 3363  5936ab2c-5385b3c
│ │ │ │ -00045850: 2d35 3337 3562 3263 322d 3130 3330 3561  -5375b2c2-10305a
│ │ │ │ -00045860: 6232 642b 3832 3937 6233 642b 3130 3636  b2d+8297b3d+1066
│ │ │ │ -00045870: 3362 327c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d  3b2|.|----------
│ │ │ │ +000457c0: 2020 2020 2020 2020 7c0a 7c35 6162 6364          |.|5abcd
│ │ │ │ +000457d0: 2b36 3239 3762 3263 642b 3135 3337 3961  +6297b2cd+15379a
│ │ │ │ +000457e0: 6332 642d 3132 3838 3162 6332 642d 3633  c2d-12881bc2d-63
│ │ │ │ +000457f0: 3431 6132 6432 2b38 3239 3261 6264 322b  41a2d2+8292abd2+
│ │ │ │ +00045800: 3131 3836 3262 3264 322d 3130 3531 3661  11862b2d2-10516a
│ │ │ │ +00045810: 6364 322d 3132 3439 7c0a 7c36 3934 3561  cd2-1249|.|6945a
│ │ │ │ +00045820: 3262 322d 3132 3039 3861 6233 2d37 3234  2b2-12098ab3-724
│ │ │ │ +00045830: 3762 342b 3135 3933 3661 6232 632d 3533  7b4+15936ab2c-53
│ │ │ │ +00045840: 3835 6233 632d 3533 3735 6232 6332 2d31  85b3c-5375b2c2-1
│ │ │ │ +00045850: 3033 3035 6162 3264 2b38 3239 3762 3364  0305ab2d+8297b3d
│ │ │ │ +00045860: 2b31 3036 3633 6232 7c0a 7c2d 2d2d 2d2d  +10663b2|.|-----
│ │ │ │ +00045870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000458a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000458b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000458c0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ +000458b0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +000458c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000458d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000458e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045910: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │  00045930: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045af0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00045ae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00045bd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045c30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │  00045c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045cd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00045cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045d70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +00045d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │  00045df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045e10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -00045e60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -00045eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │  00045ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00045ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00045f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00045f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00045f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00045f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045ff0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00045fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046030: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00046040: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00046020: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00046030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ -00046080: 2031 3233 3661 332b 3839 3232 6132 622d   1236a3+8922a2b-
│ │ │ │ -00046090: 3335 387c 0a7c 6232 642b 3630 3430 6163  358|.|b2d+6040ac
│ │ │ │ -000460a0: 642d 3830 3232 6263 642d 3339 3638 6332  d-8022bcd-3968c2
│ │ │ │ -000460b0: 642b 3331 3634 6164 322d 3239 3562 6432  d+3164ad2-295bd2
│ │ │ │ -000460c0: 2b37 3635 3063 6432 2d31 3433 3838 6433  +7650cd2-14388d3
│ │ │ │ -000460d0: 2031 3033 3730 6162 322d 3730 3932 6233   10370ab2-7092b3
│ │ │ │ -000460e0: 2d39 377c 0a7c 2020 2020 2020 2020 2020  -97|.|          
│ │ │ │ +00046070: 2020 2020 2020 3132 3336 6133 2b38 3932        1236a3+892
│ │ │ │ +00046080: 3261 3262 2d33 3538 7c0a 7c62 3264 2b36  2a2b-358|.|b2d+6
│ │ │ │ +00046090: 3034 3061 6364 2d38 3032 3262 6364 2d33  040acd-8022bcd-3
│ │ │ │ +000460a0: 3936 3863 3264 2b33 3136 3461 6432 2d32  968c2d+3164ad2-2
│ │ │ │ +000460b0: 3935 6264 322b 3736 3530 6364 322d 3134  95bd2+7650cd2-14
│ │ │ │ +000460c0: 3338 3864 3320 3130 3337 3061 6232 2d37  388d3 10370ab2-7
│ │ │ │ +000460d0: 3039 3262 332d 3937 7c0a 7c20 2020 2020  092b3-97|.|     
│ │ │ │ +000460e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000460f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046120: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00046130: 2020 207c 0a7c 6164 322d 3632 3435 6264     |.|ad2-6245bd
│ │ │ │ -00046140: 322b 3836 3339 6364 322d 3934 3236 6433  2+8639cd2-9426d3
│ │ │ │ +00046110: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00046120: 2020 2020 2020 2020 7c0a 7c61 6432 2d36          |.|ad2-6
│ │ │ │ +00046130: 3234 3562 6432 2b38 3633 3963 6432 2d39  245bd2+8639cd2-9
│ │ │ │ +00046140: 3432 3664 3320 2020 2020 2020 2020 2020  426d3           
│ │ │ │  00046150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046170: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00046180: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00046160: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00046170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00046180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046190: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000461c0: 2039 3934 6133 2b31 3934 3661 3262 2b36   994a3+1946a2b+6
│ │ │ │ -000461d0: 3732 337c 0a7c 3135 3235 3063 6432 2d31  723|.|15250cd2-1
│ │ │ │ -000461e0: 3035 3637 6433 2031 3233 3661 332b 3839  0567d3 1236a3+89
│ │ │ │ -000461f0: 3232 6132 622d 3335 3839 6162 322b 3839  22a2b-3589ab2+89
│ │ │ │ -00046200: 3731 6233 2d35 3030 3661 3263 2d35 3539  71b3-5006a2c-559
│ │ │ │ -00046210: 3961 6263 2d31 3431 3635 6232 632b 3838  9abc-14165b2c+88
│ │ │ │ -00046220: 3830 617c 0a7c 2020 2020 2020 2020 2020  80a|.|          
│ │ │ │ -00046230: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +000461b0: 2020 2020 2020 3939 3461 332b 3139 3436        994a3+1946
│ │ │ │ +000461c0: 6132 622b 3637 3233 7c0a 7c31 3532 3530  a2b+6723|.|15250
│ │ │ │ +000461d0: 6364 322d 3130 3536 3764 3320 3132 3336  cd2-10567d3 1236
│ │ │ │ +000461e0: 6133 2b38 3932 3261 3262 2d33 3538 3961  a3+8922a2b-3589a
│ │ │ │ +000461f0: 6232 2b38 3937 3162 332d 3530 3036 6132  b2+8971b3-5006a2
│ │ │ │ +00046200: 632d 3535 3939 6162 632d 3134 3136 3562  c-5599abc-14165b
│ │ │ │ +00046210: 3263 2b38 3838 3061 7c0a 7c20 2020 2020  2c+8880a|.|     
│ │ │ │ +00046220: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00046230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046270: 2020 207c 0a7c 3320 2020 2020 2020 2020     |.|3         
│ │ │ │ -00046280: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00046260: 2020 2020 2020 2020 7c0a 7c33 2020 2020          |.|3    
│ │ │ │ +00046270: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00046280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000462d0: 2020 2020 2020 202d 3939 3461 332d 3139         -994a3-19
│ │ │ │ -000462e0: 3436 6132 622d 3637 3233 6162 322d 3534  46a2b-6723ab2-54
│ │ │ │ -000462f0: 3833 6233 2d36 3435 3361 3263 2d31 3139  83b3-6453a2c-119
│ │ │ │ -00046300: 3261 6263 2d31 3532 3530 6232 632d 3131  2abc-15250b2c-11
│ │ │ │ -00046310: 3034 357c 0a7c 2b31 3433 3838 6433 202d  045|.|+14388d3 -
│ │ │ │ -00046320: 3130 3337 3061 6232 2b37 3039 3262 332b  10370ab2+7092b3+
│ │ │ │ -00046330: 3937 3032 6162 632b 3636 3237 6232 632d  9702abc+6627b2c-
│ │ │ │ -00046340: 3937 3937 6263 322d 3538 3734 6333 2d38  9797bc2-5874c3-8
│ │ │ │ -00046350: 3838 3661 6264 2d34 3730 3062 3264 2b31  886abd-4700b2d+1
│ │ │ │ -00046360: 3561 637c 0a7c 2020 2020 2020 2020 2020  5ac|.|          
│ │ │ │ +000462b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000462c0: 2020 2020 2020 2020 2020 2020 2d39 3934              -994
│ │ │ │ +000462d0: 6133 2d31 3934 3661 3262 2d36 3732 3361  a3-1946a2b-6723a
│ │ │ │ +000462e0: 6232 2d35 3438 3362 332d 3634 3533 6132  b2-5483b3-6453a2
│ │ │ │ +000462f0: 632d 3131 3932 6162 632d 3135 3235 3062  c-1192abc-15250b
│ │ │ │ +00046300: 3263 2d31 3130 3435 7c0a 7c2b 3134 3338  2c-11045|.|+1438
│ │ │ │ +00046310: 3864 3320 2d31 3033 3730 6162 322b 3730  8d3 -10370ab2+70
│ │ │ │ +00046320: 3932 6233 2b39 3730 3261 6263 2b36 3632  92b3+9702abc+662
│ │ │ │ +00046330: 3762 3263 2d39 3739 3762 6332 2d35 3837  7b2c-9797bc2-587
│ │ │ │ +00046340: 3463 332d 3838 3836 6162 642d 3437 3030  4c3-8886abd-4700
│ │ │ │ +00046350: 6232 642b 3135 6163 7c0a 7c20 2020 2020  b2d+15ac|.|     
│ │ │ │ +00046360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000463a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000463b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +000463b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000463c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000463d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000463e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046400: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +00046400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046450: 2020 207c 0a7c 6364 322d 3932 6332 6432     |.|cd2-92c2d2
│ │ │ │ -00046460: 2b31 3539 3638 6164 332b 3134 3836 3062  +15968ad3+14860b
│ │ │ │ -00046470: 6433 2d38 3832 3963 6433 2d31 3132 3734  d3-8829cd3-11274
│ │ │ │ -00046480: 6434 207c 2020 2020 2020 2020 2020 2020  d4 |            
│ │ │ │ -00046490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000464a0: 2020 207c 0a7c 322b 3939 3761 6364 322b     |.|2+997acd2+
│ │ │ │ -000464b0: 3330 3135 6263 6432 2b31 3132 3734 6332  3015bcd2+11274c2
│ │ │ │ -000464c0: 6432 2020 2020 2020 2020 2020 2020 2020  d2              
│ │ │ │ -000464d0: 2020 207c 2020 2020 2020 2020 2020 2020     |            
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│ │ │ │ +00046440: 2020 2020 2020 2020 7c0a 7c63 6432 2d39          |.|cd2-9
│ │ │ │ +00046450: 3263 3264 322b 3135 3936 3861 6433 2b31  2c2d2+15968ad3+1
│ │ │ │ +00046460: 3438 3630 6264 332d 3838 3239 6364 332d  4860bd3-8829cd3-
│ │ │ │ +00046470: 3131 3237 3464 3420 7c20 2020 2020 2020  11274d4 |       
│ │ │ │ +00046480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046490: 2020 2020 2020 2020 7c0a 7c32 2b39 3937          |.|2+997
│ │ │ │ +000464a0: 6163 6432 2b33 3031 3562 6364 322b 3131  acd2+3015bcd2+11
│ │ │ │ +000464b0: 3237 3463 3264 3220 2020 2020 2020 2020  274c2d2         
│ │ │ │ +000464c0: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +000464d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00046510: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +00046520: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046590: 2020 207c 0a7c 3462 3264 3220 2d37 3634     |.|4b2d2 -764
│ │ │ │ -000465a0: 3861 3362 2d31 3332 3631 6132 6232 2b33  8a3b-13261a2b2+3
│ │ │ │ -000465b0: 3335 3861 6233 2d36 3138 6234 2d37 3637  358ab3-618b4-767
│ │ │ │ -000465c0: 6133 632d 3535 3830 6132 6263 2d31 3535  a3c-5580a2bc-155
│ │ │ │ -000465d0: 3630 6162 3263 2d34 3839 6233 632b 3135  60ab2c-489b3c+15
│ │ │ │ -000465e0: 3338 337c 0a7c 2020 2020 2020 2020 2020  383|.|          
│ │ │ │ +00046560: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +00046570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046580: 2020 2020 2020 2020 7c0a 7c34 6232 6432          |.|4b2d2
│ │ │ │ +00046590: 202d 3736 3438 6133 622d 3133 3236 3161   -7648a3b-13261a
│ │ │ │ +000465a0: 3262 322b 3333 3538 6162 332d 3631 3862  2b2+3358ab3-618b
│ │ │ │ +000465b0: 342d 3736 3761 3363 2d35 3538 3061 3262  4-767a3c-5580a2b
│ │ │ │ +000465c0: 632d 3135 3536 3061 6232 632d 3438 3962  c-15560ab2c-489b
│ │ │ │ +000465d0: 3363 2b31 3533 3833 7c0a 7c20 2020 2020  3c+15383|.|     
│ │ │ │ +000465e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000465f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00046670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00046680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000466a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000466b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000466d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000466c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000466d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000466e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000466f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046700: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00046720: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046770: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +00046770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000467c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00046860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000468b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +000468b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000468f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046900: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000468f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00046900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046940: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -00046950: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00046930: 2020 2020 2020 2020 2020 3020 2020 2020            0     
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│ │ │ │ +00046950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046960: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046990: 2020 2020 2031 3037 6133 2b34 3337 3661       107a3+4376a
│ │ │ │ -000469a0: 3262 2b7c 0a7c 3534 3962 3264 2d39 3033  2b+|.|549b2d-903
│ │ │ │ -000469b0: 3361 6364 2d32 3939 3862 6364 2b32 3033  3acd-2998bcd+203
│ │ │ │ -000469c0: 3663 3264 2d35 3130 3761 6432 2d35 3637  6c2d-5107ad2-567
│ │ │ │ -000469d0: 3962 6432 2b36 3332 3563 6432 2b31 3137  9bd2+6325cd2+117
│ │ │ │ -000469e0: 3430 6433 2038 3233 3161 6232 2b31 3331  40d3 8231ab2+131
│ │ │ │ -000469f0: 3737 627c 0a7c 2020 2020 2020 2020 2020  77b|.|          
│ │ │ │ +00046980: 2020 2020 2020 2020 2020 3130 3761 332b            107a3+
│ │ │ │ +00046990: 3433 3736 6132 622b 7c0a 7c35 3439 6232  4376a2b+|.|549b2
│ │ │ │ +000469a0: 642d 3930 3333 6163 642d 3239 3938 6263  d-9033acd-2998bc
│ │ │ │ +000469b0: 642b 3230 3336 6332 642d 3531 3037 6164  d+2036c2d-5107ad
│ │ │ │ +000469c0: 322d 3536 3739 6264 322b 3633 3235 6364  2-5679bd2+6325cd
│ │ │ │ +000469d0: 322b 3131 3734 3064 3320 3832 3331 6162  2+11740d3 8231ab
│ │ │ │ +000469e0: 322b 3133 3137 3762 7c0a 7c20 2020 2020  2+13177b|.|     
│ │ │ │ +000469f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046a30: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -00046a40: 2020 207c 0a7c 3938 6164 322d 3135 3331     |.|98ad2-1531
│ │ │ │ -00046a50: 3762 6432 2b36 3932 3263 6432 2b35 3038  7bd2+6922cd2+508
│ │ │ │ -00046a60: 3064 3320 2020 2020 2020 2020 2020 2020  0d3             
│ │ │ │ -00046a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046a80: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -00046a90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00046a20: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +00046a30: 2020 2020 2020 2020 7c0a 7c39 3861 6432          |.|98ad2
│ │ │ │ +00046a40: 2d31 3533 3137 6264 322b 3639 3232 6364  -15317bd2+6922cd
│ │ │ │ +00046a50: 322b 3530 3830 6433 2020 2020 2020 2020  2+5080d3        
│ │ │ │ +00046a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046a70: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +00046a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00046a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046ad0: 2020 2020 2031 3034 3830 6133 2b36 3230       10480a3+620
│ │ │ │ -00046ae0: 3361 327c 0a7c 3130 3836 3662 6432 2b37  3a2|.|10866bd2+7
│ │ │ │ -00046af0: 3036 3163 6432 2d32 3632 3764 3320 3130  061cd2-2627d3 10
│ │ │ │ -00046b00: 3761 332b 3433 3736 6132 622b 3337 3833  7a3+4376a2b+3783
│ │ │ │ -00046b10: 6162 322b 3130 3335 3962 332d 3535 3730  ab2+10359b3-5570
│ │ │ │ -00046b20: 6132 632d 3533 3037 6162 632d 3734 3634  a2c-5307abc-7464
│ │ │ │ -00046b30: 6232 637c 0a7c 2020 2020 2020 2020 2020  b2c|.|          
│ │ │ │ -00046b40: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00046ac0: 2020 2020 2020 2020 2020 3130 3438 3061            10480a
│ │ │ │ +00046ad0: 332b 3632 3033 6132 7c0a 7c31 3038 3636  3+6203a2|.|10866
│ │ │ │ +00046ae0: 6264 322b 3730 3631 6364 322d 3236 3237  bd2+7061cd2-2627
│ │ │ │ +00046af0: 6433 2031 3037 6133 2b34 3337 3661 3262  d3 107a3+4376a2b
│ │ │ │ +00046b00: 2b33 3738 3361 6232 2b31 3033 3539 6233  +3783ab2+10359b3
│ │ │ │ +00046b10: 2d35 3537 3061 3263 2d35 3330 3761 6263  -5570a2c-5307abc
│ │ │ │ +00046b20: 2d37 3436 3462 3263 7c0a 7c20 2020 2020  -7464b2c|.|     
│ │ │ │ +00046b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046b40: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  00046b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00046b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b80: 2020 207c 0a7c 3530 3830 6433 2020 2020     |.|5080d3    
│ │ │ │ -00046b90: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00046b70: 2020 2020 2020 2020 7c0a 7c35 3038 3064          |.|5080d
│ │ │ │ +00046b80: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00046b90: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  00046ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046bd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -00048360: 3037 6162 632d 3734 3634 6232 632b 3331  07abc-7464b2c+31
│ │ │ │ -00048370: 3837 6132 642b 3835 3730 6162 642d 3832  87a2d+8570abd-82
│ │ │ │ -00048380: 3531 6232 642b 3834 3434 6163 642b 3530  51b2d+8444acd+50
│ │ │ │ -00048390: 3731 627c 0a7c 332b 3538 3634 6162 632b  71b|.|3+5864abc+
│ │ │ │ -000483a0: 3133 3939 3062 3263 2b35 3032 3661 6264  13990b2c+5026abd
│ │ │ │ -000483b0: 2d31 3135 3231 6232 642d 3735 3031 6163  -11521b2d-7501ac
│ │ │ │ -000483c0: 642d 3137 3739 6263 6420 2020 2020 2020  d-1779bcd       
│ │ │ │ -000483d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000483e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048330: 2020 2020 2020 2020 7c0a 7c33 3738 3361          |.|3783a
│ │ │ │ +00048340: 6232 2b31 3033 3539 6233 2d35 3537 3061  b2+10359b3-5570a
│ │ │ │ +00048350: 3263 2d35 3330 3761 6263 2d37 3436 3462  2c-5307abc-7464b
│ │ │ │ +00048360: 3263 2b33 3138 3761 3264 2b38 3537 3061  2c+3187a2d+8570a
│ │ │ │ +00048370: 6264 2d38 3235 3162 3264 2b38 3434 3461  bd-8251b2d+8444a
│ │ │ │ +00048380: 6364 2b35 3037 3162 7c0a 7c33 2b35 3836  cd+5071b|.|3+586
│ │ │ │ +00048390: 3461 6263 2b31 3339 3930 6232 632b 3530  4abc+13990b2c+50
│ │ │ │ +000483a0: 3236 6162 642d 3131 3532 3162 3264 2d37  26abd-11521b2d-7
│ │ │ │ +000483b0: 3530 3161 6364 2d31 3737 3962 6364 2020  501acd-1779bcd  
│ │ │ │ +000483c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000483d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000483e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000483f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048430: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048480: 2020 207c 0a7c 622d 3935 3334 6162 322d     |.|b-9534ab2-
│ │ │ │ -00048490: 3130 3836 3662 332b 3934 3830 6132 632d  10866b3+9480a2c-
│ │ │ │ -000484a0: 3732 3536 6162 632b 3730 3631 6232 632d  7256abc+7061b2c-
│ │ │ │ -000484b0: 3531 3037 6163 322d 3536 3739 6263 322b  5107ac2-5679bc2+
│ │ │ │ -000484c0: 3633 3235 6333 2b31 3139 3530 6132 642b  6325c3+11950a2d+
│ │ │ │ -000484d0: 3533 327c 0a7c 2b33 3138 3761 3264 2b38  532|.|+3187a2d+8
│ │ │ │ -000484e0: 3537 3061 6264 2d38 3235 3162 3264 2b38  570abd-8251b2d+8
│ │ │ │ -000484f0: 3434 3461 6364 2b35 3037 3162 6364 2020  444acd+5071bcd  
│ │ │ │ +00048470: 2020 2020 2020 2020 7c0a 7c62 2d39 3533          |.|b-953
│ │ │ │ +00048480: 3461 6232 2d31 3038 3636 6233 2b39 3438  4ab2-10866b3+948
│ │ │ │ +00048490: 3061 3263 2d37 3235 3661 6263 2b37 3036  0a2c-7256abc+706
│ │ │ │ +000484a0: 3162 3263 2d35 3130 3761 6332 2d35 3637  1b2c-5107ac2-567
│ │ │ │ +000484b0: 3962 6332 2b36 3332 3563 332b 3131 3935  9bc2+6325c3+1195
│ │ │ │ +000484c0: 3061 3264 2b35 3332 7c0a 7c2b 3331 3837  0a2d+532|.|+3187
│ │ │ │ +000484d0: 6132 642b 3835 3730 6162 642d 3832 3531  a2d+8570abd-8251
│ │ │ │ +000484e0: 6232 642b 3834 3434 6163 642b 3530 3731  b2d+8444acd+5071
│ │ │ │ +000484f0: 6263 6420 2020 2020 2020 2020 2020 2020  bcd             
│ │ │ │  00048500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048520: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048570: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000485a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000485b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000485c0: 2020 207c 0a7c 6232 632d 3131 3935 3061     |.|b2c-11950a
│ │ │ │ -000485d0: 3264 2d35 3332 3161 6264 2b32 3632 3762  2d-5321abd+2627b
│ │ │ │ -000485e0: 3264 2d33 3939 3661 6364 2d37 3135 3262  2d-3996acd-7152b
│ │ │ │ -000485f0: 6364 2b39 3339 3861 6432 2b31 3533 3137  cd+9398ad2+15317
│ │ │ │ -00048600: 6264 322d 3639 3232 6364 322d 3530 3830  bd2-6922cd2-5080
│ │ │ │ -00048610: 6433 207c 0a7c 6232 642b 3735 3031 6163  d3 |.|b2d+7501ac
│ │ │ │ -00048620: 642b 3137 3739 6263 642d 3535 3439 6332  d+1779bcd-5549c2
│ │ │ │ -00048630: 642d 3130 3836 3662 6432 2b37 3036 3163  d-10866bd2+7061c
│ │ │ │ -00048640: 6432 2d32 3632 3764 3320 3130 3761 332b  d2-2627d3 107a3+
│ │ │ │ -00048650: 3433 3736 6132 622b 3337 3833 6162 322b  4376a2b+3783ab2+
│ │ │ │ -00048660: 3130 337c 0a7c 2020 2020 2020 2020 2020  103|.|          
│ │ │ │ +000485b0: 2020 2020 2020 2020 7c0a 7c62 3263 2d31          |.|b2c-1
│ │ │ │ +000485c0: 3139 3530 6132 642d 3533 3231 6162 642b  1950a2d-5321abd+
│ │ │ │ +000485d0: 3236 3237 6232 642d 3339 3936 6163 642d  2627b2d-3996acd-
│ │ │ │ +000485e0: 3731 3532 6263 642b 3933 3938 6164 322b  7152bcd+9398ad2+
│ │ │ │ +000485f0: 3135 3331 3762 6432 2d36 3932 3263 6432  15317bd2-6922cd2
│ │ │ │ +00048600: 2d35 3038 3064 3320 7c0a 7c62 3264 2b37  -5080d3 |.|b2d+7
│ │ │ │ +00048610: 3530 3161 6364 2b31 3737 3962 6364 2d35  501acd+1779bcd-5
│ │ │ │ +00048620: 3534 3963 3264 2d31 3038 3636 6264 322b  549c2d-10866bd2+
│ │ │ │ +00048630: 3730 3631 6364 322d 3236 3237 6433 2031  7061cd2-2627d3 1
│ │ │ │ +00048640: 3037 6133 2b34 3337 3661 3262 2b33 3738  07a3+4376a2b+378
│ │ │ │ +00048650: 3361 6232 2b31 3033 7c0a 7c20 2020 2020  3ab2+103|.|     
│ │ │ │ +00048660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000486a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000486b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000486a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000486b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000486c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000486d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000486e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000486f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048700: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000486f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048750: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048740: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000487a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000487a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000487e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000487f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000487e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000487f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048890: 2020 207c 0a7c 6233 632b 3134 3931 3961     |.|b3c+14919a
│ │ │ │ -000488a0: 3263 322b 3131 3334 3661 6263 322d 3134  2c2+11346abc2-14
│ │ │ │ -000488b0: 3137 3262 3263 322d 3139 3130 6163 332d  172b2c2-1910ac3-
│ │ │ │ -000488c0: 3131 3635 3662 6333 2b38 3532 3763 342b  11656bc3+8527c4+
│ │ │ │ -000488d0: 3637 3730 6133 642d 3931 3037 6132 6264  6770a3d-9107a2bd
│ │ │ │ -000488e0: 2d38 387c 0a7c 2020 2020 2020 2020 2020  -88|.|          
│ │ │ │ +00048880: 2020 2020 2020 2020 7c0a 7c62 3363 2b31          |.|b3c+1
│ │ │ │ +00048890: 3439 3139 6132 6332 2b31 3133 3436 6162  4919a2c2+11346ab
│ │ │ │ +000488a0: 6332 2d31 3431 3732 6232 6332 2d31 3931  c2-14172b2c2-191
│ │ │ │ +000488b0: 3061 6333 2d31 3136 3536 6263 332b 3835  0ac3-11656bc3+85
│ │ │ │ +000488c0: 3237 6334 2b36 3737 3061 3364 2d39 3130  27c4+6770a3d-910
│ │ │ │ +000488d0: 3761 3262 642d 3838 7c0a 7c20 2020 2020  7a2bd-88|.|     
│ │ │ │ +000488e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000488f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048930: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048970: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000489a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000489b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000489c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000489d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000489c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000489d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000489e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000489f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048a10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048a60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ac0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048b50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048bb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048ba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048c00: 2020 207c 0a7c 3733 3762 3363 2d35 3032     |.|737b3c-502
│ │ │ │ -00048c10: 3861 3263 322d 3933 3132 6162 6332 2d31  8a2c2-9312abc2-1
│ │ │ │ -00048c20: 3631 3062 3263 322b 3135 3933 3661 6333  610b2c2+15936ac3
│ │ │ │ -00048c30: 2d35 3338 3562 6333 2d35 3337 3563 342b  -5385bc3-5375c4+
│ │ │ │ -00048c40: 3639 3338 6133 642d 3933 3731 6132 6264  6938a3d-9371a2bd
│ │ │ │ -00048c50: 2d39 397c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d  -99|.|----------
│ │ │ │ +00048bf0: 2020 2020 2020 2020 7c0a 7c37 3337 6233          |.|737b3
│ │ │ │ +00048c00: 632d 3530 3238 6132 6332 2d39 3331 3261  c-5028a2c2-9312a
│ │ │ │ +00048c10: 6263 322d 3136 3130 6232 6332 2b31 3539  bc2-1610b2c2+159
│ │ │ │ +00048c20: 3336 6163 332d 3533 3835 6263 332d 3533  36ac3-5385bc3-53
│ │ │ │ +00048c30: 3735 6334 2b36 3933 3861 3364 2d39 3337  75c4+6938a3d-937
│ │ │ │ +00048c40: 3161 3262 642d 3939 7c0a 7c2d 2d2d 2d2d  1a2bd-99|.|-----
│ │ │ │ +00048c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048ca0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ +00048c90: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00048ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048ce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048d80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048de0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048e20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00048e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00048e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00049ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00049be0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00049bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00049c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00049c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00049c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ce0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00049cd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00049ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00049d20: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00049d30: 2020 207c 0a7c 6364 2020 2020 2020 2020     |.|cd        
│ │ │ │ +00049d20: 2030 2020 2020 2020 7c0a 7c63 6420 2020   0      |.|cd   
│ │ │ │ +00049d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00049d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049d70: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00049d80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00049d70: 2030 2020 2020 2020 7c0a 7c20 2020 2020   0      |.|     
│ │ │ │ +00049d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00049db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049dc0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00049dd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00049dc0: 2030 2020 2020 2020 7c0a 7c20 2020 2020   0      |.|     
│ │ │ │ +00049dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00049e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049e10: 2020 2020 2020 2020 2020 2020 3130 3761              107a
│ │ │ │ -00049e20: 332b 347c 0a7c 2020 2020 2020 2020 2020  3+4|.|          
│ │ │ │ +00049e10: 2031 3037 6133 2b34 7c0a 7c20 2020 2020   107a3+4|.|     
│ │ │ │ +00049e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049e60: 2020 2020 2020 2020 2020 2020 3832 3331              8231
│ │ │ │ -00049e70: 6162 327c 0a7c 3161 6264 2d32 3632 3762  ab2|.|1abd-2627b
│ │ │ │ -00049e80: 3264 2b33 3939 3661 6364 2b37 3135 3262  2d+3996acd+7152b
│ │ │ │ -00049e90: 6364 2b31 3137 3430 6332 642d 3933 3938  cd+11740c2d-9398
│ │ │ │ -00049ea0: 6164 322d 3135 3331 3762 6432 2b36 3932  ad2-15317bd2+692
│ │ │ │ -00049eb0: 3263 6432 2b35 3038 3064 3320 2d31 3533  2cd2+5080d3 -153
│ │ │ │ -00049ec0: 3434 617c 0a7c 2d31 3032 3539 6164 3220  44a|.|-10259ad2 
│ │ │ │ +00049e60: 2038 3233 3161 6232 7c0a 7c31 6162 642d   8231ab2|.|1abd-
│ │ │ │ +00049e70: 3236 3237 6232 642b 3339 3936 6163 642b  2627b2d+3996acd+
│ │ │ │ +00049e80: 3731 3532 6263 642b 3131 3734 3063 3264  7152bcd+11740c2d
│ │ │ │ +00049e90: 2d39 3339 3861 6432 2d31 3533 3137 6264  -9398ad2-15317bd
│ │ │ │ +00049ea0: 322b 3639 3232 6364 322b 3530 3830 6433  2+6922cd2+5080d3
│ │ │ │ +00049eb0: 202d 3135 3334 3461 7c0a 7c2d 3130 3235   -15344a|.|-1025
│ │ │ │ +00049ec0: 3961 6432 2020 2020 2020 2020 2020 2020  9ad2            
│ │ │ │  00049ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049f10: 2020 207c 0a7c 2d38 3233 3161 6232 2d31     |.|-8231ab2-1
│ │ │ │ -00049f20: 3331 3737 6233 2d35 3836 3461 6263 2d31  3177b3-5864abc-1
│ │ │ │ -00049f30: 3339 3930 6232 632b 3133 3039 6263 322d  3990b2c+1309bc2-
│ │ │ │ -00049f40: 3533 3938 6333 2d35 3032 3661 6264 2b31  5398c3-5026abd+1
│ │ │ │ -00049f50: 3135 3231 6232 642b 3735 3031 6163 642b  1521b2d+7501acd+
│ │ │ │ -00049f60: 3137 377c 0a7c 3135 3334 3461 332d 3236  177|.|15344a3-26
│ │ │ │ -00049f70: 3533 6132 622d 3130 3235 3961 6232 2d31  53a2b-10259ab2-1
│ │ │ │ -00049f80: 3233 3635 6132 632b 3732 3136 6162 632d  2365a2c+7216abc-
│ │ │ │ -00049f90: 3632 3330 6163 322b 3533 3236 6263 322d  6230ac2+5326bc2-
│ │ │ │ -00049fa0: 3130 3331 6333 2b31 3335 3038 6132 642b  1031c3+13508a2d+
│ │ │ │ -00049fb0: 3130 317c 0a7c 3133 3039 6132 622d 3533  101|.|1309a2b-53
│ │ │ │ -00049fc0: 3938 6132 632d 3535 3439 6132 6420 2020  98a2c-5549a2d   
│ │ │ │ +00049f00: 2020 2020 2020 2020 7c0a 7c2d 3832 3331          |.|-8231
│ │ │ │ +00049f10: 6162 322d 3133 3137 3762 332d 3538 3634  ab2-13177b3-5864
│ │ │ │ +00049f20: 6162 632d 3133 3939 3062 3263 2b31 3330  abc-13990b2c+130
│ │ │ │ +00049f30: 3962 6332 2d35 3339 3863 332d 3530 3236  9bc2-5398c3-5026
│ │ │ │ +00049f40: 6162 642b 3131 3532 3162 3264 2b37 3530  abd+11521b2d+750
│ │ │ │ +00049f50: 3161 6364 2b31 3737 7c0a 7c31 3533 3434  1acd+177|.|15344
│ │ │ │ +00049f60: 6133 2d32 3635 3361 3262 2d31 3032 3539  a3-2653a2b-10259
│ │ │ │ +00049f70: 6162 322d 3132 3336 3561 3263 2b37 3231  ab2-12365a2c+721
│ │ │ │ +00049f80: 3661 6263 2d36 3233 3061 6332 2b35 3332  6abc-6230ac2+532
│ │ │ │ +00049f90: 3662 6332 2d31 3033 3163 332b 3133 3530  6bc2-1031c3+1350
│ │ │ │ +00049fa0: 3861 3264 2b31 3031 7c0a 7c31 3330 3961  8a2d+101|.|1309a
│ │ │ │ +00049fb0: 3262 2d35 3339 3861 3263 2d35 3534 3961  2b-5398a2c-5549a
│ │ │ │ +00049fc0: 3264 2020 2020 2020 2020 2020 2020 2020  2d              
│ │ │ │  00049fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a000: 2020 207c 0a7c 3539 6233 2d35 3537 3061     |.|59b3-5570a
│ │ │ │ -0004a010: 3263 2d35 3330 3761 6263 2d37 3436 3462  2c-5307abc-7464b
│ │ │ │ -0004a020: 3263 2b33 3138 3761 3264 2b38 3537 3061  2c+3187a2d+8570a
│ │ │ │ -0004a030: 6264 2d38 3235 3162 3264 2b38 3434 3461  bd-8251b2d+8444a
│ │ │ │ -0004a040: 6364 2b35 3037 3162 6364 207c 2020 2020  cd+5071bcd |    
│ │ │ │ -0004a050: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00049ff0: 2020 2020 2020 2020 7c0a 7c35 3962 332d          |.|59b3-
│ │ │ │ +0004a000: 3535 3730 6132 632d 3533 3037 6162 632d  5570a2c-5307abc-
│ │ │ │ +0004a010: 3734 3634 6232 632b 3331 3837 6132 642b  7464b2c+3187a2d+
│ │ │ │ +0004a020: 3835 3730 6162 642d 3832 3531 6232 642b  8570abd-8251b2d+
│ │ │ │ +0004a030: 3834 3434 6163 642b 3530 3731 6263 6420  8444acd+5071bcd 
│ │ │ │ +0004a040: 7c20 2020 2020 2020 7c0a 7c20 2020 2020  |       |.|     
│ │ │ │ +0004a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a0e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a180: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a1d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a220: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a280: 2020 207c 0a7c 3436 6162 3264 2d35 3231     |.|46ab2d-521
│ │ │ │ -0004a290: 3862 3364 2d38 3032 6132 6364 2d36 3333  8b3d-802a2cd-633
│ │ │ │ -0004a2a0: 3261 6263 642b 3537 3238 6232 6364 2d35  2abcd+5728b2cd-5
│ │ │ │ -0004a2b0: 3334 3761 6332 642d 3133 3339 3462 6332  347ac2d-13394bc2
│ │ │ │ -0004a2c0: 642d 3239 3663 3364 2d37 3434 3661 3264  d-296c3d-7446a2d
│ │ │ │ -0004a2d0: 322d 317c 0a7c 2020 2020 2020 2020 2020  2-1|.|          
│ │ │ │ +0004a270: 2020 2020 2020 2020 7c0a 7c34 3661 6232          |.|46ab2
│ │ │ │ +0004a280: 642d 3532 3138 6233 642d 3830 3261 3263  d-5218b3d-802a2c
│ │ │ │ +0004a290: 642d 3633 3332 6162 6364 2b35 3732 3862  d-6332abcd+5728b
│ │ │ │ +0004a2a0: 3263 642d 3533 3437 6163 3264 2d31 3333  2cd-5347ac2d-133
│ │ │ │ +0004a2b0: 3934 6263 3264 2d32 3936 6333 642d 3734  94bc2d-296c3d-74
│ │ │ │ +0004a2c0: 3436 6132 6432 2d31 7c0a 7c20 2020 2020  46a2d2-1|.|     
│ │ │ │ +0004a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a310: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a370: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a360: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a3b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004a400: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a460: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a450: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a4b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a4a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004a500: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a4f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0004a590: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0004a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5f0: 2020 207c 0a7c 3031 6162 3264 2b34 3432     |.|01ab2d+442
│ │ │ │ -0004a600: 3062 3364 2b31 3233 3137 6132 6364 2b31  0b3d+12317a2cd+1
│ │ │ │ -0004a610: 3533 3738 6162 6364 2b31 3439 3231 6232  5378abcd+14921b2
│ │ │ │ -0004a620: 6364 2d38 3831 3061 6332 642b 3332 3731  cd-8810ac2d+3271
│ │ │ │ -0004a630: 6263 3264 2b31 3036 3633 6333 642b 3132  bc2d+10663c3d+12
│ │ │ │ -0004a640: 3533 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d  537|.|----------
│ │ │ │ +0004a5e0: 2020 2020 2020 2020 7c0a 7c30 3161 6232          |.|01ab2
│ │ │ │ +0004a5f0: 642b 3434 3230 6233 642b 3132 3331 3761  d+4420b3d+12317a
│ │ │ │ +0004a600: 3263 642b 3135 3337 3861 6263 642b 3134  2cd+15378abcd+14
│ │ │ │ +0004a610: 3932 3162 3263 642d 3838 3130 6163 3264  921b2cd-8810ac2d
│ │ │ │ +0004a620: 2b33 3237 3162 6332 642b 3130 3636 3363  +3271bc2d+10663c
│ │ │ │ +0004a630: 3364 2b31 3235 3337 7c0a 7c2d 2d2d 2d2d  3d+12537|.|-----
│ │ │ │ +0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004a690: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ +0004a680: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0004a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a6e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -00052110: 2020 207c 0a7c 642b 3530 3731 6263 6420     |.|d+5071bcd 
│ │ │ │ -00052120: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052100: 2020 2020 2020 2020 7c0a 7c64 2b35 3037          |.|d+507
│ │ │ │ +00052110: 3162 6364 207c 2020 2020 2020 2020 2020  1bcd |          
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│ │ │ │ -00052430: 2020 207c 0a7c 3834 6434 207c 2020 2020     |.|84d4 |    
│ │ │ │ +00052420: 2020 2020 2020 2020 7c0a 7c38 3464 3420          |.|84d4 
│ │ │ │ +00052430: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +00052480: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000525c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00052610: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00052600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  00052630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00052680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000526c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000526d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000526e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000526f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052700: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000526f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00052700: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000527a0: 2020 207c 0a7c 3633 6364 332b 3633 3431     |.|63cd3+6341
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│ │ │ │ +00052af0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00052b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052b50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20  -------+.|i15 : 
│ │ │ │ -00052b60: 234c 2020 2020 2020 2020 2020 2020 2020  #L              
│ │ │ │ +00052b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00052b50: 3135 203a 2023 4c20 2020 2020 2020 2020  15 : #L         
│ │ │ │ +00052b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ba0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00052b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052bf0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20         |.|o15 = 
│ │ │ │ -00052c00: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00052be0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00052bf0: 3135 203d 2036 2020 2020 2020 2020 2020  15 = 6          
│ │ │ │ +00052c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00052c30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00052c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052c90: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │ -00052ca0: 6e65 744c 6973 7420 4c20 2020 2020 2020  netList L       
│ │ │ │ +00052c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00052c90: 3136 203a 206e 6574 4c69 7374 204c 2020  16 : netList L  
│ │ │ │ +00052ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ce0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00052cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052d40: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +00052d20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052d30: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00052d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d80: 2020 2020 2020 207c 0a7c 6f31 3620 3d20         |.|o16 = 
│ │ │ │ -00052d90: 7c7b 302c 2032 7d7c 307c 2020 2020 2020  |{0, 2}|0|      
│ │ │ │ +00052d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00052d80: 3136 203d 207c 7b30 2c20 327d 7c30 7c20  16 = |{0, 2}|0| 
│ │ │ │ +00052d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052de0: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +00052dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052dd0: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00052de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052e30: 7c7b 302c 2032 7d7c 317c 2020 2020 2020  |{0, 2}|1|      
│ │ │ │ +00052e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052e20: 2020 2020 207c 7b30 2c20 327d 7c31 7c20       |{0, 2}|1| 
│ │ │ │ +00052e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052e80: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +00052e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052e70: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00052e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052ed0: 7c7b 312c 2031 7d7c 307c 2020 2020 2020  |{1, 1}|0|      
│ │ │ │ +00052eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052ec0: 2020 2020 207c 7b31 2c20 317d 7c30 7c20       |{1, 1}|0| 
│ │ │ │ +00052ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052f20: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +00052f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052f10: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00052f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052f70: 7c7b 312c 2031 7d7c 317c 2020 2020 2020  |{1, 1}|1|      
│ │ │ │ +00052f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052f60: 2020 2020 207c 7b31 2c20 317d 7c31 7c20       |{1, 1}|1| 
│ │ │ │ +00052f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052fb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00052fc0: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +00052fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052fb0: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00052fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00053010: 7c7b 322c 2030 7d7c 307c 2020 2020 2020  |{2, 0}|0|      
│ │ │ │ +00052ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00053000: 2020 2020 207c 7b32 2c20 307d 7c30 7c20       |{2, 0}|0| 
│ │ │ │ +00053010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053050: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00053060: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +00053040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00053050: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00053060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000530a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000530b0: 7c7b 322c 2030 7d7c 317c 2020 2020 2020  |{2, 0}|1|      
│ │ │ │ +00053090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000530a0: 2020 2020 207c 7b32 2c20 307d 7c31 7c20       |{2, 0}|1| 
│ │ │ │ +000530b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000530c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000530d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000530e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000530f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00053100: 2b2d 2d2d 2d2d 2d2b 2d2b 2020 2020 2020  +------+-+      
│ │ │ │ +000530e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000530f0: 2020 2020 202b 2d2d 2d2d 2d2d 2b2d 2b20       +------+-+ 
│ │ │ │ +00053100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00053130: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00053140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053190: 2d2d 2d2d 2d2d 2d2b 0a0a 466f 7220 6578  -------+..For ex
│ │ │ │ -000531a0: 616d 706c 6520 7765 2068 6176 653a 0a0a  ample we have:..
│ │ │ │ -000531b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00053180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46  ------------+..F
│ │ │ │ +00053190: 6f72 2065 7861 6d70 6c65 2077 6520 6861  or example we ha
│ │ │ │ +000531a0: 7665 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ve:..+----------
│ │ │ │ +000531b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000531c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000531d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000531e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000531f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00053200: 7c69 3137 203a 2068 6f6d 6f74 2328 4c5f  |i17 : homot#(L_
│ │ │ │ -00053210: 3029 2020 2020 2020 2020 2020 2020 2020  0)              
│ │ │ │ +000531f0: 2d2d 2d2b 0a7c 6931 3720 3a20 686f 6d6f  ---+.|i17 : homo
│ │ │ │ +00053200: 7423 284c 5f30 2920 2020 2020 2020 2020  t#(L_0)         
│ │ │ │ +00053210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00053250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000532a0: 7c6f 3137 203d 207b 367d 207c 202d 3133  |o17 = {6} | -13
│ │ │ │ -000532b0: 3739 3561 342b 3230 3139 6133 622b 3133  795a4+2019a3b+13
│ │ │ │ -000532c0: 3736 3961 3262 322b 3735 3836 6162 332b  769a2b2+7586ab3+
│ │ │ │ -000532d0: 3836 3439 6234 2b36 3435 3461 3363 2d31  8649b4+6454a3c-1
│ │ │ │ -000532e0: 3031 3837 6132 6263 2d31 3738 3361 7c0a  0187a2bc-1783a|.
│ │ │ │ -000532f0: 7c20 2020 2020 207b 367d 207c 2031 3131  |      {6} | 111
│ │ │ │ -00053300: 3532 6134 2d31 3333 3661 3362 2b31 3138  52a4-1336a3b+118
│ │ │ │ -00053310: 3436 6132 6232 2b31 3032 3634 6162 332b  46a2b2+10264ab3+
│ │ │ │ -00053320: 3631 3862 342d 3131 3035 3161 3363 2b31  618b4-11051a3c+1
│ │ │ │ -00053330: 3231 3239 6132 6263 2b35 3932 3761 7c0a  2129a2bc+5927a|.
│ │ │ │ -00053340: 7c20 2020 2020 207b 367d 207c 202d 3633  |      {6} | -63
│ │ │ │ -00053350: 3338 6134 2b31 3030 3235 6133 622b 3134  38a4+10025a3b+14
│ │ │ │ -00053360: 3938 3761 3363 2d39 3935 3961 3262 632d  987a3c-9959a2bc-
│ │ │ │ -00053370: 3131 3639 3161 3263 322b 3132 3333 3661  11691a2c2+12336a
│ │ │ │ -00053380: 6263 322d 3737 3836 6133 642d 3131 7c0a  bc2-7786a3d-11|.
│ │ │ │ -00053390: 7c20 2020 2020 207b 367d 207c 2032 3237  |      {6} | 227
│ │ │ │ -000533a0: 3561 342d 3233 3961 3362 2b31 3435 3934  5a4-239a3b+14594
│ │ │ │ -000533b0: 6132 6232 2d38 3135 3361 6233 2d31 3139  a2b2-8153ab3-119
│ │ │ │ -000533c0: 3435 6234 2d38 3431 3661 3363 2b36 3235  45b4-8416a3c+625
│ │ │ │ -000533d0: 3161 3262 632d 3330 3233 6162 3263 7c0a  1a2bc-3023ab2c|.
│ │ │ │ -000533e0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00053290: 2020 207c 0a7c 6f31 3720 3d20 7b36 7d20     |.|o17 = {6} 
│ │ │ │ +000532a0: 7c20 2d31 3337 3935 6134 2b32 3031 3961  | -13795a4+2019a
│ │ │ │ +000532b0: 3362 2b31 3337 3639 6132 6232 2b37 3538  3b+13769a2b2+758
│ │ │ │ +000532c0: 3661 6233 2b38 3634 3962 342b 3634 3534  6ab3+8649b4+6454
│ │ │ │ +000532d0: 6133 632d 3130 3138 3761 3262 632d 3137  a3c-10187a2bc-17
│ │ │ │ +000532e0: 3833 617c 0a7c 2020 2020 2020 7b36 7d20  83a|.|      {6} 
│ │ │ │ +000532f0: 7c20 3131 3135 3261 342d 3133 3336 6133  | 11152a4-1336a3
│ │ │ │ +00053300: 622b 3131 3834 3661 3262 322b 3130 3236  b+11846a2b2+1026
│ │ │ │ +00053310: 3461 6233 2b36 3138 6234 2d31 3130 3531  4ab3+618b4-11051
│ │ │ │ +00053320: 6133 632b 3132 3132 3961 3262 632b 3539  a3c+12129a2bc+59
│ │ │ │ +00053330: 3237 617c 0a7c 2020 2020 2020 7b36 7d20  27a|.|      {6} 
│ │ │ │ +00053340: 7c20 2d36 3333 3861 342b 3130 3032 3561  | -6338a4+10025a
│ │ │ │ +00053350: 3362 2b31 3439 3837 6133 632d 3939 3539  3b+14987a3c-9959
│ │ │ │ +00053360: 6132 6263 2d31 3136 3931 6132 6332 2b31  a2bc-11691a2c2+1
│ │ │ │ +00053370: 3233 3336 6162 6332 2d37 3738 3661 3364  2336abc2-7786a3d
│ │ │ │ +00053380: 2d31 317c 0a7c 2020 2020 2020 7b36 7d20  -11|.|      {6} 
│ │ │ │ +00053390: 7c20 3232 3735 6134 2d32 3339 6133 622b  | 2275a4-239a3b+
│ │ │ │ +000533a0: 3134 3539 3461 3262 322d 3831 3533 6162  14594a2b2-8153ab
│ │ │ │ +000533b0: 332d 3131 3934 3562 342d 3834 3136 6133  3-11945b4-8416a3
│ │ │ │ +000533c0: 632b 3632 3531 6132 6263 2d33 3032 3361  c+6251a2bc-3023a
│ │ │ │ +000533d0: 6232 637c 0a7c 2020 2020 2020 2d2d 2d2d  b2c|.|      ----
│ │ │ │ +000533e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000533f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00053430: 7c20 2020 2020 2062 3263 2b39 3231 3962  |      b2c+9219b
│ │ │ │ -00053440: 3363 2b35 3531 3361 3263 322b 3130 3535  3c+5513a2c2+1055
│ │ │ │ -00053450: 3861 6263 322b 3235 3930 6232 6332 2b31  8abc2+2590b2c2+1
│ │ │ │ -00053460: 3136 3234 6133 642d 3536 3033 6132 6264  1624a3d-5603a2bd
│ │ │ │ -00053470: 2b31 3430 3538 6162 3264 2d31 3236 7c0a  +14058ab2d-126|.
│ │ │ │ -00053480: 7c20 2020 2020 2062 3263 2b34 3839 6233  |      b2c+489b3
│ │ │ │ -00053490: 632d 3135 3338 3361 3263 322b 3530 3761  c-15383a2c2+507a
│ │ │ │ -000534a0: 6263 322d 3133 3830 3462 3263 322d 3834  bc2-13804b2c2-84
│ │ │ │ -000534b0: 3136 6163 332b 3932 6334 2d31 3130 3537  16ac3+92c4-11057
│ │ │ │ -000534c0: 6133 642d 3531 3133 6132 6264 2d32 7c0a  a3d-5113a2bd-2|.
│ │ │ │ -000534d0: 7c20 2020 2020 2035 3661 3262 642b 3439  |      56a2bd+49
│ │ │ │ -000534e0: 3630 6132 6364 2d35 3538 3961 6263 642d  60a2cd-5589abcd-
│ │ │ │ -000534f0: 3831 3633 6163 3264 2d31 3839 3562 6332  8163ac2d-1895bc2
│ │ │ │ -00053500: 642b 3934 3634 6132 6432 2d37 3235 3361  d+9464a2d2-7253a
│ │ │ │ -00053510: 6264 322b 3132 3634 3261 6364 322d 7c0a  bd2+12642acd2-|.
│ │ │ │ -00053520: 7c20 2020 2020 202b 3539 3333 6233 632b  |      +5933b3c+
│ │ │ │ -00053530: 3932 6132 6332 2b35 3334 3361 6263 322b  92a2c2+5343abc2+
│ │ │ │ -00053540: 3337 3938 6232 6332 2d31 3539 3638 6133  3798b2c2-15968a3
│ │ │ │ -00053550: 642b 3437 3361 3262 642b 3133 3239 3361  d+473a2bd+13293a
│ │ │ │ -00053560: 6232 642d 3337 3631 6233 642d 3737 7c0a  b2d-3761b3d-77|.
│ │ │ │ -00053570: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00053420: 2d2d 2d7c 0a7c 2020 2020 2020 6232 632b  ---|.|      b2c+
│ │ │ │ +00053430: 3932 3139 6233 632b 3535 3133 6132 6332  9219b3c+5513a2c2
│ │ │ │ +00053440: 2b31 3035 3538 6162 6332 2b32 3539 3062  +10558abc2+2590b
│ │ │ │ +00053450: 3263 322b 3131 3632 3461 3364 2d35 3630  2c2+11624a3d-560
│ │ │ │ +00053460: 3361 3262 642b 3134 3035 3861 6232 642d  3a2bd+14058ab2d-
│ │ │ │ +00053470: 3132 367c 0a7c 2020 2020 2020 6232 632b  126|.|      b2c+
│ │ │ │ +00053480: 3438 3962 3363 2d31 3533 3833 6132 6332  489b3c-15383a2c2
│ │ │ │ +00053490: 2b35 3037 6162 6332 2d31 3338 3034 6232  +507abc2-13804b2
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│ │ │ │ +000534c0: 642d 327c 0a7c 2020 2020 2020 3536 6132  d-2|.|      56a2
│ │ │ │ +000534d0: 6264 2b34 3936 3061 3263 642d 3535 3839  bd+4960a2cd-5589
│ │ │ │ +000534e0: 6162 6364 2d38 3136 3361 6332 642d 3138  abcd-8163ac2d-18
│ │ │ │ +000534f0: 3935 6263 3264 2b39 3436 3461 3264 322d  95bc2d+9464a2d2-
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│ │ │ │ +00053510: 6432 2d7c 0a7c 2020 2020 2020 2b35 3933  d2-|.|      +593
│ │ │ │ +00053520: 3362 3363 2b39 3261 3263 322b 3533 3433  3b3c+92a2c2+5343
│ │ │ │ +00053530: 6162 6332 2b33 3739 3862 3263 322d 3135  abc2+3798b2c2-15
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│ │ │ │ +00053550: 3332 3933 6162 3264 2d33 3736 3162 3364  3293ab2d-3761b3d
│ │ │ │ +00053560: 2d37 377c 0a7c 2020 2020 2020 2d2d 2d2d  -77|.|      ----
│ │ │ │ +00053570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000535a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000535b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000535c0: 7c20 2020 2020 2031 3562 3364 2b37 3836  |      15b3d+786
│ │ │ │ -000535d0: 3961 3263 642d 3230 3532 6162 6364 2d31  9a2cd-2052abcd-1
│ │ │ │ -000535e0: 3833 3162 3263 642b 3630 3432 6163 3264  831b2cd+6042ac2d
│ │ │ │ -000535f0: 2d32 3536 3162 6332 642d 3837 3039 6132  -2561bc2d-8709a2
│ │ │ │ -00053600: 6432 2d31 3332 3139 6162 6432 2b34 7c0a  d2-13219abd2+4|.
│ │ │ │ -00053610: 7c20 2020 2020 2037 3632 6162 3264 2b31  |      762ab2d+1
│ │ │ │ -00053620: 3430 3935 6233 642d 3135 3838 6132 6364  4095b3d-1588a2cd
│ │ │ │ -00053630: 2b32 3030 3061 6263 642d 3230 3830 6232  +2000abcd-2080b2
│ │ │ │ -00053640: 6364 2b39 3137 3561 6332 642d 3634 3962  cd+9175ac2d-649b
│ │ │ │ -00053650: 6332 642b 3838 3239 6333 642b 3231 7c0a  c2d+8829c3d+21|.
│ │ │ │ -00053660: 7c20 2020 2020 2031 3935 3862 6364 3220  |      1958bcd2 
│ │ │ │ +000535b0: 2d2d 2d7c 0a7c 2020 2020 2020 3135 6233  ---|.|      15b3
│ │ │ │ +000535c0: 642b 3738 3639 6132 6364 2d32 3035 3261  d+7869a2cd-2052a
│ │ │ │ +000535d0: 6263 642d 3138 3331 6232 6364 2b36 3034  bcd-1831b2cd+604
│ │ │ │ +000535e0: 3261 6332 642d 3235 3631 6263 3264 2d38  2ac2d-2561bc2d-8
│ │ │ │ +000535f0: 3730 3961 3264 322d 3133 3231 3961 6264  709a2d2-13219abd
│ │ │ │ +00053600: 322b 347c 0a7c 2020 2020 2020 3736 3261  2+4|.|      762a
│ │ │ │ +00053610: 6232 642b 3134 3039 3562 3364 2d31 3538  b2d+14095b3d-158
│ │ │ │ +00053620: 3861 3263 642b 3230 3030 6162 6364 2d32  8a2cd+2000abcd-2
│ │ │ │ +00053630: 3038 3062 3263 642b 3931 3735 6163 3264  080b2cd+9175ac2d
│ │ │ │ +00053640: 2d36 3439 6263 3264 2b38 3832 3963 3364  -649bc2d+8829c3d
│ │ │ │ +00053650: 2b32 317c 0a7c 2020 2020 2020 3139 3538  +21|.|      1958
│ │ │ │ +00053660: 6263 6432 2020 2020 2020 2020 2020 2020  bcd2            
│ │ │ │  00053670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000536a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000536b0: 7c20 2020 2020 2031 3761 3263 642d 3733  |      17a2cd-73
│ │ │ │ -000536c0: 3839 6162 6364 2b34 3732 3362 3263 642d  89abcd+4723b2cd-
│ │ │ │ -000536d0: 3133 3236 3261 6332 642b 3534 3331 6263  13262ac2d+5431bc
│ │ │ │ -000536e0: 3264 2b31 3132 3734 6132 6432 2d32 3137  2d+11274a2d2-217
│ │ │ │ -000536f0: 6162 6432 2b31 3236 3162 3264 322b 7c0a  abd2+1261b2d2+|.
│ │ │ │ -00053700: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +000536a0: 2020 207c 0a7c 2020 2020 2020 3137 6132     |.|      17a2
│ │ │ │ +000536b0: 6364 2d37 3338 3961 6263 642b 3437 3233  cd-7389abcd+4723
│ │ │ │ +000536c0: 6232 6364 2d31 3332 3632 6163 3264 2b35  b2cd-13262ac2d+5
│ │ │ │ +000536d0: 3433 3162 6332 642b 3131 3237 3461 3264  431bc2d+11274a2d
│ │ │ │ +000536e0: 322d 3231 3761 6264 322b 3132 3631 6232  2-217abd2+1261b2
│ │ │ │ +000536f0: 6432 2b7c 0a7c 2020 2020 2020 2d2d 2d2d  d2+|.|      ----
│ │ │ │ +00053700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00053750: 7c20 2020 2020 2032 3039 6232 6432 2b31  |      209b2d2+1
│ │ │ │ -00053760: 3232 3235 6163 6432 2d32 3630 3562 6364  2225acd2-2605bcd
│ │ │ │ -00053770: 322d 3932 6332 6432 2b31 3539 3638 6164  2-92c2d2+15968ad
│ │ │ │ -00053780: 332b 3134 3836 3062 6433 2d38 3832 3963  3+14860bd3-8829c
│ │ │ │ -00053790: 6433 2d31 3132 3734 6434 207c 2020 7c0a  d3-11274d4 |  |.
│ │ │ │ -000537a0: 7c20 2020 2020 2036 3461 3264 322b 3836  |      64a2d2+86
│ │ │ │ -000537b0: 3335 6162 6432 2d37 3136 3162 3264 322b  35abd2-7161b2d2+
│ │ │ │ -000537c0: 3939 3761 6364 322b 3330 3135 6263 6432  997acd2+3015bcd2
│ │ │ │ -000537d0: 2b31 3132 3734 6332 6432 2020 2020 2020  +11274c2d2      
│ │ │ │ -000537e0: 2020 2020 2020 2020 2020 207c 2020 7c0a             |  |.
│ │ │ │ -000537f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053740: 2d2d 2d7c 0a7c 2020 2020 2020 3230 3962  ---|.|      209b
│ │ │ │ +00053750: 3264 322b 3132 3232 3561 6364 322d 3236  2d2+12225acd2-26
│ │ │ │ +00053760: 3035 6263 6432 2d39 3263 3264 322b 3135  05bcd2-92c2d2+15
│ │ │ │ +00053770: 3936 3861 6433 2b31 3438 3630 6264 332d  968ad3+14860bd3-
│ │ │ │ +00053780: 3838 3239 6364 332d 3131 3237 3464 3420  8829cd3-11274d4 
│ │ │ │ +00053790: 7c20 207c 0a7c 2020 2020 2020 3634 6132  |  |.|      64a2
│ │ │ │ +000537a0: 6432 2b38 3633 3561 6264 322d 3731 3631  d2+8635abd2-7161
│ │ │ │ +000537b0: 6232 6432 2b39 3937 6163 6432 2b33 3031  b2d2+997acd2+301
│ │ │ │ +000537c0: 3562 6364 322b 3131 3237 3463 3264 3220  5bcd2+11274c2d2 
│ │ │ │ +000537d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000537e0: 7c20 207c 0a7c 2020 2020 2020 2020 2020  |  |.|          
│ │ │ │ +000537f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053830: 2020 2020 2020 2020 2020 207c 2020 7c0a             |  |.
│ │ │ │ -00053840: 7c20 2020 2020 2038 3230 3161 6364 322d  |      8201acd2-
│ │ │ │ -00053850: 3134 3038 3062 6364 3220 2020 2020 2020  14080bcd2       
│ │ │ │ +00053830: 7c20 207c 0a7c 2020 2020 2020 3832 3031  |  |.|      8201
│ │ │ │ +00053840: 6163 6432 2d31 3430 3830 6263 6432 2020  acd2-14080bcd2  
│ │ │ │ +00053850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053880: 2020 2020 2020 2020 2020 207c 2020 7c0a             |  |.
│ │ │ │ -00053890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053880: 7c20 207c 0a7c 2020 2020 2020 2020 2020  |  |.|          
│ │ │ │ +00053890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000538e0: 7c20 2020 2020 2020 2020 2020 2020 2034  |              4
│ │ │ │ -000538f0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +000538d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000538e0: 2020 2020 3420 2020 2020 2031 2020 2020      4      1    
│ │ │ │ +000538f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053930: 7c6f 3137 203a 204d 6174 7269 7820 5320  |o17 : Matrix S 
│ │ │ │ -00053940: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00053920: 2020 207c 0a7c 6f31 3720 3a20 4d61 7472     |.|o17 : Matr
│ │ │ │ +00053930: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +00053940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053980: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00053970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00053980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000539a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000539b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000539c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000539d0: 0a42 7574 2061 6c6c 2074 6865 2068 6f6d  .But all the hom
│ │ │ │ -000539e0: 6f74 6f70 6965 7320 6172 6520 6d69 6e69  otopies are mini
│ │ │ │ -000539f0: 6d61 6c20 696e 2074 6869 7320 6361 7365  mal in this case
│ │ │ │ -00053a00: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │ +000539c0: 2d2d 2d2b 0a0a 4275 7420 616c 6c20 7468  ---+..But all th
│ │ │ │ +000539d0: 6520 686f 6d6f 746f 7069 6573 2061 7265  e homotopies are
│ │ │ │ +000539e0: 206d 696e 696d 616c 2069 6e20 7468 6973   minimal in this
│ │ │ │ +000539f0: 2063 6173 653a 0a0a 2b2d 2d2d 2d2d 2d2d   case:..+-------
│ │ │ │ +00053a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053a30: 2b0a 7c69 3138 203a 206b 3120 3d20 535e  +.|i18 : k1 = S^
│ │ │ │ -00053a40: 312f 2869 6465 616c 2076 6172 7320 5329  1/(ideal vars S)
│ │ │ │ -00053a50: 3b20 2020 2020 2020 2020 2020 2020 207c  ;              |
│ │ │ │ -00053a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00053a20: 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 6b31  -----+.|i18 : k1
│ │ │ │ +00053a30: 203d 2053 5e31 2f28 6964 6561 6c20 7661   = S^1/(ideal va
│ │ │ │ +00053a40: 7273 2053 293b 2020 2020 2020 2020 2020  rs S);          
│ │ │ │ +00053a50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00053a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00053a90: 7c69 3139 203a 2073 656c 6563 7428 6b65  |i19 : select(ke
│ │ │ │ -00053aa0: 7973 2068 6f6d 6f74 2c6b 2d3e 286b 312a  ys homot,k->(k1*
│ │ │ │ -00053ab0: 2a68 6f6d 6f74 236b 2921 3d30 297c 0a7c  *homot#k)!=0)|.|
│ │ │ │ +00053a80: 2d2d 2d2b 0a7c 6931 3920 3a20 7365 6c65  ---+.|i19 : sele
│ │ │ │ +00053a90: 6374 286b 6579 7320 686f 6d6f 742c 6b2d  ct(keys homot,k-
│ │ │ │ +00053aa0: 3e28 6b31 2a2a 686f 6d6f 7423 6b29 213d  >(k1**homot#k)!=
│ │ │ │ +00053ab0: 3029 7c0a 7c20 2020 2020 2020 2020 2020  0)|.|           
│ │ │ │  00053ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00053af0: 3139 203d 207b 7d20 2020 2020 2020 2020  19 = {}         
│ │ │ │ +00053ae0: 207c 0a7c 6f31 3920 3d20 7b7d 2020 2020   |.|o19 = {}    
│ │ │ │ +00053af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00053b10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00053b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b40: 2020 2020 2020 2020 2020 7c0a 7c6f 3139            |.|o19
│ │ │ │ -00053b50: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ -00053b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00053b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00053b40: 0a7c 6f31 3920 3a20 4c69 7374 2020 2020  .|o19 : List    
│ │ │ │ +00053b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053b60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00053b70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00053b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ba0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00053bb0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00053bc0: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f  * *note makeHomo
│ │ │ │ -00053bd0: 746f 7069 6573 313a 206d 616b 6548 6f6d  topies1: makeHom
│ │ │ │ -00053be0: 6f74 6f70 6965 7331 2c20 2d2d 2072 6574  otopies1, -- ret
│ │ │ │ -00053bf0: 7572 6e73 2061 2073 7973 7465 6d20 6f66  urns a system of
│ │ │ │ -00053c00: 2066 6972 7374 0a20 2020 2068 6f6d 6f74   first.    homot
│ │ │ │ -00053c10: 6f70 6965 730a 0a57 6179 7320 746f 2075  opies..Ways to u
│ │ │ │ -00053c20: 7365 206d 616b 6548 6f6d 6f74 6f70 6965  se makeHomotopie
│ │ │ │ -00053c30: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
│ │ │ │ -00053c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00053c50: 2020 2a20 226d 616b 6548 6f6d 6f74 6f70    * "makeHomotop
│ │ │ │ -00053c60: 6965 7328 4d61 7472 6978 2c43 6861 696e  ies(Matrix,Chain
│ │ │ │ -00053c70: 436f 6d70 6c65 7829 220a 2020 2a20 226d  Complex)".  * "m
│ │ │ │ -00053c80: 616b 6548 6f6d 6f74 6f70 6965 7328 4d61  akeHomotopies(Ma
│ │ │ │ -00053c90: 7472 6978 2c43 6861 696e 436f 6d70 6c65  trix,ChainComple
│ │ │ │ -00053ca0: 782c 5a5a 2922 0a0a 466f 7220 7468 6520  x,ZZ)"..For the 
│ │ │ │ -00053cb0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00053cc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00053cd0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00053ce0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20  makeHomotopies: 
│ │ │ │ -00053cf0: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -00053d00: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -00053d10: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -00053d20: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -00053d30: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ -00053d40: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00053d50: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00053d60: 6e73 2e69 6e66 6f2c 204e 6f64 653a 206d  ns.info, Node: m
│ │ │ │ -00053d70: 616b 6548 6f6d 6f74 6f70 6965 7331 2c20  akeHomotopies1, 
│ │ │ │ -00053d80: 4e65 7874 3a20 6d61 6b65 486f 6d6f 746f  Next: makeHomoto
│ │ │ │ -00053d90: 7069 6573 4f6e 486f 6d6f 6c6f 6779 2c20  piesOnHomology, 
│ │ │ │ -00053da0: 5072 6576 3a20 6d61 6b65 486f 6d6f 746f  Prev: makeHomoto
│ │ │ │ -00053db0: 7069 6573 2c20 5570 3a20 546f 700a 0a6d  pies, Up: Top..m
│ │ │ │ -00053dc0: 616b 6548 6f6d 6f74 6f70 6965 7331 202d  akeHomotopies1 -
│ │ │ │ -00053dd0: 2d20 7265 7475 726e 7320 6120 7379 7374  - returns a syst
│ │ │ │ -00053de0: 656d 206f 6620 6669 7273 7420 686f 6d6f  em of first homo
│ │ │ │ -00053df0: 746f 7069 6573 0a2a 2a2a 2a2a 2a2a 2a2a  topies.*********
│ │ │ │ +00053b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00053ba0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +00053bb0: 3d0a 0a20 202a 202a 6e6f 7465 206d 616b  =..  * *note mak
│ │ │ │ +00053bc0: 6548 6f6d 6f74 6f70 6965 7331 3a20 6d61  eHomotopies1: ma
│ │ │ │ +00053bd0: 6b65 486f 6d6f 746f 7069 6573 312c 202d  keHomotopies1, -
│ │ │ │ +00053be0: 2d20 7265 7475 726e 7320 6120 7379 7374  - returns a syst
│ │ │ │ +00053bf0: 656d 206f 6620 6669 7273 740a 2020 2020  em of first.    
│ │ │ │ +00053c00: 686f 6d6f 746f 7069 6573 0a0a 5761 7973  homotopies..Ways
│ │ │ │ +00053c10: 2074 6f20 7573 6520 6d61 6b65 486f 6d6f   to use makeHomo
│ │ │ │ +00053c20: 746f 7069 6573 3a0a 3d3d 3d3d 3d3d 3d3d  topies:.========
│ │ │ │ +00053c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00053c40: 3d3d 3d0a 0a20 202a 2022 6d61 6b65 486f  ===..  * "makeHo
│ │ │ │ +00053c50: 6d6f 746f 7069 6573 284d 6174 7269 782c  motopies(Matrix,
│ │ │ │ +00053c60: 4368 6169 6e43 6f6d 706c 6578 2922 0a20  ChainComplex)". 
│ │ │ │ +00053c70: 202a 2022 6d61 6b65 486f 6d6f 746f 7069   * "makeHomotopi
│ │ │ │ +00053c80: 6573 284d 6174 7269 782c 4368 6169 6e43  es(Matrix,ChainC
│ │ │ │ +00053c90: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72  omplex,ZZ)"..For
│ │ │ │ +00053ca0: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00053cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00053cc0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00053cd0: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70  note makeHomotop
│ │ │ │ +00053ce0: 6965 733a 206d 616b 6548 6f6d 6f74 6f70  ies: makeHomotop
│ │ │ │ +00053cf0: 6965 732c 2069 7320 6120 2a6e 6f74 6520  ies, is a *note 
│ │ │ │ +00053d00: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +00053d10: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00053d20: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +00053d30: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +00053d40: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +00053d50: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +00053d60: 6465 3a20 6d61 6b65 486f 6d6f 746f 7069  de: makeHomotopi
│ │ │ │ +00053d70: 6573 312c 204e 6578 743a 206d 616b 6548  es1, Next: makeH
│ │ │ │ +00053d80: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c  omotopiesOnHomol
│ │ │ │ +00053d90: 6f67 792c 2050 7265 763a 206d 616b 6548  ogy, Prev: makeH
│ │ │ │ +00053da0: 6f6d 6f74 6f70 6965 732c 2055 703a 2054  omotopies, Up: T
│ │ │ │ +00053db0: 6f70 0a0a 6d61 6b65 486f 6d6f 746f 7069  op..makeHomotopi
│ │ │ │ +00053dc0: 6573 3120 2d2d 2072 6574 7572 6e73 2061  es1 -- returns a
│ │ │ │ +00053dd0: 2073 7973 7465 6d20 6f66 2066 6972 7374   system of first
│ │ │ │ +00053de0: 2068 6f6d 6f74 6f70 6965 730a 2a2a 2a2a   homotopies.****
│ │ │ │ +00053df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00053e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00053e10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00053e20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00053e30: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ -00053e40: 3d0a 0a20 202a 2055 7361 6765 3a20 0a20  =..  * Usage: . 
│ │ │ │ -00053e50: 2020 2020 2020 2048 203d 206d 616b 6548         H = makeH
│ │ │ │ -00053e60: 6f6d 6f74 6f70 6965 7331 2866 2c46 2c64  omotopies1(f,F,d
│ │ │ │ -00053e70: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -00053e80: 2020 2020 2a20 662c 2061 202a 6e6f 7465      * f, a *note
│ │ │ │ -00053e90: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ -00053ea0: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ -00053eb0: 3178 6e20 6d61 7472 6978 206f 6620 656c  1xn matrix of el
│ │ │ │ -00053ec0: 656d 656e 7473 206f 6620 530a 2020 2020  ements of S.    
│ │ │ │ -00053ed0: 2020 2a20 462c 2061 202a 6e6f 7465 2063    * F, a *note c
│ │ │ │ -00053ee0: 6861 696e 2063 6f6d 706c 6578 3a20 284d  hain complex: (M
│ │ │ │ -00053ef0: 6163 6175 6c61 7932 446f 6329 4368 6169  acaulay2Doc)Chai
│ │ │ │ -00053f00: 6e43 6f6d 706c 6578 2c2c 2061 646d 6974  nComplex,, admit
│ │ │ │ -00053f10: 7469 6e67 0a20 2020 2020 2020 2068 6f6d  ting.        hom
│ │ │ │ -00053f20: 6f74 6f70 6965 7320 666f 7220 7468 6520  otopies for the 
│ │ │ │ -00053f30: 656e 7472 6965 7320 6f66 2066 0a20 2020  entries of f.   
│ │ │ │ -00053f40: 2020 202a 2064 2c20 616e 202a 6e6f 7465     * d, an *note
│ │ │ │ -00053f50: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -00053f60: 6c61 7932 446f 6329 5a5a 2c2c 2068 6f77  lay2Doc)ZZ,, how
│ │ │ │ -00053f70: 2066 6172 2062 6163 6b20 746f 2063 6f6d   far back to com
│ │ │ │ -00053f80: 7075 7465 2074 6865 0a20 2020 2020 2020  pute the.       
│ │ │ │ -00053f90: 2068 6f6d 6f74 6f70 6965 7320 2864 6566   homotopies (def
│ │ │ │ -00053fa0: 6175 6c74 7320 746f 206c 656e 6774 6820  aults to length 
│ │ │ │ -00053fb0: 6f66 2046 290a 2020 2a20 4f75 7470 7574  of F).  * Output
│ │ │ │ -00053fc0: 733a 0a20 2020 2020 202a 2048 2c20 6120  s:.      * H, a 
│ │ │ │ -00053fd0: 2a6e 6f74 6520 6861 7368 2074 6162 6c65  *note hash table
│ │ │ │ -00053fe0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00053ff0: 4861 7368 5461 626c 652c 2c20 6769 7665  HashTable,, give
│ │ │ │ -00054000: 7320 7468 6520 686f 6d6f 746f 7079 0a20  s the homotopy. 
│ │ │ │ -00054010: 2020 2020 2020 2066 726f 6d20 465f 6920         from F_i 
│ │ │ │ -00054020: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ -00054030: 2066 5f6a 2061 7320 7468 6520 7661 6c75   f_j as the valu
│ │ │ │ -00054040: 6520 2448 235c 7b6a 2c69 5c7d 240a 0a44  e $H#\{j,i\}$..D
│ │ │ │ -00054050: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00054060: 3d3d 3d3d 3d3d 0a0a 5361 6d65 2061 7320  ======..Same as 
│ │ │ │ -00054070: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -00054080: 6275 7420 6f6e 6c79 2063 6f6d 7075 7465  but only compute
│ │ │ │ -00054090: 7320 7468 6520 6f72 6469 6e61 7279 2068  s the ordinary h
│ │ │ │ -000540a0: 6f6d 6f74 6f70 6965 732c 206e 6f74 2074  omotopies, not t
│ │ │ │ -000540b0: 6865 0a68 6967 6865 7220 6f6e 6573 2e20  he.higher ones. 
│ │ │ │ -000540c0: 5573 6564 2069 6e20 6578 7465 7269 6f72  Used in exterior
│ │ │ │ -000540d0: 546f 724d 6f64 756c 650a 0a53 6565 2061  TorModule..See a
│ │ │ │ -000540e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -000540f0: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f  * *note makeHomo
│ │ │ │ -00054100: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f  topies: makeHomo
│ │ │ │ -00054110: 746f 7069 6573 2c20 2d2d 2072 6574 7572  topies, -- retur
│ │ │ │ -00054120: 6e73 2061 2073 7973 7465 6d20 6f66 2068  ns a system of h
│ │ │ │ -00054130: 6967 6865 720a 2020 2020 686f 6d6f 746f  igher.    homoto
│ │ │ │ -00054140: 7069 6573 0a20 202a 202a 6e6f 7465 2065  pies.  * *note e
│ │ │ │ -00054150: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -00054160: 3a20 6578 7465 7269 6f72 546f 724d 6f64  : exteriorTorMod
│ │ │ │ -00054170: 756c 652c 202d 2d20 546f 7220 6173 2061  ule, -- Tor as a
│ │ │ │ -00054180: 206d 6f64 756c 6520 6f76 6572 2061 6e0a   module over an.
│ │ │ │ -00054190: 2020 2020 6578 7465 7269 6f72 2061 6c67      exterior alg
│ │ │ │ -000541a0: 6562 7261 206f 7220 6269 6772 6164 6564  ebra or bigraded
│ │ │ │ -000541b0: 2061 6c67 6562 7261 0a0a 5761 7973 2074   algebra..Ways t
│ │ │ │ -000541c0: 6f20 7573 6520 6d61 6b65 486f 6d6f 746f  o use makeHomoto
│ │ │ │ -000541d0: 7069 6573 313a 0a3d 3d3d 3d3d 3d3d 3d3d  pies1:.=========
│ │ │ │ -000541e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000541f0: 3d3d 3d0a 0a20 202a 2022 6d61 6b65 486f  ===..  * "makeHo
│ │ │ │ -00054200: 6d6f 746f 7069 6573 3128 4d61 7472 6978  motopies1(Matrix
│ │ │ │ -00054210: 2c43 6861 696e 436f 6d70 6c65 7829 220a  ,ChainComplex)".
│ │ │ │ -00054220: 2020 2a20 226d 616b 6548 6f6d 6f74 6f70    * "makeHomotop
│ │ │ │ -00054230: 6965 7331 284d 6174 7269 782c 4368 6169  ies1(Matrix,Chai
│ │ │ │ -00054240: 6e43 6f6d 706c 6578 2c5a 5a29 220a 0a46  nComplex,ZZ)"..F
│ │ │ │ -00054250: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00054260: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00054270: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00054280: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74   *note makeHomot
│ │ │ │ -00054290: 6f70 6965 7331 3a20 6d61 6b65 486f 6d6f  opies1: makeHomo
│ │ │ │ -000542a0: 746f 7069 6573 312c 2069 7320 6120 2a6e  topies1, is a *n
│ │ │ │ -000542b0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -000542c0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -000542d0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -000542e0: 6e2c 2e0a 1f0a 4669 6c65 3a20 436f 6d70  n,....File: Comp
│ │ │ │ -000542f0: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -00054300: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ -00054310: 2c20 4e6f 6465 3a20 6d61 6b65 486f 6d6f  , Node: makeHomo
│ │ │ │ -00054320: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779  topiesOnHomology
│ │ │ │ -00054330: 2c20 4e65 7874 3a20 6d61 6b65 4d6f 6475  , Next: makeModu
│ │ │ │ -00054340: 6c65 2c20 5072 6576 3a20 6d61 6b65 486f  le, Prev: makeHo
│ │ │ │ -00054350: 6d6f 746f 7069 6573 312c 2055 703a 2054  motopies1, Up: T
│ │ │ │ -00054360: 6f70 0a0a 6d61 6b65 486f 6d6f 746f 7069  op..makeHomotopi
│ │ │ │ -00054370: 6573 4f6e 486f 6d6f 6c6f 6779 202d 2d20  esOnHomology -- 
│ │ │ │ -00054380: 486f 6d6f 6c6f 6779 206f 6620 6120 636f  Homology of a co
│ │ │ │ -00054390: 6d70 6c65 7820 6173 2065 7874 6572 696f  mplex as exterio
│ │ │ │ -000543a0: 7220 6d6f 6475 6c65 0a2a 2a2a 2a2a 2a2a  r module.*******
│ │ │ │ +00053e20: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +00053e30: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ +00053e40: 653a 200a 2020 2020 2020 2020 4820 3d20  e: .        H = 
│ │ │ │ +00053e50: 6d61 6b65 486f 6d6f 746f 7069 6573 3128  makeHomotopies1(
│ │ │ │ +00053e60: 662c 462c 6429 0a20 202a 2049 6e70 7574  f,F,d).  * Input
│ │ │ │ +00053e70: 733a 0a20 2020 2020 202a 2066 2c20 6120  s:.      * f, a 
│ │ │ │ +00053e80: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d  *note matrix: (M
│ │ │ │ +00053e90: 6163 6175 6c61 7932 446f 6329 4d61 7472  acaulay2Doc)Matr
│ │ │ │ +00053ea0: 6978 2c2c 2031 786e 206d 6174 7269 7820  ix,, 1xn matrix 
│ │ │ │ +00053eb0: 6f66 2065 6c65 6d65 6e74 7320 6f66 2053  of elements of S
│ │ │ │ +00053ec0: 0a20 2020 2020 202a 2046 2c20 6120 2a6e  .      * F, a *n
│ │ │ │ +00053ed0: 6f74 6520 6368 6169 6e20 636f 6d70 6c65  ote chain comple
│ │ │ │ +00053ee0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +00053ef0: 2943 6861 696e 436f 6d70 6c65 782c 2c20  )ChainComplex,, 
│ │ │ │ +00053f00: 6164 6d69 7474 696e 670a 2020 2020 2020  admitting.      
│ │ │ │ +00053f10: 2020 686f 6d6f 746f 7069 6573 2066 6f72    homotopies for
│ │ │ │ +00053f20: 2074 6865 2065 6e74 7269 6573 206f 6620   the entries of 
│ │ │ │ +00053f30: 660a 2020 2020 2020 2a20 642c 2061 6e20  f.      * d, an 
│ │ │ │ +00053f40: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ +00053f50: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ +00053f60: 2c20 686f 7720 6661 7220 6261 636b 2074  , how far back t
│ │ │ │ +00053f70: 6f20 636f 6d70 7574 6520 7468 650a 2020  o compute the.  
│ │ │ │ +00053f80: 2020 2020 2020 686f 6d6f 746f 7069 6573        homotopies
│ │ │ │ +00053f90: 2028 6465 6661 756c 7473 2074 6f20 6c65   (defaults to le
│ │ │ │ +00053fa0: 6e67 7468 206f 6620 4629 0a20 202a 204f  ngth of F).  * O
│ │ │ │ +00053fb0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +00053fc0: 482c 2061 202a 6e6f 7465 2068 6173 6820  H, a *note hash 
│ │ │ │ +00053fd0: 7461 626c 653a 2028 4d61 6361 756c 6179  table: (Macaulay
│ │ │ │ +00053fe0: 3244 6f63 2948 6173 6854 6162 6c65 2c2c  2Doc)HashTable,,
│ │ │ │ +00053ff0: 2067 6976 6573 2074 6865 2068 6f6d 6f74   gives the homot
│ │ │ │ +00054000: 6f70 790a 2020 2020 2020 2020 6672 6f6d  opy.        from
│ │ │ │ +00054010: 2046 5f69 2063 6f72 7265 7370 6f6e 6469   F_i correspondi
│ │ │ │ +00054020: 6e67 2074 6f20 665f 6a20 6173 2074 6865  ng to f_j as the
│ │ │ │ +00054030: 2076 616c 7565 2024 4823 5c7b 6a2c 695c   value $H#\{j,i\
│ │ │ │ +00054040: 7d24 0a0a 4465 7363 7269 7074 696f 6e0a  }$..Description.
│ │ │ │ +00054050: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 616d  ===========..Sam
│ │ │ │ +00054060: 6520 6173 206d 616b 6548 6f6d 6f74 6f70  e as makeHomotop
│ │ │ │ +00054070: 6965 732c 2062 7574 206f 6e6c 7920 636f  ies, but only co
│ │ │ │ +00054080: 6d70 7574 6573 2074 6865 206f 7264 696e  mputes the ordin
│ │ │ │ +00054090: 6172 7920 686f 6d6f 746f 7069 6573 2c20  ary homotopies, 
│ │ │ │ +000540a0: 6e6f 7420 7468 650a 6869 6768 6572 206f  not the.higher o
│ │ │ │ +000540b0: 6e65 732e 2055 7365 6420 696e 2065 7874  nes. Used in ext
│ │ │ │ +000540c0: 6572 696f 7254 6f72 4d6f 6475 6c65 0a0a  eriorTorModule..
│ │ │ │ +000540d0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +000540e0: 3d0a 0a20 202a 202a 6e6f 7465 206d 616b  =..  * *note mak
│ │ │ │ +000540f0: 6548 6f6d 6f74 6f70 6965 733a 206d 616b  eHomotopies: mak
│ │ │ │ +00054100: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20  eHomotopies, -- 
│ │ │ │ +00054110: 7265 7475 726e 7320 6120 7379 7374 656d  returns a system
│ │ │ │ +00054120: 206f 6620 6869 6768 6572 0a20 2020 2068   of higher.    h
│ │ │ │ +00054130: 6f6d 6f74 6f70 6965 730a 2020 2a20 2a6e  omotopies.  * *n
│ │ │ │ +00054140: 6f74 6520 6578 7465 7269 6f72 546f 724d  ote exteriorTorM
│ │ │ │ +00054150: 6f64 756c 653a 2065 7874 6572 696f 7254  odule: exteriorT
│ │ │ │ +00054160: 6f72 4d6f 6475 6c65 2c20 2d2d 2054 6f72  orModule, -- Tor
│ │ │ │ +00054170: 2061 7320 6120 6d6f 6475 6c65 206f 7665   as a module ove
│ │ │ │ +00054180: 7220 616e 0a20 2020 2065 7874 6572 696f  r an.    exterio
│ │ │ │ +00054190: 7220 616c 6765 6272 6120 6f72 2062 6967  r algebra or big
│ │ │ │ +000541a0: 7261 6465 6420 616c 6765 6272 610a 0a57  raded algebra..W
│ │ │ │ +000541b0: 6179 7320 746f 2075 7365 206d 616b 6548  ays to use makeH
│ │ │ │ +000541c0: 6f6d 6f74 6f70 6965 7331 3a0a 3d3d 3d3d  omotopies1:.====
│ │ │ │ +000541d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000541e0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ +000541f0: 616b 6548 6f6d 6f74 6f70 6965 7331 284d  akeHomotopies1(M
│ │ │ │ +00054200: 6174 7269 782c 4368 6169 6e43 6f6d 706c  atrix,ChainCompl
│ │ │ │ +00054210: 6578 2922 0a20 202a 2022 6d61 6b65 486f  ex)".  * "makeHo
│ │ │ │ +00054220: 6d6f 746f 7069 6573 3128 4d61 7472 6978  motopies1(Matrix
│ │ │ │ +00054230: 2c43 6861 696e 436f 6d70 6c65 782c 5a5a  ,ChainComplex,ZZ
│ │ │ │ +00054240: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00054250: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00054260: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00054270: 626a 6563 7420 2a6e 6f74 6520 6d61 6b65  bject *note make
│ │ │ │ +00054280: 486f 6d6f 746f 7069 6573 313a 206d 616b  Homotopies1: mak
│ │ │ │ +00054290: 6548 6f6d 6f74 6f70 6965 7331 2c20 6973  eHomotopies1, is
│ │ │ │ +000542a0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +000542b0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ +000542c0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +000542d0: 6e63 7469 6f6e 2c2e 0a1f 0a46 696c 653a  nction,....File:
│ │ │ │ +000542e0: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +000542f0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +00054300: 2e69 6e66 6f2c 204e 6f64 653a 206d 616b  .info, Node: mak
│ │ │ │ +00054310: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d  eHomotopiesOnHom
│ │ │ │ +00054320: 6f6c 6f67 792c 204e 6578 743a 206d 616b  ology, Next: mak
│ │ │ │ +00054330: 654d 6f64 756c 652c 2050 7265 763a 206d  eModule, Prev: m
│ │ │ │ +00054340: 616b 6548 6f6d 6f74 6f70 6965 7331 2c20  akeHomotopies1, 
│ │ │ │ +00054350: 5570 3a20 546f 700a 0a6d 616b 6548 6f6d  Up: Top..makeHom
│ │ │ │ +00054360: 6f74 6f70 6965 734f 6e48 6f6d 6f6c 6f67  otopiesOnHomolog
│ │ │ │ +00054370: 7920 2d2d 2048 6f6d 6f6c 6f67 7920 6f66  y -- Homology of
│ │ │ │ +00054380: 2061 2063 6f6d 706c 6578 2061 7320 6578   a complex as ex
│ │ │ │ +00054390: 7465 7269 6f72 206d 6f64 756c 650a 2a2a  terior module.**
│ │ │ │ +000543a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000543b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000543c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000543d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000543e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -000543f0: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -00054400: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00054410: 2020 2020 2020 2848 2c68 2920 3d20 6d61        (H,h) = ma
│ │ │ │ -00054420: 6b65 486f 6d6f 746f 7069 6573 4f6e 486f  keHomotopiesOnHo
│ │ │ │ -00054430: 6d6f 6c6f 6779 2866 662c 2043 290a 2020  mology(ff, C).  
│ │ │ │ -00054440: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00054450: 2a20 6666 2c20 6120 2a6e 6f74 6520 6d61  * ff, a *note ma
│ │ │ │ -00054460: 7472 6978 3a20 284d 6163 6175 6c61 7932  trix: (Macaulay2
│ │ │ │ -00054470: 446f 6329 4d61 7472 6978 2c2c 206d 6174  Doc)Matrix,, mat
│ │ │ │ -00054480: 7269 7820 6f66 2065 6c65 6d65 6e74 7320  rix of elements 
│ │ │ │ -00054490: 686f 6d6f 746f 7069 630a 2020 2020 2020  homotopic.      
│ │ │ │ -000544a0: 2020 746f 2030 206f 6e20 430a 2020 2020    to 0 on C.    
│ │ │ │ -000544b0: 2020 2a20 432c 2061 202a 6e6f 7465 2063    * C, a *note c
│ │ │ │ -000544c0: 6861 696e 2063 6f6d 706c 6578 3a20 284d  hain complex: (M
│ │ │ │ -000544d0: 6163 6175 6c61 7932 446f 6329 4368 6169  acaulay2Doc)Chai
│ │ │ │ -000544e0: 6e43 6f6d 706c 6578 2c2c 200a 2020 2a20  nComplex,, .  * 
│ │ │ │ -000544f0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00054500: 2048 2c20 6120 2a6e 6f74 6520 6861 7368   H, a *note hash
│ │ │ │ -00054510: 2074 6162 6c65 3a20 284d 6163 6175 6c61   table: (Macaula
│ │ │ │ -00054520: 7932 446f 6329 4861 7368 5461 626c 652c  y2Doc)HashTable,
│ │ │ │ -00054530: 2c20 486f 6d6f 6c6f 6779 206f 6620 432c  , Homology of C,
│ │ │ │ -00054540: 2069 6e64 6578 6564 0a20 2020 2020 2020   indexed.       
│ │ │ │ -00054550: 2062 7920 706c 6163 6573 2069 6e20 7468   by places in th
│ │ │ │ -00054560: 6520 430a 2020 2020 2020 2a20 682c 2061  e C.      * h, a
│ │ │ │ -00054570: 202a 6e6f 7465 2068 6173 6820 7461 626c   *note hash tabl
│ │ │ │ -00054580: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00054590: 2948 6173 6854 6162 6c65 2c2c 2068 6f6d  )HashTable,, hom
│ │ │ │ -000545a0: 6f74 6f70 6965 7320 666f 720a 2020 2020  otopies for.    
│ │ │ │ -000545b0: 2020 2020 656c 656d 656e 7473 206f 6620      elements of 
│ │ │ │ -000545c0: 6620 6f6e 2074 6865 2068 6f6d 6f6c 6f67  f on the homolog
│ │ │ │ -000545d0: 7920 6f66 2043 0a0a 4465 7363 7269 7074  y of C..Descript
│ │ │ │ -000545e0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -000545f0: 0a54 6865 2073 6372 6970 7420 6361 6c6c  .The script call
│ │ │ │ -00054600: 7320 6d61 6b65 486f 6d6f 746f 7069 6573  s makeHomotopies
│ │ │ │ -00054610: 3120 746f 2070 726f 6475 6365 2068 6f6d  1 to produce hom
│ │ │ │ -00054620: 6f74 6f70 6965 7320 666f 7220 7468 6520  otopies for the 
│ │ │ │ -00054630: 6666 5f69 206f 6e20 432c 2061 6e64 0a74  ff_i on C, and.t
│ │ │ │ -00054640: 6865 6e20 636f 6d70 7574 6573 2074 6865  hen computes the
│ │ │ │ -00054650: 6972 2061 6374 696f 6e20 6f6e 2074 6865  ir action on the
│ │ │ │ -00054660: 2048 6f6d 6f6c 6f67 7920 6f66 2043 2e0a   Homology of C..
│ │ │ │ -00054670: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00054680: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6578  ==..  * *note ex
│ │ │ │ -00054690: 7465 7269 6f72 546f 724d 6f64 756c 653a  teriorTorModule:
│ │ │ │ -000546a0: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ -000546b0: 6c65 2c20 2d2d 2054 6f72 2061 7320 6120  le, -- Tor as a 
│ │ │ │ -000546c0: 6d6f 6475 6c65 206f 7665 7220 616e 0a20  module over an. 
│ │ │ │ -000546d0: 2020 2065 7874 6572 696f 7220 616c 6765     exterior alge
│ │ │ │ -000546e0: 6272 6120 6f72 2062 6967 7261 6465 6420  bra or bigraded 
│ │ │ │ -000546f0: 616c 6765 6272 610a 2020 2a20 2a6e 6f74  algebra.  * *not
│ │ │ │ -00054700: 6520 6578 7465 7269 6f72 4578 744d 6f64  e exteriorExtMod
│ │ │ │ -00054710: 756c 653a 2065 7874 6572 696f 7245 7874  ule: exteriorExt
│ │ │ │ -00054720: 4d6f 6475 6c65 2c20 2d2d 2045 7874 284d  Module, -- Ext(M
│ │ │ │ -00054730: 2c6b 2920 6f72 2045 7874 284d 2c4e 2920  ,k) or Ext(M,N) 
│ │ │ │ -00054740: 6173 2061 0a20 2020 206d 6f64 756c 6520  as a.    module 
│ │ │ │ -00054750: 6f76 6572 2061 6e20 6578 7465 7269 6f72  over an exterior
│ │ │ │ -00054760: 2061 6c67 6562 7261 0a0a 5761 7973 2074   algebra..Ways t
│ │ │ │ -00054770: 6f20 7573 6520 6d61 6b65 486f 6d6f 746f  o use makeHomoto
│ │ │ │ -00054780: 7069 6573 4f6e 486f 6d6f 6c6f 6779 3a0a  piesOnHomology:.
│ │ │ │ +000543e0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +000543f0: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +00054400: 3a20 0a20 2020 2020 2020 2028 482c 6829  : .        (H,h)
│ │ │ │ +00054410: 203d 206d 616b 6548 6f6d 6f74 6f70 6965   = makeHomotopie
│ │ │ │ +00054420: 734f 6e48 6f6d 6f6c 6f67 7928 6666 2c20  sOnHomology(ff, 
│ │ │ │ +00054430: 4329 0a20 202a 2049 6e70 7574 733a 0a20  C).  * Inputs:. 
│ │ │ │ +00054440: 2020 2020 202a 2066 662c 2061 202a 6e6f       * ff, a *no
│ │ │ │ +00054450: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +00054460: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ +00054470: 2c20 6d61 7472 6978 206f 6620 656c 656d  , matrix of elem
│ │ │ │ +00054480: 656e 7473 2068 6f6d 6f74 6f70 6963 0a20  ents homotopic. 
│ │ │ │ +00054490: 2020 2020 2020 2074 6f20 3020 6f6e 2043         to 0 on C
│ │ │ │ +000544a0: 0a20 2020 2020 202a 2043 2c20 6120 2a6e  .      * C, a *n
│ │ │ │ +000544b0: 6f74 6520 6368 6169 6e20 636f 6d70 6c65  ote chain comple
│ │ │ │ +000544c0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +000544d0: 2943 6861 696e 436f 6d70 6c65 782c 2c20  )ChainComplex,, 
│ │ │ │ +000544e0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +000544f0: 2020 2020 2a20 482c 2061 202a 6e6f 7465      * H, a *note
│ │ │ │ +00054500: 2068 6173 6820 7461 626c 653a 2028 4d61   hash table: (Ma
│ │ │ │ +00054510: 6361 756c 6179 3244 6f63 2948 6173 6854  caulay2Doc)HashT
│ │ │ │ +00054520: 6162 6c65 2c2c 2048 6f6d 6f6c 6f67 7920  able,, Homology 
│ │ │ │ +00054530: 6f66 2043 2c20 696e 6465 7865 640a 2020  of C, indexed.  
│ │ │ │ +00054540: 2020 2020 2020 6279 2070 6c61 6365 7320        by places 
│ │ │ │ +00054550: 696e 2074 6865 2043 0a20 2020 2020 202a  in the C.      *
│ │ │ │ +00054560: 2068 2c20 6120 2a6e 6f74 6520 6861 7368   h, a *note hash
│ │ │ │ +00054570: 2074 6162 6c65 3a20 284d 6163 6175 6c61   table: (Macaula
│ │ │ │ +00054580: 7932 446f 6329 4861 7368 5461 626c 652c  y2Doc)HashTable,
│ │ │ │ +00054590: 2c20 686f 6d6f 746f 7069 6573 2066 6f72  , homotopies for
│ │ │ │ +000545a0: 0a20 2020 2020 2020 2065 6c65 6d65 6e74  .        element
│ │ │ │ +000545b0: 7320 6f66 2066 206f 6e20 7468 6520 686f  s of f on the ho
│ │ │ │ +000545c0: 6d6f 6c6f 6779 206f 6620 430a 0a44 6573  mology of C..Des
│ │ │ │ +000545d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +000545e0: 3d3d 3d3d 0a0a 5468 6520 7363 7269 7074  ====..The script
│ │ │ │ +000545f0: 2063 616c 6c73 206d 616b 6548 6f6d 6f74   calls makeHomot
│ │ │ │ +00054600: 6f70 6965 7331 2074 6f20 7072 6f64 7563  opies1 to produc
│ │ │ │ +00054610: 6520 686f 6d6f 746f 7069 6573 2066 6f72  e homotopies for
│ │ │ │ +00054620: 2074 6865 2066 665f 6920 6f6e 2043 2c20   the ff_i on C, 
│ │ │ │ +00054630: 616e 640a 7468 656e 2063 6f6d 7075 7465  and.then compute
│ │ │ │ +00054640: 7320 7468 6569 7220 6163 7469 6f6e 206f  s their action o
│ │ │ │ +00054650: 6e20 7468 6520 486f 6d6f 6c6f 6779 206f  n the Homology o
│ │ │ │ +00054660: 6620 432e 0a0a 5365 6520 616c 736f 0a3d  f C...See also.=
│ │ │ │ +00054670: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00054680: 7465 2065 7874 6572 696f 7254 6f72 4d6f  te exteriorTorMo
│ │ │ │ +00054690: 6475 6c65 3a20 6578 7465 7269 6f72 546f  dule: exteriorTo
│ │ │ │ +000546a0: 724d 6f64 756c 652c 202d 2d20 546f 7220  rModule, -- Tor 
│ │ │ │ +000546b0: 6173 2061 206d 6f64 756c 6520 6f76 6572  as a module over
│ │ │ │ +000546c0: 2061 6e0a 2020 2020 6578 7465 7269 6f72   an.    exterior
│ │ │ │ +000546d0: 2061 6c67 6562 7261 206f 7220 6269 6772   algebra or bigr
│ │ │ │ +000546e0: 6164 6564 2061 6c67 6562 7261 0a20 202a  aded algebra.  *
│ │ │ │ +000546f0: 202a 6e6f 7465 2065 7874 6572 696f 7245   *note exteriorE
│ │ │ │ +00054700: 7874 4d6f 6475 6c65 3a20 6578 7465 7269  xtModule: exteri
│ │ │ │ +00054710: 6f72 4578 744d 6f64 756c 652c 202d 2d20  orExtModule, -- 
│ │ │ │ +00054720: 4578 7428 4d2c 6b29 206f 7220 4578 7428  Ext(M,k) or Ext(
│ │ │ │ +00054730: 4d2c 4e29 2061 7320 610a 2020 2020 6d6f  M,N) as a.    mo
│ │ │ │ +00054740: 6475 6c65 206f 7665 7220 616e 2065 7874  dule over an ext
│ │ │ │ +00054750: 6572 696f 7220 616c 6765 6272 610a 0a57  erior algebra..W
│ │ │ │ +00054760: 6179 7320 746f 2075 7365 206d 616b 6548  ays to use makeH
│ │ │ │ +00054770: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c  omotopiesOnHomol
│ │ │ │ +00054780: 6f67 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ogy:.===========
│ │ │ │  00054790: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000547a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000547b0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65  =====..  * "make
│ │ │ │ -000547c0: 486f 6d6f 746f 7069 6573 4f6e 486f 6d6f  HomotopiesOnHomo
│ │ │ │ -000547d0: 6c6f 6779 284d 6174 7269 782c 4368 6169  logy(Matrix,Chai
│ │ │ │ -000547e0: 6e43 6f6d 706c 6578 2922 0a0a 466f 7220  nComplex)"..For 
│ │ │ │ -000547f0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -00054800: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00054810: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00054820: 6f74 6520 6d61 6b65 486f 6d6f 746f 7069  ote makeHomotopi
│ │ │ │ -00054830: 6573 4f6e 486f 6d6f 6c6f 6779 3a20 6d61  esOnHomology: ma
│ │ │ │ -00054840: 6b65 486f 6d6f 746f 7069 6573 4f6e 486f  keHomotopiesOnHo
│ │ │ │ -00054850: 6d6f 6c6f 6779 2c20 6973 2061 202a 6e6f  mology, is a *no
│ │ │ │ -00054860: 7465 0a6d 6574 686f 6420 6675 6e63 7469  te.method functi
│ │ │ │ -00054870: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
│ │ │ │ -00054880: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -00054890: 2c2e 0a1f 0a46 696c 653a 2043 6f6d 706c  ,....File: Compl
│ │ │ │ -000548a0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -000548b0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -000548c0: 204e 6f64 653a 206d 616b 654d 6f64 756c   Node: makeModul
│ │ │ │ -000548d0: 652c 204e 6578 743a 206d 616b 6554 2c20  e, Next: makeT, 
│ │ │ │ -000548e0: 5072 6576 3a20 6d61 6b65 486f 6d6f 746f  Prev: makeHomoto
│ │ │ │ -000548f0: 7069 6573 4f6e 486f 6d6f 6c6f 6779 2c20  piesOnHomology, 
│ │ │ │ -00054900: 5570 3a20 546f 700a 0a6d 616b 654d 6f64  Up: Top..makeMod
│ │ │ │ -00054910: 756c 6520 2d2d 206d 616b 6573 2061 204d  ule -- makes a M
│ │ │ │ -00054920: 6f64 756c 6520 6f75 7420 6f66 2061 2063  odule out of a c
│ │ │ │ -00054930: 6f6c 6c65 6374 696f 6e20 6f66 206d 6f64  ollection of mod
│ │ │ │ -00054940: 756c 6573 2061 6e64 206d 6170 730a 2a2a  ules and maps.**
│ │ │ │ +000547a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +000547b0: 226d 616b 6548 6f6d 6f74 6f70 6965 734f  "makeHomotopiesO
│ │ │ │ +000547c0: 6e48 6f6d 6f6c 6f67 7928 4d61 7472 6978  nHomology(Matrix
│ │ │ │ +000547d0: 2c43 6861 696e 436f 6d70 6c65 7829 220a  ,ChainComplex)".
│ │ │ │ +000547e0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +000547f0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +00054800: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +00054810: 6374 202a 6e6f 7465 206d 616b 6548 6f6d  ct *note makeHom
│ │ │ │ +00054820: 6f74 6f70 6965 734f 6e48 6f6d 6f6c 6f67  otopiesOnHomolog
│ │ │ │ +00054830: 793a 206d 616b 6548 6f6d 6f74 6f70 6965  y: makeHomotopie
│ │ │ │ +00054840: 734f 6e48 6f6d 6f6c 6f67 792c 2069 7320  sOnHomology, is 
│ │ │ │ +00054850: 6120 2a6e 6f74 650a 6d65 7468 6f64 2066  a *note.method f
│ │ │ │ +00054860: 756e 6374 696f 6e3a 2028 4d61 6361 756c  unction: (Macaul
│ │ │ │ +00054870: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +00054880: 6374 696f 6e2c 2e0a 1f0a 4669 6c65 3a20  ction,....File: 
│ │ │ │ +00054890: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +000548a0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +000548b0: 696e 666f 2c20 4e6f 6465 3a20 6d61 6b65  info, Node: make
│ │ │ │ +000548c0: 4d6f 6475 6c65 2c20 4e65 7874 3a20 6d61  Module, Next: ma
│ │ │ │ +000548d0: 6b65 542c 2050 7265 763a 206d 616b 6548  keT, Prev: makeH
│ │ │ │ +000548e0: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c  omotopiesOnHomol
│ │ │ │ +000548f0: 6f67 792c 2055 703a 2054 6f70 0a0a 6d61  ogy, Up: Top..ma
│ │ │ │ +00054900: 6b65 4d6f 6475 6c65 202d 2d20 6d61 6b65  keModule -- make
│ │ │ │ +00054910: 7320 6120 4d6f 6475 6c65 206f 7574 206f  s a Module out o
│ │ │ │ +00054920: 6620 6120 636f 6c6c 6563 7469 6f6e 206f  f a collection o
│ │ │ │ +00054930: 6620 6d6f 6475 6c65 7320 616e 6420 6d61  f modules and ma
│ │ │ │ +00054940: 7073 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ps.*************
│ │ │ │  00054950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00054960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00054970: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054980: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054990: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ -000549a0: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ -000549b0: 3a20 0a20 2020 2020 2020 204d 203d 206d  : .        M = m
│ │ │ │ -000549c0: 616b 654d 6f64 756c 6528 482c 452c 7068  akeModule(H,E,ph
│ │ │ │ -000549d0: 6929 0a20 202a 2049 6e70 7574 733a 0a20  i).  * Inputs:. 
│ │ │ │ -000549e0: 2020 2020 202a 2048 2c20 6120 2a6e 6f74       * H, a *not
│ │ │ │ -000549f0: 6520 6861 7368 2074 6162 6c65 3a20 284d  e hash table: (M
│ │ │ │ -00054a00: 6163 6175 6c61 7932 446f 6329 4861 7368  acaulay2Doc)Hash
│ │ │ │ -00054a10: 5461 626c 652c 2c20 6772 6164 6564 2063  Table,, graded c
│ │ │ │ -00054a20: 6f6d 706f 6e65 6e74 7320 7468 6174 0a20  omponents that. 
│ │ │ │ -00054a30: 2020 2020 2020 2061 7265 206d 6f64 756c         are modul
│ │ │ │ -00054a40: 6573 2c20 746f 206d 616b 6520 696e 746f  es, to make into
│ │ │ │ -00054a50: 2061 7320 7369 6e67 6c65 206d 6f64 756c   as single modul
│ │ │ │ -00054a60: 650a 2020 2020 2020 2a20 452c 2061 202a  e.      * E, a *
│ │ │ │ -00054a70: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -00054a80: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -00054a90: 782c 2c20 4d61 7472 6978 206f 6620 7661  x,, Matrix of va
│ │ │ │ -00054aa0: 7269 6162 6c65 7320 7768 6f73 650a 2020  riables whose.  
│ │ │ │ -00054ab0: 2020 2020 2020 6163 7469 6f6e 2077 696c        action wil
│ │ │ │ -00054ac0: 6c20 6465 6669 6e65 640a 2020 2020 2020  l defined.      
│ │ │ │ -00054ad0: 2a20 7068 692c 2061 202a 6e6f 7465 2068  * phi, a *note h
│ │ │ │ -00054ae0: 6173 6820 7461 626c 653a 2028 4d61 6361  ash table: (Maca
│ │ │ │ -00054af0: 756c 6179 3244 6f63 2948 6173 6854 6162  ulay2Doc)HashTab
│ │ │ │ -00054b00: 6c65 2c2c 206d 6170 7320 6265 7477 6565  le,, maps betwee
│ │ │ │ -00054b10: 6e20 7468 650a 2020 2020 2020 2020 6772  n the.        gr
│ │ │ │ -00054b20: 6164 6564 2063 6f6d 706f 6e65 6e74 7320  aded components 
│ │ │ │ -00054b30: 7468 6174 2077 696c 6c20 6265 2074 6865  that will be the
│ │ │ │ -00054b40: 2061 6374 696f 6e20 6f66 2074 6865 2076   action of the v
│ │ │ │ -00054b50: 6172 6961 626c 6573 2069 6e20 450a 2020  ariables in E.  
│ │ │ │ -00054b60: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00054b70: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f   * M, a *note mo
│ │ │ │ -00054b80: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -00054b90: 446f 6329 4d6f 6475 6c65 2c2c 2067 7261  Doc)Module,, gra
│ │ │ │ -00054ba0: 6465 6420 6d6f 6475 6c65 7320 7768 6f73  ded modules whos
│ │ │ │ -00054bb0: 650a 2020 2020 2020 2020 636f 6d70 6f6e  e.        compon
│ │ │ │ -00054bc0: 656e 7473 2061 7265 2067 6976 656e 2062  ents are given b
│ │ │ │ -00054bd0: 7920 480a 0a44 6573 6372 6970 7469 6f6e  y H..Description
│ │ │ │ -00054be0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -00054bf0: 6520 4861 7368 7461 626c 6520 4820 7368  e Hashtable H sh
│ │ │ │ -00054c00: 6f75 6c64 2068 6176 6520 636f 6e73 6563  ould have consec
│ │ │ │ -00054c10: 7574 6976 6520 696e 7465 6765 7220 6b65  utive integer ke
│ │ │ │ -00054c20: 7973 2069 5f30 2e2e 695f 302c 2073 6179  ys i_0..i_0, say
│ │ │ │ -00054c30: 2c20 7769 7468 2076 616c 7565 730a 4823  , with values.H#
│ │ │ │ -00054c40: 6920 7468 6174 2061 7265 206d 6f64 756c  i that are modul
│ │ │ │ -00054c50: 6573 206f 7665 7220 6120 7269 6e67 2053  es over a ring S
│ │ │ │ -00054c60: 4520 7768 6f73 6520 7661 7269 6162 6c65  E whose variable
│ │ │ │ -00054c70: 7320 696e 636c 7564 6520 7468 6520 656c  s include the el
│ │ │ │ -00054c80: 656d 656e 7473 206f 6620 452e 0a45 3a20  ements of E..E: 
│ │ │ │ -00054c90: 5c6f 706c 7573 2053 455e 7b64 5f69 7d20  \oplus SE^{d_i} 
│ │ │ │ -00054ca0: 5c74 6f20 5345 5e31 2069 7320 6120 6d61  \to SE^1 is a ma
│ │ │ │ -00054cb0: 7472 6978 206f 6620 6320 7661 7269 6162  trix of c variab
│ │ │ │ -00054cc0: 6c65 7320 6672 6f6d 2053 4520 4820 6973  les from SE H is
│ │ │ │ -00054cd0: 2061 2068 6173 6854 6162 6c65 0a6f 6620   a hashTable.of 
│ │ │ │ -00054ce0: 6d20 7061 6972 7320 7b69 2c20 745f 697d  m pairs {i, t_i}
│ │ │ │ -00054cf0: 2c20 7768 6572 6520 7468 6520 745f 6920  , where the t_i 
│ │ │ │ -00054d00: 6172 6520 5245 2d6d 6f64 756c 6573 2c20  are RE-modules, 
│ │ │ │ -00054d10: 616e 6420 7468 6520 6920 6172 6520 636f  and the i are co
│ │ │ │ -00054d20: 6e73 6563 7574 6976 650a 696e 7465 6765  nsecutive.intege
│ │ │ │ -00054d30: 722e 2070 6869 2069 7320 6120 6861 7368  r. phi is a hash
│ │ │ │ -00054d40: 2d74 6162 6c65 206f 6620 686f 6d6f 6765  -table of homoge
│ │ │ │ -00054d50: 6e65 6f75 7320 6d61 7073 2070 6869 237b  neous maps phi#{
│ │ │ │ -00054d60: 6a2c 697d 3a20 4823 692a 2a46 5f6a 5c74  j,i}: H#i**F_j\t
│ │ │ │ -00054d70: 6f20 4823 2869 2b31 290a 7768 6572 6520  o H#(i+1).where 
│ │ │ │ -00054d80: 465f 6a20 3d20 736f 7572 6365 2028 455f  F_j = source (E_
│ │ │ │ -00054d90: 7b6a 7d20 3d20 6d61 7472 6978 207b 7b65  {j} = matrix {{e
│ │ │ │ -00054da0: 5f6a 7d7d 292e 2054 6875 7320 7468 6520  _j}}). Thus the 
│ │ │ │ -00054db0: 6d61 7073 2070 237b 6a2c 697d 203d 2028  maps p#{j,i} = (
│ │ │ │ -00054dc0: 455f 6a20 7c7c 0a2d 7068 6923 7b6a 2c69  E_j ||.-phi#{j,i
│ │ │ │ -00054dd0: 7d29 3a20 745f 692a 2a46 5f6a 205c 746f  }): t_i**F_j \to
│ │ │ │ -00054de0: 2074 5f69 2b2b 745f 7b28 692b 3129 7d2c   t_i++t_{(i+1)},
│ │ │ │ -00054df0: 2061 7265 2068 6f6d 6f67 656e 656f 7573   are homogeneous
│ │ │ │ -00054e00: 2e20 5468 6520 7363 7269 7074 2072 6574  . The script ret
│ │ │ │ -00054e10: 7572 6e73 204d 0a3d 205c 6f70 6c75 735f  urns M.= \oplus_
│ │ │ │ -00054e20: 6920 545f 2061 7320 616e 2053 452d 6d6f  i T_ as an SE-mo
│ │ │ │ -00054e30: 6475 6c65 2c20 636f 6d70 7574 6564 2061  dule, computed a
│ │ │ │ -00054e40: 7320 7468 6520 7175 6f74 6965 6e74 206f  s the quotient o
│ │ │ │ -00054e50: 6620 5020 3a3d 205c 6f70 6c75 7320 545f  f P := \oplus T_
│ │ │ │ -00054e60: 690a 6f62 7461 696e 6564 2062 7920 6661  i.obtained by fa
│ │ │ │ -00054e70: 6374 6f72 696e 6720 6f75 7420 7468 6520  ctoring out the 
│ │ │ │ -00054e80: 7375 6d20 6f66 2074 6865 2069 6d61 6765  sum of the image
│ │ │ │ -00054e90: 7320 6f66 2074 6865 206d 6170 7320 7023  s of the maps p#
│ │ │ │ -00054ea0: 7b6a 2c69 7d0a 0a54 6865 2048 6173 6874  {j,i}..The Hasht
│ │ │ │ -00054eb0: 6162 6c65 2070 6869 2068 6173 206b 6579  able phi has key
│ │ │ │ -00054ec0: 7320 6f66 2074 6865 2066 6f72 6d20 7b6a  s of the form {j
│ │ │ │ -00054ed0: 2c69 7d20 7768 6572 6520 6a20 7275 6e73  ,i} where j runs
│ │ │ │ -00054ee0: 2066 726f 6d20 3020 746f 2063 2d31 2c20   from 0 to c-1, 
│ │ │ │ -00054ef0: 6920 616e 640a 692b 3120 6172 6520 6b65  i and.i+1 are ke
│ │ │ │ -00054f00: 7973 206f 6620 482c 2061 6e64 2070 6869  ys of H, and phi
│ │ │ │ -00054f10: 237b 6a2c 697d 2069 7320 7468 6520 6d61  #{j,i} is the ma
│ │ │ │ -00054f20: 7020 6672 6f6d 2028 736f 7572 6365 2045  p from (source E
│ │ │ │ -00054f30: 5f7b 697d 292a 2a48 2369 2074 6f20 4823  _{i})**H#i to H#
│ │ │ │ -00054f40: 2869 2b31 290a 7468 6174 2077 696c 6c20  (i+1).that will 
│ │ │ │ -00054f50: 6265 2069 6465 6e74 6966 6965 6420 7769  be identified wi
│ │ │ │ -00054f60: 7468 2074 6865 2061 6374 696f 6e20 6f66  th the action of
│ │ │ │ -00054f70: 2045 5f7b 6a7d 2e0a 0a54 6865 2073 6372   E_{j}...The scr
│ │ │ │ -00054f80: 6970 7420 6973 2075 7365 6420 696e 2062  ipt is used in b
│ │ │ │ -00054f90: 6f74 6820 7468 6520 7369 6e67 6c79 2067  oth the singly g
│ │ │ │ -00054fa0: 7261 6465 6420 6361 7365 2c20 666f 7220  raded case, for 
│ │ │ │ -00054fb0: 6578 616d 706c 6520 696e 0a65 7874 6572  example in.exter
│ │ │ │ -00054fc0: 696f 7254 6f72 4d6f 6475 6c65 2866 662c  iorTorModule(ff,
│ │ │ │ -00054fd0: 4d29 2061 6e64 2069 6e20 7468 6520 6269  M) and in the bi
│ │ │ │ -00054fe0: 6772 6164 6564 2063 6173 652c 2066 6f72  graded case, for
│ │ │ │ -00054ff0: 2065 7861 6d70 6c65 2069 6e0a 6578 7465   example in.exte
│ │ │ │ -00055000: 7269 6f72 546f 724d 6f64 756c 6528 6666  riorTorModule(ff
│ │ │ │ -00055010: 2c4d 2c4e 292e 0a0a 496e 2074 6865 2066  ,M,N)...In the f
│ │ │ │ -00055020: 6f6c 6c6f 7769 6e67 2077 6520 7573 6520  ollowing we use 
│ │ │ │ -00055030: 6d61 6b65 4d6f 6475 6c65 2074 6f20 636f  makeModule to co
│ │ │ │ -00055040: 6e73 7472 7563 7420 6279 2068 616e 6420  nstruct by hand 
│ │ │ │ -00055050: 6120 6672 6565 206d 6f64 756c 6520 6f66  a free module of
│ │ │ │ -00055060: 2072 616e 6b20 310a 6f76 6572 2074 6865   rank 1.over the
│ │ │ │ -00055070: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ -00055080: 6120 6f6e 2078 2c79 2c20 7374 6172 7469  a on x,y, starti
│ │ │ │ -00055090: 6e67 2077 6974 6820 7468 6520 636f 6e73  ng with the cons
│ │ │ │ -000550a0: 7472 7563 7469 6f6e 206f 6620 6120 6d6f  truction of a mo
│ │ │ │ -000550b0: 6475 6c65 0a6f 7665 7220 6120 6269 686f  dule.over a biho
│ │ │ │ -000550c0: 6d6f 6765 6e65 6f75 7320 7269 6e67 2e0a  mogeneous ring..
│ │ │ │ -000550d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00054980: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +00054990: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +000549a0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000549b0: 4d20 3d20 6d61 6b65 4d6f 6475 6c65 2848  M = makeModule(H
│ │ │ │ +000549c0: 2c45 2c70 6869 290a 2020 2a20 496e 7075  ,E,phi).  * Inpu
│ │ │ │ +000549d0: 7473 3a0a 2020 2020 2020 2a20 482c 2061  ts:.      * H, a
│ │ │ │ +000549e0: 202a 6e6f 7465 2068 6173 6820 7461 626c   *note hash tabl
│ │ │ │ +000549f0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00054a00: 2948 6173 6854 6162 6c65 2c2c 2067 7261  )HashTable,, gra
│ │ │ │ +00054a10: 6465 6420 636f 6d70 6f6e 656e 7473 2074  ded components t
│ │ │ │ +00054a20: 6861 740a 2020 2020 2020 2020 6172 6520  hat.        are 
│ │ │ │ +00054a30: 6d6f 6475 6c65 732c 2074 6f20 6d61 6b65  modules, to make
│ │ │ │ +00054a40: 2069 6e74 6f20 6173 2073 696e 676c 6520   into as single 
│ │ │ │ +00054a50: 6d6f 6475 6c65 0a20 2020 2020 202a 2045  module.      * E
│ │ │ │ +00054a60: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ +00054a70: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00054a80: 4d61 7472 6978 2c2c 204d 6174 7269 7820  Matrix,, Matrix 
│ │ │ │ +00054a90: 6f66 2076 6172 6961 626c 6573 2077 686f  of variables who
│ │ │ │ +00054aa0: 7365 0a20 2020 2020 2020 2061 6374 696f  se.        actio
│ │ │ │ +00054ab0: 6e20 7769 6c6c 2064 6566 696e 6564 0a20  n will defined. 
│ │ │ │ +00054ac0: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ +00054ad0: 6f74 6520 6861 7368 2074 6162 6c65 3a20  ote hash table: 
│ │ │ │ +00054ae0: 284d 6163 6175 6c61 7932 446f 6329 4861  (Macaulay2Doc)Ha
│ │ │ │ +00054af0: 7368 5461 626c 652c 2c20 6d61 7073 2062  shTable,, maps b
│ │ │ │ +00054b00: 6574 7765 656e 2074 6865 0a20 2020 2020  etween the.     
│ │ │ │ +00054b10: 2020 2067 7261 6465 6420 636f 6d70 6f6e     graded compon
│ │ │ │ +00054b20: 656e 7473 2074 6861 7420 7769 6c6c 2062  ents that will b
│ │ │ │ +00054b30: 6520 7468 6520 6163 7469 6f6e 206f 6620  e the action of 
│ │ │ │ +00054b40: 7468 6520 7661 7269 6162 6c65 7320 696e  the variables in
│ │ │ │ +00054b50: 2045 0a20 202a 204f 7574 7075 7473 3a0a   E.  * Outputs:.
│ │ │ │ +00054b60: 2020 2020 2020 2a20 4d2c 2061 202a 6e6f        * M, a *no
│ │ │ │ +00054b70: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ +00054b80: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ +00054b90: 2c20 6772 6164 6564 206d 6f64 756c 6573  , graded modules
│ │ │ │ +00054ba0: 2077 686f 7365 0a20 2020 2020 2020 2063   whose.        c
│ │ │ │ +00054bb0: 6f6d 706f 6e65 6e74 7320 6172 6520 6769  omponents are gi
│ │ │ │ +00054bc0: 7665 6e20 6279 2048 0a0a 4465 7363 7269  ven by H..Descri
│ │ │ │ +00054bd0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00054be0: 3d0a 0a54 6865 2048 6173 6874 6162 6c65  =..The Hashtable
│ │ │ │ +00054bf0: 2048 2073 686f 756c 6420 6861 7665 2063   H should have c
│ │ │ │ +00054c00: 6f6e 7365 6375 7469 7665 2069 6e74 6567  onsecutive integ
│ │ │ │ +00054c10: 6572 206b 6579 7320 695f 302e 2e69 5f30  er keys i_0..i_0
│ │ │ │ +00054c20: 2c20 7361 792c 2077 6974 6820 7661 6c75  , say, with valu
│ │ │ │ +00054c30: 6573 0a48 2369 2074 6861 7420 6172 6520  es.H#i that are 
│ │ │ │ +00054c40: 6d6f 6475 6c65 7320 6f76 6572 2061 2072  modules over a r
│ │ │ │ +00054c50: 696e 6720 5345 2077 686f 7365 2076 6172  ing SE whose var
│ │ │ │ +00054c60: 6961 626c 6573 2069 6e63 6c75 6465 2074  iables include t
│ │ │ │ +00054c70: 6865 2065 6c65 6d65 6e74 7320 6f66 2045  he elements of E
│ │ │ │ +00054c80: 2e0a 453a 205c 6f70 6c75 7320 5345 5e7b  ..E: \oplus SE^{
│ │ │ │ +00054c90: 645f 697d 205c 746f 2053 455e 3120 6973  d_i} \to SE^1 is
│ │ │ │ +00054ca0: 2061 206d 6174 7269 7820 6f66 2063 2076   a matrix of c v
│ │ │ │ +00054cb0: 6172 6961 626c 6573 2066 726f 6d20 5345  ariables from SE
│ │ │ │ +00054cc0: 2048 2069 7320 6120 6861 7368 5461 626c   H is a hashTabl
│ │ │ │ +00054cd0: 650a 6f66 206d 2070 6169 7273 207b 692c  e.of m pairs {i,
│ │ │ │ +00054ce0: 2074 5f69 7d2c 2077 6865 7265 2074 6865   t_i}, where the
│ │ │ │ +00054cf0: 2074 5f69 2061 7265 2052 452d 6d6f 6475   t_i are RE-modu
│ │ │ │ +00054d00: 6c65 732c 2061 6e64 2074 6865 2069 2061  les, and the i a
│ │ │ │ +00054d10: 7265 2063 6f6e 7365 6375 7469 7665 0a69  re consecutive.i
│ │ │ │ +00054d20: 6e74 6567 6572 2e20 7068 6920 6973 2061  nteger. phi is a
│ │ │ │ +00054d30: 2068 6173 682d 7461 626c 6520 6f66 2068   hash-table of h
│ │ │ │ +00054d40: 6f6d 6f67 656e 656f 7573 206d 6170 7320  omogeneous maps 
│ │ │ │ +00054d50: 7068 6923 7b6a 2c69 7d3a 2048 2369 2a2a  phi#{j,i}: H#i**
│ │ │ │ +00054d60: 465f 6a5c 746f 2048 2328 692b 3129 0a77  F_j\to H#(i+1).w
│ │ │ │ +00054d70: 6865 7265 2046 5f6a 203d 2073 6f75 7263  here F_j = sourc
│ │ │ │ +00054d80: 6520 2845 5f7b 6a7d 203d 206d 6174 7269  e (E_{j} = matri
│ │ │ │ +00054d90: 7820 7b7b 655f 6a7d 7d29 2e20 5468 7573  x {{e_j}}). Thus
│ │ │ │ +00054da0: 2074 6865 206d 6170 7320 7023 7b6a 2c69   the maps p#{j,i
│ │ │ │ +00054db0: 7d20 3d20 2845 5f6a 207c 7c0a 2d70 6869  } = (E_j ||.-phi
│ │ │ │ +00054dc0: 237b 6a2c 697d 293a 2074 5f69 2a2a 465f  #{j,i}): t_i**F_
│ │ │ │ +00054dd0: 6a20 5c74 6f20 745f 692b 2b74 5f7b 2869  j \to t_i++t_{(i
│ │ │ │ +00054de0: 2b31 297d 2c20 6172 6520 686f 6d6f 6765  +1)}, are homoge
│ │ │ │ +00054df0: 6e65 6f75 732e 2054 6865 2073 6372 6970  neous. The scrip
│ │ │ │ +00054e00: 7420 7265 7475 726e 7320 4d0a 3d20 5c6f  t returns M.= \o
│ │ │ │ +00054e10: 706c 7573 5f69 2054 5f20 6173 2061 6e20  plus_i T_ as an 
│ │ │ │ +00054e20: 5345 2d6d 6f64 756c 652c 2063 6f6d 7075  SE-module, compu
│ │ │ │ +00054e30: 7465 6420 6173 2074 6865 2071 756f 7469  ted as the quoti
│ │ │ │ +00054e40: 656e 7420 6f66 2050 203a 3d20 5c6f 706c  ent of P := \opl
│ │ │ │ +00054e50: 7573 2054 5f69 0a6f 6274 6169 6e65 6420  us T_i.obtained 
│ │ │ │ +00054e60: 6279 2066 6163 746f 7269 6e67 206f 7574  by factoring out
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│ │ │ │ +00054ea0: 4861 7368 7461 626c 6520 7068 6920 6861  Hashtable phi ha
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│ │ │ │ +00054f30: 746f 2048 2328 692b 3129 0a74 6861 7420  to H#(i+1).that 
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│ │ │ │ +00054fb0: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +00054fc0: 6528 6666 2c4d 2920 616e 6420 696e 2074  e(ff,M) and in t
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│ │ │ │ +00054fe0: 2c20 666f 7220 6578 616d 706c 6520 696e  , for example in
│ │ │ │ +00054ff0: 0a65 7874 6572 696f 7254 6f72 4d6f 6475  .exteriorTorModu
│ │ │ │ +00055000: 6c65 2866 662c 4d2c 4e29 2e0a 0a49 6e20  le(ff,M,N)...In 
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│ │ │ │ +000550b0: 2062 6968 6f6d 6f67 656e 656f 7573 2072   bihomogeneous r
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│ │ │ │ +000550d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000550e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000550f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00055120: 0a7c 6931 203a 2053 4520 3d20 5a5a 2f31  .|i1 : SE = ZZ/1
│ │ │ │ -00055130: 3031 5b61 2c62 2c63 2c78 2c79 2c44 6567  01[a,b,c,x,y,Deg
│ │ │ │ -00055140: 7265 6573 3d3e 746f 4c69 7374 2833 3a7b  rees=>toList(3:{
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│ │ │ │ -00055160: 312c 317d 292c 2020 2020 2020 2020 207c  1,1}),         |
│ │ │ │ -00055170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055110: 2d2d 2d2d 2b0a 7c69 3120 3a20 5345 203d  ----+.|i1 : SE =
│ │ │ │ +00055120: 205a 5a2f 3130 315b 612c 622c 632c 782c   ZZ/101[a,b,c,x,
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│ │ │ │ +00055140: 7428 333a 7b31 2c30 7d29 7c74 6f4c 6973  t(3:{1,0})|toLis
│ │ │ │ +00055150: 7428 323a 7b31 2c31 7d29 2c20 2020 2020  t(2:{1,1}),     
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│ │ │ │ +00055170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000551a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000551b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000551c0: 0a7c 6f31 203d 2053 4520 2020 2020 2020  .|o1 = SE       
│ │ │ │ +000551b0: 2020 2020 7c0a 7c6f 3120 3d20 5345 2020      |.|o1 = SE  
│ │ │ │ +000551c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +00055280: 6172 6961 626c 6528 7329 2020 2020 2020  ariable(s)      
│ │ │ │ +00055290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000552b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000552c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000552d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000552e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000552f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00055300: 0a7c 536b 6577 436f 6d6d 7574 6174 6976  .|SkewCommutativ
│ │ │ │ -00055310: 653d 3e7b 782c 797d 5d20 2020 2020 2020  e=>{x,y}]       
│ │ │ │ +000552f0: 2d2d 2d2d 7c0a 7c53 6b65 7743 6f6d 6d75  ----|.|SkewCommu
│ │ │ │ +00055300: 7461 7469 7665 3d3e 7b78 2c79 7d5d 2020  tative=>{x,y}]  
│ │ │ │ +00055310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055350: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │ +00055350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000553a0: 0a7c 6932 203a 2052 4520 3d20 5345 2f69  .|i2 : RE = SE/i
│ │ │ │ -000553b0: 6465 616c 2261 322c 6232 2c63 3222 2020  deal"a2,b2,c2"  
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│ │ │ │ +000553a0: 2053 452f 6964 6561 6c22 6132 2c62 322c   SE/ideal"a2,b2,
│ │ │ │ +000553b0: 6332 2220 2020 2020 2020 2020 2020 2020  c2"             
│ │ │ │  000553c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000553d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000553e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000553f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000553e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000553f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055440: 0a7c 6f32 203d 2052 4520 2020 2020 2020  .|o2 = RE       
│ │ │ │ +00055430: 2020 2020 7c0a 7c6f 3220 3d20 5245 2020      |.|o2 = RE  
│ │ │ │ +00055440: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055480: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000554b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000554c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000554d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -000554f0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
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│ │ │ │ +000554e0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +000554f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055500: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055530: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00055520: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00055530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
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│ │ │ │ -000555a0: 312c 5245 5e7b 323a 7b20 2d31 2c2d 317d  1,RE^{2:{ -1,-1}
│ │ │ │ -000555b0: 7d7d 2c20 7b32 2c52 455e 7b7b 202d 322c  }}, {2,RE^{{ -2,
│ │ │ │ -000555c0: 2d32 7d7d 7d7d 2020 2020 2020 2020 207c  -2}}}}         |
│ │ │ │ -000555d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055570: 2d2d 2d2d 2b0a 7c69 3320 3a20 5420 3d20  ----+.|i3 : T = 
│ │ │ │ +00055580: 6861 7368 5461 626c 6520 7b7b 302c 5245  hashTable {{0,RE
│ │ │ │ +00055590: 5e31 7d2c 7b31 2c52 455e 7b32 3a7b 202d  ^1},{1,RE^{2:{ -
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│ │ │ │ +000555c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  000555e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055620: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055630: 2020 2020 2020 2020 3120 2020 2020 2020          1       
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│ │ │ │ +00055620: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00055630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00055670: 5461 626c 657b 3020 3d3e 2052 4520 7d20  Table{0 => RE } 
│ │ │ │ +00055680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000556c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000556d0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000556b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000556c0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000556d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000556e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000556f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055710: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055720: 2031 203d 3e20 5245 2020 2020 2020 2020   1 => RE        
│ │ │ │ +00055700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055710: 2020 2020 2020 3120 3d3e 2052 4520 2020        1 => RE   
│ │ │ │ +00055720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055770: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00055750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055760: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00055770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000557a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000557b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000557c0: 2032 203d 3e20 5245 2020 2020 2020 2020   2 => RE        
│ │ │ │ +000557a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000557b0: 2020 2020 2020 3220 3d3e 2052 4520 2020        2 => RE   
│ │ │ │ +000557c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000557d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000557e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000557f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000557f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055810: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055850: 0a7c 6f33 203a 2048 6173 6854 6162 6c65  .|o3 : HashTable
│ │ │ │ +00055840: 2020 2020 7c0a 7c6f 3320 3a20 4861 7368      |.|o3 : Hash
│ │ │ │ +00055850: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │  00055860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +000558a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000558b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000558c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000558d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000558e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000558f0: 0a7c 6934 203a 2045 203d 206d 6174 7269  .|i4 : E = matri
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│ │ │ │ +000558e0: 2d2d 2d2d 2b0a 7c69 3420 3a20 4520 3d20  ----+.|i4 : E = 
│ │ │ │ +000558f0: 6d61 7472 6978 7b7b 782c 797d 7d20 2020  matrix{{x,y}}   
│ │ │ │ +00055900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00055920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055990: 0a7c 6f34 203d 207c 2078 2079 207c 2020  .|o4 = | x y |  
│ │ │ │ +00055980: 2020 2020 7c0a 7c6f 3420 3d20 7c20 7820      |.|o4 = | x 
│ │ │ │ +00055990: 7920 7c20 2020 2020 2020 2020 2020 2020  y |             
│ │ │ │  000559a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000559b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000559c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000559d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000559e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000559d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000559e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000559f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055a40: 3120 2020 2020 2020 3220 2020 2020 2020  1       2       
│ │ │ │ +00055a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055a30: 2020 2020 2031 2020 2020 2020 2032 2020       1       2  
│ │ │ │ +00055a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055a80: 0a7c 6f34 203a 204d 6174 7269 7820 5245  .|o4 : Matrix RE
│ │ │ │ -00055a90: 2020 3c2d 2d20 5245 2020 2020 2020 2020    <-- RE        
│ │ │ │ +00055a70: 2020 2020 7c0a 7c6f 3420 3a20 4d61 7472      |.|o4 : Matr
│ │ │ │ +00055a80: 6978 2052 4520 203c 2d2d 2052 4520 2020  ix RE  <-- RE   
│ │ │ │ +00055a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00055ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00055ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00055b20: 0a7c 6935 203a 2046 3d61 7070 6c79 2832  .|i5 : F=apply(2
│ │ │ │ -00055b30: 2c20 6a2d 3e20 736f 7572 6365 2045 5f7b  , j-> source E_{
│ │ │ │ -00055b40: 6a7d 2920 2020 2020 2020 2020 2020 2020  j})             
│ │ │ │ +00055b10: 2d2d 2d2d 2b0a 7c69 3520 3a20 463d 6170  ----+.|i5 : F=ap
│ │ │ │ +00055b20: 706c 7928 322c 206a 2d3e 2073 6f75 7263  ply(2, j-> sourc
│ │ │ │ +00055b30: 6520 455f 7b6a 7d29 2020 2020 2020 2020  e E_{j})        
│ │ │ │ +00055b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055bc0: 0a7c 2020 2020 2020 2020 3120 2020 2031  .|        1    1
│ │ │ │ +00055bb0: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ +00055bc0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00055bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055c10: 0a7c 6f35 203d 207b 5245 202c 2052 4520  .|o5 = {RE , RE 
│ │ │ │ -00055c20: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00055c00: 2020 2020 7c0a 7c6f 3520 3d20 7b52 4520      |.|o5 = {RE 
│ │ │ │ +00055c10: 2c20 5245 207d 2020 2020 2020 2020 2020  , RE }          
│ │ │ │ +00055c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055c60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055cb0: 0a7c 6f35 203a 204c 6973 7420 2020 2020  .|o5 : List     
│ │ │ │ +00055ca0: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374      |.|o5 : List
│ │ │ │ +00055cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055cf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055d00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00055cf0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00055d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00055d50: 0a7c 6936 203a 2070 6869 203d 2068 6173  .|i6 : phi = has
│ │ │ │ -00055d60: 6854 6162 6c65 7b20 7b7b 302c 307d 2c20  hTable{ {{0,0}, 
│ │ │ │ -00055d70: 6d61 7028 5423 312c 2046 5f30 2a2a 5423  map(T#1, F_0**T#
│ │ │ │ -00055d80: 302c 2054 2331 5f7b 307d 297d 2c7b 7b31  0, T#1_{0})},{{1
│ │ │ │ -00055d90: 2c30 7d2c 206d 6170 2854 2331 2c20 207c  ,0}, map(T#1,  |
│ │ │ │ -00055da0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055d40: 2d2d 2d2d 2b0a 7c69 3620 3a20 7068 6920  ----+.|i6 : phi 
│ │ │ │ +00055d50: 3d20 6861 7368 5461 626c 657b 207b 7b30  = hashTable{ {{0
│ │ │ │ +00055d60: 2c30 7d2c 206d 6170 2854 2331 2c20 465f  ,0}, map(T#1, F_
│ │ │ │ +00055d70: 302a 2a54 2330 2c20 5423 315f 7b30 7d29  0**T#0, T#1_{0})
│ │ │ │ +00055d80: 7d2c 7b7b 312c 307d 2c20 6d61 7028 5423  },{{1,0}, map(T#
│ │ │ │ +00055d90: 312c 2020 7c0a 7c20 2020 2020 2020 2020  1,  |.|         
│ │ │ │ +00055da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055df0: 0a7c 6f36 203d 2048 6173 6854 6162 6c65  .|o6 = HashTable
│ │ │ │ -00055e00: 7b7b 302c 2030 7d20 3d3e 207b 312c 2031  {{0, 0} => {1, 1
│ │ │ │ -00055e10: 7d20 7c20 3120 7c20 2020 7d20 2020 2020  } | 1 |   }     
│ │ │ │ +00055de0: 2020 2020 7c0a 7c6f 3620 3d20 4861 7368      |.|o6 = Hash
│ │ │ │ +00055df0: 5461 626c 657b 7b30 2c20 307d 203d 3e20  Table{{0, 0} => 
│ │ │ │ +00055e00: 7b31 2c20 317d 207c 2031 207c 2020 207d  {1, 1} | 1 |   }
│ │ │ │ +00055e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055e40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055e50: 2020 2020 2020 2020 2020 207b 312c 2031             {1, 1
│ │ │ │ -00055e60: 7d20 7c20 3020 7c20 2020 2020 2020 2020  } | 0 |         
│ │ │ │ +00055e30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055e50: 7b31 2c20 317d 207c 2030 207c 2020 2020  {1, 1} | 0 |    
│ │ │ │ +00055e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055ea0: 207b 302c 2031 7d20 3d3e 207b 322c 2032   {0, 1} => {2, 2
│ │ │ │ -00055eb0: 7d20 7c20 3020 3120 7c20 2020 2020 2020  } | 0 1 |       
│ │ │ │ +00055e80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055e90: 2020 2020 2020 7b30 2c20 317d 203d 3e20        {0, 1} => 
│ │ │ │ +00055ea0: 7b32 2c20 327d 207c 2030 2031 207c 2020  {2, 2} | 0 1 |  
│ │ │ │ +00055eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055ef0: 207b 312c 2030 7d20 3d3e 207b 312c 2031   {1, 0} => {1, 1
│ │ │ │ -00055f00: 7d20 7c20 3020 7c20 2020 2020 2020 2020  } | 0 |         
│ │ │ │ +00055ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055ee0: 2020 2020 2020 7b31 2c20 307d 203d 3e20        {1, 0} => 
│ │ │ │ +00055ef0: 7b31 2c20 317d 207c 2030 207c 2020 2020  {1, 1} | 0 |    
│ │ │ │ +00055f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055f30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055f40: 2020 2020 2020 2020 2020 207b 312c 2031             {1, 1
│ │ │ │ -00055f50: 7d20 7c20 3120 7c20 2020 2020 2020 2020  } | 1 |         
│ │ │ │ +00055f20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055f40: 7b31 2c20 317d 207c 2031 207c 2020 2020  {1, 1} | 1 |    
│ │ │ │ +00055f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055f80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00055f90: 207b 312c 2031 7d20 3d3e 207b 322c 2032   {1, 1} => {2, 2
│ │ │ │ -00055fa0: 7d20 7c20 2d31 2030 207c 2020 2020 2020  } | -1 0 |      
│ │ │ │ +00055f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055f80: 2020 2020 2020 7b31 2c20 317d 203d 3e20        {1, 1} => 
│ │ │ │ +00055f90: 7b32 2c20 327d 207c 202d 3120 3020 7c20  {2, 2} | -1 0 | 
│ │ │ │ +00055fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00055fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00055fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056020: 0a7c 6f36 203a 2048 6173 6854 6162 6c65  .|o6 : HashTable
│ │ │ │ +00056010: 2020 2020 7c0a 7c6f 3620 3a20 4861 7368      |.|o6 : Hash
│ │ │ │ +00056020: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │  00056030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056070: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00056060: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00056070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000560a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000560b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000560c0: 0a7c 465f 312a 2a54 2330 2c20 5423 315f  .|F_1**T#0, T#1_
│ │ │ │ -000560d0: 7b31 7d29 7d2c 7b7b 302c 317d 2c20 6d61  {1})},{{0,1}, ma
│ │ │ │ -000560e0: 7028 5423 322c 2046 5f30 2a2a 5423 312c  p(T#2, F_0**T#1,
│ │ │ │ -000560f0: 2054 2331 5e7b 317d 297d 2c20 7b7b 312c   T#1^{1})}, {{1,
│ │ │ │ -00056100: 317d 2c20 2d6d 6170 2854 2332 2c20 207c  1}, -map(T#2,  |
│ │ │ │ -00056110: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000560b0: 2d2d 2d2d 7c0a 7c46 5f31 2a2a 5423 302c  ----|.|F_1**T#0,
│ │ │ │ +000560c0: 2054 2331 5f7b 317d 297d 2c7b 7b30 2c31   T#1_{1})},{{0,1
│ │ │ │ +000560d0: 7d2c 206d 6170 2854 2332 2c20 465f 302a  }, map(T#2, F_0*
│ │ │ │ +000560e0: 2a54 2331 2c20 5423 315e 7b31 7d29 7d2c  *T#1, T#1^{1})},
│ │ │ │ +000560f0: 207b 7b31 2c31 7d2c 202d 6d61 7028 5423   {{1,1}, -map(T#
│ │ │ │ +00056100: 322c 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  2,  |.|---------
│ │ │ │ +00056110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00056160: 0a7c 465f 312a 2a54 2331 2c20 5423 315e  .|F_1**T#1, T#1^
│ │ │ │ -00056170: 7b30 7d29 7d7d 2020 2020 2020 2020 2020  {0})}}          
│ │ │ │ +00056150: 2d2d 2d2d 7c0a 7c46 5f31 2a2a 5423 312c  ----|.|F_1**T#1,
│ │ │ │ +00056160: 2054 2331 5e7b 307d 297d 7d20 2020 2020   T#1^{0})}}     
│ │ │ │ +00056170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000561a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000561b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000561a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000561b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000561c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000561d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000561e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000561f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00056200: 0a7c 6937 203a 2061 7070 6c79 286b 6579  .|i7 : apply(key
│ │ │ │ -00056210: 7320 7068 692c 206b 2d3e 6973 486f 6d6f  s phi, k->isHomo
│ │ │ │ -00056220: 6765 6e65 6f75 7320 7068 6923 6b29 2020  geneous phi#k)  
│ │ │ │ +000561f0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6170 706c  ----+.|i7 : appl
│ │ │ │ +00056200: 7928 6b65 7973 2070 6869 2c20 6b2d 3e69  y(keys phi, k->i
│ │ │ │ +00056210: 7348 6f6d 6f67 656e 656f 7573 2070 6869  sHomogeneous phi
│ │ │ │ +00056220: 236b 2920 2020 2020 2020 2020 2020 2020  #k)             
│ │ │ │  00056230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00056240: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000562a0: 0a7c 6f37 203d 207b 7472 7565 2c20 7472  .|o7 = {true, tr
│ │ │ │ -000562b0: 7565 2c20 7472 7565 2c20 7472 7565 7d20  ue, true, true} 
│ │ │ │ +00056290: 2020 2020 7c0a 7c6f 3720 3d20 7b74 7275      |.|o7 = {tru
│ │ │ │ +000562a0: 652c 2074 7275 652c 2074 7275 652c 2074  e, true, true, t
│ │ │ │ +000562b0: 7275 657d 2020 2020 2020 2020 2020 2020  rue}            
│ │ │ │  000562c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000562d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000562e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000562f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000562e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000562f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056340: 0a7c 6f37 203a 204c 6973 7420 2020 2020  .|o7 : List     
│ │ │ │ +00056330: 2020 2020 7c0a 7c6f 3720 3a20 4c69 7374      |.|o7 : List
│ │ │ │ +00056340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056390: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00056380: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00056390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000563a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000563b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000563c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000563d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000563e0: 0a7c 6938 203a 2058 203d 206d 616b 654d  .|i8 : X = makeM
│ │ │ │ -000563f0: 6f64 756c 6528 542c 452c 7068 6929 2020  odule(T,E,phi)  
│ │ │ │ +000563d0: 2d2d 2d2d 2b0a 7c69 3820 3a20 5820 3d20  ----+.|i8 : X = 
│ │ │ │ +000563e0: 6d61 6b65 4d6f 6475 6c65 2854 2c45 2c70  makeModule(T,E,p
│ │ │ │ +000563f0: 6869 2920 2020 2020 2020 2020 2020 2020  hi)             
│ │ │ │  00056400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00056420: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056480: 0a7c 6f38 203d 2063 6f6b 6572 6e65 6c20  .|o8 = cokernel 
│ │ │ │ -00056490: 7b30 2c20 307d 207c 202d 7820 3020 2030  {0, 0} | -x 0  0
│ │ │ │ -000564a0: 2020 2d79 2030 2020 3020 207c 2020 2020    -y 0  0  |    
│ │ │ │ +00056470: 2020 2020 7c0a 7c6f 3820 3d20 636f 6b65      |.|o8 = coke
│ │ │ │ +00056480: 726e 656c 207b 302c 2030 7d20 7c20 2d78  rnel {0, 0} | -x
│ │ │ │ +00056490: 2030 2020 3020 202d 7920 3020 2030 2020   0  0  -y 0  0  
│ │ │ │ +000564a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000564b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000564c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000564d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000564e0: 7b31 2c20 317d 207c 2031 2020 2d78 2030  {1, 1} | 1  -x 0
│ │ │ │ -000564f0: 2020 3020 202d 7920 3020 207c 2020 2020    0  -y 0  |    
│ │ │ │ +000564c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000564d0: 2020 2020 207b 312c 2031 7d20 7c20 3120       {1, 1} | 1 
│ │ │ │ +000564e0: 202d 7820 3020 2030 2020 2d79 2030 2020   -x 0  0  -y 0  
│ │ │ │ +000564f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00056500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00056530: 7b31 2c20 317d 207c 2030 2020 3020 202d  {1, 1} | 0  0  -
│ │ │ │ -00056540: 7820 3120 2030 2020 2d79 207c 2020 2020  x 1  0  -y |    
│ │ │ │ +00056510: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056520: 2020 2020 207b 312c 2031 7d20 7c20 3020       {1, 1} | 0 
│ │ │ │ +00056530: 2030 2020 2d78 2031 2020 3020 202d 7920   0  -x 1  0  -y 
│ │ │ │ +00056540: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00056550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00056580: 7b32 2c20 327d 207c 2030 2020 3020 2031  {2, 2} | 0  0  1
│ │ │ │ -00056590: 2020 3020 202d 3120 3020 207c 2020 2020    0  -1 0  |    
│ │ │ │ +00056560: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056570: 2020 2020 207b 322c 2032 7d20 7c20 3020       {2, 2} | 0 
│ │ │ │ +00056580: 2030 2020 3120 2030 2020 2d31 2030 2020   0  1  0  -1 0  
│ │ │ │ +00056590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000565a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000565b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000565c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000565b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000565c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000565d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000565e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000565f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056610: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00056620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056630: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00056600: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056620: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +00056630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056660: 0a7c 6f38 203a 2052 452d 6d6f 6475 6c65  .|o8 : RE-module
│ │ │ │ -00056670: 2c20 7175 6f74 6965 6e74 206f 6620 5245  , quotient of RE
│ │ │ │ +00056650: 2020 2020 7c0a 7c6f 3820 3a20 5245 2d6d      |.|o8 : RE-m
│ │ │ │ +00056660: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +00056670: 6f66 2052 4520 2020 2020 2020 2020 2020  of RE           
│ │ │ │  00056680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000566b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000566a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000566b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000566c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000566d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000566e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000566f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00056700: 0a7c 6939 203a 2069 7348 6f6d 6f67 656e  .|i9 : isHomogen
│ │ │ │ -00056710: 656f 7573 2058 2020 2020 2020 2020 2020  eous X          
│ │ │ │ +000566f0: 2d2d 2d2d 2b0a 7c69 3920 3a20 6973 486f  ----+.|i9 : isHo
│ │ │ │ +00056700: 6d6f 6765 6e65 6f75 7320 5820 2020 2020  mogeneous X     
│ │ │ │ +00056710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056750: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00056740: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000567a0: 0a7c 6f39 203d 2074 7275 6520 2020 2020  .|o9 = true     
│ │ │ │ +00056790: 2020 2020 7c0a 7c6f 3920 3d20 7472 7565      |.|o9 = true
│ │ │ │ +000567a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000567b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000567c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000567d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000567e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000567f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000567e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000567f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00056840: 0a7c 6931 3020 3a20 7120 3d20 6d61 7028  .|i10 : q = map(
│ │ │ │ -00056850: 5a5a 2f31 3031 5b78 2c79 2c20 536b 6577  ZZ/101[x,y, Skew
│ │ │ │ -00056860: 436f 6d6d 7574 6174 6976 6520 3d3e 2074  Commutative => t
│ │ │ │ -00056870: 7275 652c 2044 6567 7265 654d 6170 203d  rue, DegreeMap =
│ │ │ │ -00056880: 3e20 642d 3e7b 645f 317d 5d2c 2020 207c  > d->{d_1}],   |
│ │ │ │ -00056890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00056830: 2d2d 2d2d 2b0a 7c69 3130 203a 2071 203d  ----+.|i10 : q =
│ │ │ │ +00056840: 206d 6170 285a 5a2f 3130 315b 782c 792c   map(ZZ/101[x,y,
│ │ │ │ +00056850: 2053 6b65 7743 6f6d 6d75 7461 7469 7665   SkewCommutative
│ │ │ │ +00056860: 203d 3e20 7472 7565 2c20 4465 6772 6565   => true, Degree
│ │ │ │ +00056870: 4d61 7020 3d3e 2064 2d3e 7b64 5f31 7d5d  Map => d->{d_1}]
│ │ │ │ +00056880: 2c20 2020 7c0a 7c20 2020 2020 2020 2020  ,   |.|         
│ │ │ │ +00056890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000568d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000568e0: 0a7c 2020 2020 2020 2020 2020 2020 5a5a  .|            ZZ
│ │ │ │ +000568d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000568e0: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  000568f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056930: 0a7c 6f31 3020 3d20 6d61 7020 282d 2d2d  .|o10 = map (---
│ │ │ │ -00056940: 5b78 2e2e 795d 2c20 5245 2c20 7b30 2c20  [x..y], RE, {0, 
│ │ │ │ -00056950: 302c 2030 2c20 782c 2079 7d29 2020 2020  0, 0, x, y})    
│ │ │ │ +00056920: 2020 2020 7c0a 7c6f 3130 203d 206d 6170      |.|o10 = map
│ │ │ │ +00056930: 2028 2d2d 2d5b 782e 2e79 5d2c 2052 452c   (---[x..y], RE,
│ │ │ │ +00056940: 207b 302c 2030 2c20 302c 2078 2c20 797d   {0, 0, 0, x, y}
│ │ │ │ +00056950: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00056960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056980: 0a7c 2020 2020 2020 2020 2020 2031 3031  .|           101
│ │ │ │ +00056970: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056980: 2020 3130 3120 2020 2020 2020 2020 2020    101           
│ │ │ │  00056990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000569c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000569d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000569c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000569d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00056a30: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00056a10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056a20: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +00056a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056a70: 0a7c 6f31 3020 3a20 5269 6e67 4d61 7020  .|o10 : RingMap 
│ │ │ │ -00056a80: 2d2d 2d5b 782e 2e79 5d20 3c2d 2d20 5245  ---[x..y] <-- RE
│ │ │ │ +00056a60: 2020 2020 7c0a 7c6f 3130 203a 2052 696e      |.|o10 : Rin
│ │ │ │ +00056a70: 674d 6170 202d 2d2d 5b78 2e2e 795d 203c  gMap ---[x..y] <
│ │ │ │ +00056a80: 2d2d 2052 4520 2020 2020 2020 2020 2020  -- RE           
│ │ │ │  00056a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056ac0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00056ad0: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +00056ab0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056ac0: 2020 2020 2031 3031 2020 2020 2020 2020       101        
│ │ │ │ +00056ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056b10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00056b00: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00056b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00056b60: 0a7c 7269 6e67 2058 2c20 7b33 3a30 2c78  .|ring X, {3:0,x
│ │ │ │ -00056b70: 2c79 7d29 2020 2020 2020 2020 2020 2020  ,y})            
│ │ │ │ +00056b50: 2d2d 2d2d 7c0a 7c72 696e 6720 582c 207b  ----|.|ring X, {
│ │ │ │ +00056b60: 333a 302c 782c 797d 2920 2020 2020 2020  3:0,x,y})       
│ │ │ │ +00056b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056bb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00056ba0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00056bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00056c00: 0a7c 6931 3120 3a20 7072 756e 6520 636f  .|i11 : prune co
│ │ │ │ -00056c10: 6b65 7220 7120 7072 6573 656e 7461 7469  ker q presentati
│ │ │ │ -00056c20: 6f6e 2058 2020 2020 2020 2020 2020 2020  on X            
│ │ │ │ +00056bf0: 2d2d 2d2d 2b0a 7c69 3131 203a 2070 7275  ----+.|i11 : pru
│ │ │ │ +00056c00: 6e65 2063 6f6b 6572 2071 2070 7265 7365  ne coker q prese
│ │ │ │ +00056c10: 6e74 6174 696f 6e20 5820 2020 2020 2020  ntation X       
│ │ │ │ +00056c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056c50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00056c40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056ca0: 0a7c 2020 2020 2020 2020 5a5a 2020 2020  .|        ZZ    
│ │ │ │ -00056cb0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00056c90: 2020 2020 7c0a 7c20 2020 2020 2020 205a      |.|        Z
│ │ │ │ +00056ca0: 5a20 2020 2020 2020 3120 2020 2020 2020  Z       1       
│ │ │ │ +00056cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ce0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056cf0: 0a7c 6f31 3120 3d20 282d 2d2d 5b78 2e2e  .|o11 = (---[x..
│ │ │ │ -00056d00: 795d 2920 2020 2020 2020 2020 2020 2020  y])             
│ │ │ │ +00056ce0: 2020 2020 7c0a 7c6f 3131 203d 2028 2d2d      |.|o11 = (--
│ │ │ │ +00056cf0: 2d5b 782e 2e79 5d29 2020 2020 2020 2020  -[x..y])        
│ │ │ │ +00056d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056d40: 0a7c 2020 2020 2020 2031 3031 2020 2020  .|       101    
│ │ │ │ +00056d30: 2020 2020 7c0a 7c20 2020 2020 2020 3130      |.|       10
│ │ │ │ +00056d40: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00056d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056d90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00056d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00056d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056de0: 0a7c 2020 2020 2020 205a 5a20 2020 2020  .|       ZZ     
│ │ │ │ +00056dd0: 2020 2020 7c0a 7c20 2020 2020 2020 5a5a      |.|       ZZ
│ │ │ │ +00056de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00056e30: 0a7c 6f31 3120 3a20 2d2d 2d5b 782e 2e79  .|o11 : ---[x..y
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│ │ │ │ +00056e30: 5b78 2e2e 795d 2d6d 6f64 756c 652c 2066  [x..y]-module, f
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│ │ │ │  00056e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00056e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056e80: 0a7c 2020 2020 2020 3130 3120 2020 2020  .|      101     
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│ │ │ │ +00056e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00056ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00056ed0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │ +00056ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00056f20: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00056f30: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065  ===..  * *note e
│ │ │ │ -00056f40: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d  xteriorHomologyM
│ │ │ │ -00056f50: 6f64 756c 653a 2065 7874 6572 696f 7248  odule: exteriorH
│ │ │ │ -00056f60: 6f6d 6f6c 6f67 794d 6f64 756c 652c 202d  omologyModule, -
│ │ │ │ -00056f70: 2d20 4d61 6b65 2074 6865 2068 6f6d 6f6c  - Make the homol
│ │ │ │ -00056f80: 6f67 790a 2020 2020 6f66 2061 2063 6f6d  ogy.    of a com
│ │ │ │ -00056f90: 706c 6578 2069 6e74 6f20 6120 6d6f 6475  plex into a modu
│ │ │ │ -00056fa0: 6c65 206f 7665 7220 616e 2065 7874 6572  le over an exter
│ │ │ │ -00056fb0: 696f 7220 616c 6765 6272 610a 2020 2a20  ior algebra.  * 
│ │ │ │ -00056fc0: 2a6e 6f74 6520 6578 7465 7269 6f72 546f  *note exteriorTo
│ │ │ │ -00056fd0: 724d 6f64 756c 653a 2065 7874 6572 696f  rModule: exterio
│ │ │ │ -00056fe0: 7254 6f72 4d6f 6475 6c65 2c20 2d2d 2054  rTorModule, -- T
│ │ │ │ -00056ff0: 6f72 2061 7320 6120 6d6f 6475 6c65 206f  or as a module o
│ │ │ │ -00057000: 7665 7220 616e 0a20 2020 2065 7874 6572  ver an.    exter
│ │ │ │ -00057010: 696f 7220 616c 6765 6272 6120 6f72 2062  ior algebra or b
│ │ │ │ -00057020: 6967 7261 6465 6420 616c 6765 6272 610a  igraded algebra.
│ │ │ │ -00057030: 2020 2a20 2a6e 6f74 6520 6578 7465 7269    * *note exteri
│ │ │ │ -00057040: 6f72 4578 744d 6f64 756c 653a 2065 7874  orExtModule: ext
│ │ │ │ -00057050: 6572 696f 7245 7874 4d6f 6475 6c65 2c20  eriorExtModule, 
│ │ │ │ -00057060: 2d2d 2045 7874 284d 2c6b 2920 6f72 2045  -- Ext(M,k) or E
│ │ │ │ -00057070: 7874 284d 2c4e 2920 6173 2061 0a20 2020  xt(M,N) as a.   
│ │ │ │ -00057080: 206d 6f64 756c 6520 6f76 6572 2061 6e20   module over an 
│ │ │ │ -00057090: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ -000570a0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61  ..Ways to use ma
│ │ │ │ -000570b0: 6b65 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d  keModule:.======
│ │ │ │ -000570c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000570d0: 3d0a 0a20 202a 2022 6d61 6b65 4d6f 6475  =..  * "makeModu
│ │ │ │ -000570e0: 6c65 2848 6173 6854 6162 6c65 2c4d 6174  le(HashTable,Mat
│ │ │ │ -000570f0: 7269 782c 4861 7368 5461 626c 6529 220a  rix,HashTable)".
│ │ │ │ -00057100: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -00057110: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -00057120: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -00057130: 6374 202a 6e6f 7465 206d 616b 654d 6f64  ct *note makeMod
│ │ │ │ -00057140: 756c 653a 206d 616b 654d 6f64 756c 652c  ule: makeModule,
│ │ │ │ -00057150: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -00057160: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ -00057170: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00057180: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ -00057190: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -000571a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
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│ │ │ │ -000571c0: 6d61 6b65 542c 204e 6578 743a 206d 6174  makeT, Next: mat
│ │ │ │ -000571d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -000571e0: 2c20 5072 6576 3a20 6d61 6b65 4d6f 6475  , Prev: makeModu
│ │ │ │ -000571f0: 6c65 2c20 5570 3a20 546f 700a 0a6d 616b  le, Up: Top..mak
│ │ │ │ -00057200: 6554 202d 2d20 6d61 6b65 2074 6865 2043  eT -- make the C
│ │ │ │ -00057210: 4920 6f70 6572 6174 6f72 7320 6f6e 2061  I operators on a
│ │ │ │ -00057220: 2063 6f6d 706c 6578 0a2a 2a2a 2a2a 2a2a   complex.*******
│ │ │ │ +00056f10: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ +00056f20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00056f30: 6f74 6520 6578 7465 7269 6f72 486f 6d6f  ote exteriorHomo
│ │ │ │ +00056f40: 6c6f 6779 4d6f 6475 6c65 3a20 6578 7465  logyModule: exte
│ │ │ │ +00056f50: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475  riorHomologyModu
│ │ │ │ +00056f60: 6c65 2c20 2d2d 204d 616b 6520 7468 6520  le, -- Make the 
│ │ │ │ +00056f70: 686f 6d6f 6c6f 6779 0a20 2020 206f 6620  homology.    of 
│ │ │ │ +00056f80: 6120 636f 6d70 6c65 7820 696e 746f 2061  a complex into a
│ │ │ │ +00056f90: 206d 6f64 756c 6520 6f76 6572 2061 6e20   module over an 
│ │ │ │ +00056fa0: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ +00056fb0: 0a20 202a 202a 6e6f 7465 2065 7874 6572  .  * *note exter
│ │ │ │ +00056fc0: 696f 7254 6f72 4d6f 6475 6c65 3a20 6578  iorTorModule: ex
│ │ │ │ +00056fd0: 7465 7269 6f72 546f 724d 6f64 756c 652c  teriorTorModule,
│ │ │ │ +00056fe0: 202d 2d20 546f 7220 6173 2061 206d 6f64   -- Tor as a mod
│ │ │ │ +00056ff0: 756c 6520 6f76 6572 2061 6e0a 2020 2020  ule over an.    
│ │ │ │ +00057000: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ +00057010: 206f 7220 6269 6772 6164 6564 2061 6c67   or bigraded alg
│ │ │ │ +00057020: 6562 7261 0a20 202a 202a 6e6f 7465 2065  ebra.  * *note e
│ │ │ │ +00057030: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ +00057040: 3a20 6578 7465 7269 6f72 4578 744d 6f64  : exteriorExtMod
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│ │ │ │ +00057060: 206f 7220 4578 7428 4d2c 4e29 2061 7320   or Ext(M,N) as 
│ │ │ │ +00057070: 610a 2020 2020 6d6f 6475 6c65 206f 7665  a.    module ove
│ │ │ │ +00057080: 7220 616e 2065 7874 6572 696f 7220 616c  r an exterior al
│ │ │ │ +00057090: 6765 6272 610a 0a57 6179 7320 746f 2075  gebra..Ways to u
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│ │ │ │ +000570b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +000570d0: 654d 6f64 756c 6528 4861 7368 5461 626c  eModule(HashTabl
│ │ │ │ +000570e0: 652c 4d61 7472 6978 2c48 6173 6854 6162  e,Matrix,HashTab
│ │ │ │ +000570f0: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ +00057100: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00057110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +00057120: 206f 626a 6563 7420 2a6e 6f74 6520 6d61   object *note ma
│ │ │ │ +00057130: 6b65 4d6f 6475 6c65 3a20 6d61 6b65 4d6f  keModule: makeMo
│ │ │ │ +00057140: 6475 6c65 2c20 6973 2061 202a 6e6f 7465  dule, is a *note
│ │ │ │ +00057150: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00057160: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00057170: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +00057180: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00057190: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +000571a0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +000571b0: 6f64 653a 206d 616b 6554 2c20 4e65 7874  ode: makeT, Next
│ │ │ │ +000571c0: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ +000571d0: 6174 696f 6e2c 2050 7265 763a 206d 616b  ation, Prev: mak
│ │ │ │ +000571e0: 654d 6f64 756c 652c 2055 703a 2054 6f70  eModule, Up: Top
│ │ │ │ +000571f0: 0a0a 6d61 6b65 5420 2d2d 206d 616b 6520  ..makeT -- make 
│ │ │ │ +00057200: 7468 6520 4349 206f 7065 7261 746f 7273  the CI operators
│ │ │ │ +00057210: 206f 6e20 6120 636f 6d70 6c65 780a 2a2a   on a complex.**
│ │ │ │ +00057220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00057240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00057250: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ -00057260: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055 7361  =======..  * Usa
│ │ │ │ -00057270: 6765 3a20 0a20 2020 2020 2020 2054 203d  ge: .        T =
│ │ │ │ -00057280: 206d 616b 6554 2866 662c 462c 6929 0a20   makeT(ff,F,i). 
│ │ │ │ -00057290: 2020 2020 2020 2054 203d 206d 616b 6554         T = makeT
│ │ │ │ -000572a0: 2866 662c 462c 7430 2c69 290a 2020 2a20  (ff,F,t0,i).  * 
│ │ │ │ -000572b0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -000572c0: 6666 2c20 6120 2a6e 6f74 6520 6d61 7472  ff, a *note matr
│ │ │ │ -000572d0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -000572e0: 6329 4d61 7472 6978 2c2c 2031 7863 206d  c)Matrix,, 1xc m
│ │ │ │ -000572f0: 6174 7269 7820 7768 6f73 6520 656e 7472  atrix whose entr
│ │ │ │ -00057300: 6965 7320 6172 650a 2020 2020 2020 2020  ies are.        
│ │ │ │ -00057310: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -00057320: 7365 6374 696f 6e20 696e 2053 0a20 2020  section in S.   
│ │ │ │ -00057330: 2020 202a 2046 2c20 6120 2a6e 6f74 6520     * F, a *note 
│ │ │ │ -00057340: 6368 6169 6e20 636f 6d70 6c65 783a 2028  chain complex: (
│ │ │ │ -00057350: 4d61 6361 756c 6179 3244 6f63 2943 6861  Macaulay2Doc)Cha
│ │ │ │ -00057360: 696e 436f 6d70 6c65 782c 2c20 6f76 6572  inComplex,, over
│ │ │ │ -00057370: 2053 2f69 6465 616c 2066 660a 2020 2020   S/ideal ff.    
│ │ │ │ -00057380: 2020 2a20 7430 2c20 6120 2a6e 6f74 6520    * t0, a *note 
│ │ │ │ -00057390: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ -000573a0: 7932 446f 6329 4d61 7472 6978 2c2c 2043  y2Doc)Matrix,, C
│ │ │ │ -000573b0: 492d 6f70 6572 6174 6f72 206f 6e20 4620  I-operator on F 
│ │ │ │ -000573c0: 666f 7220 6666 5f30 2074 6f0a 2020 2020  for ff_0 to.    
│ │ │ │ -000573d0: 2020 2020 6265 2070 7265 7365 7276 6564      be preserved
│ │ │ │ -000573e0: 0a20 2020 2020 202a 2069 2c20 616e 202a  .      * i, an *
│ │ │ │ -000573f0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ -00057400: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ -00057410: 2064 6566 696e 6520 4349 206f 7065 7261   define CI opera
│ │ │ │ -00057420: 746f 7273 2066 726f 6d20 465f 690a 2020  tors from F_i.  
│ │ │ │ -00057430: 2020 2020 2020 5c74 6f20 465f 7b69 2d32        \to F_{i-2
│ │ │ │ -00057440: 7d0a 2020 2a20 4f75 7470 7574 733a 0a20  }.  * Outputs:. 
│ │ │ │ -00057450: 2020 2020 202a 204c 2c20 6120 2a6e 6f74       * L, a *not
│ │ │ │ -00057460: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ -00057470: 7932 446f 6329 4c69 7374 2c2c 206f 6620  y2Doc)List,, of 
│ │ │ │ -00057480: 4349 206f 7065 7261 746f 7273 2046 5f69  CI operators F_i
│ │ │ │ -00057490: 205c 746f 2046 5f7b 692d 327d 0a20 2020   \to F_{i-2}.   
│ │ │ │ -000574a0: 2020 2020 2063 6f72 7265 7370 6f6e 6469       correspondi
│ │ │ │ -000574b0: 6e67 2074 6f20 656e 7472 6965 7320 6f66  ng to entries of
│ │ │ │ -000574c0: 2066 660a 0a44 6573 6372 6970 7469 6f6e   ff..Description
│ │ │ │ -000574d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 7375  .===========..su
│ │ │ │ -000574e0: 6273 7469 7475 7465 206d 6174 7269 6365  bstitute matrice
│ │ │ │ -000574f0: 7320 6f66 2074 776f 2064 6966 6665 7265  s of two differe
│ │ │ │ -00057500: 6e74 6961 6c73 206f 6620 4620 696e 746f  ntials of F into
│ │ │ │ -00057510: 2053 203d 2072 696e 6720 6666 2c20 636f   S = ring ff, co
│ │ │ │ -00057520: 6d70 6f73 6520 7468 656d 2c0a 616e 6420  mpose them,.and 
│ │ │ │ -00057530: 6469 7669 6465 2062 7920 656e 7472 6965  divide by entrie
│ │ │ │ -00057540: 7320 6f66 2066 662c 2069 6e20 6f72 6465  s of ff, in orde
│ │ │ │ -00057550: 722e 2049 6620 7468 6520 7365 636f 6e64  r. If the second
│ │ │ │ -00057560: 204d 6174 7269 7820 6172 6775 6d65 6e74   Matrix argument
│ │ │ │ -00057570: 2074 3020 6973 0a70 7265 7365 6e74 2c20   t0 is.present, 
│ │ │ │ -00057580: 7573 6520 6974 2061 7320 7468 6520 6669  use it as the fi
│ │ │ │ -00057590: 7273 7420 4349 206f 7065 7261 746f 722e  rst CI operator.
│ │ │ │ -000575a0: 0a0a 5468 6520 6465 6772 6565 7320 6f66  ..The degrees of
│ │ │ │ -000575b0: 2074 6865 2074 6172 6765 7473 206f 6620   the targets of 
│ │ │ │ -000575c0: 7468 6520 545f 6a20 6172 6520 6368 616e  the T_j are chan
│ │ │ │ -000575d0: 6765 6420 6279 2074 6865 2064 6567 7265  ged by the degre
│ │ │ │ -000575e0: 6573 206f 6620 7468 6520 665f 6a20 746f  es of the f_j to
│ │ │ │ -000575f0: 0a6d 616b 6520 7468 6520 545f 6a20 686f  .make the T_j ho
│ │ │ │ -00057600: 6d6f 6765 6e65 6f75 732e 0a0a 2b2d 2d2d  mogeneous...+---
│ │ │ │ +00057240: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +00057250: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +00057260: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00057270: 2020 5420 3d20 6d61 6b65 5428 6666 2c46    T = makeT(ff,F
│ │ │ │ +00057280: 2c69 290a 2020 2020 2020 2020 5420 3d20  ,i).        T = 
│ │ │ │ +00057290: 6d61 6b65 5428 6666 2c46 2c74 302c 6929  makeT(ff,F,t0,i)
│ │ │ │ +000572a0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +000572b0: 2020 202a 2066 662c 2061 202a 6e6f 7465     * ff, a *note
│ │ │ │ +000572c0: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +000572d0: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +000572e0: 3178 6320 6d61 7472 6978 2077 686f 7365  1xc matrix whose
│ │ │ │ +000572f0: 2065 6e74 7269 6573 2061 7265 0a20 2020   entries are.   
│ │ │ │ +00057300: 2020 2020 2061 2063 6f6d 706c 6574 6520       a complete 
│ │ │ │ +00057310: 696e 7465 7273 6563 7469 6f6e 2069 6e20  intersection in 
│ │ │ │ +00057320: 530a 2020 2020 2020 2a20 462c 2061 202a  S.      * F, a *
│ │ │ │ +00057330: 6e6f 7465 2063 6861 696e 2063 6f6d 706c  note chain compl
│ │ │ │ +00057340: 6578 3a20 284d 6163 6175 6c61 7932 446f  ex: (Macaulay2Do
│ │ │ │ +00057350: 6329 4368 6169 6e43 6f6d 706c 6578 2c2c  c)ChainComplex,,
│ │ │ │ +00057360: 206f 7665 7220 532f 6964 6561 6c20 6666   over S/ideal ff
│ │ │ │ +00057370: 0a20 2020 2020 202a 2074 302c 2061 202a  .      * t0, a *
│ │ │ │ +00057380: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ +00057390: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ +000573a0: 782c 2c20 4349 2d6f 7065 7261 746f 7220  x,, CI-operator 
│ │ │ │ +000573b0: 6f6e 2046 2066 6f72 2066 665f 3020 746f  on F for ff_0 to
│ │ │ │ +000573c0: 0a20 2020 2020 2020 2062 6520 7072 6573  .        be pres
│ │ │ │ +000573d0: 6572 7665 640a 2020 2020 2020 2a20 692c  erved.      * i,
│ │ │ │ +000573e0: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ +000573f0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00057400: 295a 5a2c 2c20 6465 6669 6e65 2043 4920  )ZZ,, define CI 
│ │ │ │ +00057410: 6f70 6572 6174 6f72 7320 6672 6f6d 2046  operators from F
│ │ │ │ +00057420: 5f69 0a20 2020 2020 2020 205c 746f 2046  _i.        \to F
│ │ │ │ +00057430: 5f7b 692d 327d 0a20 202a 204f 7574 7075  _{i-2}.  * Outpu
│ │ │ │ +00057440: 7473 3a0a 2020 2020 2020 2a20 4c2c 2061  ts:.      * L, a
│ │ │ │ +00057450: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ +00057460: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ +00057470: 2c20 6f66 2043 4920 6f70 6572 6174 6f72  , of CI operator
│ │ │ │ +00057480: 7320 465f 6920 5c74 6f20 465f 7b69 2d32  s F_i \to F_{i-2
│ │ │ │ +00057490: 7d0a 2020 2020 2020 2020 636f 7272 6573  }.        corres
│ │ │ │ +000574a0: 706f 6e64 696e 6720 746f 2065 6e74 7269  ponding to entri
│ │ │ │ +000574b0: 6573 206f 6620 6666 0a0a 4465 7363 7269  es of ff..Descri
│ │ │ │ +000574c0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +000574d0: 3d0a 0a73 7562 7374 6974 7574 6520 6d61  =..substitute ma
│ │ │ │ +000574e0: 7472 6963 6573 206f 6620 7477 6f20 6469  trices of two di
│ │ │ │ +000574f0: 6666 6572 656e 7469 616c 7320 6f66 2046  fferentials of F
│ │ │ │ +00057500: 2069 6e74 6f20 5320 3d20 7269 6e67 2066   into S = ring f
│ │ │ │ +00057510: 662c 2063 6f6d 706f 7365 2074 6865 6d2c  f, compose them,
│ │ │ │ +00057520: 0a61 6e64 2064 6976 6964 6520 6279 2065  .and divide by e
│ │ │ │ +00057530: 6e74 7269 6573 206f 6620 6666 2c20 696e  ntries of ff, in
│ │ │ │ +00057540: 206f 7264 6572 2e20 4966 2074 6865 2073   order. If the s
│ │ │ │ +00057550: 6563 6f6e 6420 4d61 7472 6978 2061 7267  econd Matrix arg
│ │ │ │ +00057560: 756d 656e 7420 7430 2069 730a 7072 6573  ument t0 is.pres
│ │ │ │ +00057570: 656e 742c 2075 7365 2069 7420 6173 2074  ent, use it as t
│ │ │ │ +00057580: 6865 2066 6972 7374 2043 4920 6f70 6572  he first CI oper
│ │ │ │ +00057590: 6174 6f72 2e0a 0a54 6865 2064 6567 7265  ator...The degre
│ │ │ │ +000575a0: 6573 206f 6620 7468 6520 7461 7267 6574  es of the target
│ │ │ │ +000575b0: 7320 6f66 2074 6865 2054 5f6a 2061 7265  s of the T_j are
│ │ │ │ +000575c0: 2063 6861 6e67 6564 2062 7920 7468 6520   changed by the 
│ │ │ │ +000575d0: 6465 6772 6565 7320 6f66 2074 6865 2066  degrees of the f
│ │ │ │ +000575e0: 5f6a 2074 6f0a 6d61 6b65 2074 6865 2054  _j to.make the T
│ │ │ │ +000575f0: 5f6a 2068 6f6d 6f67 656e 656f 7573 2e0a  _j homogeneous..
│ │ │ │ +00057600: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00057610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057630: 2b0a 7c69 3120 3a20 5320 3d20 5a5a 2f31  +.|i1 : S = ZZ/1
│ │ │ │ -00057640: 3031 5b78 2c79 2c7a 5d3b 2020 2020 2020  01[x,y,z];      
│ │ │ │ -00057650: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057620: 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 203d  -----+.|i1 : S =
│ │ │ │ +00057630: 205a 5a2f 3130 315b 782c 792c 7a5d 3b20   ZZ/101[x,y,z]; 
│ │ │ │ +00057640: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00057650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00057680: 3220 3a20 6666 203d 206d 6174 7269 7822  2 : ff = matrix"
│ │ │ │ -00057690: 7833 2c79 332c 7a33 223b 2020 2020 2020  x3,y3,z3";      
│ │ │ │ -000576a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000576b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000576d0: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ -000576e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000576f0: 7c6f 3220 3a20 4d61 7472 6978 2053 2020  |o2 : Matrix S  
│ │ │ │ -00057700: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -00057710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00057720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057730: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -00057740: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ -00057750: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -00057760: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00057770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057780: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 4d20  ------+.|i4 : M 
│ │ │ │ -00057790: 3d20 636f 6b65 7220 6d61 7472 6978 2278  = coker matrix"x
│ │ │ │ -000577a0: 2c79 2c7a 3b79 2c7a 2c78 223b 7c0a 2b2d  ,y,z;y,z,x";|.+-
│ │ │ │ +00057670: 2d2b 0a7c 6932 203a 2066 6620 3d20 6d61  -+.|i2 : ff = ma
│ │ │ │ +00057680: 7472 6978 2278 332c 7933 2c7a 3322 3b20  trix"x3,y3,z3"; 
│ │ │ │ +00057690: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000576a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000576b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000576c0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +000576d0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +000576e0: 2020 207c 0a7c 6f32 203a 204d 6174 7269     |.|o2 : Matri
│ │ │ │ +000576f0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +00057700: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00057710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00057730: 0a7c 6933 203a 2052 203d 2053 2f69 6465  .|i3 : R = S/ide
│ │ │ │ +00057740: 616c 2066 663b 2020 2020 2020 2020 2020  al ff;          
│ │ │ │ +00057750: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00057760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00057780: 203a 204d 203d 2063 6f6b 6572 206d 6174   : M = coker mat
│ │ │ │ +00057790: 7269 7822 782c 792c 7a3b 792c 7a2c 7822  rix"x,y,z;y,z,x"
│ │ │ │ +000577a0: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ;|.+------------
│ │ │ │  000577b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000577c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000577d0: 2d2d 2b0a 7c69 3520 3a20 6265 7474 6920  --+.|i5 : betti 
│ │ │ │ -000577e0: 2846 203d 2072 6573 204d 2920 2020 2020  (F = res M)     
│ │ │ │ -000577f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000577c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2062  -------+.|i5 : b
│ │ │ │ +000577d0: 6574 7469 2028 4620 3d20 7265 7320 4d29  etti (F = res M)
│ │ │ │ +000577e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000577f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057810: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00057820: 7c20 2020 2020 2020 2020 2020 2030 2031  |            0 1
│ │ │ │ -00057830: 2032 2033 2034 2020 2020 2020 2020 2020   2 3 4          
│ │ │ │ -00057840: 2020 2020 7c0a 7c6f 3520 3d20 746f 7461      |.|o5 = tota
│ │ │ │ -00057850: 6c3a 2032 2033 2035 2036 2038 2020 2020  l: 2 3 5 6 8    
│ │ │ │ -00057860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00057870: 2020 2020 2020 303a 2032 2033 202e 202e        0: 2 3 . .
│ │ │ │ -00057880: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -00057890: 7c0a 7c20 2020 2020 2020 2020 313a 202e  |.|         1: .
│ │ │ │ -000578a0: 202e 2035 2036 202e 2020 2020 2020 2020   . 5 6 .        
│ │ │ │ -000578b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000578c0: 2020 323a 202e 202e 202e 202e 2038 2020    2: . . . . 8  
│ │ │ │ -000578d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00057810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00057820: 2020 3020 3120 3220 3320 3420 2020 2020    0 1 2 3 4     
│ │ │ │ +00057830: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +00057840: 2074 6f74 616c 3a20 3220 3320 3520 3620   total: 2 3 5 6 
│ │ │ │ +00057850: 3820 2020 2020 2020 2020 2020 2020 207c  8              |
│ │ │ │ +00057860: 0a7c 2020 2020 2020 2020 2030 3a20 3220  .|         0: 2 
│ │ │ │ +00057870: 3320 2e20 2e20 2e20 2020 2020 2020 2020  3 . . .         
│ │ │ │ +00057880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057890: 2031 3a20 2e20 2e20 3520 3620 2e20 2020   1: . . 5 6 .   
│ │ │ │ +000578a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000578b0: 2020 2020 2020 2032 3a20 2e20 2e20 2e20         2: . . . 
│ │ │ │ +000578c0: 2e20 3820 2020 2020 2020 2020 2020 2020  . 8             
│ │ │ │ +000578d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000578e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000578f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057900: 2020 7c0a 7c6f 3520 3a20 4265 7474 6954    |.|o5 : BettiT
│ │ │ │ -00057910: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │ -00057920: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000578f0: 2020 2020 2020 207c 0a7c 6f35 203a 2042         |.|o5 : B
│ │ │ │ +00057900: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00057910: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00057920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00057950: 7c69 3620 3a20 5420 3d20 6d61 6b65 5428  |i6 : T = makeT(
│ │ │ │ -00057960: 6666 2c46 2c33 293b 2020 2020 2020 2020  ff,F,3);        
│ │ │ │ -00057970: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00057980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -000579a0: 3a20 6e65 744c 6973 7420 5420 2020 2020  : netList T     
│ │ │ │ -000579b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000579c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000579d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000579e0: 2020 2020 2020 7c0a 7c20 2020 2020 2b2d        |.|     +-
│ │ │ │ -000579f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057a00: 2d2d 2d2d 2d2d 2d2b 2020 2020 7c0a 7c6f  -------+    |.|o
│ │ │ │ -00057a10: 3720 3d20 7c7b 347d 207c 2030 2030 2030  7 = |{4} | 0 0 0
│ │ │ │ -00057a20: 2030 2020 3120 3020 7c20 2020 207c 2020   0  1 0 |    |  
│ │ │ │ -00057a30: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c    |.|     |{4} |
│ │ │ │ -00057a40: 2030 2030 2030 202d 3120 3020 3020 7c20   0 0 0 -1 0 0 | 
│ │ │ │ -00057a50: 2020 207c 2020 2020 7c0a 7c20 2020 2020     |    |.|     
│ │ │ │ -00057a60: 7c7b 347d 207c 2030 2030 2030 2030 2020  |{4} | 0 0 0 0  
│ │ │ │ -00057a70: 3020 3120 7c20 2020 207c 2020 2020 7c0a  0 1 |    |    |.
│ │ │ │ -00057a80: 7c20 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d  |     +---------
│ │ │ │ -00057a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00057aa0: 2020 2020 7c0a 7c20 2020 2020 7c7b 347d      |.|     |{4}
│ │ │ │ -00057ab0: 207c 2030 2031 2030 2030 2030 2030 207c   | 0 1 0 0 0 0 |
│ │ │ │ -00057ac0: 2020 2020 207c 2020 2020 7c0a 7c20 2020       |    |.|   
│ │ │ │ -00057ad0: 2020 7c7b 347d 207c 2031 2030 2030 2030    |{4} | 1 0 0 0
│ │ │ │ -00057ae0: 2030 2030 207c 2020 2020 207c 2020 2020   0 0 |     |    
│ │ │ │ -00057af0: 7c0a 7c20 2020 2020 7c7b 347d 207c 2030  |.|     |{4} | 0
│ │ │ │ -00057b00: 2030 2031 2030 2030 2030 207c 2020 2020   0 1 0 0 0 |    
│ │ │ │ -00057b10: 207c 2020 2020 7c0a 7c20 2020 2020 2b2d   |    |.|     +-
│ │ │ │ -00057b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057b30: 2d2d 2d2d 2d2d 2d2b 2020 2020 7c0a 7c20  -------+    |.| 
│ │ │ │ -00057b40: 2020 2020 7c7b 347d 207c 2030 2020 2d31      |{4} | 0  -1
│ │ │ │ -00057b50: 2030 2020 3020 2d31 2030 2020 7c7c 2020   0  0 -1 0  ||  
│ │ │ │ -00057b60: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c    |.|     |{4} |
│ │ │ │ -00057b70: 202d 3120 3020 2030 2020 3120 3020 2030   -1 0  0  1 0  0
│ │ │ │ -00057b80: 2020 7c7c 2020 2020 7c0a 7c20 2020 2020    ||    |.|     
│ │ │ │ -00057b90: 7c7b 347d 207c 2030 2020 3020 202d 3120  |{4} | 0  0  -1 
│ │ │ │ -00057ba0: 3020 3020 202d 3120 7c7c 2020 2020 7c0a  0 0  -1 ||    |.
│ │ │ │ -00057bb0: 7c20 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d  |     +---------
│ │ │ │ -00057bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00057bd0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00057be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -00057c00: 3a20 6973 486f 6d6f 6765 6e65 6f75 7320  : isHomogeneous 
│ │ │ │ -00057c10: 545f 3220 2020 2020 2020 2020 2020 2020  T_2             
│ │ │ │ -00057c20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00057c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057c40: 2020 2020 2020 7c0a 7c6f 3820 3d20 7472        |.|o8 = tr
│ │ │ │ -00057c50: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -00057c60: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00057940: 2d2d 2d2b 0a7c 6936 203a 2054 203d 206d  ---+.|i6 : T = m
│ │ │ │ +00057950: 616b 6554 2866 662c 462c 3329 3b20 2020  akeT(ff,F,3);   
│ │ │ │ +00057960: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00057970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00057990: 0a7c 6937 203a 206e 6574 4c69 7374 2054  .|i7 : netList T
│ │ │ │ +000579a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000579b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000579c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000579d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000579e0: 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     +------------
│ │ │ │ +000579f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020  ------------+   
│ │ │ │ +00057a00: 207c 0a7c 6f37 203d 207c 7b34 7d20 7c20   |.|o7 = |{4} | 
│ │ │ │ +00057a10: 3020 3020 3020 3020 2031 2030 207c 2020  0 0 0 0  1 0 |  
│ │ │ │ +00057a20: 2020 7c20 2020 207c 0a7c 2020 2020 207c    |    |.|     |
│ │ │ │ +00057a30: 7b34 7d20 7c20 3020 3020 3020 2d31 2030  {4} | 0 0 0 -1 0
│ │ │ │ +00057a40: 2030 207c 2020 2020 7c20 2020 207c 0a7c   0 |    |    |.|
│ │ │ │ +00057a50: 2020 2020 207c 7b34 7d20 7c20 3020 3020       |{4} | 0 0 
│ │ │ │ +00057a60: 3020 3020 2030 2031 207c 2020 2020 7c20  0 0  0 1 |    | 
│ │ │ │ +00057a70: 2020 207c 0a7c 2020 2020 202b 2d2d 2d2d     |.|     +----
│ │ │ │ +00057a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057a90: 2d2d 2d2d 2b20 2020 207c 0a7c 2020 2020  ----+    |.|    
│ │ │ │ +00057aa0: 207c 7b34 7d20 7c20 3020 3120 3020 3020   |{4} | 0 1 0 0 
│ │ │ │ +00057ab0: 3020 3020 7c20 2020 2020 7c20 2020 207c  0 0 |     |    |
│ │ │ │ +00057ac0: 0a7c 2020 2020 207c 7b34 7d20 7c20 3120  .|     |{4} | 1 
│ │ │ │ +00057ad0: 3020 3020 3020 3020 3020 7c20 2020 2020  0 0 0 0 0 |     
│ │ │ │ +00057ae0: 7c20 2020 207c 0a7c 2020 2020 207c 7b34  |    |.|     |{4
│ │ │ │ +00057af0: 7d20 7c20 3020 3020 3120 3020 3020 3020  } | 0 0 1 0 0 0 
│ │ │ │ +00057b00: 7c20 2020 2020 7c20 2020 207c 0a7c 2020  |     |    |.|  
│ │ │ │ +00057b10: 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     +------------
│ │ │ │ +00057b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020  ------------+   
│ │ │ │ +00057b30: 207c 0a7c 2020 2020 207c 7b34 7d20 7c20   |.|     |{4} | 
│ │ │ │ +00057b40: 3020 202d 3120 3020 2030 202d 3120 3020  0  -1 0  0 -1 0 
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│ │ │ │ +00057ba0: 2020 207c 0a7c 2020 2020 202b 2d2d 2d2d     |.|     +----
│ │ │ │ +00057bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057bc0: 2d2d 2d2d 2b20 2020 207c 0a2b 2d2d 2d2d  ----+    |.+----
│ │ │ │ +00057bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00057bf0: 0a7c 6938 203a 2069 7348 6f6d 6f67 656e  .|i8 : isHomogen
│ │ │ │ +00057c00: 656f 7573 2054 5f32 2020 2020 2020 2020  eous T_2        
│ │ │ │ +00057c10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057c30: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00057c40: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00057c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057c60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00057c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057c90: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d  --+..Caveat.====
│ │ │ │ -00057ca0: 3d3d 0a0a 5363 7269 7074 2061 7373 756d  ==..Script assum
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│ │ │ │ -00057dc0: 616b 6554 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  akeT:.==========
│ │ │ │ -00057dd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ -00057de0: 616b 6554 284d 6174 7269 782c 4368 6169  akeT(Matrix,Chai
│ │ │ │ -00057df0: 6e43 6f6d 706c 6578 2c5a 5a29 220a 0a46  nComplex,ZZ)"..F
│ │ │ │ -00057e00: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00057e10: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00057e20: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00057e30: 202a 6e6f 7465 206d 616b 6554 3a20 6d61   *note makeT: ma
│ │ │ │ -00057e40: 6b65 542c 2069 7320 6120 2a6e 6f74 6520  keT, is a *note 
│ │ │ │ -00057e50: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ -00057e60: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ -00057e70: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ -00057e80: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ -00057e90: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -00057ea0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ -00057eb0: 6465 3a20 6d61 7472 6978 4661 6374 6f72  de: matrixFactor
│ │ │ │ -00057ec0: 697a 6174 696f 6e2c 204e 6578 743a 206d  ization, Next: m
│ │ │ │ -00057ed0: 6642 6f75 6e64 2c20 5072 6576 3a20 6d61  fBound, Prev: ma
│ │ │ │ -00057ee0: 6b65 542c 2055 703a 2054 6f70 0a0a 6d61  keT, Up: Top..ma
│ │ │ │ -00057ef0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
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│ │ │ │ -00057f10: 6967 6865 7220 636f 6469 6d65 6e73 696f  igher codimensio
│ │ │ │ -00057f20: 6e20 6d61 7472 6978 2066 6163 746f 7269  n matrix factori
│ │ │ │ -00057f30: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a  zation.*********
│ │ │ │ +00057c80: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174  -------+..Caveat
│ │ │ │ +00057c90: 0a3d 3d3d 3d3d 3d0a 0a53 6372 6970 7420  .======..Script 
│ │ │ │ +00057ca0: 6173 7375 6d65 7320 7468 6174 2072 696e  assumes that rin
│ │ │ │ +00057cb0: 6720 4620 3d3d 2028 7269 6e67 2066 6629  g F == (ring ff)
│ │ │ │ +00057cc0: 2f28 6964 6561 6c20 6666 292e 2049 7420  /(ideal ff). It 
│ │ │ │ +00057cd0: 6d69 6768 7420 6265 206d 6f72 6520 7573  might be more us
│ │ │ │ +00057ce0: 6566 756c 2074 6f0a 7265 7475 726e 2074  eful to.return t
│ │ │ │ +00057cf0: 6865 206f 7065 7261 746f 7273 2061 7320  he operators as 
│ │ │ │ +00057d00: 6d61 7472 6963 6573 206f 7665 7220 5320  matrices over S 
│ │ │ │ +00057d10: 7261 7468 6572 2074 6861 6e20 6f76 6572  rather than over
│ │ │ │ +00057d20: 2052 2c20 7369 6e63 6520 7468 6973 2069   R, since this i
│ │ │ │ +00057d30: 7320 7768 6174 0a77 6527 6420 6e65 6564  s what.we'd need
│ │ │ │ +00057d40: 2066 6f72 2074 6869 6e67 7320 6c69 6b65   for things like
│ │ │ │ +00057d50: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +00057d60: 7469 6f6e 2028 7768 6572 6520 7468 6973  tion (where this
│ │ │ │ +00057d70: 2070 726f 6365 7373 2063 7572 7265 6e74   process current
│ │ │ │ +00057d80: 6c79 0a64 6f6e 6520 6f6e 2074 6865 2066  ly.done on the f
│ │ │ │ +00057d90: 6c79 2c20 6e6f 7420 6361 6c6c 696e 6720  ly, not calling 
│ │ │ │ +00057da0: 6d61 6b65 5429 0a0a 5761 7973 2074 6f20  makeT)..Ways to 
│ │ │ │ +00057db0: 7573 6520 6d61 6b65 543a 0a3d 3d3d 3d3d  use makeT:.=====
│ │ │ │ +00057dc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00057dd0: 202a 2022 6d61 6b65 5428 4d61 7472 6978   * "makeT(Matrix
│ │ │ │ +00057de0: 2c43 6861 696e 436f 6d70 6c65 782c 5a5a  ,ChainComplex,ZZ
│ │ │ │ +00057df0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00057e00: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00057e10: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00057e20: 626a 6563 7420 2a6e 6f74 6520 6d61 6b65  bject *note make
│ │ │ │ +00057e30: 543a 206d 616b 6554 2c20 6973 2061 202a  T: makeT, is a *
│ │ │ │ +00057e40: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00057e50: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +00057e60: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00057e70: 6f6e 2c2e 0a1f 0a46 696c 653a 2043 6f6d  on,....File: Com
│ │ │ │ +00057e80: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +00057e90: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ +00057ea0: 6f2c 204e 6f64 653a 206d 6174 7269 7846  o, Node: matrixF
│ │ │ │ +00057eb0: 6163 746f 7269 7a61 7469 6f6e 2c20 4e65  actorization, Ne
│ │ │ │ +00057ec0: 7874 3a20 6d66 426f 756e 642c 2050 7265  xt: mfBound, Pre
│ │ │ │ +00057ed0: 763a 206d 616b 6554 2c20 5570 3a20 546f  v: makeT, Up: To
│ │ │ │ +00057ee0: 700a 0a6d 6174 7269 7846 6163 746f 7269  p..matrixFactori
│ │ │ │ +00057ef0: 7a61 7469 6f6e 202d 2d20 4d61 7073 2069  zation -- Maps i
│ │ │ │ +00057f00: 6e20 6120 6869 6768 6572 2063 6f64 696d  n a higher codim
│ │ │ │ +00057f10: 656e 7369 6f6e 206d 6174 7269 7820 6661  ension matrix fa
│ │ │ │ +00057f20: 6374 6f72 697a 6174 696f 6e0a 2a2a 2a2a  ctorization.****
│ │ │ │ +00057f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057f60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00057f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00057f80: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ -00057f90: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ -00057fa0: 2020 2020 2020 2020 4d46 203d 206d 6174          MF = mat
│ │ │ │ -00057fb0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00057fc0: 2866 662c 4d29 0a20 202a 2049 6e70 7574  (ff,M).  * Input
│ │ │ │ -00057fd0: 733a 0a20 2020 2020 202a 2066 662c 2061  s:.      * ff, a
│ │ │ │ -00057fe0: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ -00057ff0: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ -00058000: 7269 782c 2c20 6120 7375 6666 6963 6965  rix,, a sufficie
│ │ │ │ -00058010: 6e74 6c79 2067 656e 6572 616c 0a20 2020  ntly general.   
│ │ │ │ -00058020: 2020 2020 2072 6567 756c 6172 2073 6571       regular seq
│ │ │ │ -00058030: 7565 6e63 6520 696e 2061 2072 696e 6720  uence in a ring 
│ │ │ │ -00058040: 530a 2020 2020 2020 2a20 4d2c 2061 202a  S.      * M, a *
│ │ │ │ -00058050: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00058060: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00058070: 652c 2c20 6120 6d61 7869 6d61 6c20 436f  e,, a maximal Co
│ │ │ │ -00058080: 6865 6e2d 4d61 6361 756c 6179 0a20 2020  hen-Macaulay.   
│ │ │ │ -00058090: 2020 2020 206d 6f64 756c 6520 6f76 6572       module over
│ │ │ │ -000580a0: 2053 2f69 6465 616c 2066 660a 2020 2a20   S/ideal ff.  * 
│ │ │ │ -000580b0: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ -000580c0: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ -000580d0: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
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│ │ │ │ -00058100: 2020 2a20 4175 676d 656e 7461 7469 6f6e    * Augmentation
│ │ │ │ -00058110: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00058120: 2076 616c 7565 2074 7275 650a 2020 2020   value true.    
│ │ │ │ -00058130: 2020 2a20 4368 6563 6b20 3d3e 202e 2e2e    * Check => ...
│ │ │ │ -00058140: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00058150: 6661 6c73 650a 2020 2020 2020 2a20 4c61  false.      * La
│ │ │ │ -00058160: 7965 7265 6420 3d3e 202e 2e2e 2c20 6465  yered => ..., de
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│ │ │ │ -00058190: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -000581a0: 2076 616c 7565 2066 616c 7365 0a20 202a   value false.  *
│ │ │ │ -000581b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -000581c0: 2a20 4d46 2c20 6120 2a6e 6f74 6520 6c69  * MF, a *note li
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│ │ │ │ -000581f0: 616d 6d61 5c7d 2c20 7768 6572 6520 643a  amma\}, where d:
│ │ │ │ -00058200: 415f 3120 5c74 6f0a 2020 2020 2020 2020  A_1 \to.        
│ │ │ │ -00058210: 415f 3020 616e 6420 683a 205c 6f70 6c75  A_0 and h: \oplu
│ │ │ │ -00058220: 7320 415f 3028 7029 205c 746f 2041 5f31  s A_0(p) \to A_1
│ │ │ │ -00058230: 2069 7320 7468 6520 6469 7265 6374 2073   is the direct s
│ │ │ │ -00058240: 756d 206f 6620 7061 7274 6961 6c0a 2020  um of partial.  
│ │ │ │ -00058250: 2020 2020 2020 686f 6d6f 746f 7069 6573        homotopies
│ │ │ │ -00058260: 2c20 616e 6420 6761 6d6d 613a 2041 5f30  , and gamma: A_0
│ │ │ │ -00058270: 202d 3e4d 2069 7320 7468 6520 6175 676d   ->M is the augm
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│ │ │ │ -00058290: 6564 206f 6e6c 7920 6966 0a20 2020 2020  ed only if.     
│ │ │ │ -000582a0: 2020 2041 7567 6d65 6e74 6174 696f 6e20     Augmentation 
│ │ │ │ -000582b0: 3d3e 7472 7565 290a 0a44 6573 6372 6970  =>true)..Descrip
│ │ │ │ -000582c0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -000582d0: 0a0a 5468 6520 696e 7075 7420 6d6f 6475  ..The input modu
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│ │ │ │ -000582f0: 206d 6178 696d 616c 2043 6f68 656e 2d4d   maximal Cohen-M
│ │ │ │ -00058300: 6163 6175 6c61 7920 6d6f 6475 6c65 206f  acaulay module o
│ │ │ │ -00058310: 7665 7220 5220 3d20 532f 6964 6561 6c0a  ver R = S/ideal.
│ │ │ │ -00058320: 6666 2e20 2049 6620 4d20 6973 2069 6e20  ff.  If M is in 
│ │ │ │ -00058330: 6661 6374 2061 2022 6869 6768 2073 797a  fact a "high syz
│ │ │ │ -00058340: 7967 7922 2c20 7468 656e 2074 6865 2066  ygy", then the f
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│ │ │ │ -00058360: 6374 6f72 697a 6174 696f 6e28 6666 2c4d  ctorization(ff,M
│ │ │ │ -00058370: 2c4c 6179 6572 6564 3d3e 6661 6c73 6529  ,Layered=>false)
│ │ │ │ -00058380: 2075 7365 7320 6120 6469 6666 6572 656e   uses a differen
│ │ │ │ -00058390: 742c 2066 6173 7465 7220 616c 676f 7269  t, faster algori
│ │ │ │ -000583a0: 7468 6d0a 7768 6963 6820 6f6e 6c79 2077  thm.which only w
│ │ │ │ -000583b0: 6f72 6b73 2069 6e20 7468 6520 6869 6768  orks in the high
│ │ │ │ -000583c0: 2073 797a 7967 7920 6361 7365 2e0a 0a49   syzygy case...I
│ │ │ │ -000583d0: 6e20 616c 6c20 6578 616d 706c 6573 2077  n all examples w
│ │ │ │ -000583e0: 6520 6b6e 6f77 2c20 4d20 6361 6e20 6265  e know, M can be
│ │ │ │ -000583f0: 2063 6f6e 7369 6465 7265 6420 6120 2268   considered a "h
│ │ │ │ -00058400: 6967 6820 7379 7a79 6779 2220 6173 206c  igh syzygy" as l
│ │ │ │ -00058410: 6f6e 6720 6173 0a45 7874 5e7b 6576 656e  ong as.Ext^{even
│ │ │ │ -00058420: 7d5f 5228 4d2c 6b29 2061 6e64 2045 7874  }_R(M,k) and Ext
│ │ │ │ -00058430: 5e7b 6f64 647d 5f52 284d 2c6b 2920 6861  ^{odd}_R(M,k) ha
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│ │ │ │ -00058460: 7269 6e67 0a6f 6620 4349 206f 7065 7261  ring.of CI opera
│ │ │ │ -00058470: 746f 7273 2028 7265 6772 6164 6564 2077  tors (regraded w
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│ │ │ │ -000584c0: 2069 7320 7468 6174 2069 7420 7375 6666   is that it suff
│ │ │ │ -000584d0: 6963 6573 2074 6f20 6861 7665 2072 6567  ices to have reg
│ │ │ │ -000584e0: 756c 6172 6974 7920 3c20 2d28 322a 6469  ularity < -(2*di
│ │ │ │ -000584f0: 6d20 522b 3129 2e0a 0a57 6865 6e20 7468  m R+1)...When th
│ │ │ │ -00058500: 6520 6f70 7469 6f6e 616c 2069 6e70 7574  e optional input
│ │ │ │ -00058510: 2043 6865 636b 3d3d 7472 7565 2028 7468   Check==true (th
│ │ │ │ -00058520: 6520 6465 6661 756c 7420 6973 2043 6865  e default is Che
│ │ │ │ -00058530: 636b 3d3d 6661 6c73 6529 2c20 7468 650a  ck==false), the.
│ │ │ │ -00058540: 7072 6f70 6572 7469 6573 2069 6e20 7468  properties in th
│ │ │ │ -00058550: 6520 6465 6669 6e69 7469 6f6e 206f 6620  e definition of 
│ │ │ │ -00058560: 4d61 7472 6978 2046 6163 746f 7269 7a61  Matrix Factoriza
│ │ │ │ -00058570: 7469 6f6e 2061 7265 2076 6572 6966 6965  tion are verifie
│ │ │ │ -00058580: 640a 0a54 6865 206f 7574 7075 7420 6973  d..The output is
│ │ │ │ -00058590: 2061 206c 6973 7420 6f66 206d 6170 7320   a list of maps 
│ │ │ │ -000585a0: 5c7b 642c 685c 7d20 6f72 205c 7b64 2c68  \{d,h\} or \{d,h
│ │ │ │ -000585b0: 2c67 616d 6d61 5c7d 2c20 7768 6572 6520  ,gamma\}, where 
│ │ │ │ -000585c0: 6761 6d6d 6120 6973 2061 6e0a 6175 676d  gamma is an.augm
│ │ │ │ -000585d0: 656e 7461 7469 6f6e 2c20 7468 6174 2069  entation, that i
│ │ │ │ -000585e0: 732c 2061 206d 6170 2066 726f 6d20 7461  s, a map from ta
│ │ │ │ -000585f0: 7267 6574 2064 2074 6f20 4d2e 0a0a 5468  rget d to M...Th
│ │ │ │ -00058600: 6520 6d61 7020 6420 6973 2061 2073 7065  e map d is a spe
│ │ │ │ -00058610: 6369 616c 206c 6966 7469 6e67 2074 6f20  cial lifting to 
│ │ │ │ -00058620: 5320 6f66 2061 2070 7265 7365 6e74 6174  S of a presentat
│ │ │ │ -00058630: 696f 6e20 6f66 204d 206f 7665 7220 522e  ion of M over R.
│ │ │ │ -00058640: 2054 6f20 6578 706c 6169 6e0a 7468 6520   To explain.the 
│ │ │ │ -00058650: 636f 6e74 656e 7473 2c20 7765 2069 6e74  contents, we int
│ │ │ │ -00058660: 726f 6475 6365 2073 6f6d 6520 6e6f 7461  roduce some nota
│ │ │ │ -00058670: 7469 6f6e 2028 6672 6f6d 2045 6973 656e  tion (from Eisen
│ │ │ │ -00058680: 6275 6420 616e 6420 5065 6576 612c 2022  bud and Peeva, "
│ │ │ │ -00058690: 4d69 6e69 6d61 6c0a 6672 6565 2072 6573  Minimal.free res
│ │ │ │ -000586a0: 6f6c 7574 696f 6e73 206f 7665 7220 636f  olutions over co
│ │ │ │ -000586b0: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ -000586c0: 696f 6e73 2220 4c65 6374 7572 6520 4e6f  ions" Lecture No
│ │ │ │ -000586d0: 7465 7320 696e 204d 6174 6865 6d61 7469  tes in Mathemati
│ │ │ │ -000586e0: 6373 2c0a 3231 3532 2e20 5370 7269 6e67  cs,.2152. Spring
│ │ │ │ -000586f0: 6572 2c20 4368 616d 2c20 3230 3136 2e20  er, Cham, 2016. 
│ │ │ │ -00058700: 782b 3130 3720 7070 2e20 4953 424e 3a20  x+107 pp. ISBN: 
│ │ │ │ -00058710: 3937 382d 332d 3331 392d 3236 3433 362d  978-3-319-26436-
│ │ │ │ -00058720: 333b 0a39 3738 2d33 2d33 3139 2d32 3634  3;.978-3-319-264
│ │ │ │ -00058730: 3337 2d30 292e 0a0a 5228 6929 203d 2053  37-0)...R(i) = S
│ │ │ │ -00058740: 2f28 6666 5f30 2c2e 2e2c 6666 5f7b 692d  /(ff_0,..,ff_{i-
│ │ │ │ -00058750: 317d 292e 2048 6572 6520 303c 3d20 6920  1}). Here 0<= i 
│ │ │ │ -00058760: 3c3d 2063 2c20 616e 6420 5220 3d20 5228  <= c, and R = R(
│ │ │ │ -00058770: 6329 2061 6e64 2053 203d 2052 2830 292e  c) and S = R(0).
│ │ │ │ -00058780: 0a0a 4228 6929 203d 2074 6865 206d 6174  ..B(i) = the mat
│ │ │ │ -00058790: 7269 7820 286f 7665 7220 5329 2072 6570  rix (over S) rep
│ │ │ │ -000587a0: 7265 7365 6e74 696e 6720 645f 693a 2042  resenting d_i: B
│ │ │ │ -000587b0: 5f31 2869 2920 5c74 6f20 425f 3028 6929  _1(i) \to B_0(i)
│ │ │ │ -000587c0: 0a0a 6428 6929 3a20 415f 3128 6929 205c  ..d(i): A_1(i) \
│ │ │ │ -000587d0: 746f 2041 5f30 2869 2920 7468 6520 7265  to A_0(i) the re
│ │ │ │ -000587e0: 7374 7269 6374 696f 6e20 6f66 2064 203d  striction of d =
│ │ │ │ -000587f0: 2064 2863 292e 2077 6865 7265 2041 2869   d(c). where A(i
│ │ │ │ -00058800: 2920 3d0a 5c6f 706c 7573 5f7b 693d 317d  ) =.\oplus_{i=1}
│ │ │ │ -00058810: 5e70 2042 2869 290a 0a0a 0a54 6865 206d  ^p B(i)....The m
│ │ │ │ -00058820: 6170 2068 2069 7320 6120 6469 7265 6374  ap h is a direct
│ │ │ │ -00058830: 2073 756d 206f 6620 6d61 7073 2074 6172   sum of maps tar
│ │ │ │ -00058840: 6765 7420 6428 7029 205c 746f 2073 6f75  get d(p) \to sou
│ │ │ │ -00058850: 7263 6520 6428 7029 2074 6861 7420 6172  rce d(p) that ar
│ │ │ │ -00058860: 650a 686f 6d6f 746f 7069 6573 2066 6f72  e.homotopies for
│ │ │ │ -00058870: 2066 665f 7020 6f6e 2074 6865 2072 6573   ff_p on the res
│ │ │ │ -00058880: 7472 6963 7469 6f6e 2064 2870 293a 206f  triction d(p): o
│ │ │ │ -00058890: 7665 7220 7468 6520 7269 6e67 2052 2328  ver the ring R#(
│ │ │ │ -000588a0: 702d 3129 203d 0a53 2f28 6666 2331 2e2e  p-1) =.S/(ff#1..
│ │ │ │ -000588b0: 6666 2328 702d 3129 2c20 736f 2064 2870  ff#(p-1), so d(p
│ │ │ │ -000588c0: 2920 2a20 6823 7020 3d20 6666 2370 206d  ) * h#p = ff#p m
│ │ │ │ -000588d0: 6f64 2028 6666 2331 2e2e 6666 2328 702d  od (ff#1..ff#(p-
│ │ │ │ -000588e0: 3129 2e0a 0a49 6e20 6164 6469 7469 6f6e  1)...In addition
│ │ │ │ -000588f0: 2c20 6823 7020 2a20 6428 7029 2069 6e64  , h#p * d(p) ind
│ │ │ │ -00058900: 7563 6573 2066 6623 7020 6f6e 2042 3123  uces ff#p on B1#
│ │ │ │ -00058910: 7020 6d6f 6420 2866 6623 312e 2e66 6623  p mod (ff#1..ff#
│ │ │ │ -00058920: 2870 2d31 292e 0a0a 4865 7265 2069 7320  (p-1)...Here is 
│ │ │ │ -00058930: 6120 7369 6d70 6c65 2065 7861 6d70 6c65  a simple example
│ │ │ │ -00058940: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │ +00057f70: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ +00057f80: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055 7361  =======..  * Usa
│ │ │ │ +00057f90: 6765 3a20 0a20 2020 2020 2020 204d 4620  ge: .        MF 
│ │ │ │ +00057fa0: 3d20 6d61 7472 6978 4661 6374 6f72 697a  = matrixFactoriz
│ │ │ │ +00057fb0: 6174 696f 6e28 6666 2c4d 290a 2020 2a20  ation(ff,M).  * 
│ │ │ │ +00057fc0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00057fd0: 6666 2c20 6120 2a6e 6f74 6520 6d61 7472  ff, a *note matr
│ │ │ │ +00057fe0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +00057ff0: 6329 4d61 7472 6978 2c2c 2061 2073 7566  c)Matrix,, a suf
│ │ │ │ +00058000: 6669 6369 656e 746c 7920 6765 6e65 7261  ficiently genera
│ │ │ │ +00058010: 6c0a 2020 2020 2020 2020 7265 6775 6c61  l.        regula
│ │ │ │ +00058020: 7220 7365 7175 656e 6365 2069 6e20 6120  r sequence in a 
│ │ │ │ +00058030: 7269 6e67 2053 0a20 2020 2020 202a 204d  ring S.      * M
│ │ │ │ +00058040: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +00058050: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00058060: 4d6f 6475 6c65 2c2c 2061 206d 6178 696d  Module,, a maxim
│ │ │ │ +00058070: 616c 2043 6f68 656e 2d4d 6163 6175 6c61  al Cohen-Macaula
│ │ │ │ +00058080: 790a 2020 2020 2020 2020 6d6f 6475 6c65  y.        module
│ │ │ │ +00058090: 206f 7665 7220 532f 6964 6561 6c20 6666   over S/ideal ff
│ │ │ │ +000580a0: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ +000580b0: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ +000580c0: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ +000580d0: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ +000580e0: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ +000580f0: 0a20 2020 2020 202a 2041 7567 6d65 6e74  .      * Augment
│ │ │ │ +00058100: 6174 696f 6e20 3d3e 202e 2e2e 2c20 6465  ation => ..., de
│ │ │ │ +00058110: 6661 756c 7420 7661 6c75 6520 7472 7565  fault value true
│ │ │ │ +00058120: 0a20 2020 2020 202a 2043 6865 636b 203d  .      * Check =
│ │ │ │ +00058130: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00058140: 616c 7565 2066 616c 7365 0a20 2020 2020  alue false.     
│ │ │ │ +00058150: 202a 204c 6179 6572 6564 203d 3e20 2e2e   * Layered => ..
│ │ │ │ +00058160: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +00058170: 2074 7275 650a 2020 2020 2020 2a20 5665   true.      * Ve
│ │ │ │ +00058180: 7262 6f73 6520 3d3e 202e 2e2e 2c20 6465  rbose => ..., de
│ │ │ │ +00058190: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ +000581a0: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ +000581b0: 2020 2020 202a 204d 462c 2061 202a 6e6f       * MF, a *no
│ │ │ │ +000581c0: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ +000581d0: 6179 3244 6f63 294c 6973 742c 2c20 5c7b  ay2Doc)List,, \{
│ │ │ │ +000581e0: 642c 682c 6761 6d6d 615c 7d2c 2077 6865  d,h,gamma\}, whe
│ │ │ │ +000581f0: 7265 2064 3a41 5f31 205c 746f 0a20 2020  re d:A_1 \to.   
│ │ │ │ +00058200: 2020 2020 2041 5f30 2061 6e64 2068 3a20       A_0 and h: 
│ │ │ │ +00058210: 5c6f 706c 7573 2041 5f30 2870 2920 5c74  \oplus A_0(p) \t
│ │ │ │ +00058220: 6f20 415f 3120 6973 2074 6865 2064 6972  o A_1 is the dir
│ │ │ │ +00058230: 6563 7420 7375 6d20 6f66 2070 6172 7469  ect sum of parti
│ │ │ │ +00058240: 616c 0a20 2020 2020 2020 2068 6f6d 6f74  al.        homot
│ │ │ │ +00058250: 6f70 6965 732c 2061 6e64 2067 616d 6d61  opies, and gamma
│ │ │ │ +00058260: 3a20 415f 3020 2d3e 4d20 6973 2074 6865  : A_0 ->M is the
│ │ │ │ +00058270: 2061 7567 6d65 6e74 6174 696f 6e20 2872   augmentation (r
│ │ │ │ +00058280: 6574 7572 6e65 6420 6f6e 6c79 2069 660a  eturned only if.
│ │ │ │ +00058290: 2020 2020 2020 2020 4175 676d 656e 7461          Augmenta
│ │ │ │ +000582a0: 7469 6f6e 203d 3e74 7275 6529 0a0a 4465  tion =>true)..De
│ │ │ │ +000582b0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +000582c0: 3d3d 3d3d 3d0a 0a54 6865 2069 6e70 7574  =====..The input
│ │ │ │ +000582d0: 206d 6f64 756c 6520 4d20 7368 6f75 6c64   module M should
│ │ │ │ +000582e0: 2062 6520 6120 6d61 7869 6d61 6c20 436f   be a maximal Co
│ │ │ │ +000582f0: 6865 6e2d 4d61 6361 756c 6179 206d 6f64  hen-Macaulay mod
│ │ │ │ +00058300: 756c 6520 6f76 6572 2052 203d 2053 2f69  ule over R = S/i
│ │ │ │ +00058310: 6465 616c 0a66 662e 2020 4966 204d 2069  deal.ff.  If M i
│ │ │ │ +00058320: 7320 696e 2066 6163 7420 6120 2268 6967  s in fact a "hig
│ │ │ │ +00058330: 6820 7379 7a79 6779 222c 2074 6865 6e20  h syzygy", then 
│ │ │ │ +00058340: 7468 6520 6675 6e63 7469 6f6e 0a6d 6174  the function.mat
│ │ │ │ +00058350: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +00058360: 2866 662c 4d2c 4c61 7965 7265 643d 3e66  (ff,M,Layered=>f
│ │ │ │ +00058370: 616c 7365 2920 7573 6573 2061 2064 6966  alse) uses a dif
│ │ │ │ +00058380: 6665 7265 6e74 2c20 6661 7374 6572 2061  ferent, faster a
│ │ │ │ +00058390: 6c67 6f72 6974 686d 0a77 6869 6368 206f  lgorithm.which o
│ │ │ │ +000583a0: 6e6c 7920 776f 726b 7320 696e 2074 6865  nly works in the
│ │ │ │ +000583b0: 2068 6967 6820 7379 7a79 6779 2063 6173   high syzygy cas
│ │ │ │ +000583c0: 652e 0a0a 496e 2061 6c6c 2065 7861 6d70  e...In all examp
│ │ │ │ +000583d0: 6c65 7320 7765 206b 6e6f 772c 204d 2063  les we know, M c
│ │ │ │ +000583e0: 616e 2062 6520 636f 6e73 6964 6572 6564  an be considered
│ │ │ │ +000583f0: 2061 2022 6869 6768 2073 797a 7967 7922   a "high syzygy"
│ │ │ │ +00058400: 2061 7320 6c6f 6e67 2061 730a 4578 745e   as long as.Ext^
│ │ │ │ +00058410: 7b65 7665 6e7d 5f52 284d 2c6b 2920 616e  {even}_R(M,k) an
│ │ │ │ +00058420: 6420 4578 745e 7b6f 6464 7d5f 5228 4d2c  d Ext^{odd}_R(M,
│ │ │ │ +00058430: 6b29 2068 6176 6520 6e65 6761 7469 7665  k) have negative
│ │ │ │ +00058440: 2072 6567 756c 6172 6974 7920 6f76 6572   regularity over
│ │ │ │ +00058450: 2074 6865 2072 696e 670a 6f66 2043 4920   the ring.of CI 
│ │ │ │ +00058460: 6f70 6572 6174 6f72 7320 2872 6567 7261  operators (regra
│ │ │ │ +00058470: 6465 6420 7769 7468 2076 6172 6961 626c  ded with variabl
│ │ │ │ +00058480: 6573 206f 6620 6465 6772 6565 2031 2e20  es of degree 1. 
│ │ │ │ +00058490: 486f 7765 7665 722c 2074 6865 2062 6573  However, the bes
│ │ │ │ +000584a0: 7420 7265 7375 6c74 0a77 6520 6361 6e20  t result.we can 
│ │ │ │ +000584b0: 7072 6f76 6520 6973 2074 6861 7420 6974  prove is that it
│ │ │ │ +000584c0: 2073 7566 6669 6365 7320 746f 2068 6176   suffices to hav
│ │ │ │ +000584d0: 6520 7265 6775 6c61 7269 7479 203c 202d  e regularity < -
│ │ │ │ +000584e0: 2832 2a64 696d 2052 2b31 292e 0a0a 5768  (2*dim R+1)...Wh
│ │ │ │ +000584f0: 656e 2074 6865 206f 7074 696f 6e61 6c20  en the optional 
│ │ │ │ +00058500: 696e 7075 7420 4368 6563 6b3d 3d74 7275  input Check==tru
│ │ │ │ +00058510: 6520 2874 6865 2064 6566 6175 6c74 2069  e (the default i
│ │ │ │ +00058520: 7320 4368 6563 6b3d 3d66 616c 7365 292c  s Check==false),
│ │ │ │ +00058530: 2074 6865 0a70 726f 7065 7274 6965 7320   the.properties 
│ │ │ │ +00058540: 696e 2074 6865 2064 6566 696e 6974 696f  in the definitio
│ │ │ │ +00058550: 6e20 6f66 204d 6174 7269 7820 4661 6374  n of Matrix Fact
│ │ │ │ +00058560: 6f72 697a 6174 696f 6e20 6172 6520 7665  orization are ve
│ │ │ │ +00058570: 7269 6669 6564 0a0a 5468 6520 6f75 7470  rified..The outp
│ │ │ │ +00058580: 7574 2069 7320 6120 6c69 7374 206f 6620  ut is a list of 
│ │ │ │ +00058590: 6d61 7073 205c 7b64 2c68 5c7d 206f 7220  maps \{d,h\} or 
│ │ │ │ +000585a0: 5c7b 642c 682c 6761 6d6d 615c 7d2c 2077  \{d,h,gamma\}, w
│ │ │ │ +000585b0: 6865 7265 2067 616d 6d61 2069 7320 616e  here gamma is an
│ │ │ │ +000585c0: 0a61 7567 6d65 6e74 6174 696f 6e2c 2074  .augmentation, t
│ │ │ │ +000585d0: 6861 7420 6973 2c20 6120 6d61 7020 6672  hat is, a map fr
│ │ │ │ +000585e0: 6f6d 2074 6172 6765 7420 6420 746f 204d  om target d to M
│ │ │ │ +000585f0: 2e0a 0a54 6865 206d 6170 2064 2069 7320  ...The map d is 
│ │ │ │ +00058600: 6120 7370 6563 6961 6c20 6c69 6674 696e  a special liftin
│ │ │ │ +00058610: 6720 746f 2053 206f 6620 6120 7072 6573  g to S of a pres
│ │ │ │ +00058620: 656e 7461 7469 6f6e 206f 6620 4d20 6f76  entation of M ov
│ │ │ │ +00058630: 6572 2052 2e20 546f 2065 7870 6c61 696e  er R. To explain
│ │ │ │ +00058640: 0a74 6865 2063 6f6e 7465 6e74 732c 2077  .the contents, w
│ │ │ │ +00058650: 6520 696e 7472 6f64 7563 6520 736f 6d65  e introduce some
│ │ │ │ +00058660: 206e 6f74 6174 696f 6e20 2866 726f 6d20   notation (from 
│ │ │ │ +00058670: 4569 7365 6e62 7564 2061 6e64 2050 6565  Eisenbud and Pee
│ │ │ │ +00058680: 7661 2c20 224d 696e 696d 616c 0a66 7265  va, "Minimal.fre
│ │ │ │ +00058690: 6520 7265 736f 6c75 7469 6f6e 7320 6f76  e resolutions ov
│ │ │ │ +000586a0: 6572 2063 6f6d 706c 6574 6520 696e 7465  er complete inte
│ │ │ │ +000586b0: 7273 6563 7469 6f6e 7322 204c 6563 7475  rsections" Lectu
│ │ │ │ +000586c0: 7265 204e 6f74 6573 2069 6e20 4d61 7468  re Notes in Math
│ │ │ │ +000586d0: 656d 6174 6963 732c 0a32 3135 322e 2053  ematics,.2152. S
│ │ │ │ +000586e0: 7072 696e 6765 722c 2043 6861 6d2c 2032  pringer, Cham, 2
│ │ │ │ +000586f0: 3031 362e 2078 2b31 3037 2070 702e 2049  016. x+107 pp. I
│ │ │ │ +00058700: 5342 4e3a 2039 3738 2d33 2d33 3139 2d32  SBN: 978-3-319-2
│ │ │ │ +00058710: 3634 3336 2d33 3b0a 3937 382d 332d 3331  6436-3;.978-3-31
│ │ │ │ +00058720: 392d 3236 3433 372d 3029 2e0a 0a52 2869  9-26437-0)...R(i
│ │ │ │ +00058730: 2920 3d20 532f 2866 665f 302c 2e2e 2c66  ) = S/(ff_0,..,f
│ │ │ │ +00058740: 665f 7b69 2d31 7d29 2e20 4865 7265 2030  f_{i-1}). Here 0
│ │ │ │ +00058750: 3c3d 2069 203c 3d20 632c 2061 6e64 2052  <= i <= c, and R
│ │ │ │ +00058760: 203d 2052 2863 2920 616e 6420 5320 3d20   = R(c) and S = 
│ │ │ │ +00058770: 5228 3029 2e0a 0a42 2869 2920 3d20 7468  R(0)...B(i) = th
│ │ │ │ +00058780: 6520 6d61 7472 6978 2028 6f76 6572 2053  e matrix (over S
│ │ │ │ +00058790: 2920 7265 7072 6573 656e 7469 6e67 2064  ) representing d
│ │ │ │ +000587a0: 5f69 3a20 425f 3128 6929 205c 746f 2042  _i: B_1(i) \to B
│ │ │ │ +000587b0: 5f30 2869 290a 0a64 2869 293a 2041 5f31  _0(i)..d(i): A_1
│ │ │ │ +000587c0: 2869 2920 5c74 6f20 415f 3028 6929 2074  (i) \to A_0(i) t
│ │ │ │ +000587d0: 6865 2072 6573 7472 6963 7469 6f6e 206f  he restriction o
│ │ │ │ +000587e0: 6620 6420 3d20 6428 6329 2e20 7768 6572  f d = d(c). wher
│ │ │ │ +000587f0: 6520 4128 6929 203d 0a5c 6f70 6c75 735f  e A(i) =.\oplus_
│ │ │ │ +00058800: 7b69 3d31 7d5e 7020 4228 6929 0a0a 0a0a  {i=1}^p B(i)....
│ │ │ │ +00058810: 5468 6520 6d61 7020 6820 6973 2061 2064  The map h is a d
│ │ │ │ +00058820: 6972 6563 7420 7375 6d20 6f66 206d 6170  irect sum of map
│ │ │ │ +00058830: 7320 7461 7267 6574 2064 2870 2920 5c74  s target d(p) \t
│ │ │ │ +00058840: 6f20 736f 7572 6365 2064 2870 2920 7468  o source d(p) th
│ │ │ │ +00058850: 6174 2061 7265 0a68 6f6d 6f74 6f70 6965  at are.homotopie
│ │ │ │ +00058860: 7320 666f 7220 6666 5f70 206f 6e20 7468  s for ff_p on th
│ │ │ │ +00058870: 6520 7265 7374 7269 6374 696f 6e20 6428  e restriction d(
│ │ │ │ +00058880: 7029 3a20 6f76 6572 2074 6865 2072 696e  p): over the rin
│ │ │ │ +00058890: 6720 5223 2870 2d31 2920 3d0a 532f 2866  g R#(p-1) =.S/(f
│ │ │ │ +000588a0: 6623 312e 2e66 6623 2870 2d31 292c 2073  f#1..ff#(p-1), s
│ │ │ │ +000588b0: 6f20 6428 7029 202a 2068 2370 203d 2066  o d(p) * h#p = f
│ │ │ │ +000588c0: 6623 7020 6d6f 6420 2866 6623 312e 2e66  f#p mod (ff#1..f
│ │ │ │ +000588d0: 6623 2870 2d31 292e 0a0a 496e 2061 6464  f#(p-1)...In add
│ │ │ │ +000588e0: 6974 696f 6e2c 2068 2370 202a 2064 2870  ition, h#p * d(p
│ │ │ │ +000588f0: 2920 696e 6475 6365 7320 6666 2370 206f  ) induces ff#p o
│ │ │ │ +00058900: 6e20 4231 2370 206d 6f64 2028 6666 2331  n B1#p mod (ff#1
│ │ │ │ +00058910: 2e2e 6666 2328 702d 3129 2e0a 0a48 6572  ..ff#(p-1)...Her
│ │ │ │ +00058920: 6520 6973 2061 2073 696d 706c 6520 6578  e is a simple ex
│ │ │ │ +00058930: 616d 706c 653a 0a0a 2b2d 2d2d 2d2d 2d2d  ample:..+-------
│ │ │ │ +00058940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00058970: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ -00058980: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00058990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058960: 2b0a 7c69 3120 3a20 7365 7452 616e 646f  +.|i1 : setRando
│ │ │ │ +00058970: 6d53 6565 6420 3020 2020 2020 2020 2020  mSeed 0         
│ │ │ │ +00058980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00058990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000589a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000589b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000589c0: 0a7c 6f31 203d 2030 2020 2020 2020 2020  .|o1 = 0        
│ │ │ │ -000589d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000589e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000589b0: 2020 2020 7c0a 7c6f 3120 3d20 3020 2020      |.|o1 = 0   
│ │ │ │ +000589c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000589d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000589e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000589f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058a10: 2d2d 2d2b 0a7c 6932 203a 206b 6b20 3d20  ---+.|i2 : kk = 
│ │ │ │ -00058a20: 5a5a 2f31 3031 2020 2020 2020 2020 2020  ZZ/101          
│ │ │ │ -00058a30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058a00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00058a10: 6b6b 203d 205a 5a2f 3130 3120 2020 2020  kk = ZZ/101     
│ │ │ │ +00058a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058a30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00058a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a60: 2020 2020 2020 207c 0a7c 6f32 203d 206b         |.|o2 = k
│ │ │ │ -00058a70: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ -00058a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00058a50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00058a60: 3220 3d20 6b6b 2020 2020 2020 2020 2020  2 = kk          
│ │ │ │ +00058a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058a80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00058a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ab0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00058ac0: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ -00058ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ae0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00058ab0: 7c0a 7c6f 3220 3a20 5175 6f74 6965 6e74  |.|o2 : Quotient
│ │ │ │ +00058ac0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00058ad0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00058ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00058b10: 0a7c 6933 203a 2053 203d 206b 6b5b 612c  .|i3 : S = kk[a,
│ │ │ │ -00058b20: 622c 752c 765d 2020 2020 2020 2020 2020  b,u,v]          
│ │ │ │ -00058b30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00058b00: 2d2d 2d2d 2b0a 7c69 3320 3a20 5320 3d20  ----+.|i3 : S = 
│ │ │ │ +00058b10: 6b6b 5b61 2c62 2c75 2c76 5d20 2020 2020  kk[a,b,u,v]     
│ │ │ │ +00058b20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058b30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00058b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b60: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ +00058b50: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +00058b60: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  00058b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00058b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058bb0: 2020 2020 2020 207c 0a7c 6f33 203a 2050         |.|o3 : P
│ │ │ │ -00058bc0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ -00058bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058be0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00058ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00058bb0: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +00058bc0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00058bd0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00058be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00058c10: 203a 2066 6620 3d20 6d61 7472 6978 2261   : ff = matrix"a
│ │ │ │ -00058c20: 752c 6276 2220 2020 2020 2020 2020 2020  u,bv"           
│ │ │ │ -00058c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058c00: 2b0a 7c69 3420 3a20 6666 203d 206d 6174  +.|i4 : ff = mat
│ │ │ │ +00058c10: 7269 7822 6175 2c62 7622 2020 2020 2020  rix"au,bv"      
│ │ │ │ +00058c20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00058c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00058c60: 0a7c 6f34 203d 207c 2061 7520 6276 207c  .|o4 = | au bv |
│ │ │ │ -00058c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058c80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00058c50: 2020 2020 7c0a 7c6f 3420 3d20 7c20 6175      |.|o4 = | au
│ │ │ │ +00058c60: 2062 7620 7c20 2020 2020 2020 2020 2020   bv |           
│ │ │ │ +00058c70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058c80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00058c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00058cc0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -00058cd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00058ce0: 6f34 203a 204d 6174 7269 7820 5320 203c  o4 : Matrix S  <
│ │ │ │ -00058cf0: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ -00058d00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00058ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00058cb0: 2020 2020 2020 2020 3120 2020 2020 2032          1      2
│ │ │ │ +00058cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058cd0: 2020 7c0a 7c6f 3420 3a20 4d61 7472 6978    |.|o4 : Matrix
│ │ │ │ +00058ce0: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ +00058cf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00058d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058d30: 2d2b 0a7c 6935 203a 2052 203d 2053 2f69  -+.|i5 : R = S/i
│ │ │ │ -00058d40: 6465 616c 2066 6620 2020 2020 2020 2020  deal ff         
│ │ │ │ -00058d50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00058d20: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 5220  ------+.|i5 : R 
│ │ │ │ +00058d30: 3d20 532f 6964 6561 6c20 6666 2020 2020  = S/ideal ff    
│ │ │ │ +00058d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058d50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00058d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058d80: 2020 2020 207c 0a7c 6f35 203d 2052 2020       |.|o5 = R  
│ │ │ │ +00058d70: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00058d80: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │  00058d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00058db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00058dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058dd0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -00058de0: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ -00058df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058e00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00058da0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00058db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058dd0: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ +00058de0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00058df0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00058e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00058e30: 6936 203a 204d 3020 3d20 525e 312f 6964  i6 : M0 = R^1/id
│ │ │ │ -00058e40: 6561 6c22 612c 6222 2020 2020 2020 2020  eal"a,b"        
│ │ │ │ -00058e50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00058e20: 2d2d 2b0a 7c69 3620 3a20 4d30 203d 2052  --+.|i6 : M0 = R
│ │ │ │ +00058e30: 5e31 2f69 6465 616c 2261 2c62 2220 2020  ^1/ideal"a,b"   
│ │ │ │ +00058e40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00058e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058e80: 207c 0a7c 6f36 203d 2063 6f6b 6572 6e65   |.|o6 = cokerne
│ │ │ │ -00058e90: 6c20 7c20 6120 6220 7c20 2020 2020 2020  l | a b |       
│ │ │ │ -00058ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00058e70: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ +00058e80: 6b65 726e 656c 207c 2061 2062 207c 2020  kernel | a b |  
│ │ │ │ +00058e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058ea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00058eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00058ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ef0: 2020 2020 3120 2020 2020 2020 2020 207c      1          |
│ │ │ │ -00058f00: 0a7c 6f36 203a 2052 2d6d 6f64 756c 652c  .|o6 : R-module,
│ │ │ │ -00058f10: 2071 756f 7469 656e 7420 6f66 2052 2020   quotient of R  
│ │ │ │ -00058f20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00058ec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00058ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058ee0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00058ef0: 2020 2020 7c0a 7c6f 3620 3a20 522d 6d6f      |.|o6 : R-mo
│ │ │ │ +00058f00: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +00058f10: 6620 5220 2020 2020 2020 2020 2020 7c0a  f R           |.
│ │ │ │ +00058f20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00058f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058f50: 2d2d 2d2b 0a7c 6937 203a 204d 203d 2068  ---+.|i7 : M = h
│ │ │ │ -00058f60: 6967 6853 797a 7967 7920 4d30 2020 2020  ighSyzygy M0    
│ │ │ │ -00058f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058f40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +00058f50: 4d20 3d20 6869 6768 5379 7a79 6779 204d  M = highSyzygy M
│ │ │ │ +00058f60: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00058f70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00058f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058fa0: 2020 2020 2020 207c 0a7c 6f37 203d 2063         |.|o7 = c
│ │ │ │ -00058fb0: 6f6b 6572 6e65 6c20 7b32 7d20 7c20 6220  okernel {2} | b 
│ │ │ │ -00058fc0: 2d61 2030 2030 207c 2020 2020 2020 2020  -a 0 0 |        
│ │ │ │ -00058fd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00058fe0: 2020 7b32 7d20 7c20 3020 3020 2061 2062    {2} | 0 0  a b
│ │ │ │ -00058ff0: 207c 2020 2020 2020 2020 207c 0a7c 2020   |         |.|  
│ │ │ │ -00059000: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -00059010: 7c20 3020 7620 2030 2075 207c 2020 2020  | 0 v  0 u |    
│ │ │ │ -00059020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058f90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00058fa0: 3720 3d20 636f 6b65 726e 656c 207b 327d  7 = cokernel {2}
│ │ │ │ +00058fb0: 207c 2062 202d 6120 3020 3020 7c20 2020   | b -a 0 0 |   
│ │ │ │ +00058fc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00058fd0: 2020 2020 2020 207b 327d 207c 2030 2030         {2} | 0 0
│ │ │ │ +00058fe0: 2020 6120 6220 7c20 2020 2020 2020 2020    a b |         
│ │ │ │ +00058ff0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00059000: 207b 327d 207c 2030 2076 2020 3020 7520   {2} | 0 v  0 u 
│ │ │ │ +00059010: 7c20 2020 2020 2020 2020 7c0a 7c20 2020  |         |.|   
│ │ │ │ +00059020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00059050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00059060: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00059070: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -00059080: 2052 2d6d 6f64 756c 652c 2071 756f 7469   R-module, quoti
│ │ │ │ -00059090: 656e 7420 6f66 2052 2020 2020 2020 2020  ent of R        
│ │ │ │ -000590a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00059040: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00059050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059060: 2020 2033 2020 2020 2020 2020 2020 7c0a     3          |.
│ │ │ │ +00059070: 7c6f 3720 3a20 522d 6d6f 6475 6c65 2c20  |o7 : R-module, 
│ │ │ │ +00059080: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ +00059090: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000590a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000590b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000590c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000590d0: 6938 203a 204d 4620 3d20 6d61 7472 6978  i8 : MF = matrix
│ │ │ │ -000590e0: 4661 6374 6f72 697a 6174 696f 6e28 6666  Factorization(ff
│ │ │ │ -000590f0: 2c4d 293b 2020 207c 0a2b 2d2d 2d2d 2d2d  ,M);   |.+------
│ │ │ │ +000590c0: 2d2d 2b0a 7c69 3820 3a20 4d46 203d 206d  --+.|i8 : MF = m
│ │ │ │ +000590d0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +000590e0: 6f6e 2866 662c 4d29 3b20 2020 7c0a 2b2d  on(ff,M);   |.+-
│ │ │ │ +000590f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059120: 2d2b 0a7c 6939 203a 206e 6574 4c69 7374  -+.|i9 : netList
│ │ │ │ -00059130: 2042 5261 6e6b 7320 4d46 2020 2020 2020   BRanks MF      
│ │ │ │ -00059140: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00059110: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 6e65  ------+.|i9 : ne
│ │ │ │ +00059120: 744c 6973 7420 4252 616e 6b73 204d 4620  tList BRanks MF 
│ │ │ │ +00059130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059140: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00059150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059170: 2020 2020 207c 0a7c 2020 2020 202b 2d2b       |.|     +-+
│ │ │ │ -00059180: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │ -00059190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000591a0: 0a7c 6f39 203d 207c 327c 327c 2020 2020  .|o9 = |2|2|    
│ │ │ │ -000591b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000591c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000591d0: 202b 2d2b 2d2b 2020 2020 2020 2020 2020   +-+-+          
│ │ │ │ -000591e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000591f0: 2020 207c 0a7c 2020 2020 207c 317c 327c     |.|     |1|2|
│ │ │ │ +00059160: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00059170: 2020 2b2d 2b2d 2b20 2020 2020 2020 2020    +-+-+         
│ │ │ │ +00059180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059190: 2020 2020 7c0a 7c6f 3920 3d20 7c32 7c32      |.|o9 = |2|2
│ │ │ │ +000591a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000591b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000591c0: 7c20 2020 2020 2b2d 2b2d 2b20 2020 2020  |     +-+-+     
│ │ │ │ +000591d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000591e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000591f0: 7c31 7c32 7c20 2020 2020 2020 2020 2020  |1|2|           
│ │ │ │  00059200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00059220: 2020 2020 202b 2d2b 2d2b 2020 2020 2020       +-+-+      
│ │ │ │ -00059230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059240: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00059210: 2020 7c0a 7c20 2020 2020 2b2d 2b2d 2b20    |.|     +-+-+ 
│ │ │ │ +00059220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059230: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00059240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059270: 2d2b 0a7c 6931 3020 3a20 6e65 744c 6973  -+.|i10 : netLis
│ │ │ │ -00059280: 7420 624d 6170 7320 4d46 2020 2020 2020  t bMaps MF      
│ │ │ │ -00059290: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00059260: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 206e  ------+.|i10 : n
│ │ │ │ +00059270: 6574 4c69 7374 2062 4d61 7073 204d 4620  etList bMaps MF 
│ │ │ │ +00059280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059290: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000592a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000592b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000592c0: 2020 2020 207c 0a7c 2020 2020 2020 2b2d       |.|      +-
│ │ │ │ -000592d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020  ----------+     
│ │ │ │ -000592e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000592f0: 0a7c 6f31 3020 3d20 7c7b 327d 207c 2061  .|o10 = |{2} | a
│ │ │ │ -00059300: 2062 207c 7c20 2020 2020 2020 2020 2020   b ||           
│ │ │ │ -00059310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00059320: 2020 7c7b 327d 207c 2030 2075 207c 7c20    |{2} | 0 u || 
│ │ │ │ -00059330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059340: 2020 207c 0a7c 2020 2020 2020 2b2d 2d2d     |.|      +---
│ │ │ │ -00059350: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020  --------+       
│ │ │ │ -00059360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00059370: 2020 2020 2020 7c7b 327d 207c 2062 2061        |{2} | b a
│ │ │ │ -00059380: 207c 7c20 2020 2020 2020 2020 2020 2020   ||             
│ │ │ │ -00059390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000593a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020  +-----------+   
│ │ │ │ -000593b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000593c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000592b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000592c0: 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b     +-----------+
│ │ │ │ +000592d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000592e0: 2020 2020 7c0a 7c6f 3130 203d 207c 7b32      |.|o10 = |{2
│ │ │ │ +000592f0: 7d20 7c20 6120 6220 7c7c 2020 2020 2020  } | a b ||      
│ │ │ │ +00059300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00059310: 7c20 2020 2020 207c 7b32 7d20 7c20 3020  |      |{2} | 0 
│ │ │ │ +00059320: 7520 7c7c 2020 2020 2020 2020 2020 2020  u ||            
│ │ │ │ +00059330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00059340: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020   +-----------+  
│ │ │ │ +00059350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059360: 2020 7c0a 7c20 2020 2020 207c 7b32 7d20    |.|      |{2} 
│ │ │ │ +00059370: 7c20 6220 6120 7c7c 2020 2020 2020 2020  | b a ||        
│ │ │ │ +00059380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00059390: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d       +----------
│ │ │ │ +000593a0: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │ +000593b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000593c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000593d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000593e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000593f0: 3120 3a20 6265 7474 6920 7265 7328 4d2c  1 : betti res(M,
│ │ │ │ -00059400: 204c 656e 6774 684c 696d 6974 203d 3e20   LengthLimit => 
│ │ │ │ -00059410: 3729 2020 207c 0a7c 2020 2020 2020 2020  7)   |.|        
│ │ │ │ +000593e0: 2b0a 7c69 3131 203a 2062 6574 7469 2072  +.|i11 : betti r
│ │ │ │ +000593f0: 6573 284d 2c20 4c65 6e67 7468 4c69 6d69  es(M, LengthLimi
│ │ │ │ +00059400: 7420 3d3e 2037 2920 2020 7c0a 7c20 2020  t => 7)   |.|   
│ │ │ │ +00059410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00059440: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -00059450: 2031 2032 2033 2034 2035 2036 2020 3720   1 2 3 4 5 6  7 
│ │ │ │ -00059460: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -00059470: 3d20 746f 7461 6c3a 2033 2034 2035 2036  = total: 3 4 5 6
│ │ │ │ -00059480: 2037 2038 2039 2031 3020 2020 2020 2020   7 8 9 10       
│ │ │ │ -00059490: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000594a0: 323a 2033 2034 2035 2036 2037 2038 2039  2: 3 4 5 6 7 8 9
│ │ │ │ -000594b0: 2031 3020 2020 2020 2020 2020 207c 0a7c   10          |.|
│ │ │ │ +00059430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00059440: 2020 2020 3020 3120 3220 3320 3420 3520      0 1 2 3 4 5 
│ │ │ │ +00059450: 3620 2037 2020 2020 2020 2020 2020 7c0a  6  7          |.
│ │ │ │ +00059460: 7c6f 3131 203d 2074 6f74 616c 3a20 3320  |o11 = total: 3 
│ │ │ │ +00059470: 3420 3520 3620 3720 3820 3920 3130 2020  4 5 6 7 8 9 10  
│ │ │ │ +00059480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00059490: 2020 2020 2032 3a20 3320 3420 3520 3620       2: 3 4 5 6 
│ │ │ │ +000594a0: 3720 3820 3920 3130 2020 2020 2020 2020  7 8 9 10        
│ │ │ │ +000594b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000594c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000594d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000594e0: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ -000594f0: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ -00059500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000594d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000594e0: 3131 203a 2042 6574 7469 5461 6c6c 7920  11 : BettiTally 
│ │ │ │ +000594f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059500: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00059510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00059540: 3220 3a20 696e 6669 6e69 7465 4265 7474  2 : infiniteBett
│ │ │ │ -00059550: 694e 756d 6265 7273 2028 4d46 2c37 2920  iNumbers (MF,7) 
│ │ │ │ -00059560: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059530: 2b0a 7c69 3132 203a 2069 6e66 696e 6974  +.|i12 : infinit
│ │ │ │ +00059540: 6542 6574 7469 4e75 6d62 6572 7320 284d  eBettiNumbers (M
│ │ │ │ +00059550: 462c 3729 2020 2020 2020 7c0a 7c20 2020  F,7)      |.|   
│ │ │ │ +00059560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00059590: 0a7c 6f31 3220 3d20 7b33 2c20 342c 2035  .|o12 = {3, 4, 5
│ │ │ │ -000595a0: 2c20 362c 2037 2c20 382c 2039 2c20 3130  , 6, 7, 8, 9, 10
│ │ │ │ -000595b0: 7d20 2020 2020 2020 207c 0a7c 2020 2020  }        |.|    
│ │ │ │ +00059580: 2020 2020 7c0a 7c6f 3132 203d 207b 332c      |.|o12 = {3,
│ │ │ │ +00059590: 2034 2c20 352c 2036 2c20 372c 2038 2c20   4, 5, 6, 7, 8, 
│ │ │ │ +000595a0: 392c 2031 307d 2020 2020 2020 2020 7c0a  9, 10}        |.
│ │ │ │ +000595b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000595c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000595d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000595e0: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374     |.|o12 : List
│ │ │ │ +000595d0: 2020 2020 2020 2020 7c0a 7c6f 3132 203a          |.|o12 :
│ │ │ │ +000595e0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  000595f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059600: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00059600: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00059610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059630: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -00059640: 6265 7474 6920 7265 7320 7075 7368 466f  betti res pushFo
│ │ │ │ -00059650: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -00059660: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +00059620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00059630: 3133 203a 2062 6574 7469 2072 6573 2070  13 : betti res p
│ │ │ │ +00059640: 7573 6846 6f72 7761 7264 286d 6170 2852  ushForward(map(R
│ │ │ │ +00059650: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020  ,S),M)|.|       
│ │ │ │ +00059660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059680: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00059690: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ -000596a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000596b0: 2020 2020 207c 0a7c 6f31 3320 3d20 746f       |.|o13 = to
│ │ │ │ -000596c0: 7461 6c3a 2033 2035 2032 2020 2020 2020  tal: 3 5 2      
│ │ │ │ -000596d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000596e0: 0a7c 2020 2020 2020 2020 2020 323a 2033  .|          2: 3
│ │ │ │ -000596f0: 2034 202e 2020 2020 2020 2020 2020 2020   4 .            
│ │ │ │ -00059700: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00059710: 2020 2020 2020 333a 202e 2031 2032 2020        3: . 1 2  
│ │ │ │ -00059720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059730: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00059680: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00059690: 3020 3120 3220 2020 2020 2020 2020 2020  0 1 2           
│ │ │ │ +000596a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ +000596b0: 203d 2074 6f74 616c 3a20 3320 3520 3220   = total: 3 5 2 
│ │ │ │ +000596c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000596d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000596e0: 2032 3a20 3320 3420 2e20 2020 2020 2020   2: 3 4 .       
│ │ │ │ +000596f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00059700: 7c20 2020 2020 2020 2020 2033 3a20 2e20  |          3: . 
│ │ │ │ +00059710: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │ +00059720: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00059730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00059760: 6f31 3320 3a20 4265 7474 6954 616c 6c79  o13 : BettiTally
│ │ │ │ -00059770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059780: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00059750: 2020 7c0a 7c6f 3133 203a 2042 6574 7469    |.|o13 : Betti
│ │ │ │ +00059760: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +00059770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00059780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000597a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000597b0: 2d2b 0a7c 6931 3420 3a20 6669 6e69 7465  -+.|i14 : finite
│ │ │ │ -000597c0: 4265 7474 694e 756d 6265 7273 204d 4620  BettiNumbers MF 
│ │ │ │ -000597d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000597a0: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2066  ------+.|i14 : f
│ │ │ │ +000597b0: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +000597c0: 7320 4d46 2020 2020 2020 2020 2020 2020  s MF            
│ │ │ │ +000597d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000597e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000597f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059800: 2020 2020 207c 0a7c 6f31 3420 3d20 7b33       |.|o14 = {3
│ │ │ │ -00059810: 2c20 352c 2032 7d20 2020 2020 2020 2020  , 5, 2}         
│ │ │ │ -00059820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00059830: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00059840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059850: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -00059860: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ -00059870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059880: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000597f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ +00059800: 203d 207b 332c 2035 2c20 327d 2020 2020   = {3, 5, 2}    
│ │ │ │ +00059810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059820: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00059830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00059850: 7c6f 3134 203a 204c 6973 7420 2020 2020  |o14 : List     
│ │ │ │ +00059860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059870: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00059880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000598a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -000598b0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -000598c0: 3d0a 0a20 202a 202a 6e6f 7465 2066 696e  =..  * *note fin
│ │ │ │ -000598d0: 6974 6542 6574 7469 4e75 6d62 6572 733a  iteBettiNumbers:
│ │ │ │ -000598e0: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ -000598f0: 6572 732c 202d 2d20 6265 7474 6920 6e75  ers, -- betti nu
│ │ │ │ -00059900: 6d62 6572 7320 6f66 2066 696e 6974 650a  mbers of finite.
│ │ │ │ -00059910: 2020 2020 7265 736f 6c75 7469 6f6e 2063      resolution c
│ │ │ │ -00059920: 6f6d 7075 7465 6420 6672 6f6d 2061 206d  omputed from a m
│ │ │ │ -00059930: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ -00059940: 696f 6e0a 2020 2a20 2a6e 6f74 6520 696e  ion.  * *note in
│ │ │ │ -00059950: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -00059960: 7273 3a20 696e 6669 6e69 7465 4265 7474  rs: infiniteBett
│ │ │ │ -00059970: 694e 756d 6265 7273 2c20 2d2d 2062 6574  iNumbers, -- bet
│ │ │ │ -00059980: 7469 206e 756d 6265 7273 206f 660a 2020  ti numbers of.  
│ │ │ │ -00059990: 2020 6669 6e69 7465 2072 6573 6f6c 7574    finite resolut
│ │ │ │ -000599a0: 696f 6e20 636f 6d70 7574 6564 2066 726f  ion computed fro
│ │ │ │ -000599b0: 6d20 6120 6d61 7472 6978 2066 6163 746f  m a matrix facto
│ │ │ │ -000599c0: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f  rization.  * *no
│ │ │ │ -000599d0: 7465 2068 6967 6853 797a 7967 793a 2068  te highSyzygy: h
│ │ │ │ -000599e0: 6967 6853 797a 7967 792c 202d 2d20 5265  ighSyzygy, -- Re
│ │ │ │ -000599f0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ -00059a00: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ -00059a10: 2074 6865 0a20 2020 2072 6567 756c 6172   the.    regular
│ │ │ │ -00059a20: 6974 7920 6f66 2045 7874 284d 2c6b 290a  ity of Ext(M,k).
│ │ │ │ -00059a30: 2020 2a20 2a6e 6f74 6520 624d 6170 733a    * *note bMaps:
│ │ │ │ -00059a40: 2062 4d61 7073 2c20 2d2d 206c 6973 7420   bMaps, -- list 
│ │ │ │ -00059a50: 7468 6520 6d61 7073 2020 645f 703a 425f  the maps  d_p:B_
│ │ │ │ -00059a60: 3128 7029 2d2d 3e42 5f30 2870 2920 696e  1(p)-->B_0(p) in
│ │ │ │ -00059a70: 2061 0a20 2020 206d 6174 7269 7846 6163   a.    matrixFac
│ │ │ │ -00059a80: 746f 7269 7a61 7469 6f6e 0a20 202a 202a  torization.  * *
│ │ │ │ -00059a90: 6e6f 7465 2042 5261 6e6b 733a 2042 5261  note BRanks: BRa
│ │ │ │ -00059aa0: 6e6b 732c 202d 2d20 7261 6e6b 7320 6f66  nks, -- ranks of
│ │ │ │ -00059ab0: 2074 6865 206d 6f64 756c 6573 2042 5f69   the modules B_i
│ │ │ │ -00059ac0: 2864 2920 696e 2061 0a20 2020 206d 6174  (d) in a.    mat
│ │ │ │ -00059ad0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00059ae0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61  ..Ways to use ma
│ │ │ │ -00059af0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -00059b00: 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  n:.=============
│ │ │ │ -00059b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00059b20: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978  ===..  * "matrix
│ │ │ │ -00059b30: 4661 6374 6f72 697a 6174 696f 6e28 4d61  Factorization(Ma
│ │ │ │ -00059b40: 7472 6978 2c4d 6f64 756c 6529 220a 0a46  trix,Module)"..F
│ │ │ │ -00059b50: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00059b60: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00059b70: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00059b80: 202a 6e6f 7465 206d 6174 7269 7846 6163   *note matrixFac
│ │ │ │ -00059b90: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472  torization: matr
│ │ │ │ -00059ba0: 6978 4661 6374 6f72 697a 6174 696f 6e2c  ixFactorization,
│ │ │ │ -00059bb0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -00059bc0: 6f64 0a66 756e 6374 696f 6e20 7769 7468  od.function with
│ │ │ │ -00059bd0: 206f 7074 696f 6e73 3a20 284d 6163 6175   options: (Macau
│ │ │ │ -00059be0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -00059bf0: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e  nctionWithOption
│ │ │ │ -00059c00: 732c 2e0a 1f0a 4669 6c65 3a20 436f 6d70  s,....File: Comp
│ │ │ │ -00059c10: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -00059c20: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ -00059c30: 2c20 4e6f 6465 3a20 6d66 426f 756e 642c  , Node: mfBound,
│ │ │ │ -00059c40: 204e 6578 743a 206d 6f64 756c 6541 7345   Next: moduleAsE
│ │ │ │ -00059c50: 7874 2c20 5072 6576 3a20 6d61 7472 6978  xt, Prev: matrix
│ │ │ │ -00059c60: 4661 6374 6f72 697a 6174 696f 6e2c 2055  Factorization, U
│ │ │ │ -00059c70: 703a 2054 6f70 0a0a 6d66 426f 756e 6420  p: Top..mfBound 
│ │ │ │ -00059c80: 2d2d 2064 6574 6572 6d69 6e65 7320 686f  -- determines ho
│ │ │ │ -00059c90: 7720 6869 6768 2061 2073 797a 7967 7920  w high a syzygy 
│ │ │ │ -00059ca0: 746f 2074 616b 6520 666f 7220 226d 6174  to take for "mat
│ │ │ │ -00059cb0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00059cc0: 220a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ".**************
│ │ │ │ +000598a0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +000598b0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +000598c0: 6520 6669 6e69 7465 4265 7474 694e 756d  e finiteBettiNum
│ │ │ │ +000598d0: 6265 7273 3a20 6669 6e69 7465 4265 7474  bers: finiteBett
│ │ │ │ +000598e0: 694e 756d 6265 7273 2c20 2d2d 2062 6574  iNumbers, -- bet
│ │ │ │ +000598f0: 7469 206e 756d 6265 7273 206f 6620 6669  ti numbers of fi
│ │ │ │ +00059900: 6e69 7465 0a20 2020 2072 6573 6f6c 7574  nite.    resolut
│ │ │ │ +00059910: 696f 6e20 636f 6d70 7574 6564 2066 726f  ion computed fro
│ │ │ │ +00059920: 6d20 6120 6d61 7472 6978 2066 6163 746f  m a matrix facto
│ │ │ │ +00059930: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f  rization.  * *no
│ │ │ │ +00059940: 7465 2069 6e66 696e 6974 6542 6574 7469  te infiniteBetti
│ │ │ │ +00059950: 4e75 6d62 6572 733a 2069 6e66 696e 6974  Numbers: infinit
│ │ │ │ +00059960: 6542 6574 7469 4e75 6d62 6572 732c 202d  eBettiNumbers, -
│ │ │ │ +00059970: 2d20 6265 7474 6920 6e75 6d62 6572 7320  - betti numbers 
│ │ │ │ +00059980: 6f66 0a20 2020 2066 696e 6974 6520 7265  of.    finite re
│ │ │ │ +00059990: 736f 6c75 7469 6f6e 2063 6f6d 7075 7465  solution compute
│ │ │ │ +000599a0: 6420 6672 6f6d 2061 206d 6174 7269 7820  d from a matrix 
│ │ │ │ +000599b0: 6661 6374 6f72 697a 6174 696f 6e0a 2020  factorization.  
│ │ │ │ +000599c0: 2a20 2a6e 6f74 6520 6869 6768 5379 7a79  * *note highSyzy
│ │ │ │ +000599d0: 6779 3a20 6869 6768 5379 7a79 6779 2c20  gy: highSyzygy, 
│ │ │ │ +000599e0: 2d2d 2052 6574 7572 6e73 2061 2073 797a  -- Returns a syz
│ │ │ │ +000599f0: 7967 7920 6d6f 6475 6c65 206f 6e65 2062  ygy module one b
│ │ │ │ +00059a00: 6579 6f6e 6420 7468 650a 2020 2020 7265  eyond the.    re
│ │ │ │ +00059a10: 6775 6c61 7269 7479 206f 6620 4578 7428  gularity of Ext(
│ │ │ │ +00059a20: 4d2c 6b29 0a20 202a 202a 6e6f 7465 2062  M,k).  * *note b
│ │ │ │ +00059a30: 4d61 7073 3a20 624d 6170 732c 202d 2d20  Maps: bMaps, -- 
│ │ │ │ +00059a40: 6c69 7374 2074 6865 206d 6170 7320 2064  list the maps  d
│ │ │ │ +00059a50: 5f70 3a42 5f31 2870 292d 2d3e 425f 3028  _p:B_1(p)-->B_0(
│ │ │ │ +00059a60: 7029 2069 6e20 610a 2020 2020 6d61 7472  p) in a.    matr
│ │ │ │ +00059a70: 6978 4661 6374 6f72 697a 6174 696f 6e0a  ixFactorization.
│ │ │ │ +00059a80: 2020 2a20 2a6e 6f74 6520 4252 616e 6b73    * *note BRanks
│ │ │ │ +00059a90: 3a20 4252 616e 6b73 2c20 2d2d 2072 616e  : BRanks, -- ran
│ │ │ │ +00059aa0: 6b73 206f 6620 7468 6520 6d6f 6475 6c65  ks of the module
│ │ │ │ +00059ab0: 7320 425f 6928 6429 2069 6e20 610a 2020  s B_i(d) in a.  
│ │ │ │ +00059ac0: 2020 6d61 7472 6978 4661 6374 6f72 697a    matrixFactoriz
│ │ │ │ +00059ad0: 6174 696f 6e0a 0a57 6179 7320 746f 2075  ation..Ways to u
│ │ │ │ +00059ae0: 7365 206d 6174 7269 7846 6163 746f 7269  se matrixFactori
│ │ │ │ +00059af0: 7a61 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d  zation:.========
│ │ │ │ +00059b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00059b10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ +00059b20: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +00059b30: 6f6e 284d 6174 7269 782c 4d6f 6475 6c65  on(Matrix,Module
│ │ │ │ +00059b40: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00059b50: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00059b60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00059b70: 626a 6563 7420 2a6e 6f74 6520 6d61 7472  bject *note matr
│ │ │ │ +00059b80: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ +00059b90: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +00059ba0: 7469 6f6e 2c20 6973 2061 202a 6e6f 7465  tion, is a *note
│ │ │ │ +00059bb0: 206d 6574 686f 640a 6675 6e63 7469 6f6e   method.function
│ │ │ │ +00059bc0: 2077 6974 6820 6f70 7469 6f6e 733a 2028   with options: (
│ │ │ │ +00059bd0: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +00059be0: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
│ │ │ │ +00059bf0: 7074 696f 6e73 2c2e 0a1f 0a46 696c 653a  ptions,....File:
│ │ │ │ +00059c00: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +00059c10: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +00059c20: 2e69 6e66 6f2c 204e 6f64 653a 206d 6642  .info, Node: mfB
│ │ │ │ +00059c30: 6f75 6e64 2c20 4e65 7874 3a20 6d6f 6475  ound, Next: modu
│ │ │ │ +00059c40: 6c65 4173 4578 742c 2050 7265 763a 206d  leAsExt, Prev: m
│ │ │ │ +00059c50: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +00059c60: 6f6e 2c20 5570 3a20 546f 700a 0a6d 6642  on, Up: Top..mfB
│ │ │ │ +00059c70: 6f75 6e64 202d 2d20 6465 7465 726d 696e  ound -- determin
│ │ │ │ +00059c80: 6573 2068 6f77 2068 6967 6820 6120 7379  es how high a sy
│ │ │ │ +00059c90: 7a79 6779 2074 6f20 7461 6b65 2066 6f72  zygy to take for
│ │ │ │ +00059ca0: 2022 6d61 7472 6978 4661 6374 6f72 697a   "matrixFactoriz
│ │ │ │ +00059cb0: 6174 696f 6e22 0a2a 2a2a 2a2a 2a2a 2a2a  ation".*********
│ │ │ │ +00059cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00059cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00059ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00059cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00059d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ -00059d10: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ -00059d20: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00059d30: 2020 2020 7020 3d20 6d66 426f 756e 6420      p = mfBound 
│ │ │ │ -00059d40: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020  M.  * Inputs:.  
│ │ │ │ -00059d50: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ -00059d60: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ -00059d70: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ -00059d80: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ -00059d90: 696e 7465 7273 6563 7469 6f6e 0a20 202a  intersection.  *
│ │ │ │ -00059da0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -00059db0: 2a20 702c 2061 6e20 2a6e 6f74 6520 696e  * p, an *note in
│ │ │ │ -00059dc0: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ -00059dd0: 3244 6f63 295a 5a2c 2c20 0a0a 4465 7363  2Doc)ZZ,, ..Desc
│ │ │ │ -00059de0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -00059df0: 3d3d 3d0a 0a49 6620 7020 3d20 6d66 426f  ===..If p = mfBo
│ │ │ │ -00059e00: 756e 6420 4d2c 2074 6865 6e20 7468 6520  und M, then the 
│ │ │ │ -00059e10: 702d 7468 2073 797a 7967 7920 6f66 204d  p-th syzygy of M
│ │ │ │ -00059e20: 2c20 7768 6963 6820 6973 2063 6f6d 7075  , which is compu
│ │ │ │ -00059e30: 7465 6420 6279 0a68 6967 6853 797a 7967  ted by.highSyzyg
│ │ │ │ -00059e40: 7928 4d29 2c20 7368 6f75 6c64 2028 7468  y(M), should (th
│ │ │ │ -00059e50: 6973 2069 7320 6120 636f 6e6a 6563 7475  is is a conjectu
│ │ │ │ -00059e60: 7265 2920 6265 2061 2022 6869 6768 2053  re) be a "high S
│ │ │ │ -00059e70: 797a 7967 7922 2069 6e20 7468 6520 7365  yzygy" in the se
│ │ │ │ -00059e80: 6e73 650a 7265 7175 6972 6564 2066 6f72  nse.required for
│ │ │ │ -00059e90: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -00059ea0: 7469 6f6e 2e20 496e 2065 7861 6d70 6c65  tion. In example
│ │ │ │ -00059eb0: 732c 2074 6865 2065 7374 696d 6174 6520  s, the estimate 
│ │ │ │ -00059ec0: 7365 656d 7320 7368 6172 7020 2865 7863  seems sharp (exc
│ │ │ │ -00059ed0: 6570 740a 7768 656e 204d 2069 7320 616c  ept.when M is al
│ │ │ │ -00059ee0: 7265 6164 7920 6120 6869 6768 2073 797a  ready a high syz
│ │ │ │ -00059ef0: 7967 7929 2e0a 0a54 6865 2061 6374 7561  ygy)...The actua
│ │ │ │ -00059f00: 6c20 666f 726d 756c 6120 7573 6564 2069  l formula used i
│ │ │ │ -00059f10: 733a 0a0a 6d66 426f 756e 6420 4d20 3d20  s:..mfBound M = 
│ │ │ │ -00059f20: 6d61 7828 322a 725f 7b65 7665 6e7d 2c20  max(2*r_{even}, 
│ │ │ │ -00059f30: 312b 322a 725f 7b6f 6464 7d29 0a0a 7768  1+2*r_{odd})..wh
│ │ │ │ -00059f40: 6572 6520 725f 7b65 7665 6e7d 203d 2072  ere r_{even} = r
│ │ │ │ -00059f50: 6567 756c 6172 6974 7920 6576 656e 4578  egularity evenEx
│ │ │ │ -00059f60: 744d 6f64 756c 6520 4d20 616e 6420 725f  tModule M and r_
│ │ │ │ -00059f70: 7b6f 6464 7d20 3d20 7265 6775 6c61 7269  {odd} = regulari
│ │ │ │ -00059f80: 7479 0a6f 6464 4578 744d 6f64 756c 6520  ty.oddExtModule 
│ │ │ │ -00059f90: 4d2e 2048 6572 6520 6576 656e 4578 744d  M. Here evenExtM
│ │ │ │ -00059fa0: 6f64 756c 6520 4d20 6973 2074 6865 2065  odule M is the e
│ │ │ │ -00059fb0: 7665 6e20 6465 6772 6565 2070 6172 7420  ven degree part 
│ │ │ │ -00059fc0: 6f66 2045 7874 284d 2c20 2872 6573 6964  of Ext(M, (resid
│ │ │ │ -00059fd0: 7565 0a63 6c61 7373 2066 6965 6c64 2929  ue.class field))
│ │ │ │ -00059fe0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ -00059ff0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -0005a000: 6869 6768 5379 7a79 6779 3a20 6869 6768  highSyzygy: high
│ │ │ │ -0005a010: 5379 7a79 6779 2c20 2d2d 2052 6574 7572  Syzygy, -- Retur
│ │ │ │ -0005a020: 6e73 2061 2073 797a 7967 7920 6d6f 6475  ns a syzygy modu
│ │ │ │ -0005a030: 6c65 206f 6e65 2062 6579 6f6e 6420 7468  le one beyond th
│ │ │ │ -0005a040: 650a 2020 2020 7265 6775 6c61 7269 7479  e.    regularity
│ │ │ │ -0005a050: 206f 6620 4578 7428 4d2c 6b29 0a20 202a   of Ext(M,k).  *
│ │ │ │ -0005a060: 202a 6e6f 7465 2065 7665 6e45 7874 4d6f   *note evenExtMo
│ │ │ │ -0005a070: 6475 6c65 3a20 6576 656e 4578 744d 6f64  dule: evenExtMod
│ │ │ │ -0005a080: 756c 652c 202d 2d20 6576 656e 2070 6172  ule, -- even par
│ │ │ │ -0005a090: 7420 6f66 2045 7874 5e2a 284d 2c6b 2920  t of Ext^*(M,k) 
│ │ │ │ -0005a0a0: 6f76 6572 2061 0a20 2020 2063 6f6d 706c  over a.    compl
│ │ │ │ -0005a0b0: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ -0005a0c0: 2061 7320 6d6f 6475 6c65 206f 7665 7220   as module over 
│ │ │ │ -0005a0d0: 4349 206f 7065 7261 746f 7220 7269 6e67  CI operator ring
│ │ │ │ -0005a0e0: 0a20 202a 202a 6e6f 7465 206f 6464 4578  .  * *note oddEx
│ │ │ │ -0005a0f0: 744d 6f64 756c 653a 206f 6464 4578 744d  tModule: oddExtM
│ │ │ │ -0005a100: 6f64 756c 652c 202d 2d20 6f64 6420 7061  odule, -- odd pa
│ │ │ │ -0005a110: 7274 206f 6620 4578 745e 2a28 4d2c 6b29  rt of Ext^*(M,k)
│ │ │ │ -0005a120: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ -0005a130: 0a20 2020 2069 6e74 6572 7365 6374 696f  .    intersectio
│ │ │ │ -0005a140: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ -0005a150: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ -0005a160: 670a 2020 2a20 2a6e 6f74 6520 6d61 7472  g.  * *note matr
│ │ │ │ -0005a170: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ -0005a180: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0005a190: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ -0005a1a0: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ -0005a1b0: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ -0005a1c0: 2066 6163 746f 7269 7a61 7469 6f6e 0a0a   factorization..
│ │ │ │ -0005a1d0: 5761 7973 2074 6f20 7573 6520 6d66 426f  Ways to use mfBo
│ │ │ │ -0005a1e0: 756e 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  und:.===========
│ │ │ │ -0005a1f0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -0005a200: 6d66 426f 756e 6428 4d6f 6475 6c65 2922  mfBound(Module)"
│ │ │ │ -0005a210: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0005a220: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0005a230: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0005a240: 6563 7420 2a6e 6f74 6520 6d66 426f 756e  ect *note mfBoun
│ │ │ │ -0005a250: 643a 206d 6642 6f75 6e64 2c20 6973 2061  d: mfBound, is a
│ │ │ │ -0005a260: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ -0005a270: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61  nction:.(Macaula
│ │ │ │ -0005a280: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -0005a290: 7469 6f6e 2c2e 0a1f 0a46 696c 653a 2043  tion,....File: C
│ │ │ │ -0005a2a0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0005a2b0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0005a2c0: 6e66 6f2c 204e 6f64 653a 206d 6f64 756c  nfo, Node: modul
│ │ │ │ -0005a2d0: 6541 7345 7874 2c20 4e65 7874 3a20 6e65  eAsExt, Next: ne
│ │ │ │ -0005a2e0: 7745 7874 2c20 5072 6576 3a20 6d66 426f  wExt, Prev: mfBo
│ │ │ │ -0005a2f0: 756e 642c 2055 703a 2054 6f70 0a0a 6d6f  und, Up: Top..mo
│ │ │ │ -0005a300: 6475 6c65 4173 4578 7420 2d2d 2046 696e  duleAsExt -- Fin
│ │ │ │ -0005a310: 6420 6120 6d6f 6475 6c65 2077 6974 6820  d a module with 
│ │ │ │ -0005a320: 6769 7665 6e20 6173 796d 7074 6f74 6963  given asymptotic
│ │ │ │ -0005a330: 2072 6573 6f6c 7574 696f 6e0a 2a2a 2a2a   resolution.****
│ │ │ │ +00059d00: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ +00059d10: 3d3d 3d0a 0a20 202a 2055 7361 6765 3a20  ===..  * Usage: 
│ │ │ │ +00059d20: 0a20 2020 2020 2020 2070 203d 206d 6642  .        p = mfB
│ │ │ │ +00059d30: 6f75 6e64 204d 0a20 202a 2049 6e70 7574  ound M.  * Input
│ │ │ │ +00059d40: 733a 0a20 2020 2020 202a 204d 2c20 6120  s:.      * M, a 
│ │ │ │ +00059d50: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +00059d60: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +00059d70: 6c65 2c2c 206f 7665 7220 6120 636f 6d70  le,, over a comp
│ │ │ │ +00059d80: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +00059d90: 6e0a 2020 2a20 4f75 7470 7574 733a 0a20  n.  * Outputs:. 
│ │ │ │ +00059da0: 2020 2020 202a 2070 2c20 616e 202a 6e6f       * p, an *no
│ │ │ │ +00059db0: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00059dc0: 6175 6c61 7932 446f 6329 5a5a 2c2c 200a  aulay2Doc)ZZ,, .
│ │ │ │ +00059dd0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00059de0: 3d3d 3d3d 3d3d 3d3d 0a0a 4966 2070 203d  ========..If p =
│ │ │ │ +00059df0: 206d 6642 6f75 6e64 204d 2c20 7468 656e   mfBound M, then
│ │ │ │ +00059e00: 2074 6865 2070 2d74 6820 7379 7a79 6779   the p-th syzygy
│ │ │ │ +00059e10: 206f 6620 4d2c 2077 6869 6368 2069 7320   of M, which is 
│ │ │ │ +00059e20: 636f 6d70 7574 6564 2062 790a 6869 6768  computed by.high
│ │ │ │ +00059e30: 5379 7a79 6779 284d 292c 2073 686f 756c  Syzygy(M), shoul
│ │ │ │ +00059e40: 6420 2874 6869 7320 6973 2061 2063 6f6e  d (this is a con
│ │ │ │ +00059e50: 6a65 6374 7572 6529 2062 6520 6120 2268  jecture) be a "h
│ │ │ │ +00059e60: 6967 6820 5379 7a79 6779 2220 696e 2074  igh Syzygy" in t
│ │ │ │ +00059e70: 6865 2073 656e 7365 0a72 6571 7569 7265  he sense.require
│ │ │ │ +00059e80: 6420 666f 7220 6d61 7472 6978 4661 6374  d for matrixFact
│ │ │ │ +00059e90: 6f72 697a 6174 696f 6e2e 2049 6e20 6578  orization. In ex
│ │ │ │ +00059ea0: 616d 706c 6573 2c20 7468 6520 6573 7469  amples, the esti
│ │ │ │ +00059eb0: 6d61 7465 2073 6565 6d73 2073 6861 7270  mate seems sharp
│ │ │ │ +00059ec0: 2028 6578 6365 7074 0a77 6865 6e20 4d20   (except.when M 
│ │ │ │ +00059ed0: 6973 2061 6c72 6561 6479 2061 2068 6967  is already a hig
│ │ │ │ +00059ee0: 6820 7379 7a79 6779 292e 0a0a 5468 6520  h syzygy)...The 
│ │ │ │ +00059ef0: 6163 7475 616c 2066 6f72 6d75 6c61 2075  actual formula u
│ │ │ │ +00059f00: 7365 6420 6973 3a0a 0a6d 6642 6f75 6e64  sed is:..mfBound
│ │ │ │ +00059f10: 204d 203d 206d 6178 2832 2a72 5f7b 6576   M = max(2*r_{ev
│ │ │ │ +00059f20: 656e 7d2c 2031 2b32 2a72 5f7b 6f64 647d  en}, 1+2*r_{odd}
│ │ │ │ +00059f30: 290a 0a77 6865 7265 2072 5f7b 6576 656e  )..where r_{even
│ │ │ │ +00059f40: 7d20 3d20 7265 6775 6c61 7269 7479 2065  } = regularity e
│ │ │ │ +00059f50: 7665 6e45 7874 4d6f 6475 6c65 204d 2061  venExtModule M a
│ │ │ │ +00059f60: 6e64 2072 5f7b 6f64 647d 203d 2072 6567  nd r_{odd} = reg
│ │ │ │ +00059f70: 756c 6172 6974 790a 6f64 6445 7874 4d6f  ularity.oddExtMo
│ │ │ │ +00059f80: 6475 6c65 204d 2e20 4865 7265 2065 7665  dule M. Here eve
│ │ │ │ +00059f90: 6e45 7874 4d6f 6475 6c65 204d 2069 7320  nExtModule M is 
│ │ │ │ +00059fa0: 7468 6520 6576 656e 2064 6567 7265 6520  the even degree 
│ │ │ │ +00059fb0: 7061 7274 206f 6620 4578 7428 4d2c 2028  part of Ext(M, (
│ │ │ │ +00059fc0: 7265 7369 6475 650a 636c 6173 7320 6669  residue.class fi
│ │ │ │ +00059fd0: 656c 6429 292e 0a0a 5365 6520 616c 736f  eld))...See also
│ │ │ │ +00059fe0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00059ff0: 6e6f 7465 2068 6967 6853 797a 7967 793a  note highSyzygy:
│ │ │ │ +0005a000: 2068 6967 6853 797a 7967 792c 202d 2d20   highSyzygy, -- 
│ │ │ │ +0005a010: 5265 7475 726e 7320 6120 7379 7a79 6779  Returns a syzygy
│ │ │ │ +0005a020: 206d 6f64 756c 6520 6f6e 6520 6265 796f   module one beyo
│ │ │ │ +0005a030: 6e64 2074 6865 0a20 2020 2072 6567 756c  nd the.    regul
│ │ │ │ +0005a040: 6172 6974 7920 6f66 2045 7874 284d 2c6b  arity of Ext(M,k
│ │ │ │ +0005a050: 290a 2020 2a20 2a6e 6f74 6520 6576 656e  ).  * *note even
│ │ │ │ +0005a060: 4578 744d 6f64 756c 653a 2065 7665 6e45  ExtModule: evenE
│ │ │ │ +0005a070: 7874 4d6f 6475 6c65 2c20 2d2d 2065 7665  xtModule, -- eve
│ │ │ │ +0005a080: 6e20 7061 7274 206f 6620 4578 745e 2a28  n part of Ext^*(
│ │ │ │ +0005a090: 4d2c 6b29 206f 7665 7220 610a 2020 2020  M,k) over a.    
│ │ │ │ +0005a0a0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +0005a0b0: 6374 696f 6e20 6173 206d 6f64 756c 6520  ction as module 
│ │ │ │ +0005a0c0: 6f76 6572 2043 4920 6f70 6572 6174 6f72  over CI operator
│ │ │ │ +0005a0d0: 2072 696e 670a 2020 2a20 2a6e 6f74 6520   ring.  * *note 
│ │ │ │ +0005a0e0: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64  oddExtModule: od
│ │ │ │ +0005a0f0: 6445 7874 4d6f 6475 6c65 2c20 2d2d 206f  dExtModule, -- o
│ │ │ │ +0005a100: 6464 2070 6172 7420 6f66 2045 7874 5e2a  dd part of Ext^*
│ │ │ │ +0005a110: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d  (M,k) over a com
│ │ │ │ +0005a120: 706c 6574 650a 2020 2020 696e 7465 7273  plete.    inters
│ │ │ │ +0005a130: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65  ection as module
│ │ │ │ +0005a140: 206f 7665 7220 4349 206f 7065 7261 746f   over CI operato
│ │ │ │ +0005a150: 7220 7269 6e67 0a20 202a 202a 6e6f 7465  r ring.  * *note
│ │ │ │ +0005a160: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0005a170: 7469 6f6e 3a20 6d61 7472 6978 4661 6374  tion: matrixFact
│ │ │ │ +0005a180: 6f72 697a 6174 696f 6e2c 202d 2d20 4d61  orization, -- Ma
│ │ │ │ +0005a190: 7073 2069 6e20 6120 6869 6768 6572 0a20  ps in a higher. 
│ │ │ │ +0005a1a0: 2020 2063 6f64 696d 656e 7369 6f6e 206d     codimension m
│ │ │ │ +0005a1b0: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ +0005a1c0: 696f 6e0a 0a57 6179 7320 746f 2075 7365  ion..Ways to use
│ │ │ │ +0005a1d0: 206d 6642 6f75 6e64 3a0a 3d3d 3d3d 3d3d   mfBound:.======
│ │ │ │ +0005a1e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0005a1f0: 2020 2a20 226d 6642 6f75 6e64 284d 6f64    * "mfBound(Mod
│ │ │ │ +0005a200: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ +0005a210: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0005a220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0005a230: 6520 6f62 6a65 6374 202a 6e6f 7465 206d  e object *note m
│ │ │ │ +0005a240: 6642 6f75 6e64 3a20 6d66 426f 756e 642c  fBound: mfBound,
│ │ │ │ +0005a250: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +0005a260: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ +0005a270: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +0005a280: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ +0005a290: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0005a2a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0005a2b0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0005a2c0: 6d6f 6475 6c65 4173 4578 742c 204e 6578  moduleAsExt, Nex
│ │ │ │ +0005a2d0: 743a 206e 6577 4578 742c 2050 7265 763a  t: newExt, Prev:
│ │ │ │ +0005a2e0: 206d 6642 6f75 6e64 2c20 5570 3a20 546f   mfBound, Up: To
│ │ │ │ +0005a2f0: 700a 0a6d 6f64 756c 6541 7345 7874 202d  p..moduleAsExt -
│ │ │ │ +0005a300: 2d20 4669 6e64 2061 206d 6f64 756c 6520  - Find a module 
│ │ │ │ +0005a310: 7769 7468 2067 6976 656e 2061 7379 6d70  with given asymp
│ │ │ │ +0005a320: 746f 7469 6320 7265 736f 6c75 7469 6f6e  totic resolution
│ │ │ │ +0005a330: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  0005a340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005a350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005a360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005a370: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ -0005a380: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ -0005a390: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0005a3a0: 2020 4d20 3d20 6d6f 6475 6c65 4173 4578    M = moduleAsEx
│ │ │ │ -0005a3b0: 7428 4d4d 2c52 290a 2020 2a20 496e 7075  t(MM,R).  * Inpu
│ │ │ │ -0005a3c0: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061  ts:.      * M, a
│ │ │ │ -0005a3d0: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ -0005a3e0: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ -0005a3f0: 756c 652c 2c20 6d6f 6475 6c65 206f 7665  ule,, module ove
│ │ │ │ -0005a400: 7220 706f 6c79 6e6f 6d69 616c 2072 696e  r polynomial rin
│ │ │ │ -0005a410: 670a 2020 2020 2020 2020 7769 7468 2063  g.        with c
│ │ │ │ -0005a420: 2076 6172 6961 626c 6573 0a20 2020 2020   variables.     
│ │ │ │ -0005a430: 202a 2052 2c20 6120 2a6e 6f74 6520 7269   * R, a *note ri
│ │ │ │ -0005a440: 6e67 3a20 284d 6163 6175 6c61 7932 446f  ng: (Macaulay2Do
│ │ │ │ -0005a450: 6329 5269 6e67 2c2c 2028 6772 6164 6564  c)Ring,, (graded
│ │ │ │ -0005a460: 2920 636f 6d70 6c65 7465 2069 6e74 6572  ) complete inter
│ │ │ │ -0005a470: 7365 6374 696f 6e0a 2020 2020 2020 2020  section.        
│ │ │ │ -0005a480: 7269 6e67 206f 6620 636f 6469 6d65 6e73  ring of codimens
│ │ │ │ -0005a490: 696f 6e20 632c 2065 6d62 6564 6469 6e67  ion c, embedding
│ │ │ │ -0005a4a0: 2064 696d 656e 7369 6f6e 206e 0a20 202a   dimension n.  *
│ │ │ │ -0005a4b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0005a4c0: 2a20 4e2c 2061 202a 6e6f 7465 206d 6f64  * N, a *note mod
│ │ │ │ -0005a4d0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0005a4e0: 6f63 294d 6f64 756c 652c 2c20 6d6f 6475  oc)Module,, modu
│ │ │ │ -0005a4f0: 6c65 206f 7665 7220 5220 7375 6368 2074  le over R such t
│ │ │ │ -0005a500: 6861 740a 2020 2020 2020 2020 4578 745f  hat.        Ext_
│ │ │ │ -0005a510: 5228 4e2c 6b29 203d 204d 5c6f 7469 6d65  R(N,k) = M\otime
│ │ │ │ -0005a520: 7320 5c77 6564 6765 286b 5e6e 2920 696e  s \wedge(k^n) in
│ │ │ │ -0005a530: 206c 6172 6765 2068 6f6d 6f6c 6f67 6963   large homologic
│ │ │ │ -0005a540: 616c 2064 6567 7265 652e 0a0a 4465 7363  al degree...Desc
│ │ │ │ -0005a550: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0005a560: 3d3d 3d0a 0a54 6865 2072 6f75 7469 6e65  ===..The routine
│ │ │ │ -0005a570: 2060 606d 6f64 756c 6541 7345 7874 2727   ``moduleAsExt''
│ │ │ │ -0005a580: 2069 7320 6120 7061 7274 6961 6c20 696e   is a partial in
│ │ │ │ -0005a590: 7665 7273 6520 746f 2074 6865 2072 6f75  verse to the rou
│ │ │ │ -0005a5a0: 7469 6e65 2045 7874 4d6f 6475 6c65 2c0a  tine ExtModule,.
│ │ │ │ -0005a5b0: 636f 6d70 7574 6564 2066 6f6c 6c6f 7769  computed followi
│ │ │ │ -0005a5c0: 6e67 2069 6465 6173 206f 6620 4176 7261  ng ideas of Avra
│ │ │ │ -0005a5d0: 6d6f 7620 616e 6420 4a6f 7267 656e 7365  mov and Jorgense
│ │ │ │ -0005a5e0: 6e3a 2067 6976 656e 2061 206d 6f64 756c  n: given a modul
│ │ │ │ -0005a5f0: 6520 4520 6f76 6572 2061 0a70 6f6c 796e  e E over a.polyn
│ │ │ │ -0005a600: 6f6d 6961 6c20 7269 6e67 206b 5b78 5f31  omial ring k[x_1
│ │ │ │ -0005a610: 2e2e 785f 635d 2c20 6974 2070 726f 7669  ..x_c], it provi
│ │ │ │ -0005a620: 6465 7320 6120 6d6f 6475 6c65 204e 206f  des a module N o
│ │ │ │ -0005a630: 7665 7220 6120 7370 6563 6966 6965 6420  ver a specified 
│ │ │ │ -0005a640: 706f 6c79 6e6f 6d69 616c 0a72 696e 6720  polynomial.ring 
│ │ │ │ -0005a650: 696e 206e 2076 6172 6961 626c 6573 2073  in n variables s
│ │ │ │ -0005a660: 7563 6820 7468 6174 2045 7874 284e 2c6b  uch that Ext(N,k
│ │ │ │ -0005a670: 2920 6167 7265 6573 2077 6974 6820 2445  ) agrees with $E
│ │ │ │ -0005a680: 273d 455c 6f74 696d 6573 205c 7765 6467  '=E\otimes \wedg
│ │ │ │ -0005a690: 6528 6b5e 6e29 240a 6166 7465 7220 7472  e(k^n)$.after tr
│ │ │ │ -0005a6a0: 756e 6361 7469 6f6e 2e20 4865 7265 2074  uncation. Here t
│ │ │ │ -0005a6b0: 6865 2067 7261 6469 6e67 206f 6e20 4520  he grading on E 
│ │ │ │ -0005a6c0: 6973 2074 616b 656e 2074 6f20 6265 2065  is taken to be e
│ │ │ │ -0005a6d0: 7665 6e2c 2077 6869 6c65 0a24 5c77 6564  ven, while.$\wed
│ │ │ │ -0005a6e0: 6765 286b 5e6e 2924 2068 6173 2067 656e  ge(k^n)$ has gen
│ │ │ │ -0005a6f0: 6572 6174 6f72 7320 696e 2064 6567 7265  erators in degre
│ │ │ │ -0005a700: 6520 312e 2054 6865 2072 6f75 7469 6e65  e 1. The routine
│ │ │ │ -0005a710: 2068 664d 6f64 756c 6541 7345 7874 2063   hfModuleAsExt c
│ │ │ │ -0005a720: 6f6d 7075 7465 730a 7468 6520 7265 7375  omputes.the resu
│ │ │ │ -0005a730: 6c74 696e 6720 6869 6c62 6572 7420 6675  lting hilbert fu
│ │ │ │ -0005a740: 6e63 7469 6f6e 2066 6f72 2045 272e 2054  nction for E'. T
│ │ │ │ -0005a750: 6869 7320 7573 6573 2069 6465 6173 206f  his uses ideas o
│ │ │ │ -0005a760: 6620 4176 7261 6d6f 7620 616e 640a 4a6f  f Avramov and.Jo
│ │ │ │ -0005a770: 7267 656e 7365 6e2e 204e 6f74 6520 7468  rgensen. Note th
│ │ │ │ -0005a780: 6174 2074 6865 206d 6f64 756c 6520 4578  at the module Ex
│ │ │ │ -0005a790: 7428 4e2c 6b29 2028 7472 756e 6361 7465  t(N,k) (truncate
│ │ │ │ -0005a7a0: 6429 2077 696c 6c20 6175 746f 6d61 7469  d) will automati
│ │ │ │ -0005a7b0: 6361 6c6c 7920 6265 2066 7265 650a 6f76  cally be free.ov
│ │ │ │ -0005a7c0: 6572 2074 6865 2065 7874 6572 696f 7220  er the exterior 
│ │ │ │ -0005a7d0: 616c 6765 6272 6120 245c 7765 6467 6528  algebra $\wedge(
│ │ │ │ -0005a7e0: 6b5e 6e29 2420 6765 6e65 7261 7465 6420  k^n)$ generated 
│ │ │ │ -0005a7f0: 6279 2045 7874 5e31 286b 2c6b 293b 206e  by Ext^1(k,k); n
│ │ │ │ -0005a800: 6f74 2061 2074 7970 6963 616c 0a45 7874  ot a typical.Ext
│ │ │ │ -0005a810: 206d 6f64 756c 652e 0a0a 4d6f 7265 2070   module...More p
│ │ │ │ -0005a820: 7265 6369 7365 6c79 3a0a 0a53 7570 706f  recisely:..Suppo
│ │ │ │ -0005a830: 7365 2074 6861 7420 2452 203d 206b 5b61  se that $R = k[a
│ │ │ │ -0005a840: 5f31 2c5c 646f 7473 2c20 615f 6e5d 2f28  _1,\dots, a_n]/(
│ │ │ │ -0005a850: 665f 312c 5c64 6f74 732c 665f 6329 2420  f_1,\dots,f_c)$ 
│ │ │ │ -0005a860: 6c65 7420 244b 4b20 3d0a 6b5b 785f 312c  let $KK =.k[x_1,
│ │ │ │ -0005a870: 5c64 6f74 732c 785f 635d 242c 2061 6e64  \dots,x_c]$, and
│ │ │ │ -0005a880: 206c 6574 2024 5c4c 616d 6264 6120 3d20   let $\Lambda = 
│ │ │ │ -0005a890: 5c77 6564 6765 206b 5e6e 242e 2024 4520  \wedge k^n$. $E 
│ │ │ │ -0005a8a0: 3d20 4b4b 5c6f 7469 6d65 735c 4c61 6d62  = KK\otimes\Lamb
│ │ │ │ -0005a8b0: 6461 242c 2073 6f0a 7468 6174 2074 6865  da$, so.that the
│ │ │ │ -0005a8c0: 206d 696e 696d 616c 2024 5224 2d66 7265   minimal $R$-fre
│ │ │ │ -0005a8d0: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ -0005a8e0: 246b 2420 6861 7320 756e 6465 726c 7969  $k$ has underlyi
│ │ │ │ -0005a8f0: 6e67 206d 6f64 756c 6520 2452 5c6f 7469  ng module $R\oti
│ │ │ │ -0005a900: 6d65 730a 455e 2a24 2c20 7768 6572 6520  mes.E^*$, where 
│ │ │ │ -0005a910: 2445 5e2a 2420 6973 2074 6865 2067 7261  $E^*$ is the gra
│ │ │ │ -0005a920: 6465 6420 7665 6374 6f72 2073 7061 6365  ded vector space
│ │ │ │ -0005a930: 2064 7561 6c20 6f66 2024 4524 2e0a 0a4c   dual of $E$...L
│ │ │ │ -0005a940: 6574 204d 4d20 6265 2074 6865 2072 6573  et MM be the res
│ │ │ │ -0005a950: 756c 7420 6f66 2074 7275 6e63 6174 696e  ult of truncatin
│ │ │ │ -0005a960: 6720 4d20 6174 2069 7473 2072 6567 756c  g M at its regul
│ │ │ │ -0005a970: 6172 6974 7920 616e 6420 7368 6966 7469  arity and shifti
│ │ │ │ -0005a980: 6e67 2069 7420 736f 2074 6861 740a 6974  ng it so that.it
│ │ │ │ -0005a990: 2069 7320 6765 6e65 7261 7465 6420 696e   is generated in
│ │ │ │ -0005a9a0: 2064 6567 7265 6520 302e 204c 6574 2024   degree 0. Let $
│ │ │ │ -0005a9b0: 4624 2062 6520 6120 244b 4b24 2d66 7265  F$ be a $KK$-fre
│ │ │ │ -0005a9c0: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ -0005a9d0: 244d 4d24 2c20 616e 640a 7772 6974 6520  $MM$, and.write 
│ │ │ │ -0005a9e0: 2446 5f69 203d 204b 4b5c 6f74 696d 6573  $F_i = KK\otimes
│ │ │ │ -0005a9f0: 2056 5f69 2e24 2053 696e 6365 206c 696e   V_i.$ Since lin
│ │ │ │ -0005aa00: 6561 7220 666f 726d 7320 6f76 6572 2024  ear forms over $
│ │ │ │ -0005aa10: 4b4b 2420 636f 7272 6573 706f 6e64 2074  KK$ correspond t
│ │ │ │ -0005aa20: 6f20 4349 0a6f 7065 7261 746f 7273 206f  o CI.operators o
│ │ │ │ -0005aa30: 6620 6465 6772 6565 202d 3220 6f6e 2074  f degree -2 on t
│ │ │ │ -0005aa40: 6865 2072 6573 6f6c 7574 696f 6e20 4720  he resolution G 
│ │ │ │ -0005aa50: 6f66 206b 206f 7665 7220 522c 2077 6520  of k over R, we 
│ │ │ │ -0005aa60: 6d61 7920 666f 726d 2061 206d 6170 2024  may form a map $
│ │ │ │ -0005aa70: 240a 645f 312b 645f 323a 205c 7375 6d5f  $.d_1+d_2: \sum_
│ │ │ │ -0005aa80: 7b69 3d30 7d5e 6d20 475f 7b69 2b31 7d5c  {i=0}^m G_{i+1}\
│ │ │ │ -0005aa90: 6f74 696d 6573 2056 5f7b 6d2d 697d 5e2a  otimes V_{m-i}^*
│ │ │ │ -0005aaa0: 205c 746f 205c 7375 6d5f 7b69 3d30 7d5e   \to \sum_{i=0}^
│ │ │ │ -0005aab0: 6d20 475f 695c 6f74 696d 6573 0a56 5f7b  m G_i\otimes.V_{
│ │ │ │ -0005aac0: 6d2d 697d 5e2a 2024 2420 7768 6572 6520  m-i}^* $$ where 
│ │ │ │ -0005aad0: 2464 5f31 2420 6973 2074 6865 2064 6972  $d_1$ is the dir
│ │ │ │ -0005aae0: 6563 7420 7375 6d20 6f66 2074 6865 2064  ect sum of the d
│ │ │ │ -0005aaf0: 6966 6665 7265 6e74 6961 6c73 2024 2847  ifferentials $(G
│ │ │ │ -0005ab00: 5f7b 692b 317d 5c74 6f0a 475f 6929 5c6f  _{i+1}\to.G_i)\o
│ │ │ │ -0005ab10: 7469 6d65 7320 565f 695e 2a24 2061 6e64  times V_i^*$ and
│ │ │ │ -0005ab20: 2024 645f 3224 2069 7320 7468 6520 6469   $d_2$ is the di
│ │ │ │ -0005ab30: 7265 6374 2073 756d 206f 6620 7468 6520  rect sum of the 
│ │ │ │ -0005ab40: 6d61 7073 2024 5c70 6869 5f69 2420 6465  maps $\phi_i$ de
│ │ │ │ -0005ab50: 6669 6e65 640a 6672 6f6d 2074 6865 2064  fined.from the d
│ │ │ │ -0005ab60: 6966 6665 7265 6e74 6961 6c73 206f 6620  ifferentials of 
│ │ │ │ -0005ab70: 2446 2420 6279 2073 7562 7374 6974 7574  $F$ by substitut
│ │ │ │ -0005ab80: 696e 6720 4349 206f 7065 7261 746f 7273  ing CI operators
│ │ │ │ -0005ab90: 2066 6f72 206c 696e 6561 7220 666f 726d   for linear form
│ │ │ │ -0005aba0: 732c 0a24 5c70 6869 5f69 3a20 475f 7b69  s,.$\phi_i: G_{i
│ │ │ │ -0005abb0: 2b31 7d5c 6f74 696d 6573 2056 5f69 205c  +1}\otimes V_i \
│ │ │ │ -0005abc0: 746f 2047 5f7b 692d 317d 5c6f 7469 6d65  to G_{i-1}\otime
│ │ │ │ -0005abd0: 7320 565f 7b69 2d31 7d24 2e20 5468 6520  s V_{i-1}$. The 
│ │ │ │ -0005abe0: 7363 7269 7074 2072 6574 7572 6e73 2074  script returns t
│ │ │ │ -0005abf0: 6865 0a6d 6f64 756c 6520 4e20 7468 6174  he.module N that
│ │ │ │ -0005ac00: 2069 7320 7468 6520 636f 6b65 726e 656c   is the cokernel
│ │ │ │ -0005ac10: 206f 6620 2464 5f31 2b64 5f32 242e 0a0a   of $d_1+d_2$...
│ │ │ │ -0005ac20: 5468 6520 6d6f 6475 6c65 2024 4578 745f  The module $Ext_
│ │ │ │ -0005ac30: 5228 4e2c 6b29 2420 6167 7265 6573 2c20  R(N,k)$ agrees, 
│ │ │ │ -0005ac40: 6166 7465 7220 6120 6665 7720 7374 6570  after a few step
│ │ │ │ -0005ac50: 732c 2077 6974 6820 7468 6520 6d6f 6475  s, with the modu
│ │ │ │ -0005ac60: 6c65 2064 6572 6976 6564 2066 726f 6d0a  le derived from.
│ │ │ │ -0005ac70: 244d 4d24 2062 7920 7465 6e73 6f72 696e  $MM$ by tensorin
│ │ │ │ -0005ac80: 6720 6974 2077 6974 6820 245c 4c61 6d62  g it with $\Lamb
│ │ │ │ -0005ac90: 6461 242c 2074 6861 7420 6973 2c20 7769  da$, that is, wi
│ │ │ │ -0005aca0: 7468 2074 6865 206d 6f64 756c 65c3 9f20  th the module.. 
│ │ │ │ -0005acb0: 2424 204d 4d27 203d 205c 7375 6d5f 6a0a  $$ MM' = \sum_j.
│ │ │ │ -0005acc0: 284d 4d27 286a 295c 6f74 696d 6573 205c  (MM'(j)\otimes \
│ │ │ │ -0005acd0: 4c61 6d62 6461 5f6a 2920 2424 2073 6f20  Lambda_j) $$ so 
│ │ │ │ -0005ace0: 7468 6174 2024 4d4d 275f 7020 3d20 284d  that $MM'_p = (M
│ │ │ │ -0005acf0: 4d5f 705c 6f74 696d 6573 204c 616d 6264  M_p\otimes Lambd
│ │ │ │ -0005ad00: 615f 3029 205c 6f70 6c75 730a 284d 4d5f  a_0) \oplus.(MM_
│ │ │ │ -0005ad10: 7b70 2d31 7d5c 6f74 696d 6573 204c 616d  {p-1}\otimes Lam
│ │ │ │ -0005ad20: 6264 615f 3129 205c 6f70 6c75 735c 6364  bda_1) \oplus\cd
│ │ │ │ -0005ad30: 6f74 7324 2e0a 0a54 6865 2066 756e 6374  ots$...The funct
│ │ │ │ -0005ad40: 696f 6e20 6866 4d6f 6475 6c65 4173 4578  ion hfModuleAsEx
│ │ │ │ -0005ad50: 7420 636f 6d70 7574 6573 2074 6865 2048  t computes the H
│ │ │ │ -0005ad60: 696c 6265 7274 2066 756e 6374 696f 6e20  ilbert function 
│ │ │ │ -0005ad70: 6f66 204d 4d27 206e 756d 6572 6963 616c  of MM' numerical
│ │ │ │ -0005ad80: 6c79 0a66 726f 6d20 7468 6174 206f 6620  ly.from that of 
│ │ │ │ -0005ad90: 4d4d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  MM...+----------
│ │ │ │ +0005a360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0005a370: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ +0005a380: 3d0a 0a20 202a 2055 7361 6765 3a20 0a20  =..  * Usage: . 
│ │ │ │ +0005a390: 2020 2020 2020 204d 203d 206d 6f64 756c         M = modul
│ │ │ │ +0005a3a0: 6541 7345 7874 284d 4d2c 5229 0a20 202a  eAsExt(MM,R).  *
│ │ │ │ +0005a3b0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +0005a3c0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ +0005a3d0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +0005a3e0: 6329 4d6f 6475 6c65 2c2c 206d 6f64 756c  c)Module,, modul
│ │ │ │ +0005a3f0: 6520 6f76 6572 2070 6f6c 796e 6f6d 6961  e over polynomia
│ │ │ │ +0005a400: 6c20 7269 6e67 0a20 2020 2020 2020 2077  l ring.        w
│ │ │ │ +0005a410: 6974 6820 6320 7661 7269 6162 6c65 730a  ith c variables.
│ │ │ │ +0005a420: 2020 2020 2020 2a20 522c 2061 202a 6e6f        * R, a *no
│ │ │ │ +0005a430: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ +0005a440: 6179 3244 6f63 2952 696e 672c 2c20 2867  ay2Doc)Ring,, (g
│ │ │ │ +0005a450: 7261 6465 6429 2063 6f6d 706c 6574 6520  raded) complete 
│ │ │ │ +0005a460: 696e 7465 7273 6563 7469 6f6e 0a20 2020  intersection.   
│ │ │ │ +0005a470: 2020 2020 2072 696e 6720 6f66 2063 6f64       ring of cod
│ │ │ │ +0005a480: 696d 656e 7369 6f6e 2063 2c20 656d 6265  imension c, embe
│ │ │ │ +0005a490: 6464 696e 6720 6469 6d65 6e73 696f 6e20  dding dimension 
│ │ │ │ +0005a4a0: 6e0a 2020 2a20 4f75 7470 7574 733a 0a20  n.  * Outputs:. 
│ │ │ │ +0005a4b0: 2020 2020 202a 204e 2c20 6120 2a6e 6f74       * N, a *not
│ │ │ │ +0005a4c0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +0005a4d0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +0005a4e0: 206d 6f64 756c 6520 6f76 6572 2052 2073   module over R s
│ │ │ │ +0005a4f0: 7563 6820 7468 6174 0a20 2020 2020 2020  uch that.       
│ │ │ │ +0005a500: 2045 7874 5f52 284e 2c6b 2920 3d20 4d5c   Ext_R(N,k) = M\
│ │ │ │ +0005a510: 6f74 696d 6573 205c 7765 6467 6528 6b5e  otimes \wedge(k^
│ │ │ │ +0005a520: 6e29 2069 6e20 6c61 7267 6520 686f 6d6f  n) in large homo
│ │ │ │ +0005a530: 6c6f 6769 6361 6c20 6465 6772 6565 2e0a  logical degree..
│ │ │ │ +0005a540: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0005a550: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 726f  ========..The ro
│ │ │ │ +0005a560: 7574 696e 6520 6060 6d6f 6475 6c65 4173  utine ``moduleAs
│ │ │ │ +0005a570: 4578 7427 2720 6973 2061 2070 6172 7469  Ext'' is a parti
│ │ │ │ +0005a580: 616c 2069 6e76 6572 7365 2074 6f20 7468  al inverse to th
│ │ │ │ +0005a590: 6520 726f 7574 696e 6520 4578 744d 6f64  e routine ExtMod
│ │ │ │ +0005a5a0: 756c 652c 0a63 6f6d 7075 7465 6420 666f  ule,.computed fo
│ │ │ │ +0005a5b0: 6c6c 6f77 696e 6720 6964 6561 7320 6f66  llowing ideas of
│ │ │ │ +0005a5c0: 2041 7672 616d 6f76 2061 6e64 204a 6f72   Avramov and Jor
│ │ │ │ +0005a5d0: 6765 6e73 656e 3a20 6769 7665 6e20 6120  gensen: given a 
│ │ │ │ +0005a5e0: 6d6f 6475 6c65 2045 206f 7665 7220 610a  module E over a.
│ │ │ │ +0005a5f0: 706f 6c79 6e6f 6d69 616c 2072 696e 6720  polynomial ring 
│ │ │ │ +0005a600: 6b5b 785f 312e 2e78 5f63 5d2c 2069 7420  k[x_1..x_c], it 
│ │ │ │ +0005a610: 7072 6f76 6964 6573 2061 206d 6f64 756c  provides a modul
│ │ │ │ +0005a620: 6520 4e20 6f76 6572 2061 2073 7065 6369  e N over a speci
│ │ │ │ +0005a630: 6669 6564 2070 6f6c 796e 6f6d 6961 6c0a  fied polynomial.
│ │ │ │ +0005a640: 7269 6e67 2069 6e20 6e20 7661 7269 6162  ring in n variab
│ │ │ │ +0005a650: 6c65 7320 7375 6368 2074 6861 7420 4578  les such that Ex
│ │ │ │ +0005a660: 7428 4e2c 6b29 2061 6772 6565 7320 7769  t(N,k) agrees wi
│ │ │ │ +0005a670: 7468 2024 4527 3d45 5c6f 7469 6d65 7320  th $E'=E\otimes 
│ │ │ │ +0005a680: 5c77 6564 6765 286b 5e6e 2924 0a61 6674  \wedge(k^n)$.aft
│ │ │ │ +0005a690: 6572 2074 7275 6e63 6174 696f 6e2e 2048  er truncation. H
│ │ │ │ +0005a6a0: 6572 6520 7468 6520 6772 6164 696e 6720  ere the grading 
│ │ │ │ +0005a6b0: 6f6e 2045 2069 7320 7461 6b65 6e20 746f  on E is taken to
│ │ │ │ +0005a6c0: 2062 6520 6576 656e 2c20 7768 696c 650a   be even, while.
│ │ │ │ +0005a6d0: 245c 7765 6467 6528 6b5e 6e29 2420 6861  $\wedge(k^n)$ ha
│ │ │ │ +0005a6e0: 7320 6765 6e65 7261 746f 7273 2069 6e20  s generators in 
│ │ │ │ +0005a6f0: 6465 6772 6565 2031 2e20 5468 6520 726f  degree 1. The ro
│ │ │ │ +0005a700: 7574 696e 6520 6866 4d6f 6475 6c65 4173  utine hfModuleAs
│ │ │ │ +0005a710: 4578 7420 636f 6d70 7574 6573 0a74 6865  Ext computes.the
│ │ │ │ +0005a720: 2072 6573 756c 7469 6e67 2068 696c 6265   resulting hilbe
│ │ │ │ +0005a730: 7274 2066 756e 6374 696f 6e20 666f 7220  rt function for 
│ │ │ │ +0005a740: 4527 2e20 5468 6973 2075 7365 7320 6964  E'. This uses id
│ │ │ │ +0005a750: 6561 7320 6f66 2041 7672 616d 6f76 2061  eas of Avramov a
│ │ │ │ +0005a760: 6e64 0a4a 6f72 6765 6e73 656e 2e20 4e6f  nd.Jorgensen. No
│ │ │ │ +0005a770: 7465 2074 6861 7420 7468 6520 6d6f 6475  te that the modu
│ │ │ │ +0005a780: 6c65 2045 7874 284e 2c6b 2920 2874 7275  le Ext(N,k) (tru
│ │ │ │ +0005a790: 6e63 6174 6564 2920 7769 6c6c 2061 7574  ncated) will aut
│ │ │ │ +0005a7a0: 6f6d 6174 6963 616c 6c79 2062 6520 6672  omatically be fr
│ │ │ │ +0005a7b0: 6565 0a6f 7665 7220 7468 6520 6578 7465  ee.over the exte
│ │ │ │ +0005a7c0: 7269 6f72 2061 6c67 6562 7261 2024 5c77  rior algebra $\w
│ │ │ │ +0005a7d0: 6564 6765 286b 5e6e 2924 2067 656e 6572  edge(k^n)$ gener
│ │ │ │ +0005a7e0: 6174 6564 2062 7920 4578 745e 3128 6b2c  ated by Ext^1(k,
│ │ │ │ +0005a7f0: 6b29 3b20 6e6f 7420 6120 7479 7069 6361  k); not a typica
│ │ │ │ +0005a800: 6c0a 4578 7420 6d6f 6475 6c65 2e0a 0a4d  l.Ext module...M
│ │ │ │ +0005a810: 6f72 6520 7072 6563 6973 656c 793a 0a0a  ore precisely:..
│ │ │ │ +0005a820: 5375 7070 6f73 6520 7468 6174 2024 5220  Suppose that $R 
│ │ │ │ +0005a830: 3d20 6b5b 615f 312c 5c64 6f74 732c 2061  = k[a_1,\dots, a
│ │ │ │ +0005a840: 5f6e 5d2f 2866 5f31 2c5c 646f 7473 2c66  _n]/(f_1,\dots,f
│ │ │ │ +0005a850: 5f63 2924 206c 6574 2024 4b4b 203d 0a6b  _c)$ let $KK =.k
│ │ │ │ +0005a860: 5b78 5f31 2c5c 646f 7473 2c78 5f63 5d24  [x_1,\dots,x_c]$
│ │ │ │ +0005a870: 2c20 616e 6420 6c65 7420 245c 4c61 6d62  , and let $\Lamb
│ │ │ │ +0005a880: 6461 203d 205c 7765 6467 6520 6b5e 6e24  da = \wedge k^n$
│ │ │ │ +0005a890: 2e20 2445 203d 204b 4b5c 6f74 696d 6573  . $E = KK\otimes
│ │ │ │ +0005a8a0: 5c4c 616d 6264 6124 2c20 736f 0a74 6861  \Lambda$, so.tha
│ │ │ │ +0005a8b0: 7420 7468 6520 6d69 6e69 6d61 6c20 2452  t the minimal $R
│ │ │ │ +0005a8c0: 242d 6672 6565 2072 6573 6f6c 7574 696f  $-free resolutio
│ │ │ │ +0005a8d0: 6e20 6f66 2024 6b24 2068 6173 2075 6e64  n of $k$ has und
│ │ │ │ +0005a8e0: 6572 6c79 696e 6720 6d6f 6475 6c65 2024  erlying module $
│ │ │ │ +0005a8f0: 525c 6f74 696d 6573 0a45 5e2a 242c 2077  R\otimes.E^*$, w
│ │ │ │ +0005a900: 6865 7265 2024 455e 2a24 2069 7320 7468  here $E^*$ is th
│ │ │ │ +0005a910: 6520 6772 6164 6564 2076 6563 746f 7220  e graded vector 
│ │ │ │ +0005a920: 7370 6163 6520 6475 616c 206f 6620 2445  space dual of $E
│ │ │ │ +0005a930: 242e 0a0a 4c65 7420 4d4d 2062 6520 7468  $...Let MM be th
│ │ │ │ +0005a940: 6520 7265 7375 6c74 206f 6620 7472 756e  e result of trun
│ │ │ │ +0005a950: 6361 7469 6e67 204d 2061 7420 6974 7320  cating M at its 
│ │ │ │ +0005a960: 7265 6775 6c61 7269 7479 2061 6e64 2073  regularity and s
│ │ │ │ +0005a970: 6869 6674 696e 6720 6974 2073 6f20 7468  hifting it so th
│ │ │ │ +0005a980: 6174 0a69 7420 6973 2067 656e 6572 6174  at.it is generat
│ │ │ │ +0005a990: 6564 2069 6e20 6465 6772 6565 2030 2e20  ed in degree 0. 
│ │ │ │ +0005a9a0: 4c65 7420 2446 2420 6265 2061 2024 4b4b  Let $F$ be a $KK
│ │ │ │ +0005a9b0: 242d 6672 6565 2072 6573 6f6c 7574 696f  $-free resolutio
│ │ │ │ +0005a9c0: 6e20 6f66 2024 4d4d 242c 2061 6e64 0a77  n of $MM$, and.w
│ │ │ │ +0005a9d0: 7269 7465 2024 465f 6920 3d20 4b4b 5c6f  rite $F_i = KK\o
│ │ │ │ +0005a9e0: 7469 6d65 7320 565f 692e 2420 5369 6e63  times V_i.$ Sinc
│ │ │ │ +0005a9f0: 6520 6c69 6e65 6172 2066 6f72 6d73 206f  e linear forms o
│ │ │ │ +0005aa00: 7665 7220 244b 4b24 2063 6f72 7265 7370  ver $KK$ corresp
│ │ │ │ +0005aa10: 6f6e 6420 746f 2043 490a 6f70 6572 6174  ond to CI.operat
│ │ │ │ +0005aa20: 6f72 7320 6f66 2064 6567 7265 6520 2d32  ors of degree -2
│ │ │ │ +0005aa30: 206f 6e20 7468 6520 7265 736f 6c75 7469   on the resoluti
│ │ │ │ +0005aa40: 6f6e 2047 206f 6620 6b20 6f76 6572 2052  on G of k over R
│ │ │ │ +0005aa50: 2c20 7765 206d 6179 2066 6f72 6d20 6120  , we may form a 
│ │ │ │ +0005aa60: 6d61 7020 2424 0a64 5f31 2b64 5f32 3a20  map $$.d_1+d_2: 
│ │ │ │ +0005aa70: 5c73 756d 5f7b 693d 307d 5e6d 2047 5f7b  \sum_{i=0}^m G_{
│ │ │ │ +0005aa80: 692b 317d 5c6f 7469 6d65 7320 565f 7b6d  i+1}\otimes V_{m
│ │ │ │ +0005aa90: 2d69 7d5e 2a20 5c74 6f20 5c73 756d 5f7b  -i}^* \to \sum_{
│ │ │ │ +0005aaa0: 693d 307d 5e6d 2047 5f69 5c6f 7469 6d65  i=0}^m G_i\otime
│ │ │ │ +0005aab0: 730a 565f 7b6d 2d69 7d5e 2a20 2424 2077  s.V_{m-i}^* $$ w
│ │ │ │ +0005aac0: 6865 7265 2024 645f 3124 2069 7320 7468  here $d_1$ is th
│ │ │ │ +0005aad0: 6520 6469 7265 6374 2073 756d 206f 6620  e direct sum of 
│ │ │ │ +0005aae0: 7468 6520 6469 6666 6572 656e 7469 616c  the differential
│ │ │ │ +0005aaf0: 7320 2428 475f 7b69 2b31 7d5c 746f 0a47  s $(G_{i+1}\to.G
│ │ │ │ +0005ab00: 5f69 295c 6f74 696d 6573 2056 5f69 5e2a  _i)\otimes V_i^*
│ │ │ │ +0005ab10: 2420 616e 6420 2464 5f32 2420 6973 2074  $ and $d_2$ is t
│ │ │ │ +0005ab20: 6865 2064 6972 6563 7420 7375 6d20 6f66  he direct sum of
│ │ │ │ +0005ab30: 2074 6865 206d 6170 7320 245c 7068 695f   the maps $\phi_
│ │ │ │ +0005ab40: 6924 2064 6566 696e 6564 0a66 726f 6d20  i$ defined.from 
│ │ │ │ +0005ab50: 7468 6520 6469 6666 6572 656e 7469 616c  the differential
│ │ │ │ +0005ab60: 7320 6f66 2024 4624 2062 7920 7375 6273  s of $F$ by subs
│ │ │ │ +0005ab70: 7469 7475 7469 6e67 2043 4920 6f70 6572  tituting CI oper
│ │ │ │ +0005ab80: 6174 6f72 7320 666f 7220 6c69 6e65 6172  ators for linear
│ │ │ │ +0005ab90: 2066 6f72 6d73 2c0a 245c 7068 695f 693a   forms,.$\phi_i:
│ │ │ │ +0005aba0: 2047 5f7b 692b 317d 5c6f 7469 6d65 7320   G_{i+1}\otimes 
│ │ │ │ +0005abb0: 565f 6920 5c74 6f20 475f 7b69 2d31 7d5c  V_i \to G_{i-1}\
│ │ │ │ +0005abc0: 6f74 696d 6573 2056 5f7b 692d 317d 242e  otimes V_{i-1}$.
│ │ │ │ +0005abd0: 2054 6865 2073 6372 6970 7420 7265 7475   The script retu
│ │ │ │ +0005abe0: 726e 7320 7468 650a 6d6f 6475 6c65 204e  rns the.module N
│ │ │ │ +0005abf0: 2074 6861 7420 6973 2074 6865 2063 6f6b   that is the cok
│ │ │ │ +0005ac00: 6572 6e65 6c20 6f66 2024 645f 312b 645f  ernel of $d_1+d_
│ │ │ │ +0005ac10: 3224 2e0a 0a54 6865 206d 6f64 756c 6520  2$...The module 
│ │ │ │ +0005ac20: 2445 7874 5f52 284e 2c6b 2924 2061 6772  $Ext_R(N,k)$ agr
│ │ │ │ +0005ac30: 6565 732c 2061 6674 6572 2061 2066 6577  ees, after a few
│ │ │ │ +0005ac40: 2073 7465 7073 2c20 7769 7468 2074 6865   steps, with the
│ │ │ │ +0005ac50: 206d 6f64 756c 6520 6465 7269 7665 6420   module derived 
│ │ │ │ +0005ac60: 6672 6f6d 0a24 4d4d 2420 6279 2074 656e  from.$MM$ by ten
│ │ │ │ +0005ac70: 736f 7269 6e67 2069 7420 7769 7468 2024  soring it with $
│ │ │ │ +0005ac80: 5c4c 616d 6264 6124 2c20 7468 6174 2069  \Lambda$, that i
│ │ │ │ +0005ac90: 732c 2077 6974 6820 7468 6520 6d6f 6475  s, with the modu
│ │ │ │ +0005aca0: 6c65 c39f 2024 2420 4d4d 2720 3d20 5c73  le.. $$ MM' = \s
│ │ │ │ +0005acb0: 756d 5f6a 0a28 4d4d 2728 6a29 5c6f 7469  um_j.(MM'(j)\oti
│ │ │ │ +0005acc0: 6d65 7320 5c4c 616d 6264 615f 6a29 2024  mes \Lambda_j) $
│ │ │ │ +0005acd0: 2420 736f 2074 6861 7420 244d 4d27 5f70  $ so that $MM'_p
│ │ │ │ +0005ace0: 203d 2028 4d4d 5f70 5c6f 7469 6d65 7320   = (MM_p\otimes 
│ │ │ │ +0005acf0: 4c61 6d62 6461 5f30 2920 5c6f 706c 7573  Lambda_0) \oplus
│ │ │ │ +0005ad00: 0a28 4d4d 5f7b 702d 317d 5c6f 7469 6d65  .(MM_{p-1}\otime
│ │ │ │ +0005ad10: 7320 4c61 6d62 6461 5f31 2920 5c6f 706c  s Lambda_1) \opl
│ │ │ │ +0005ad20: 7573 5c63 646f 7473 242e 0a0a 5468 6520  us\cdots$...The 
│ │ │ │ +0005ad30: 6675 6e63 7469 6f6e 2068 664d 6f64 756c  function hfModul
│ │ │ │ +0005ad40: 6541 7345 7874 2063 6f6d 7075 7465 7320  eAsExt computes 
│ │ │ │ +0005ad50: 7468 6520 4869 6c62 6572 7420 6675 6e63  the Hilbert func
│ │ │ │ +0005ad60: 7469 6f6e 206f 6620 4d4d 2720 6e75 6d65  tion of MM' nume
│ │ │ │ +0005ad70: 7269 6361 6c6c 790a 6672 6f6d 2074 6861  rically.from tha
│ │ │ │ +0005ad80: 7420 6f66 204d 4d2e 0a0a 2b2d 2d2d 2d2d  t of MM...+-----
│ │ │ │ +0005ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005adc0: 2d2d 2d2b 0a7c 6931 203a 206b 6b20 3d20  ---+.|i1 : kk = 
│ │ │ │ -0005add0: 5a5a 2f31 3031 3b20 2020 2020 2020 2020  ZZ/101;         
│ │ │ │ -0005ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005adf0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005adb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0005adc0: 6b6b 203d 205a 5a2f 3130 313b 2020 2020  kk = ZZ/101;    
│ │ │ │ +0005add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ade0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ae00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae20: 2d2d 2d2b 0a7c 6932 203a 2053 203d 206b  ---+.|i2 : S = k
│ │ │ │ -0005ae30: 6b5b 612c 622c 635d 3b20 2020 2020 2020  k[a,b,c];       
│ │ │ │ -0005ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005ae10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0005ae20: 5320 3d20 6b6b 5b61 2c62 2c63 5d3b 2020  S = kk[a,b,c];  
│ │ │ │ +0005ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ae40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae80: 2d2d 2d2b 0a7c 6933 203a 2066 6620 3d20  ---+.|i3 : ff = 
│ │ │ │ -0005ae90: 6d61 7472 6978 7b7b 615e 342c 2062 5e34  matrix{{a^4, b^4
│ │ │ │ -0005aea0: 2c63 5e34 7d7d 3b20 2020 2020 2020 2020  ,c^4}};         
│ │ │ │ -0005aeb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005ae70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0005ae80: 6666 203d 206d 6174 7269 787b 7b61 5e34  ff = matrix{{a^4
│ │ │ │ +0005ae90: 2c20 625e 342c 635e 347d 7d3b 2020 2020  , b^4,c^4}};    
│ │ │ │ +0005aea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aee0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0005aef0: 2020 2031 2020 2020 2020 3320 2020 2020     1      3     
│ │ │ │ -0005af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af10: 2020 207c 0a7c 6f33 203a 204d 6174 7269     |.|o3 : Matri
│ │ │ │ -0005af20: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ -0005af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005aed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005aee0: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +0005aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af00: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0005af10: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +0005af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005af40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005af70: 2d2d 2d2b 0a7c 6934 203a 2052 203d 2053  ---+.|i4 : R = S
│ │ │ │ -0005af80: 2f69 6465 616c 2066 663b 2020 2020 2020  /ideal ff;      
│ │ │ │ -0005af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005afa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005af60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +0005af70: 5220 3d20 532f 6964 6561 6c20 6666 3b20  R = S/ideal ff; 
│ │ │ │ +0005af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005afd0: 2d2d 2d2b 0a7c 6935 203a 204f 7073 203d  ---+.|i5 : Ops =
│ │ │ │ -0005afe0: 206b 6b5b 785f 312c 785f 322c 785f 335d   kk[x_1,x_2,x_3]
│ │ │ │ -0005aff0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0005b000: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005afc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +0005afd0: 4f70 7320 3d20 6b6b 5b78 5f31 2c78 5f32  Ops = kk[x_1,x_2
│ │ │ │ +0005afe0: 2c78 5f33 5d3b 2020 2020 2020 2020 2020  ,x_3];          
│ │ │ │ +0005aff0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b030: 2d2d 2d2b 0a7c 6936 203a 204d 4d20 3d20  ---+.|i6 : MM = 
│ │ │ │ -0005b040: 4f70 735e 312f 2878 5f31 2a69 6465 616c  Ops^1/(x_1*ideal
│ │ │ │ -0005b050: 2878 5f32 5e32 2c78 5f33 2929 3b20 2020  (x_2^2,x_3));   
│ │ │ │ -0005b060: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005b020: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +0005b030: 4d4d 203d 204f 7073 5e31 2f28 785f 312a  MM = Ops^1/(x_1*
│ │ │ │ +0005b040: 6964 6561 6c28 785f 325e 322c 785f 3329  ideal(x_2^2,x_3)
│ │ │ │ +0005b050: 293b 2020 2020 2020 7c0a 2b2d 2d2d 2d2d  );      |.+-----
│ │ │ │ +0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b090: 2d2d 2d2b 0a7c 6937 203a 204e 203d 206d  ---+.|i7 : N = m
│ │ │ │ -0005b0a0: 6f64 756c 6541 7345 7874 284d 4d2c 5229  oduleAsExt(MM,R)
│ │ │ │ -0005b0b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0005b0c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005b080: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +0005b090: 4e20 3d20 6d6f 6475 6c65 4173 4578 7428  N = moduleAsExt(
│ │ │ │ +0005b0a0: 4d4d 2c52 293b 2020 2020 2020 2020 2020  MM,R);          
│ │ │ │ +0005b0b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b0f0: 2d2d 2d2b 0a7c 6938 203a 2062 6574 7469  ---+.|i8 : betti
│ │ │ │ -0005b100: 2072 6573 2820 4e2c 204c 656e 6774 684c   res( N, LengthL
│ │ │ │ -0005b110: 696d 6974 203d 3e20 3130 2920 2020 2020  imit => 10)     
│ │ │ │ -0005b120: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005b0e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +0005b0f0: 6265 7474 6920 7265 7328 204e 2c20 4c65  betti res( N, Le
│ │ │ │ +0005b100: 6e67 7468 4c69 6d69 7420 3d3e 2031 3029  ngthLimit => 10)
│ │ │ │ +0005b110: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b150: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0005b160: 2020 2030 2020 3120 2032 2020 3320 2034     0  1  2  3  4
│ │ │ │ -0005b170: 2020 3520 2036 2020 3720 2038 2020 3920    5  6  7  8  9 
│ │ │ │ -0005b180: 3130 207c 0a7c 6f38 203d 2074 6f74 616c  10 |.|o8 = total
│ │ │ │ -0005b190: 3a20 3336 2032 3720 3239 2033 3120 3333  : 36 27 29 31 33
│ │ │ │ -0005b1a0: 2033 3520 3337 2033 3920 3431 2034 3320   35 37 39 41 43 
│ │ │ │ -0005b1b0: 3435 207c 0a7c 2020 2020 2020 2020 2d36  45 |.|        -6
│ │ │ │ -0005b1c0: 3a20 3138 2020 3620 202e 2020 2e20 202e  : 18  6  .  .  .
│ │ │ │ -0005b1d0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b1e0: 202e 207c 0a7c 2020 2020 2020 2020 2d35   . |.|        -5
│ │ │ │ -0005b1f0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b200: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b210: 202e 207c 0a7c 2020 2020 2020 2020 2d34   . |.|        -4
│ │ │ │ -0005b220: 3a20 3138 2032 3120 3231 2020 3720 202e  : 18 21 21  7  .
│ │ │ │ -0005b230: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b240: 202e 207c 0a7c 2020 2020 2020 2020 2d33   . |.|        -3
│ │ │ │ -0005b250: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b260: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b270: 202e 207c 0a7c 2020 2020 2020 2020 2d32   . |.|        -2
│ │ │ │ -0005b280: 3a20 202e 2020 2e20 2038 2032 3420 3234  :  .  .  8 24 24
│ │ │ │ -0005b290: 2020 3820 202e 2020 2e20 202e 2020 2e20    8  .  .  .  . 
│ │ │ │ -0005b2a0: 202e 207c 0a7c 2020 2020 2020 2020 2d31   . |.|        -1
│ │ │ │ -0005b2b0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b2c0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b2d0: 202e 207c 0a7c 2020 2020 2020 2020 2030   . |.|         0
│ │ │ │ -0005b2e0: 3a20 202e 2020 2e20 202e 2020 2e20 2039  :  .  .  .  .  9
│ │ │ │ -0005b2f0: 2032 3720 3237 2020 3920 202e 2020 2e20   27 27  9  .  . 
│ │ │ │ -0005b300: 202e 207c 0a7c 2020 2020 2020 2020 2031   . |.|         1
│ │ │ │ -0005b310: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b320: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b330: 202e 207c 0a7c 2020 2020 2020 2020 2032   . |.|         2
│ │ │ │ -0005b340: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b350: 2020 2e20 3130 2033 3020 3330 2031 3020    . 10 30 30 10 
│ │ │ │ -0005b360: 202e 207c 0a7c 2020 2020 2020 2020 2033   . |.|         3
│ │ │ │ -0005b370: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b380: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b390: 202e 207c 0a7c 2020 2020 2020 2020 2034   . |.|         4
│ │ │ │ -0005b3a0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b3b0: 2020 2e20 202e 2020 2e20 3131 2033 3320    .  .  . 11 33 
│ │ │ │ -0005b3c0: 3333 207c 0a7c 2020 2020 2020 2020 2035  33 |.|         5
│ │ │ │ -0005b3d0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b3e0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b3f0: 202e 207c 0a7c 2020 2020 2020 2020 2036   . |.|         6
│ │ │ │ -0005b400: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005b410: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005b420: 3132 207c 0a7c 2020 2020 2020 2020 2020  12 |.|          
│ │ │ │ +0005b140: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005b150: 2020 2020 2020 2020 3020 2031 2020 3220          0  1  2 
│ │ │ │ +0005b160: 2033 2020 3420 2035 2020 3620 2037 2020   3  4  5  6  7  
│ │ │ │ +0005b170: 3820 2039 2031 3020 7c0a 7c6f 3820 3d20  8  9 10 |.|o8 = 
│ │ │ │ +0005b180: 746f 7461 6c3a 2033 3620 3237 2032 3920  total: 36 27 29 
│ │ │ │ +0005b190: 3331 2033 3320 3335 2033 3720 3339 2034  31 33 35 37 39 4
│ │ │ │ +0005b1a0: 3120 3433 2034 3520 7c0a 7c20 2020 2020  1 43 45 |.|     
│ │ │ │ +0005b1b0: 2020 202d 363a 2031 3820 2036 2020 2e20     -6: 18  6  . 
│ │ │ │ +0005b1c0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b1d0: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b1e0: 2020 202d 353a 2020 2e20 202e 2020 2e20     -5:  .  .  . 
│ │ │ │ +0005b1f0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b200: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b210: 2020 202d 343a 2031 3820 3231 2032 3120     -4: 18 21 21 
│ │ │ │ +0005b220: 2037 2020 2e20 202e 2020 2e20 202e 2020   7  .  .  .  .  
│ │ │ │ +0005b230: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b240: 2020 202d 333a 2020 2e20 202e 2020 2e20     -3:  .  .  . 
│ │ │ │ +0005b250: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b260: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b270: 2020 202d 323a 2020 2e20 202e 2020 3820     -2:  .  .  8 
│ │ │ │ +0005b280: 3234 2032 3420 2038 2020 2e20 202e 2020  24 24  8  .  .  
│ │ │ │ +0005b290: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b2a0: 2020 202d 313a 2020 2e20 202e 2020 2e20     -1:  .  .  . 
│ │ │ │ +0005b2b0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b2c0: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b2d0: 2020 2020 303a 2020 2e20 202e 2020 2e20      0:  .  .  . 
│ │ │ │ +0005b2e0: 202e 2020 3920 3237 2032 3720 2039 2020   .  9 27 27  9  
│ │ │ │ +0005b2f0: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b300: 2020 2020 313a 2020 2e20 202e 2020 2e20      1:  .  .  . 
│ │ │ │ +0005b310: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b320: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b330: 2020 2020 323a 2020 2e20 202e 2020 2e20      2:  .  .  . 
│ │ │ │ +0005b340: 202e 2020 2e20 202e 2031 3020 3330 2033   .  .  . 10 30 3
│ │ │ │ +0005b350: 3020 3130 2020 2e20 7c0a 7c20 2020 2020  0 10  . |.|     
│ │ │ │ +0005b360: 2020 2020 333a 2020 2e20 202e 2020 2e20      3:  .  .  . 
│ │ │ │ +0005b370: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b380: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b390: 2020 2020 343a 2020 2e20 202e 2020 2e20      4:  .  .  . 
│ │ │ │ +0005b3a0: 202e 2020 2e20 202e 2020 2e20 202e 2031   .  .  .  .  . 1
│ │ │ │ +0005b3b0: 3120 3333 2033 3320 7c0a 7c20 2020 2020  1 33 33 |.|     
│ │ │ │ +0005b3c0: 2020 2020 353a 2020 2e20 202e 2020 2e20      5:  .  .  . 
│ │ │ │ +0005b3d0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b3e0: 2e20 202e 2020 2e20 7c0a 7c20 2020 2020  .  .  . |.|     
│ │ │ │ +0005b3f0: 2020 2020 363a 2020 2e20 202e 2020 2e20      6:  .  .  . 
│ │ │ │ +0005b400: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005b410: 2e20 202e 2031 3220 7c0a 7c20 2020 2020  .  . 12 |.|     
│ │ │ │ +0005b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b450: 2020 207c 0a7c 6f38 203a 2042 6574 7469     |.|o8 : Betti
│ │ │ │ -0005b460: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ -0005b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b480: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005b440: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ +0005b450: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0005b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b470: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005b480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b4b0: 2d2d 2d2b 0a7c 6939 203a 2068 664d 6f64  ---+.|i9 : hfMod
│ │ │ │ -0005b4c0: 756c 6541 7345 7874 2831 322c 4d4d 2c33  uleAsExt(12,MM,3
│ │ │ │ -0005b4d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0005b4e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005b4a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +0005b4b0: 6866 4d6f 6475 6c65 4173 4578 7428 3132  hfModuleAsExt(12
│ │ │ │ +0005b4c0: 2c4d 4d2c 3329 2020 2020 2020 2020 2020  ,MM,3)          
│ │ │ │ +0005b4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b510: 2020 207c 0a7c 6f39 203d 2028 3233 2c20     |.|o9 = (23, 
│ │ │ │ -0005b520: 3235 2c20 3237 2c20 3239 2c20 3331 2c20  25, 27, 29, 31, 
│ │ │ │ -0005b530: 3333 2c20 3335 2c20 3337 2c20 3339 2c20  33, 35, 37, 39, 
│ │ │ │ -0005b540: 3431 297c 0a7c 2020 2020 2020 2020 2020  41)|.|          
│ │ │ │ +0005b500: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ +0005b510: 2832 332c 2032 352c 2032 372c 2032 392c  (23, 25, 27, 29,
│ │ │ │ +0005b520: 2033 312c 2033 332c 2033 352c 2033 372c   31, 33, 35, 37,
│ │ │ │ +0005b530: 2033 392c 2034 3129 7c0a 7c20 2020 2020   39, 41)|.|     
│ │ │ │ +0005b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b570: 2020 207c 0a7c 6f39 203a 2053 6571 7565     |.|o9 : Seque
│ │ │ │ -0005b580: 6e63 6520 2020 2020 2020 2020 2020 2020  nce             
│ │ │ │ -0005b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b5a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005b560: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ +0005b570: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +0005b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b590: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005b5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b5d0: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d  ---+..Caveat.===
│ │ │ │ -0005b5e0: 3d3d 3d0a 0a54 6865 2065 6c65 6d65 6e74  ===..The element
│ │ │ │ -0005b5f0: 7320 665f 312e 2e66 5f63 206d 7573 7420  s f_1..f_c must 
│ │ │ │ -0005b600: 6265 2068 6f6d 6f67 656e 656f 7573 206f  be homogeneous o
│ │ │ │ -0005b610: 6620 7468 6520 7361 6d65 2064 6567 7265  f the same degre
│ │ │ │ -0005b620: 652e 2054 6865 2073 6372 6970 7420 636f  e. The script co
│ │ │ │ -0005b630: 756c 640a 6265 2072 6577 7269 7474 656e  uld.be rewritten
│ │ │ │ -0005b640: 2074 6f20 6163 636f 6d6d 6f64 6174 6520   to accommodate 
│ │ │ │ -0005b650: 6469 6666 6572 656e 7420 6465 6772 6565  different degree
│ │ │ │ -0005b660: 732c 2062 7574 206f 6e6c 7920 6279 2067  s, but only by g
│ │ │ │ -0005b670: 6f69 6e67 2074 6f20 7468 6520 6c6f 6361  oing to the loca
│ │ │ │ -0005b680: 6c0a 6361 7465 676f 7279 0a0a 5365 6520  l.category..See 
│ │ │ │ -0005b690: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -0005b6a0: 202a 202a 6e6f 7465 2045 7874 4d6f 6475   * *note ExtModu
│ │ │ │ -0005b6b0: 6c65 3a20 4578 744d 6f64 756c 652c 202d  le: ExtModule, -
│ │ │ │ -0005b6c0: 2d20 4578 745e 2a28 4d2c 6b29 206f 7665  - Ext^*(M,k) ove
│ │ │ │ -0005b6d0: 7220 6120 636f 6d70 6c65 7465 2069 6e74  r a complete int
│ │ │ │ -0005b6e0: 6572 7365 6374 696f 6e20 6173 0a20 2020  ersection as.   
│ │ │ │ -0005b6f0: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -0005b700: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -0005b710: 2a20 2a6e 6f74 6520 6576 656e 4578 744d  * *note evenExtM
│ │ │ │ -0005b720: 6f64 756c 653a 2065 7665 6e45 7874 4d6f  odule: evenExtMo
│ │ │ │ -0005b730: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061  dule, -- even pa
│ │ │ │ -0005b740: 7274 206f 6620 4578 745e 2a28 4d2c 6b29  rt of Ext^*(M,k)
│ │ │ │ -0005b750: 206f 7665 7220 610a 2020 2020 636f 6d70   over a.    comp
│ │ │ │ -0005b760: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ -0005b770: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ -0005b780: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ -0005b790: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445  g.  * *note oddE
│ │ │ │ -0005b7a0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874  xtModule: oddExt
│ │ │ │ -0005b7b0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070  Module, -- odd p
│ │ │ │ -0005b7c0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ -0005b7d0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ -0005b7e0: 650a 2020 2020 696e 7465 7273 6563 7469  e.    intersecti
│ │ │ │ -0005b7f0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -0005b800: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -0005b810: 6e67 0a20 202a 202a 6e6f 7465 2045 7874  ng.  * *note Ext
│ │ │ │ -0005b820: 4d6f 6475 6c65 4461 7461 3a20 4578 744d  ModuleData: ExtM
│ │ │ │ -0005b830: 6f64 756c 6544 6174 612c 202d 2d20 4576  oduleData, -- Ev
│ │ │ │ -0005b840: 656e 2061 6e64 206f 6464 2045 7874 206d  en and odd Ext m
│ │ │ │ -0005b850: 6f64 756c 6573 2061 6e64 2074 6865 6972  odules and their
│ │ │ │ -0005b860: 0a20 2020 2072 6567 756c 6172 6974 790a  .    regularity.
│ │ │ │ -0005b870: 2020 2a20 2a6e 6f74 6520 6866 4d6f 6475    * *note hfModu
│ │ │ │ -0005b880: 6c65 4173 4578 743a 2068 664d 6f64 756c  leAsExt: hfModul
│ │ │ │ -0005b890: 6541 7345 7874 2c20 2d2d 2070 7265 6469  eAsExt, -- predi
│ │ │ │ -0005b8a0: 6374 2062 6574 7469 206e 756d 6265 7273  ct betti numbers
│ │ │ │ -0005b8b0: 206f 660a 2020 2020 6d6f 6475 6c65 4173   of.    moduleAs
│ │ │ │ -0005b8c0: 4578 7428 4d2c 5229 0a0a 5761 7973 2074  Ext(M,R)..Ways t
│ │ │ │ -0005b8d0: 6f20 7573 6520 6d6f 6475 6c65 4173 4578  o use moduleAsEx
│ │ │ │ -0005b8e0: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │ -0005b8f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -0005b900: 2022 6d6f 6475 6c65 4173 4578 7428 4d6f   "moduleAsExt(Mo
│ │ │ │ -0005b910: 6475 6c65 2c52 696e 6729 220a 0a46 6f72  dule,Ring)"..For
│ │ │ │ -0005b920: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -0005b930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0005b940: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -0005b950: 6e6f 7465 206d 6f64 756c 6541 7345 7874  note moduleAsExt
│ │ │ │ -0005b960: 3a20 6d6f 6475 6c65 4173 4578 742c 2069  : moduleAsExt, i
│ │ │ │ -0005b970: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ -0005b980: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ -0005b990: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -0005b9a0: 756e 6374 696f 6e2c 2e0a 1f0a 4669 6c65  unction,....File
│ │ │ │ -0005b9b0: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -0005b9c0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0005b9d0: 732e 696e 666f 2c20 4e6f 6465 3a20 6e65  s.info, Node: ne
│ │ │ │ -0005b9e0: 7745 7874 2c20 4e65 7874 3a20 6f64 6445  wExt, Next: oddE
│ │ │ │ -0005b9f0: 7874 4d6f 6475 6c65 2c20 5072 6576 3a20  xtModule, Prev: 
│ │ │ │ -0005ba00: 6d6f 6475 6c65 4173 4578 742c 2055 703a  moduleAsExt, Up:
│ │ │ │ -0005ba10: 2054 6f70 0a0a 6e65 7745 7874 202d 2d20   Top..newExt -- 
│ │ │ │ -0005ba20: 476c 6f62 616c 2045 7874 2066 6f72 206d  Global Ext for m
│ │ │ │ -0005ba30: 6f64 756c 6573 206f 7665 7220 6120 636f  odules over a co
│ │ │ │ -0005ba40: 6d70 6c65 7465 2049 6e74 6572 7365 6374  mplete Intersect
│ │ │ │ -0005ba50: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ion.************
│ │ │ │ +0005b5c0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561  --------+..Cavea
│ │ │ │ +0005b5d0: 740a 3d3d 3d3d 3d3d 0a0a 5468 6520 656c  t.======..The el
│ │ │ │ +0005b5e0: 656d 656e 7473 2066 5f31 2e2e 665f 6320  ements f_1..f_c 
│ │ │ │ +0005b5f0: 6d75 7374 2062 6520 686f 6d6f 6765 6e65  must be homogene
│ │ │ │ +0005b600: 6f75 7320 6f66 2074 6865 2073 616d 6520  ous of the same 
│ │ │ │ +0005b610: 6465 6772 6565 2e20 5468 6520 7363 7269  degree. The scri
│ │ │ │ +0005b620: 7074 2063 6f75 6c64 0a62 6520 7265 7772  pt could.be rewr
│ │ │ │ +0005b630: 6974 7465 6e20 746f 2061 6363 6f6d 6d6f  itten to accommo
│ │ │ │ +0005b640: 6461 7465 2064 6966 6665 7265 6e74 2064  date different d
│ │ │ │ +0005b650: 6567 7265 6573 2c20 6275 7420 6f6e 6c79  egrees, but only
│ │ │ │ +0005b660: 2062 7920 676f 696e 6720 746f 2074 6865   by going to the
│ │ │ │ +0005b670: 206c 6f63 616c 0a63 6174 6567 6f72 790a   local.category.
│ │ │ │ +0005b680: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0005b690: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 4578  ==..  * *note Ex
│ │ │ │ +0005b6a0: 744d 6f64 756c 653a 2045 7874 4d6f 6475  tModule: ExtModu
│ │ │ │ +0005b6b0: 6c65 2c20 2d2d 2045 7874 5e2a 284d 2c6b  le, -- Ext^*(M,k
│ │ │ │ +0005b6c0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +0005b6d0: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ +0005b6e0: 730a 2020 2020 6d6f 6475 6c65 206f 7665  s.    module ove
│ │ │ │ +0005b6f0: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +0005b700: 6e67 0a20 202a 202a 6e6f 7465 2065 7665  ng.  * *note eve
│ │ │ │ +0005b710: 6e45 7874 4d6f 6475 6c65 3a20 6576 656e  nExtModule: even
│ │ │ │ +0005b720: 4578 744d 6f64 756c 652c 202d 2d20 6576  ExtModule, -- ev
│ │ │ │ +0005b730: 656e 2070 6172 7420 6f66 2045 7874 5e2a  en part of Ext^*
│ │ │ │ +0005b740: 284d 2c6b 2920 6f76 6572 2061 0a20 2020  (M,k) over a.   
│ │ │ │ +0005b750: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +0005b760: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65  ection as module
│ │ │ │ +0005b770: 206f 7665 7220 4349 206f 7065 7261 746f   over CI operato
│ │ │ │ +0005b780: 7220 7269 6e67 0a20 202a 202a 6e6f 7465  r ring.  * *note
│ │ │ │ +0005b790: 206f 6464 4578 744d 6f64 756c 653a 206f   oddExtModule: o
│ │ │ │ +0005b7a0: 6464 4578 744d 6f64 756c 652c 202d 2d20  ddExtModule, -- 
│ │ │ │ +0005b7b0: 6f64 6420 7061 7274 206f 6620 4578 745e  odd part of Ext^
│ │ │ │ +0005b7c0: 2a28 4d2c 6b29 206f 7665 7220 6120 636f  *(M,k) over a co
│ │ │ │ +0005b7d0: 6d70 6c65 7465 0a20 2020 2069 6e74 6572  mplete.    inter
│ │ │ │ +0005b7e0: 7365 6374 696f 6e20 6173 206d 6f64 756c  section as modul
│ │ │ │ +0005b7f0: 6520 6f76 6572 2043 4920 6f70 6572 6174  e over CI operat
│ │ │ │ +0005b800: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74  or ring.  * *not
│ │ │ │ +0005b810: 6520 4578 744d 6f64 756c 6544 6174 613a  e ExtModuleData:
│ │ │ │ +0005b820: 2045 7874 4d6f 6475 6c65 4461 7461 2c20   ExtModuleData, 
│ │ │ │ +0005b830: 2d2d 2045 7665 6e20 616e 6420 6f64 6420  -- Even and odd 
│ │ │ │ +0005b840: 4578 7420 6d6f 6475 6c65 7320 616e 6420  Ext modules and 
│ │ │ │ +0005b850: 7468 6569 720a 2020 2020 7265 6775 6c61  their.    regula
│ │ │ │ +0005b860: 7269 7479 0a20 202a 202a 6e6f 7465 2068  rity.  * *note h
│ │ │ │ +0005b870: 664d 6f64 756c 6541 7345 7874 3a20 6866  fModuleAsExt: hf
│ │ │ │ +0005b880: 4d6f 6475 6c65 4173 4578 742c 202d 2d20  ModuleAsExt, -- 
│ │ │ │ +0005b890: 7072 6564 6963 7420 6265 7474 6920 6e75  predict betti nu
│ │ │ │ +0005b8a0: 6d62 6572 7320 6f66 0a20 2020 206d 6f64  mbers of.    mod
│ │ │ │ +0005b8b0: 756c 6541 7345 7874 284d 2c52 290a 0a57  uleAsExt(M,R)..W
│ │ │ │ +0005b8c0: 6179 7320 746f 2075 7365 206d 6f64 756c  ays to use modul
│ │ │ │ +0005b8d0: 6541 7345 7874 3a0a 3d3d 3d3d 3d3d 3d3d  eAsExt:.========
│ │ │ │ +0005b8e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005b8f0: 0a0a 2020 2a20 226d 6f64 756c 6541 7345  ..  * "moduleAsE
│ │ │ │ +0005b900: 7874 284d 6f64 756c 652c 5269 6e67 2922  xt(Module,Ring)"
│ │ │ │ +0005b910: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0005b920: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0005b930: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +0005b940: 6563 7420 2a6e 6f74 6520 6d6f 6475 6c65  ect *note module
│ │ │ │ +0005b950: 4173 4578 743a 206d 6f64 756c 6541 7345  AsExt: moduleAsE
│ │ │ │ +0005b960: 7874 2c20 6973 2061 202a 6e6f 7465 206d  xt, is a *note m
│ │ │ │ +0005b970: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ +0005b980: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ +0005b990: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a1f  thodFunction,...
│ │ │ │ +0005b9a0: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ +0005b9b0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0005b9c0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ +0005b9d0: 653a 206e 6577 4578 742c 204e 6578 743a  e: newExt, Next:
│ │ │ │ +0005b9e0: 206f 6464 4578 744d 6f64 756c 652c 2050   oddExtModule, P
│ │ │ │ +0005b9f0: 7265 763a 206d 6f64 756c 6541 7345 7874  rev: moduleAsExt
│ │ │ │ +0005ba00: 2c20 5570 3a20 546f 700a 0a6e 6577 4578  , Up: Top..newEx
│ │ │ │ +0005ba10: 7420 2d2d 2047 6c6f 6261 6c20 4578 7420  t -- Global Ext 
│ │ │ │ +0005ba20: 666f 7220 6d6f 6475 6c65 7320 6f76 6572  for modules over
│ │ │ │ +0005ba30: 2061 2063 6f6d 706c 6574 6520 496e 7465   a complete Inte
│ │ │ │ +0005ba40: 7273 6563 7469 6f6e 0a2a 2a2a 2a2a 2a2a  rsection.*******
│ │ │ │ +0005ba50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005ba60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005ba70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005ba80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005ba90: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -0005baa0: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -0005bab0: 200a 2020 2020 2020 2020 4520 3d20 6e65   .        E = ne
│ │ │ │ -0005bac0: 7745 7874 284d 2c4e 290a 2020 2a20 496e  wExt(M,N).  * In
│ │ │ │ -0005bad0: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c  puts:.      * M,
│ │ │ │ -0005bae0: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ -0005baf0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -0005bb00: 6f64 756c 652c 2c20 6f76 6572 2061 2063  odule,, over a c
│ │ │ │ -0005bb10: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ -0005bb20: 7469 6f6e 0a20 2020 2020 2020 2052 6261  tion.        Rba
│ │ │ │ -0005bb30: 720a 2020 2020 2020 2a20 4e2c 2061 202a  r.      * N, a *
│ │ │ │ -0005bb40: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0005bb50: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0005bb60: 652c 2c20 6f76 6572 2052 6261 720a 2020  e,, over Rbar.  
│ │ │ │ -0005bb70: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ -0005bb80: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ -0005bb90: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ -0005bba0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ -0005bbb0: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ -0005bbc0: 2020 2020 2a20 4368 6563 6b20 3d3e 202e      * Check => .
│ │ │ │ -0005bbd0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -0005bbe0: 6520 6661 6c73 650a 2020 2020 2020 2a20  e false.      * 
│ │ │ │ -0005bbf0: 4772 6164 696e 6720 3d3e 202e 2e2e 2c20  Grading => ..., 
│ │ │ │ -0005bc00: 6465 6661 756c 7420 7661 6c75 6520 320a  default value 2.
│ │ │ │ -0005bc10: 2020 2020 2020 2a20 4c69 6674 203d 3e20        * Lift => 
│ │ │ │ -0005bc20: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -0005bc30: 7565 2066 616c 7365 0a20 2020 2020 202a  ue false.      *
│ │ │ │ -0005bc40: 2056 6172 6961 626c 6573 203d 3e20 2e2e   Variables => ..
│ │ │ │ -0005bc50: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0005bc60: 2073 0a20 202a 204f 7574 7075 7473 3a0a   s.  * Outputs:.
│ │ │ │ -0005bc70: 2020 2020 2020 2a20 452c 2061 202a 6e6f        * E, a *no
│ │ │ │ -0005bc80: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -0005bc90: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -0005bca0: 2c20 6f76 6572 2061 2072 696e 6720 5320  , over a ring S 
│ │ │ │ -0005bcb0: 6d61 6465 2066 726f 6d20 7269 6e67 0a20  made from ring. 
│ │ │ │ -0005bcc0: 2020 2020 2020 2070 7265 7365 6e74 6174         presentat
│ │ │ │ -0005bcd0: 696f 6e20 5262 6172 2077 6974 6820 636f  ion Rbar with co
│ │ │ │ -0005bce0: 6469 6d20 5262 6172 206e 6577 2076 6172  dim Rbar new var
│ │ │ │ -0005bcf0: 6961 626c 6573 0a0a 4465 7363 7269 7074  iables..Descript
│ │ │ │ -0005bd00: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -0005bd10: 0a4c 6574 2052 6261 7220 3d20 522f 2866  .Let Rbar = R/(f
│ │ │ │ -0005bd20: 312e 2e66 6329 2c20 6120 636f 6d70 6c65  1..fc), a comple
│ │ │ │ -0005bd30: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ -0005bd40: 6f66 2063 6f64 696d 656e 7369 6f6e 2063  of codimension c
│ │ │ │ -0005bd50: 2c20 616e 6420 6c65 7420 4d2c 4e20 6265  , and let M,N be
│ │ │ │ -0005bd60: 0a52 6261 722d 6d6f 6475 6c65 732e 2057  .Rbar-modules. W
│ │ │ │ -0005bd70: 6520 6173 7375 6d65 2074 6861 7420 7468  e assume that th
│ │ │ │ -0005bd80: 6520 7075 7368 466f 7277 6172 6420 6f66  e pushForward of
│ │ │ │ -0005bd90: 204d 2074 6f20 5220 6861 7320 6669 6e69   M to R has fini
│ │ │ │ -0005bda0: 7465 2066 7265 650a 7265 736f 6c75 7469  te free.resoluti
│ │ │ │ -0005bdb0: 6f6e 2e20 5468 6520 7363 7269 7074 2074  on. The script t
│ │ │ │ -0005bdc0: 6865 6e20 636f 6d70 7574 6573 2074 6865  hen computes the
│ │ │ │ -0005bdd0: 2074 6f74 616c 2045 7874 284d 2c4e 2920   total Ext(M,N) 
│ │ │ │ -0005bde0: 6173 2061 206d 6f64 756c 6520 6f76 6572  as a module over
│ │ │ │ -0005bdf0: 2053 203d 0a6b 6b28 735f 312e 2e73 5f63   S =.kk(s_1..s_c
│ │ │ │ -0005be00: 2c67 656e 7320 5229 2c20 7573 696e 6720  ,gens R), using 
│ │ │ │ -0005be10: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
│ │ │ │ -0005be20: 6f74 616c 2e0a 0a49 6620 4368 6563 6b20  otal...If Check 
│ │ │ │ -0005be30: 3d3e 2074 7275 652c 2074 6865 6e20 7468  => true, then th
│ │ │ │ -0005be40: 6520 7265 7375 6c74 2069 7320 636f 6d70  e result is comp
│ │ │ │ -0005be50: 6172 6564 2077 6974 6820 7468 6520 6275  ared with the bu
│ │ │ │ -0005be60: 696c 742d 696e 2067 6c6f 6261 6c20 4578  ilt-in global Ex
│ │ │ │ -0005be70: 740a 7772 6974 7465 6e20 6279 2041 7672  t.written by Avr
│ │ │ │ -0005be80: 616d 6f76 2061 6e64 2047 7261 7973 6f6e  amov and Grayson
│ │ │ │ -0005be90: 2028 6275 7420 6e6f 7465 2074 6865 2064   (but note the d
│ │ │ │ -0005bea0: 6966 6665 7265 6e63 652c 2065 7870 6c61  ifference, expla
│ │ │ │ -0005beb0: 696e 6564 2062 656c 6f77 292e 0a0a 4966  ined below)...If
│ │ │ │ -0005bec0: 204c 6966 7420 3d3e 2066 616c 7365 2074   Lift => false t
│ │ │ │ -0005bed0: 6865 2072 6573 756c 7420 6973 2072 6574  he result is ret
│ │ │ │ -0005bee0: 7572 6e65 6420 6f76 6572 2061 6e64 2065  urned over and e
│ │ │ │ -0005bef0: 7874 656e 7369 6f6e 206f 6620 5262 6172  xtension of Rbar
│ │ │ │ -0005bf00: 3b20 6966 204c 6966 7420 3d3e 0a74 7275  ; if Lift =>.tru
│ │ │ │ -0005bf10: 6520 7468 6520 7265 7375 6c74 2069 7320  e the result is 
│ │ │ │ -0005bf20: 7265 7475 726e 6564 206f 7665 7220 616e  returned over an
│ │ │ │ -0005bf30: 6420 6578 7465 6e73 696f 6e20 6f66 2052  d extension of R
│ │ │ │ -0005bf40: 2e0a 0a49 6620 4772 6164 696e 6720 3d3e  ...If Grading =>
│ │ │ │ -0005bf50: 2032 2c20 7468 6520 6465 6661 756c 742c   2, the default,
│ │ │ │ -0005bf60: 2074 6865 6e20 7468 6520 7265 7375 6c74   then the result
│ │ │ │ -0005bf70: 2069 7320 6269 6772 6164 6564 2028 7468   is bigraded (th
│ │ │ │ -0005bf80: 6973 2069 7320 6e65 6365 7373 6172 790a  is is necessary.
│ │ │ │ -0005bf90: 7768 656e 2043 6865 636b 3d3e 7472 7565  when Check=>true
│ │ │ │ -0005bfa0: 0a0a 5468 6520 6465 6661 756c 7420 5661  ..The default Va
│ │ │ │ -0005bfb0: 7269 6162 6c65 7320 3d3e 2073 796d 626f  riables => symbo
│ │ │ │ -0005bfc0: 6c20 2273 2220 6769 7665 7320 7468 6520  l "s" gives the 
│ │ │ │ -0005bfd0: 6e65 7720 7661 7269 6162 6c65 7320 7468  new variables th
│ │ │ │ -0005bfe0: 6520 6e61 6d65 2073 5f69 2c0a 693d 302e  e name s_i,.i=0.
│ │ │ │ -0005bff0: 2e63 2d31 2e20 286e 6f74 6520 7468 6174  .c-1. (note that
│ │ │ │ -0005c000: 2074 6865 2062 7569 6c74 696e 2045 7874   the builtin Ext
│ │ │ │ -0005c010: 2075 7365 7320 585f 312e 2e58 5f63 2e0a   uses X_1..X_c..
│ │ │ │ -0005c020: 0a4f 6e20 536f 6d65 2065 7861 6d70 6c65  .On Some example
│ │ │ │ -0005c030: 7320 6e65 7745 7874 2069 7320 6661 7374  s newExt is fast
│ │ │ │ -0005c040: 6572 2074 6861 6e20 4578 743b 206f 6e20  er than Ext; on 
│ │ │ │ -0005c050: 6f74 6865 7273 2069 7427 7320 736c 6f77  others it's slow
│ │ │ │ -0005c060: 6572 2e0a 0a41 2073 696d 706c 6520 6578  er...A simple ex
│ │ │ │ -0005c070: 616d 706c 653a 2069 6620 5220 3d20 6b5b  ample: if R = k[
│ │ │ │ -0005c080: 785f 312e 2e78 5f6e 5d20 616e 6420 4920  x_1..x_n] and I 
│ │ │ │ -0005c090: 6973 2063 6f6e 7461 696e 6564 2069 6e20  is contained in 
│ │ │ │ -0005c0a0: 7468 6520 6375 6265 206f 6620 7468 650a  the cube of the.
│ │ │ │ -0005c0b0: 6d61 7869 6d61 6c20 6964 6561 6c2c 2074  maximal ideal, t
│ │ │ │ -0005c0c0: 6865 6e20 4578 7428 6b2c 6b29 2069 7320  hen Ext(k,k) is 
│ │ │ │ -0005c0d0: 6120 6672 6565 2053 2f28 785f 312e 2e78  a free S/(x_1..x
│ │ │ │ -0005c0e0: 5f6e 2920 3d20 6b5b 735f 302e 2e73 5f28  _n) = k[s_0..s_(
│ │ │ │ -0005c0f0: 632d 3129 5d2d 206d 6f64 756c 650a 7769  c-1)]- module.wi
│ │ │ │ -0005c100: 7468 2062 696e 6f6d 6961 6c28 6e2c 6929  th binomial(n,i)
│ │ │ │ -0005c110: 2067 656e 6572 6174 6f72 7320 696e 2064   generators in d
│ │ │ │ -0005c120: 6567 7265 6520 690a 0a2b 2d2d 2d2d 2d2d  egree i..+------
│ │ │ │ +0005ba80: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +0005ba90: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +0005baa0: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ +0005bab0: 203d 206e 6577 4578 7428 4d2c 4e29 0a20   = newExt(M,N). 
│ │ │ │ +0005bac0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +0005bad0: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f   * M, a *note mo
│ │ │ │ +0005bae0: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ +0005baf0: 446f 6329 4d6f 6475 6c65 2c2c 206f 7665  Doc)Module,, ove
│ │ │ │ +0005bb00: 7220 6120 636f 6d70 6c65 7465 2069 6e74  r a complete int
│ │ │ │ +0005bb10: 6572 7365 6374 696f 6e0a 2020 2020 2020  ersection.      
│ │ │ │ +0005bb20: 2020 5262 6172 0a20 2020 2020 202a 204e    Rbar.      * N
│ │ │ │ +0005bb30: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0005bb40: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0005bb50: 4d6f 6475 6c65 2c2c 206f 7665 7220 5262  Module,, over Rb
│ │ │ │ +0005bb60: 6172 0a20 202a 202a 6e6f 7465 204f 7074  ar.  * *note Opt
│ │ │ │ +0005bb70: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +0005bb80: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +0005bb90: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +0005bba0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +0005bbb0: 2c3a 0a20 2020 2020 202a 2043 6865 636b  ,:.      * Check
│ │ │ │ +0005bbc0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +0005bbd0: 2076 616c 7565 2066 616c 7365 0a20 2020   value false.   
│ │ │ │ +0005bbe0: 2020 202a 2047 7261 6469 6e67 203d 3e20     * Grading => 
│ │ │ │ +0005bbf0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +0005bc00: 7565 2032 0a20 2020 2020 202a 204c 6966  ue 2.      * Lif
│ │ │ │ +0005bc10: 7420 3d3e 202e 2e2e 2c20 6465 6661 756c  t => ..., defaul
│ │ │ │ +0005bc20: 7420 7661 6c75 6520 6661 6c73 650a 2020  t value false.  
│ │ │ │ +0005bc30: 2020 2020 2a20 5661 7269 6162 6c65 7320      * Variables 
│ │ │ │ +0005bc40: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0005bc50: 7661 6c75 6520 730a 2020 2a20 4f75 7470  value s.  * Outp
│ │ │ │ +0005bc60: 7574 733a 0a20 2020 2020 202a 2045 2c20  uts:.      * E, 
│ │ │ │ +0005bc70: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ +0005bc80: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ +0005bc90: 6475 6c65 2c2c 206f 7665 7220 6120 7269  dule,, over a ri
│ │ │ │ +0005bca0: 6e67 2053 206d 6164 6520 6672 6f6d 2072  ng S made from r
│ │ │ │ +0005bcb0: 696e 670a 2020 2020 2020 2020 7072 6573  ing.        pres
│ │ │ │ +0005bcc0: 656e 7461 7469 6f6e 2052 6261 7220 7769  entation Rbar wi
│ │ │ │ +0005bcd0: 7468 2063 6f64 696d 2052 6261 7220 6e65  th codim Rbar ne
│ │ │ │ +0005bce0: 7720 7661 7269 6162 6c65 730a 0a44 6573  w variables..Des
│ │ │ │ +0005bcf0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +0005bd00: 3d3d 3d3d 0a0a 4c65 7420 5262 6172 203d  ====..Let Rbar =
│ │ │ │ +0005bd10: 2052 2f28 6631 2e2e 6663 292c 2061 2063   R/(f1..fc), a c
│ │ │ │ +0005bd20: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ +0005bd30: 7469 6f6e 206f 6620 636f 6469 6d65 6e73  tion of codimens
│ │ │ │ +0005bd40: 696f 6e20 632c 2061 6e64 206c 6574 204d  ion c, and let M
│ │ │ │ +0005bd50: 2c4e 2062 650a 5262 6172 2d6d 6f64 756c  ,N be.Rbar-modul
│ │ │ │ +0005bd60: 6573 2e20 5765 2061 7373 756d 6520 7468  es. We assume th
│ │ │ │ +0005bd70: 6174 2074 6865 2070 7573 6846 6f72 7761  at the pushForwa
│ │ │ │ +0005bd80: 7264 206f 6620 4d20 746f 2052 2068 6173  rd of M to R has
│ │ │ │ +0005bd90: 2066 696e 6974 6520 6672 6565 0a72 6573   finite free.res
│ │ │ │ +0005bda0: 6f6c 7574 696f 6e2e 2054 6865 2073 6372  olution. The scr
│ │ │ │ +0005bdb0: 6970 7420 7468 656e 2063 6f6d 7075 7465  ipt then compute
│ │ │ │ +0005bdc0: 7320 7468 6520 746f 7461 6c20 4578 7428  s the total Ext(
│ │ │ │ +0005bdd0: 4d2c 4e29 2061 7320 6120 6d6f 6475 6c65  M,N) as a module
│ │ │ │ +0005bde0: 206f 7665 7220 5320 3d0a 6b6b 2873 5f31   over S =.kk(s_1
│ │ │ │ +0005bdf0: 2e2e 735f 632c 6765 6e73 2052 292c 2075  ..s_c,gens R), u
│ │ │ │ +0005be00: 7369 6e67 2045 6973 656e 6275 6453 6861  sing EisenbudSha
│ │ │ │ +0005be10: 6d61 7368 546f 7461 6c2e 0a0a 4966 2043  mashTotal...If C
│ │ │ │ +0005be20: 6865 636b 203d 3e20 7472 7565 2c20 7468  heck => true, th
│ │ │ │ +0005be30: 656e 2074 6865 2072 6573 756c 7420 6973  en the result is
│ │ │ │ +0005be40: 2063 6f6d 7061 7265 6420 7769 7468 2074   compared with t
│ │ │ │ +0005be50: 6865 2062 7569 6c74 2d69 6e20 676c 6f62  he built-in glob
│ │ │ │ +0005be60: 616c 2045 7874 0a77 7269 7474 656e 2062  al Ext.written b
│ │ │ │ +0005be70: 7920 4176 7261 6d6f 7620 616e 6420 4772  y Avramov and Gr
│ │ │ │ +0005be80: 6179 736f 6e20 2862 7574 206e 6f74 6520  ayson (but note 
│ │ │ │ +0005be90: 7468 6520 6469 6666 6572 656e 6365 2c20  the difference, 
│ │ │ │ +0005bea0: 6578 706c 6169 6e65 6420 6265 6c6f 7729  explained below)
│ │ │ │ +0005beb0: 2e0a 0a49 6620 4c69 6674 203d 3e20 6661  ...If Lift => fa
│ │ │ │ +0005bec0: 6c73 6520 7468 6520 7265 7375 6c74 2069  lse the result i
│ │ │ │ +0005bed0: 7320 7265 7475 726e 6564 206f 7665 7220  s returned over 
│ │ │ │ +0005bee0: 616e 6420 6578 7465 6e73 696f 6e20 6f66  and extension of
│ │ │ │ +0005bef0: 2052 6261 723b 2069 6620 4c69 6674 203d   Rbar; if Lift =
│ │ │ │ +0005bf00: 3e0a 7472 7565 2074 6865 2072 6573 756c  >.true the resul
│ │ │ │ +0005bf10: 7420 6973 2072 6574 7572 6e65 6420 6f76  t is returned ov
│ │ │ │ +0005bf20: 6572 2061 6e64 2065 7874 656e 7369 6f6e  er and extension
│ │ │ │ +0005bf30: 206f 6620 522e 0a0a 4966 2047 7261 6469   of R...If Gradi
│ │ │ │ +0005bf40: 6e67 203d 3e20 322c 2074 6865 2064 6566  ng => 2, the def
│ │ │ │ +0005bf50: 6175 6c74 2c20 7468 656e 2074 6865 2072  ault, then the r
│ │ │ │ +0005bf60: 6573 756c 7420 6973 2062 6967 7261 6465  esult is bigrade
│ │ │ │ +0005bf70: 6420 2874 6869 7320 6973 206e 6563 6573  d (this is neces
│ │ │ │ +0005bf80: 7361 7279 0a77 6865 6e20 4368 6563 6b3d  sary.when Check=
│ │ │ │ +0005bf90: 3e74 7275 650a 0a54 6865 2064 6566 6175  >true..The defau
│ │ │ │ +0005bfa0: 6c74 2056 6172 6961 626c 6573 203d 3e20  lt Variables => 
│ │ │ │ +0005bfb0: 7379 6d62 6f6c 2022 7322 2067 6976 6573  symbol "s" gives
│ │ │ │ +0005bfc0: 2074 6865 206e 6577 2076 6172 6961 626c   the new variabl
│ │ │ │ +0005bfd0: 6573 2074 6865 206e 616d 6520 735f 692c  es the name s_i,
│ │ │ │ +0005bfe0: 0a69 3d30 2e2e 632d 312e 2028 6e6f 7465  .i=0..c-1. (note
│ │ │ │ +0005bff0: 2074 6861 7420 7468 6520 6275 696c 7469   that the builti
│ │ │ │ +0005c000: 6e20 4578 7420 7573 6573 2058 5f31 2e2e  n Ext uses X_1..
│ │ │ │ +0005c010: 585f 632e 0a0a 4f6e 2053 6f6d 6520 6578  X_c...On Some ex
│ │ │ │ +0005c020: 616d 706c 6573 206e 6577 4578 7420 6973  amples newExt is
│ │ │ │ +0005c030: 2066 6173 7465 7220 7468 616e 2045 7874   faster than Ext
│ │ │ │ +0005c040: 3b20 6f6e 206f 7468 6572 7320 6974 2773  ; on others it's
│ │ │ │ +0005c050: 2073 6c6f 7765 722e 0a0a 4120 7369 6d70   slower...A simp
│ │ │ │ +0005c060: 6c65 2065 7861 6d70 6c65 3a20 6966 2052  le example: if R
│ │ │ │ +0005c070: 203d 206b 5b78 5f31 2e2e 785f 6e5d 2061   = k[x_1..x_n] a
│ │ │ │ +0005c080: 6e64 2049 2069 7320 636f 6e74 6169 6e65  nd I is containe
│ │ │ │ +0005c090: 6420 696e 2074 6865 2063 7562 6520 6f66  d in the cube of
│ │ │ │ +0005c0a0: 2074 6865 0a6d 6178 696d 616c 2069 6465   the.maximal ide
│ │ │ │ +0005c0b0: 616c 2c20 7468 656e 2045 7874 286b 2c6b  al, then Ext(k,k
│ │ │ │ +0005c0c0: 2920 6973 2061 2066 7265 6520 532f 2878  ) is a free S/(x
│ │ │ │ +0005c0d0: 5f31 2e2e 785f 6e29 203d 206b 5b73 5f30  _1..x_n) = k[s_0
│ │ │ │ +0005c0e0: 2e2e 735f 2863 2d31 295d 2d20 6d6f 6475  ..s_(c-1)]- modu
│ │ │ │ +0005c0f0: 6c65 0a77 6974 6820 6269 6e6f 6d69 616c  le.with binomial
│ │ │ │ +0005c100: 286e 2c69 2920 6765 6e65 7261 746f 7273  (n,i) generators
│ │ │ │ +0005c110: 2069 6e20 6465 6772 6565 2069 0a0a 2b2d   in degree i..+-
│ │ │ │ +0005c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c170: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e  -------+.|i1 : n
│ │ │ │ -0005c180: 203d 2033 3b63 3d32 3b20 2020 2020 2020   = 3;c=2;       
│ │ │ │ +0005c160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005c170: 3120 3a20 6e20 3d20 333b 633d 323b 2020  1 : n = 3;c=2;  
│ │ │ │ +0005c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c1c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005c1b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c210: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2052  -------+.|i3 : R
│ │ │ │ -0005c220: 203d 205a 5a2f 3130 315b 785f 302e 2e78   = ZZ/101[x_0..x
│ │ │ │ -0005c230: 5f28 6e2d 3129 5d20 2020 2020 2020 2020  _(n-1)]         
│ │ │ │ +0005c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005c210: 3320 3a20 5220 3d20 5a5a 2f31 3031 5b78  3 : R = ZZ/101[x
│ │ │ │ +0005c220: 5f30 2e2e 785f 286e 2d31 295d 2020 2020  _0..x_(n-1)]    
│ │ │ │ +0005c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c260: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2b0: 2020 2020 2020 207c 0a7c 6f33 203d 2052         |.|o3 = R
│ │ │ │ +0005c2a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c2b0: 3320 3d20 5220 2020 2020 2020 2020 2020  3 = R           
│ │ │ │  0005c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c300: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c2f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c350: 2020 2020 2020 207c 0a7c 6f33 203a 2050         |.|o3 : P
│ │ │ │ -0005c360: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +0005c340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c350: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +0005c360: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  0005c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c3a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005c390: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c3f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ -0005c400: 6261 7220 3d20 522f 2869 6465 616c 2061  bar = R/(ideal a
│ │ │ │ -0005c410: 7070 6c79 2863 2c20 692d 3e20 525f 695e  pply(c, i-> R_i^
│ │ │ │ -0005c420: 3329 2920 2020 2020 2020 2020 2020 2020  3))             
│ │ │ │ -0005c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005c3f0: 3420 3a20 5262 6172 203d 2052 2f28 6964  4 : Rbar = R/(id
│ │ │ │ +0005c400: 6561 6c20 6170 706c 7928 632c 2069 2d3e  eal apply(c, i->
│ │ │ │ +0005c410: 2052 5f69 5e33 2929 2020 2020 2020 2020   R_i^3))        
│ │ │ │ +0005c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c430: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c490: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ -0005c4a0: 6261 7220 2020 2020 2020 2020 2020 2020  bar             
│ │ │ │ +0005c480: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c490: 3420 3d20 5262 6172 2020 2020 2020 2020  4 = Rbar        
│ │ │ │ +0005c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c4e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c530: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ -0005c540: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +0005c520: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c530: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +0005c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c580: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005c570: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c5d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204d  -------+.|i5 : M
│ │ │ │ -0005c5e0: 6261 7220 3d20 4e62 6172 203d 2063 6f6b  bar = Nbar = cok
│ │ │ │ -0005c5f0: 6572 2076 6172 7320 5262 6172 2020 2020  er vars Rbar    
│ │ │ │ +0005c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005c5d0: 3520 3a20 4d62 6172 203d 204e 6261 7220  5 : Mbar = Nbar 
│ │ │ │ +0005c5e0: 3d20 636f 6b65 7220 7661 7273 2052 6261  = coker vars Rba
│ │ │ │ +0005c5f0: 7220 2020 2020 2020 2020 2020 2020 2020  r               
│ │ │ │  0005c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c620: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c610: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c670: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ -0005c680: 6f6b 6572 6e65 6c20 7c20 785f 3020 785f  okernel | x_0 x_
│ │ │ │ -0005c690: 3120 785f 3220 7c20 2020 2020 2020 2020  1 x_2 |         
│ │ │ │ +0005c660: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c670: 3520 3d20 636f 6b65 726e 656c 207c 2078  5 = cokernel | x
│ │ │ │ +0005c680: 5f30 2078 5f31 2078 5f32 207c 2020 2020  _0 x_1 x_2 |    
│ │ │ │ +0005c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c6c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c710: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c730: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0005c730: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  0005c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c760: 2020 2020 2020 207c 0a7c 6f35 203a 2052         |.|o5 : R
│ │ │ │ -0005c770: 6261 722d 6d6f 6475 6c65 2c20 7175 6f74  bar-module, quot
│ │ │ │ -0005c780: 6965 6e74 206f 6620 5262 6172 2020 2020  ient of Rbar    
│ │ │ │ +0005c750: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c760: 3520 3a20 5262 6172 2d6d 6f64 756c 652c  5 : Rbar-module,
│ │ │ │ +0005c770: 2071 756f 7469 656e 7420 6f66 2052 6261   quotient of Rba
│ │ │ │ +0005c780: 7220 2020 2020 2020 2020 2020 2020 2020  r               
│ │ │ │  0005c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c7b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005c7a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c800: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2045  -------+.|i6 : E
│ │ │ │ -0005c810: 203d 206e 6577 4578 7428 4d62 6172 2c4e   = newExt(Mbar,N
│ │ │ │ -0005c820: 6261 7229 2020 2020 2020 2020 2020 2020  bar)            
│ │ │ │ +0005c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005c800: 3620 3a20 4520 3d20 6e65 7745 7874 284d  6 : E = newExt(M
│ │ │ │ +0005c810: 6261 722c 4e62 6172 2920 2020 2020 2020  bar,Nbar)       
│ │ │ │ +0005c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c850: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005c840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c8a0: 2020 2020 2020 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ -0005c8b0: 6f6b 6572 6e65 6c20 7b30 2c20 307d 2020  okernel {0, 0}  
│ │ │ │ -0005c8c0: 207c 2078 5f32 2078 5f31 2078 5f30 2030   | x_2 x_1 x_0 0
│ │ │ │ -0005c8d0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c8e0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c8f0: 2020 2030 2020 207c 0a7c 2020 2020 2020     0   |.|      
│ │ │ │ -0005c900: 2020 2020 2020 2020 7b2d 322c 202d 327d          {-2, -2}
│ │ │ │ -0005c910: 207c 2030 2020 2030 2020 2030 2020 2030   | 0   0   0   0
│ │ │ │ -0005c920: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c930: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c940: 2020 2078 5f32 207c 0a7c 2020 2020 2020     x_2 |.|      
│ │ │ │ -0005c950: 2020 2020 2020 2020 7b2d 322c 202d 327d          {-2, -2}
│ │ │ │ -0005c960: 207c 2030 2020 2030 2020 2030 2020 2030   | 0   0   0   0
│ │ │ │ -0005c970: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c980: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c990: 2020 2030 2020 207c 0a7c 2020 2020 2020     0   |.|      
│ │ │ │ -0005c9a0: 2020 2020 2020 2020 7b2d 322c 202d 327d          {-2, -2}
│ │ │ │ -0005c9b0: 207c 2030 2020 2030 2020 2030 2020 2030   | 0   0   0   0
│ │ │ │ -0005c9c0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c9d0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005c9e0: 2020 2030 2020 207c 0a7c 2020 2020 2020     0   |.|      
│ │ │ │ -0005c9f0: 2020 2020 2020 2020 7b2d 312c 202d 317d          {-1, -1}
│ │ │ │ -0005ca00: 207c 2030 2020 2030 2020 2030 2020 2078   | 0   0   0   x
│ │ │ │ -0005ca10: 5f32 2078 5f31 2078 5f30 2030 2020 2030  _2 x_1 x_0 0   0
│ │ │ │ -0005ca20: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005ca30: 2020 2030 2020 207c 0a7c 2020 2020 2020     0   |.|      
│ │ │ │ -0005ca40: 2020 2020 2020 2020 7b2d 312c 202d 317d          {-1, -1}
│ │ │ │ -0005ca50: 207c 2030 2020 2030 2020 2030 2020 2030   | 0   0   0   0
│ │ │ │ -0005ca60: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ -0005ca70: 5f31 2078 5f30 2030 2020 2030 2020 2030  _1 x_0 0   0   0
│ │ │ │ -0005ca80: 2020 2030 2020 207c 0a7c 2020 2020 2020     0   |.|      
│ │ │ │ -0005ca90: 2020 2020 2020 2020 7b2d 312c 202d 317d          {-1, -1}
│ │ │ │ -0005caa0: 207c 2030 2020 2030 2020 2030 2020 2030   | 0   0   0   0
│ │ │ │ -0005cab0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005cac0: 2020 2030 2020 2078 5f32 2078 5f31 2078     0   x_2 x_1 x
│ │ │ │ -0005cad0: 5f30 2030 2020 207c 0a7c 2020 2020 2020  _0 0   |.|      
│ │ │ │ -0005cae0: 2020 2020 2020 2020 7b2d 332c 202d 337d          {-3, -3}
│ │ │ │ -0005caf0: 207c 2030 2020 2030 2020 2030 2020 2030   | 0   0   0   0
│ │ │ │ -0005cb00: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005cb10: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005cb20: 2020 2030 2020 207c 0a7c 2020 2020 2020     0   |.|      
│ │ │ │ +0005c890: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005c8a0: 3620 3d20 636f 6b65 726e 656c 207b 302c  6 = cokernel {0,
│ │ │ │ +0005c8b0: 2030 7d20 2020 7c20 785f 3220 785f 3120   0}   | x_2 x_1 
│ │ │ │ +0005c8c0: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ +0005c8d0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c8e0: 3020 2020 3020 2020 3020 2020 7c0a 7c20  0   0   0   |.| 
│ │ │ │ +0005c8f0: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ +0005c900: 2c20 2d32 7d20 7c20 3020 2020 3020 2020  , -2} | 0   0   
│ │ │ │ +0005c910: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c920: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c930: 3020 2020 3020 2020 785f 3220 7c0a 7c20  0   0   x_2 |.| 
│ │ │ │ +0005c940: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ +0005c950: 2c20 2d32 7d20 7c20 3020 2020 3020 2020  , -2} | 0   0   
│ │ │ │ +0005c960: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c970: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c980: 3020 2020 3020 2020 3020 2020 7c0a 7c20  0   0   0   |.| 
│ │ │ │ +0005c990: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ +0005c9a0: 2c20 2d32 7d20 7c20 3020 2020 3020 2020  , -2} | 0   0   
│ │ │ │ +0005c9b0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c9c0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005c9d0: 3020 2020 3020 2020 3020 2020 7c0a 7c20  0   0   0   |.| 
│ │ │ │ +0005c9e0: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ +0005c9f0: 2c20 2d31 7d20 7c20 3020 2020 3020 2020  , -1} | 0   0   
│ │ │ │ +0005ca00: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ +0005ca10: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005ca20: 3020 2020 3020 2020 3020 2020 7c0a 7c20  0   0   0   |.| 
│ │ │ │ +0005ca30: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ +0005ca40: 2c20 2d31 7d20 7c20 3020 2020 3020 2020  , -1} | 0   0   
│ │ │ │ +0005ca50: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005ca60: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ +0005ca70: 3020 2020 3020 2020 3020 2020 7c0a 7c20  0   0   0   |.| 
│ │ │ │ +0005ca80: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ +0005ca90: 2c20 2d31 7d20 7c20 3020 2020 3020 2020  , -1} | 0   0   
│ │ │ │ +0005caa0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005cab0: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ +0005cac0: 785f 3120 785f 3020 3020 2020 7c0a 7c20  x_1 x_0 0   |.| 
│ │ │ │ +0005cad0: 2020 2020 2020 2020 2020 2020 207b 2d33               {-3
│ │ │ │ +0005cae0: 2c20 2d33 7d20 7c20 3020 2020 3020 2020  , -3} | 0   0   
│ │ │ │ +0005caf0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005cb00: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005cb10: 3020 2020 3020 2020 3020 2020 7c0a 7c20  0   0   0   |.| 
│ │ │ │ +0005cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cb70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005cb80: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ -0005cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cba0: 2020 2020 2020 202f 205a 5a20 2020 2020         / ZZ     
│ │ │ │ -0005cbb0: 2020 2020 2020 2020 2020 205c 2020 2020             \    
│ │ │ │ -0005cbc0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ -0005cbd0: 2d2d 5b73 202e 2e73 202c 2078 202e 2e78  --[s ..s , x ..x
│ │ │ │ -0005cbe0: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ -0005cbf0: 2020 2020 2020 207c 2d2d 2d5b 7320 2e2e         |---[s ..
│ │ │ │ -0005cc00: 7320 2c20 7820 2e2e 7820 5d7c 2020 2020  s , x ..x ]|    
│ │ │ │ -0005cc10: 2020 2020 2020 207c 0a7c 2020 2020 2031         |.|     1
│ │ │ │ -0005cc20: 3031 2020 3020 2020 3120 2020 3020 2020  01  0   1   0   
│ │ │ │ -0005cc30: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0005cc40: 2020 2020 2020 207c 3130 3120 2030 2020         |101  0  
│ │ │ │ -0005cc50: 2031 2020 2030 2020 2032 207c 3820 2020   1   0   2 |8   
│ │ │ │ -0005cc60: 2020 2020 2020 207c 0a7c 6f36 203a 202d         |.|o6 : -
│ │ │ │ -0005cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cc80: 2d2d 2d6d 6f64 756c 652c 2071 756f 7469  ---module, quoti
│ │ │ │ -0005cc90: 656e 7420 6f66 207c 2d2d 2d2d 2d2d 2d2d  ent of |--------
│ │ │ │ -0005cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 2020 2020  -----------|    
│ │ │ │ -0005ccb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005ccc0: 2020 2020 2020 2033 2020 2033 2020 2020         3   3    
│ │ │ │ -0005ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cce0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ -0005ccf0: 3320 2020 3320 2020 2020 207c 2020 2020  3   3      |    
│ │ │ │ -0005cd00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005cd10: 2020 2020 2028 7820 2c20 7820 2920 2020       (x , x )   
│ │ │ │ -0005cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cd30: 2020 2020 2020 207c 2020 2020 2020 2878         |      (x
│ │ │ │ -0005cd40: 202c 2078 2029 2020 2020 207c 2020 2020   , x )     |    
│ │ │ │ -0005cd50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005cd60: 2020 2020 2020 2030 2020 2031 2020 2020         0   1    
│ │ │ │ -0005cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cd80: 2020 2020 2020 205c 2020 2020 2020 2020         \        
│ │ │ │ -0005cd90: 3020 2020 3120 2020 2020 202f 2020 2020  0   1      /    
│ │ │ │ -0005cda0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +0005cb60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005cb70: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │ +0005cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cb90: 2020 2020 2020 2020 2020 2020 2f20 5a5a              / ZZ
│ │ │ │ +0005cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cbb0: 5c20 2020 2020 2020 2020 2020 7c0a 7c20  \           |.| 
│ │ │ │ +0005cbc0: 2020 2020 2d2d 2d5b 7320 2e2e 7320 2c20      ---[s ..s , 
│ │ │ │ +0005cbd0: 7820 2e2e 7820 5d20 2020 2020 2020 2020  x ..x ]         
│ │ │ │ +0005cbe0: 2020 2020 2020 2020 2020 2020 7c2d 2d2d              |---
│ │ │ │ +0005cbf0: 5b73 202e 2e73 202c 2078 202e 2e78 205d  [s ..s , x ..x ]
│ │ │ │ +0005cc00: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0005cc10: 2020 2020 3130 3120 2030 2020 2031 2020      101  0   1  
│ │ │ │ +0005cc20: 2030 2020 2032 2020 2020 2020 2020 2020   0   2          
│ │ │ │ +0005cc30: 2020 2020 2020 2020 2020 2020 7c31 3031              |101
│ │ │ │ +0005cc40: 2020 3020 2020 3120 2020 3020 2020 3220    0   1   0   2 
│ │ │ │ +0005cc50: 7c38 2020 2020 2020 2020 2020 7c0a 7c6f  |8          |.|o
│ │ │ │ +0005cc60: 3620 3a20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  6 : ------------
│ │ │ │ +0005cc70: 2d2d 2d2d 2d2d 2d2d 6d6f 6475 6c65 2c20  --------module, 
│ │ │ │ +0005cc80: 7175 6f74 6965 6e74 206f 6620 7c2d 2d2d  quotient of |---
│ │ │ │ +0005cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cca0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0005ccb0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +0005ccc0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0005ccd0: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +0005cce0: 2020 2020 2033 2020 2033 2020 2020 2020       3   3      
│ │ │ │ +0005ccf0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0005cd00: 2020 2020 2020 2020 2020 2878 202c 2078            (x , x
│ │ │ │ +0005cd10: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +0005cd20: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +0005cd30: 2020 2028 7820 2c20 7820 2920 2020 2020     (x , x )     
│ │ │ │ +0005cd40: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0005cd50: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0005cd60: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0005cd70: 2020 2020 2020 2020 2020 2020 5c20 2020              \   
│ │ │ │ +0005cd80: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +0005cd90: 2f20 2020 2020 2020 2020 2020 7c0a 7c2d  /           |.|-
│ │ │ │ +0005cda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cdf0: 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020 3020  -------|.|0   0 
│ │ │ │ -0005ce00: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005ce10: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005ce20: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ce40: 2020 2020 2020 207c 0a7c 785f 3120 785f         |.|x_1 x_
│ │ │ │ -0005ce50: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ -0005ce60: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005ce70: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ce90: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ -0005cea0: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ -0005ceb0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005cec0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cee0: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ -0005cef0: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ -0005cf00: 3220 785f 3120 785f 3020 3020 2020 3020  2 x_1 x_0 0   0 
│ │ │ │ -0005cf10: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf30: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ -0005cf40: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005cf50: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005cf60: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf80: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ -0005cf90: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005cfa0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005cfb0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cfd0: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ -0005cfe0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005cff0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005d000: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0005d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d020: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ -0005d030: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -0005d040: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ -0005d050: 3120 785f 3020 7c20 2020 2020 2020 2020  1 x_0 |         
│ │ │ │ -0005d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d070: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30  ------------|.|0
│ │ │ │ +0005cdf0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ce00: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ce10: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ce30: 2020 2020 2020 2020 2020 2020 7c0a 7c78              |.|x
│ │ │ │ +0005ce40: 5f31 2078 5f30 2030 2020 2030 2020 2030  _1 x_0 0   0   0
│ │ │ │ +0005ce50: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ce60: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ce80: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0005ce90: 2020 2030 2020 2078 5f32 2078 5f31 2078     0   x_2 x_1 x
│ │ │ │ +0005cea0: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ +0005ceb0: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ced0: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0005cee0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cef0: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ +0005cf00: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cf20: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0005cf30: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cf40: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cf50: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cf70: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0005cf80: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cf90: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cfa0: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cfc0: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0005cfd0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cfe0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005cff0: 2020 2030 2020 2030 2020 207c 2020 2020     0   0   |    
│ │ │ │ +0005d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d010: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0005d020: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005d030: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ +0005d040: 5f32 2078 5f31 2078 5f30 207c 2020 2020  _2 x_1 x_0 |    
│ │ │ │ +0005d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d060: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005d070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d0c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2074  -------+.|i7 : t
│ │ │ │ -0005d0d0: 616c 6c79 2064 6567 7265 6573 2045 2020  ally degrees E  
│ │ │ │ +0005d0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005d0c0: 3720 3a20 7461 6c6c 7920 6465 6772 6565  7 : tally degree
│ │ │ │ +0005d0d0: 7320 4520 2020 2020 2020 2020 2020 2020  s E             
│ │ │ │  0005d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d110: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005d100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d160: 2020 2020 2020 207c 0a7c 6f37 203d 2054         |.|o7 = T
│ │ │ │ -0005d170: 616c 6c79 7b7b 2d31 2c20 2d31 7d20 3d3e  ally{{-1, -1} =>
│ │ │ │ -0005d180: 2033 7d20 2020 2020 2020 2020 2020 2020   3}             
│ │ │ │ +0005d150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005d160: 3720 3d20 5461 6c6c 797b 7b2d 312c 202d  7 = Tally{{-1, -
│ │ │ │ +0005d170: 317d 203d 3e20 337d 2020 2020 2020 2020  1} => 3}        
│ │ │ │ +0005d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d1b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d1c0: 2020 2020 207b 2d32 2c20 2d32 7d20 3d3e       {-2, -2} =>
│ │ │ │ -0005d1d0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005d1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d1b0: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ +0005d1c0: 327d 203d 3e20 3320 2020 2020 2020 2020  2} => 3         
│ │ │ │ +0005d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d200: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d210: 2020 2020 207b 2d33 2c20 2d33 7d20 3d3e       {-3, -3} =>
│ │ │ │ -0005d220: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0005d1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d200: 2020 2020 2020 2020 2020 7b2d 332c 202d            {-3, -
│ │ │ │ +0005d210: 337d 203d 3e20 3120 2020 2020 2020 2020  3} => 1         
│ │ │ │ +0005d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d250: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d260: 2020 2020 207b 302c 2030 7d20 3d3e 2031       {0, 0} => 1
│ │ │ │ +0005d240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d250: 2020 2020 2020 2020 2020 7b30 2c20 307d            {0, 0}
│ │ │ │ +0005d260: 203d 3e20 3120 2020 2020 2020 2020 2020   => 1           
│ │ │ │  0005d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d2a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005d290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d2f0: 2020 2020 2020 207c 0a7c 6f37 203a 2054         |.|o7 : T
│ │ │ │ -0005d300: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │ +0005d2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005d2f0: 3720 3a20 5461 6c6c 7920 2020 2020 2020  7 : Tally       
│ │ │ │ +0005d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d340: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005d330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d390: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2061  -------+.|i8 : a
│ │ │ │ -0005d3a0: 6e6e 6968 696c 6174 6f72 2045 2020 2020  nnihilator E    
│ │ │ │ +0005d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005d390: 3820 3a20 616e 6e69 6869 6c61 746f 7220  8 : annihilator 
│ │ │ │ +0005d3a0: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │  0005d3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d3e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005d3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d430: 2020 2020 2020 207c 0a7c 6f38 203d 2069         |.|o8 = i
│ │ │ │ -0005d440: 6465 616c 2028 7820 2c20 7820 2c20 7820  deal (x , x , x 
│ │ │ │ -0005d450: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0005d420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005d430: 3820 3d20 6964 6561 6c20 2878 202c 2078  8 = ideal (x , x
│ │ │ │ +0005d440: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │ +0005d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d480: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d490: 2020 2020 2020 2032 2020 2031 2020 2030         2   1   0
│ │ │ │ +0005d470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d480: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0005d490: 3120 2020 3020 2020 2020 2020 2020 2020  1   0           
│ │ │ │  0005d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d4d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005d4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d520: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d530: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ +0005d510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d520: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +0005d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d570: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d580: 2020 2020 2020 2020 2d2d 2d5b 7320 2e2e          ---[s ..
│ │ │ │ -0005d590: 7320 2c20 7820 2e2e 7820 5d20 2020 2020  s , x ..x ]     
│ │ │ │ +0005d560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d570: 2020 2020 2020 2020 2020 2020 202d 2d2d               ---
│ │ │ │ +0005d580: 5b73 202e 2e73 202c 2078 202e 2e78 205d  [s ..s , x ..x ]
│ │ │ │ +0005d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d5c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d5d0: 2020 2020 2020 2020 3130 3120 2030 2020          101  0  
│ │ │ │ -0005d5e0: 2031 2020 2030 2020 2032 2020 2020 2020   1   0   2      
│ │ │ │ +0005d5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d5c0: 2020 2020 2020 2020 2020 2020 2031 3031               101
│ │ │ │ +0005d5d0: 2020 3020 2020 3120 2020 3020 2020 3220    0   1   0   2 
│ │ │ │ +0005d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d610: 2020 2020 2020 207c 0a7c 6f38 203a 2049         |.|o8 : I
│ │ │ │ -0005d620: 6465 616c 206f 6620 2d2d 2d2d 2d2d 2d2d  deal of --------
│ │ │ │ -0005d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020  -----------     
│ │ │ │ +0005d600: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005d610: 3820 3a20 4964 6561 6c20 6f66 202d 2d2d  8 : Ideal of ---
│ │ │ │ +0005d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d660: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d680: 3320 2020 3320 2020 2020 2020 2020 2020  3   3           
│ │ │ │ +0005d650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d670: 2020 2020 2033 2020 2033 2020 2020 2020       3   3      
│ │ │ │ +0005d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d6b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d6c0: 2020 2020 2020 2020 2020 2020 2020 2878                (x
│ │ │ │ -0005d6d0: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │ +0005d6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d6c0: 2020 2028 7820 2c20 7820 2920 2020 2020     (x , x )     
│ │ │ │ +0005d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d700: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d720: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +0005d6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d710: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +0005d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d750: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005d740: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d7a0: 2d2d 2d2d 2d2d 2d2b 0a0a 416e 2065 7861  -------+..An exa
│ │ │ │ -0005d7b0: 6d70 6c65 2077 6865 7265 2074 6865 2062  mple where the b
│ │ │ │ -0005d7c0: 7569 6c74 2d69 6e20 676c 6f62 616c 2045  uilt-in global E
│ │ │ │ -0005d7d0: 7874 2069 7320 6861 7264 2074 6f20 636f  xt is hard to co
│ │ │ │ -0005d7e0: 6d70 6172 6520 6469 7265 6374 6c79 2077  mpare directly w
│ │ │ │ -0005d7f0: 6974 6820 6f75 720a 6d65 7468 6f64 206f  ith our.method o
│ │ │ │ -0005d800: 6620 636f 6d70 7574 6174 696f 6e3a 2049  f computation: I
│ │ │ │ -0005d810: 202a 6775 6573 732a 2074 6861 7420 7468   *guess* that th
│ │ │ │ -0005d820: 6520 7369 676e 2063 686f 6963 6573 2069  e sign choices i
│ │ │ │ -0005d830: 6e20 7468 6520 6275 696c 742d 696e 2061  n the built-in a
│ │ │ │ -0005d840: 6d6f 756e 740a 6573 7365 6e74 6961 6c6c  mount.essentiall
│ │ │ │ -0005d850: 7920 746f 2061 2063 6861 6e67 6520 6f66  y to a change of
│ │ │ │ -0005d860: 2076 6172 6961 626c 6520 696e 2074 6865   variable in the
│ │ │ │ -0005d870: 206e 6577 2076 6172 6961 626c 6573 2c20   new variables, 
│ │ │ │ -0005d880: 616e 6420 7370 6f69 6c20 616e 2065 6173  and spoil an eas
│ │ │ │ -0005d890: 790a 636f 6d70 6172 6973 6f6e 2e20 4275  y.comparison. Bu
│ │ │ │ -0005d8a0: 7420 666f 7220 6578 616d 706c 6520 7468  t for example th
│ │ │ │ -0005d8b0: 6520 6269 2d67 7261 6465 6420 4265 7474  e bi-graded Bett
│ │ │ │ -0005d8c0: 6920 6e75 6d62 6572 7320 6172 6520 6571  i numbers are eq
│ │ │ │ -0005d8d0: 7561 6c2e 2074 6869 7320 7365 656d 730a  ual. this seems.
│ │ │ │ -0005d8e0: 746f 2073 7461 7274 2077 6974 6820 633d  to start with c=
│ │ │ │ -0005d8f0: 332e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  3...+-----------
│ │ │ │ +0005d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41  ------------+..A
│ │ │ │ +0005d7a0: 6e20 6578 616d 706c 6520 7768 6572 6520  n example where 
│ │ │ │ +0005d7b0: 7468 6520 6275 696c 742d 696e 2067 6c6f  the built-in glo
│ │ │ │ +0005d7c0: 6261 6c20 4578 7420 6973 2068 6172 6420  bal Ext is hard 
│ │ │ │ +0005d7d0: 746f 2063 6f6d 7061 7265 2064 6972 6563  to compare direc
│ │ │ │ +0005d7e0: 746c 7920 7769 7468 206f 7572 0a6d 6574  tly with our.met
│ │ │ │ +0005d7f0: 686f 6420 6f66 2063 6f6d 7075 7461 7469  hod of computati
│ │ │ │ +0005d800: 6f6e 3a20 4920 2a67 7565 7373 2a20 7468  on: I *guess* th
│ │ │ │ +0005d810: 6174 2074 6865 2073 6967 6e20 6368 6f69  at the sign choi
│ │ │ │ +0005d820: 6365 7320 696e 2074 6865 2062 7569 6c74  ces in the built
│ │ │ │ +0005d830: 2d69 6e20 616d 6f75 6e74 0a65 7373 656e  -in amount.essen
│ │ │ │ +0005d840: 7469 616c 6c79 2074 6f20 6120 6368 616e  tially to a chan
│ │ │ │ +0005d850: 6765 206f 6620 7661 7269 6162 6c65 2069  ge of variable i
│ │ │ │ +0005d860: 6e20 7468 6520 6e65 7720 7661 7269 6162  n the new variab
│ │ │ │ +0005d870: 6c65 732c 2061 6e64 2073 706f 696c 2061  les, and spoil a
│ │ │ │ +0005d880: 6e20 6561 7379 0a63 6f6d 7061 7269 736f  n easy.compariso
│ │ │ │ +0005d890: 6e2e 2042 7574 2066 6f72 2065 7861 6d70  n. But for examp
│ │ │ │ +0005d8a0: 6c65 2074 6865 2062 692d 6772 6164 6564  le the bi-graded
│ │ │ │ +0005d8b0: 2042 6574 7469 206e 756d 6265 7273 2061   Betti numbers a
│ │ │ │ +0005d8c0: 7265 2065 7175 616c 2e20 7468 6973 2073  re equal. this s
│ │ │ │ +0005d8d0: 6565 6d73 0a74 6f20 7374 6172 7420 7769  eems.to start wi
│ │ │ │ +0005d8e0: 7468 2063 3d33 2e0a 0a2b 2d2d 2d2d 2d2d  th c=3...+------
│ │ │ │ +0005d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d940: 2d2d 2b0a 7c69 3920 3a20 7365 7452 616e  --+.|i9 : setRan
│ │ │ │ -0005d950: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ +0005d930: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2073  -------+.|i9 : s
│ │ │ │ +0005d940: 6574 5261 6e64 6f6d 5365 6564 2030 2020  etRandomSeed 0  
│ │ │ │ +0005d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d990: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005d980: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d9e0: 2020 7c0a 7c6f 3920 3d20 3020 2020 2020    |.|o9 = 0     
│ │ │ │ +0005d9d0: 2020 2020 2020 207c 0a7c 6f39 203d 2030         |.|o9 = 0
│ │ │ │ +0005d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005da30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005da20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005da80: 2d2d 2b0a 7c69 3130 203a 206e 203d 2033  --+.|i10 : n = 3
│ │ │ │ +0005da70: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ +0005da80: 6e20 3d20 3320 2020 2020 2020 2020 2020  n = 3           
│ │ │ │  0005da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dad0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005dac0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005db20: 2020 7c0a 7c6f 3130 203d 2033 2020 2020    |.|o10 = 3    
│ │ │ │ +0005db10: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +0005db20: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0005db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005db70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005db60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dbc0: 2d2d 2b0a 7c69 3131 203a 2063 203d 2033  --+.|i11 : c = 3
│ │ │ │ +0005dbb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ +0005dbc0: 6320 3d20 3320 2020 2020 2020 2020 2020  c = 3           
│ │ │ │  0005dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dc10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005dc00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dc60: 2020 7c0a 7c6f 3131 203d 2033 2020 2020    |.|o11 = 3    
│ │ │ │ +0005dc50: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +0005dc60: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0005dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dcb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005dca0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dd00: 2d2d 2b0a 7c69 3132 203a 206b 6b20 3d20  --+.|i12 : kk = 
│ │ │ │ -0005dd10: 5a5a 2f31 3031 2020 2020 2020 2020 2020  ZZ/101          
│ │ │ │ +0005dcf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ +0005dd00: 6b6b 203d 205a 5a2f 3130 3120 2020 2020  kk = ZZ/101     
│ │ │ │ +0005dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dd50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005dd40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dda0: 2020 7c0a 7c6f 3132 203d 206b 6b20 2020    |.|o12 = kk   
│ │ │ │ +0005dd90: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +0005dda0: 6b6b 2020 2020 2020 2020 2020 2020 2020  kk              
│ │ │ │  0005ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ddf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005dde0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de40: 2020 7c0a 7c6f 3132 203a 2051 756f 7469    |.|o12 : Quoti
│ │ │ │ -0005de50: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0005de30: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ +0005de40: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0005de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005de80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dee0: 2d2d 2b0a 7c69 3133 203a 2052 203d 206b  --+.|i13 : R = k
│ │ │ │ -0005def0: 6b5b 785f 302e 2e78 5f28 6e2d 3129 5d20  k[x_0..x_(n-1)] 
│ │ │ │ +0005ded0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ +0005dee0: 5220 3d20 6b6b 5b78 5f30 2e2e 785f 286e  R = kk[x_0..x_(n
│ │ │ │ +0005def0: 2d31 295d 2020 2020 2020 2020 2020 2020  -1)]            
│ │ │ │  0005df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005df20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df80: 2020 7c0a 7c6f 3133 203d 2052 2020 2020    |.|o13 = R    
│ │ │ │ +0005df70: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ +0005df80: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0005df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dfd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005dfc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e020: 2020 7c0a 7c6f 3133 203a 2050 6f6c 796e    |.|o13 : Polyn
│ │ │ │ -0005e030: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +0005e010: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ +0005e020: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0005e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e070: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005e060: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e0c0: 2d2d 2b0a 7c69 3134 203a 2049 203d 2069  --+.|i14 : I = i
│ │ │ │ -0005e0d0: 6465 616c 2061 7070 6c79 2863 2c20 692d  deal apply(c, i-
│ │ │ │ -0005e0e0: 3e52 5f69 5e32 2920 2020 2020 2020 2020  >R_i^2)         
│ │ │ │ +0005e0b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +0005e0c0: 4920 3d20 6964 6561 6c20 6170 706c 7928  I = ideal apply(
│ │ │ │ +0005e0d0: 632c 2069 2d3e 525f 695e 3229 2020 2020  c, i->R_i^2)    
│ │ │ │ +0005e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e110: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e100: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e160: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005e170: 2020 2032 2020 2032 2020 2032 2020 2020     2   2   2    
│ │ │ │ +0005e150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e160: 2020 2020 2020 2020 3220 2020 3220 2020          2   2   
│ │ │ │ +0005e170: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0005e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e1b0: 2020 7c0a 7c6f 3134 203d 2069 6465 616c    |.|o14 = ideal
│ │ │ │ -0005e1c0: 2028 7820 2c20 7820 2c20 7820 2920 2020   (x , x , x )   
│ │ │ │ +0005e1a0: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ +0005e1b0: 6964 6561 6c20 2878 202c 2078 202c 2078  ideal (x , x , x
│ │ │ │ +0005e1c0: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │  0005e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e200: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005e210: 2020 2030 2020 2031 2020 2032 2020 2020     0   1   2    
│ │ │ │ +0005e1f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e200: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +0005e210: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0005e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e250: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e240: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2a0: 2020 7c0a 7c6f 3134 203a 2049 6465 616c    |.|o14 : Ideal
│ │ │ │ -0005e2b0: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +0005e290: 2020 2020 2020 207c 0a7c 6f31 3420 3a20         |.|o14 : 
│ │ │ │ +0005e2a0: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │ +0005e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005e2e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005e2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e340: 2d2d 2b0a 7c69 3135 203a 2066 6620 3d20  --+.|i15 : ff = 
│ │ │ │ -0005e350: 6765 6e73 2049 2020 2020 2020 2020 2020  gens I          
│ │ │ │ +0005e330: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20  -------+.|i15 : 
│ │ │ │ +0005e340: 6666 203d 2067 656e 7320 4920 2020 2020  ff = gens I     
│ │ │ │ +0005e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e390: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e380: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3e0: 2020 7c0a 7c6f 3135 203d 207c 2078 5f30    |.|o15 = | x_0
│ │ │ │ -0005e3f0: 5e32 2078 5f31 5e32 2078 5f32 5e32 207c  ^2 x_1^2 x_2^2 |
│ │ │ │ +0005e3d0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20         |.|o15 = 
│ │ │ │ +0005e3e0: 7c20 785f 305e 3220 785f 315e 3220 785f  | x_0^2 x_1^2 x_
│ │ │ │ +0005e3f0: 325e 3220 7c20 2020 2020 2020 2020 2020  2^2 |           
│ │ │ │  0005e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e430: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e420: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e480: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005e490: 2020 2031 2020 2020 2020 3320 2020 2020     1      3     
│ │ │ │ +0005e470: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e480: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +0005e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e4d0: 2020 7c0a 7c6f 3135 203a 204d 6174 7269    |.|o15 : Matri
│ │ │ │ -0005e4e0: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ +0005e4c0: 2020 2020 2020 207c 0a7c 6f31 3520 3a20         |.|o15 : 
│ │ │ │ +0005e4d0: 4d61 7472 6978 2052 2020 3c2d 2d20 5220  Matrix R  <-- R 
│ │ │ │ +0005e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e520: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005e510: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e570: 2d2d 2b0a 7c69 3136 203a 2052 6261 7220  --+.|i16 : Rbar 
│ │ │ │ -0005e580: 3d20 522f 4920 2020 2020 2020 2020 2020  = R/I           
│ │ │ │ +0005e560: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │ +0005e570: 5262 6172 203d 2052 2f49 2020 2020 2020  Rbar = R/I      
│ │ │ │ +0005e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e5c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e610: 2020 7c0a 7c6f 3136 203d 2052 6261 7220    |.|o16 = Rbar 
│ │ │ │ +0005e600: 2020 2020 2020 207c 0a7c 6f31 3620 3d20         |.|o16 = 
│ │ │ │ +0005e610: 5262 6172 2020 2020 2020 2020 2020 2020  Rbar            
│ │ │ │  0005e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e660: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e6b0: 2020 7c0a 7c6f 3136 203a 2051 756f 7469    |.|o16 : Quoti
│ │ │ │ -0005e6c0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0005e6a0: 2020 2020 2020 207c 0a7c 6f31 3620 3a20         |.|o16 : 
│ │ │ │ +0005e6b0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0005e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e700: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005e6f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e750: 2d2d 2b0a 7c69 3137 203a 2062 6172 203d  --+.|i17 : bar =
│ │ │ │ -0005e760: 206d 6170 2852 6261 722c 2052 2920 2020   map(Rbar, R)   
│ │ │ │ +0005e740: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20  -------+.|i17 : 
│ │ │ │ +0005e750: 6261 7220 3d20 6d61 7028 5262 6172 2c20  bar = map(Rbar, 
│ │ │ │ +0005e760: 5229 2020 2020 2020 2020 2020 2020 2020  R)              
│ │ │ │  0005e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e790: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7f0: 2020 7c0a 7c6f 3137 203d 206d 6170 2028    |.|o17 = map (
│ │ │ │ -0005e800: 5262 6172 2c20 522c 207b 7820 2c20 7820  Rbar, R, {x , x 
│ │ │ │ -0005e810: 2c20 7820 7d29 2020 2020 2020 2020 2020  , x })          
│ │ │ │ +0005e7e0: 2020 2020 2020 207c 0a7c 6f31 3720 3d20         |.|o17 = 
│ │ │ │ +0005e7f0: 6d61 7020 2852 6261 722c 2052 2c20 7b78  map (Rbar, R, {x
│ │ │ │ +0005e800: 202c 2078 202c 2078 207d 2920 2020 2020   , x , x })     
│ │ │ │ +0005e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e840: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005e850: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ -0005e860: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0005e830: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e850: 3020 2020 3120 2020 3220 2020 2020 2020  0   1   2       
│ │ │ │ +0005e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e880: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8e0: 2020 7c0a 7c6f 3137 203a 2052 696e 674d    |.|o17 : RingM
│ │ │ │ -0005e8f0: 6170 2052 6261 7220 3c2d 2d20 5220 2020  ap Rbar <-- R   
│ │ │ │ +0005e8d0: 2020 2020 2020 207c 0a7c 6f31 3720 3a20         |.|o17 : 
│ │ │ │ +0005e8e0: 5269 6e67 4d61 7020 5262 6172 203c 2d2d  RingMap Rbar <--
│ │ │ │ +0005e8f0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  0005e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e930: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005e920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005e930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e980: 2d2d 2b0a 7c69 3138 203a 204b 203d 2063  --+.|i18 : K = c
│ │ │ │ -0005e990: 6f6b 6572 2076 6172 7320 5262 6172 2020  oker vars Rbar  
│ │ │ │ +0005e970: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20  -------+.|i18 : 
│ │ │ │ +0005e980: 4b20 3d20 636f 6b65 7220 7661 7273 2052  K = coker vars R
│ │ │ │ +0005e990: 6261 7220 2020 2020 2020 2020 2020 2020  bar             
│ │ │ │  0005e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e9d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005e9c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea20: 2020 7c0a 7c6f 3138 203d 2063 6f6b 6572    |.|o18 = coker
│ │ │ │ -0005ea30: 6e65 6c20 7c20 785f 3020 785f 3120 785f  nel | x_0 x_1 x_
│ │ │ │ -0005ea40: 3220 7c20 2020 2020 2020 2020 2020 2020  2 |             
│ │ │ │ +0005ea10: 2020 2020 2020 207c 0a7c 6f31 3820 3d20         |.|o18 = 
│ │ │ │ +0005ea20: 636f 6b65 726e 656c 207c 2078 5f30 2078  cokernel | x_0 x
│ │ │ │ +0005ea30: 5f31 2078 5f32 207c 2020 2020 2020 2020  _1 x_2 |        
│ │ │ │ +0005ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005ea60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eac0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eae0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0005eab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ead0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0005eae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb10: 2020 7c0a 7c6f 3138 203a 2052 6261 722d    |.|o18 : Rbar-
│ │ │ │ -0005eb20: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -0005eb30: 206f 6620 5262 6172 2020 2020 2020 2020   of Rbar        
│ │ │ │ +0005eb00: 2020 2020 2020 207c 0a7c 6f31 3820 3a20         |.|o18 : 
│ │ │ │ +0005eb10: 5262 6172 2d6d 6f64 756c 652c 2071 756f  Rbar-module, quo
│ │ │ │ +0005eb20: 7469 656e 7420 6f66 2052 6261 7220 2020  tient of Rbar   
│ │ │ │ +0005eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005eb50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005eb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005eb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ebb0: 2d2d 2b0a 7c69 3139 203a 204d 6261 7220  --+.|i19 : Mbar 
│ │ │ │ -0005ebc0: 3d20 7072 756e 6520 636f 6b65 7220 7261  = prune coker ra
│ │ │ │ -0005ebd0: 6e64 6f6d 2852 6261 725e 322c 2052 6261  ndom(Rbar^2, Rba
│ │ │ │ -0005ebe0: 725e 7b2d 322c 2d32 7d29 2020 2020 2020  r^{-2,-2})      
│ │ │ │ -0005ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005eba0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20  -------+.|i19 : 
│ │ │ │ +0005ebb0: 4d62 6172 203d 2070 7275 6e65 2063 6f6b  Mbar = prune cok
│ │ │ │ +0005ebc0: 6572 2072 616e 646f 6d28 5262 6172 5e32  er random(Rbar^2
│ │ │ │ +0005ebd0: 2c20 5262 6172 5e7b 2d32 2c2d 327d 2920  , Rbar^{-2,-2}) 
│ │ │ │ +0005ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ebf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec50: 2020 7c0a 7c6f 3139 203d 2063 6f6b 6572    |.|o19 = coker
│ │ │ │ -0005ec60: 6e65 6c20 7c20 785f 3078 5f31 2b31 3578  nel | x_0x_1+15x
│ │ │ │ -0005ec70: 5f30 785f 322b 3338 785f 3178 5f32 2034  _0x_2+38x_1x_2 4
│ │ │ │ -0005ec80: 3578 5f30 785f 322b 3239 785f 3178 5f32  5x_0x_2+29x_1x_2
│ │ │ │ -0005ec90: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ -0005eca0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005ecb0: 2020 2020 7c20 3335 785f 3078 5f32 2d33      | 35x_0x_2-3
│ │ │ │ -0005ecc0: 3078 5f31 785f 3220 2020 2020 2020 2078  0x_1x_2        x
│ │ │ │ -0005ecd0: 5f30 785f 312d 3130 785f 3078 5f32 2d32  _0x_1-10x_0x_2-2
│ │ │ │ -0005ece0: 3278 5f31 785f 3220 7c20 2020 2020 2020  2x_1x_2 |       
│ │ │ │ -0005ecf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005ec40: 2020 2020 2020 207c 0a7c 6f31 3920 3d20         |.|o19 = 
│ │ │ │ +0005ec50: 636f 6b65 726e 656c 207c 2078 5f30 785f  cokernel | x_0x_
│ │ │ │ +0005ec60: 312b 3135 785f 3078 5f32 2b33 3878 5f31  1+15x_0x_2+38x_1
│ │ │ │ +0005ec70: 785f 3220 3435 785f 3078 5f32 2b32 3978  x_2 45x_0x_2+29x
│ │ │ │ +0005ec80: 5f31 785f 3220 2020 2020 2020 207c 2020  _1x_2        |  
│ │ │ │ +0005ec90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005eca0: 2020 2020 2020 2020 207c 2033 3578 5f30           | 35x_0
│ │ │ │ +0005ecb0: 785f 322d 3330 785f 3178 5f32 2020 2020  x_2-30x_1x_2    
│ │ │ │ +0005ecc0: 2020 2020 785f 3078 5f31 2d31 3078 5f30      x_0x_1-10x_0
│ │ │ │ +0005ecd0: 785f 322d 3232 785f 3178 5f32 207c 2020  x_2-22x_1x_2 |  
│ │ │ │ +0005ece0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed60: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0005ed30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ed50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0005ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed90: 2020 7c0a 7c6f 3139 203a 2052 6261 722d    |.|o19 : Rbar-
│ │ │ │ -0005eda0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -0005edb0: 206f 6620 5262 6172 2020 2020 2020 2020   of Rbar        
│ │ │ │ +0005ed80: 2020 2020 2020 207c 0a7c 6f31 3920 3a20         |.|o19 : 
│ │ │ │ +0005ed90: 5262 6172 2d6d 6f64 756c 652c 2071 756f  Rbar-module, quo
│ │ │ │ +0005eda0: 7469 656e 7420 6f66 2052 6261 7220 2020  tient of Rbar   
│ │ │ │ +0005edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ede0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005edd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ee10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ee30: 2d2d 2b0a 7c69 3230 203a 2045 5320 3d20  --+.|i20 : ES = 
│ │ │ │ -0005ee40: 6e65 7745 7874 284d 6261 722c 4b2c 4c69  newExt(Mbar,K,Li
│ │ │ │ -0005ee50: 6674 203d 3e20 7472 7565 2920 2020 2020  ft => true)     
│ │ │ │ +0005ee20: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20  -------+.|i20 : 
│ │ │ │ +0005ee30: 4553 203d 206e 6577 4578 7428 4d62 6172  ES = newExt(Mbar
│ │ │ │ +0005ee40: 2c4b 2c4c 6966 7420 3d3e 2074 7275 6529  ,K,Lift => true)
│ │ │ │ +0005ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005ee70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eed0: 2020 7c0a 7c6f 3230 203d 2063 6f6b 6572    |.|o20 = coker
│ │ │ │ -0005eee0: 6e65 6c20 7b30 2c20 307d 2020 207c 2078  nel {0, 0}   | x
│ │ │ │ -0005eef0: 5f32 2078 5f31 2078 5f30 2030 2020 2030  _2 x_1 x_0 0   0
│ │ │ │ -0005ef00: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005ef10: 2020 2030 2020 2030 2020 2030 2020 2073     0   0   0   s
│ │ │ │ -0005ef20: 5f32 7c0a 7c20 2020 2020 2020 2020 2020  _2|.|           
│ │ │ │ -0005ef30: 2020 2020 7b30 2c20 307d 2020 207c 2030      {0, 0}   | 0
│ │ │ │ -0005ef40: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ -0005ef50: 5f31 2078 5f30 2030 2020 2030 2020 2030  _1 x_0 0   0   0
│ │ │ │ -0005ef60: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005ef70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005ef80: 2020 2020 7b2d 322c 202d 337d 207c 2030      {-2, -3} | 0
│ │ │ │ -0005ef90: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005efa0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005efb0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005efc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005efd0: 2020 2020 7b2d 322c 202d 337d 207c 2030      {-2, -3} | 0
│ │ │ │ -0005efe0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005eff0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f000: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005f020: 2020 2020 7b2d 322c 202d 337d 207c 2030      {-2, -3} | 0
│ │ │ │ -0005f030: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f040: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f050: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005f070: 2020 2020 7b2d 322c 202d 337d 207c 2030      {-2, -3} | 0
│ │ │ │ -0005f080: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f090: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f0a0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f0b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005f0c0: 2020 2020 7b2d 312c 202d 327d 207c 2030      {-1, -2} | 0
│ │ │ │ -0005f0d0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f0e0: 2020 2030 2020 2078 5f32 2078 5f31 2078     0   x_2 x_1 x
│ │ │ │ -0005f0f0: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ -0005f100: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005f110: 2020 2020 7b2d 312c 202d 327d 207c 2030      {-1, -2} | 0
│ │ │ │ -0005f120: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f130: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -0005f140: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ -0005f150: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005eec0: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +0005eed0: 636f 6b65 726e 656c 207b 302c 2030 7d20  cokernel {0, 0} 
│ │ │ │ +0005eee0: 2020 7c20 785f 3220 785f 3120 785f 3020    | x_2 x_1 x_0 
│ │ │ │ +0005eef0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005ef00: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005ef10: 3020 2020 735f 327c 0a7c 2020 2020 2020  0   s_2|.|      
│ │ │ │ +0005ef20: 2020 2020 2020 2020 207b 302c 2030 7d20           {0, 0} 
│ │ │ │ +0005ef30: 2020 7c20 3020 2020 3020 2020 3020 2020    | 0   0   0   
│ │ │ │ +0005ef40: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ +0005ef50: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005ef60: 3020 2020 3020 207c 0a7c 2020 2020 2020  0   0  |.|      
│ │ │ │ +0005ef70: 2020 2020 2020 2020 207b 2d32 2c20 2d33           {-2, -3
│ │ │ │ +0005ef80: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005ef90: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005efa0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005efb0: 3020 2020 3020 207c 0a7c 2020 2020 2020  0   0  |.|      
│ │ │ │ +0005efc0: 2020 2020 2020 2020 207b 2d32 2c20 2d33           {-2, -3
│ │ │ │ +0005efd0: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005efe0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005eff0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005f000: 3020 2020 3020 207c 0a7c 2020 2020 2020  0   0  |.|      
│ │ │ │ +0005f010: 2020 2020 2020 2020 207b 2d32 2c20 2d33           {-2, -3
│ │ │ │ +0005f020: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005f030: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005f040: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005f050: 3020 2020 3020 207c 0a7c 2020 2020 2020  0   0  |.|      
│ │ │ │ +0005f060: 2020 2020 2020 2020 207b 2d32 2c20 2d33           {-2, -3
│ │ │ │ +0005f070: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005f080: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005f090: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005f0a0: 3020 2020 3020 207c 0a7c 2020 2020 2020  0   0  |.|      
│ │ │ │ +0005f0b0: 2020 2020 2020 2020 207b 2d31 2c20 2d32           {-1, -2
│ │ │ │ +0005f0c0: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005f0d0: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ +0005f0e0: 785f 3120 785f 3020 3020 2020 3020 2020  x_1 x_0 0   0   
│ │ │ │ +0005f0f0: 3020 2020 3020 207c 0a7c 2020 2020 2020  0   0  |.|      
│ │ │ │ +0005f100: 2020 2020 2020 2020 207b 2d31 2c20 2d32           {-1, -2
│ │ │ │ +0005f110: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005f120: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005f130: 3020 2020 3020 2020 785f 3220 785f 3120  0   0   x_2 x_1 
│ │ │ │ +0005f140: 785f 3020 3020 207c 0a7c 2020 2020 2020  x_0 0  |.|      
│ │ │ │ +0005f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005f190: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1e0: 2020 2020 2020 3820 2020 2020 2020 2020        8         
│ │ │ │ -0005f1f0: 2020 7c0a 7c6f 3230 203a 206b 6b5b 7320    |.|o20 : kk[s 
│ │ │ │ -0005f200: 2e2e 7320 2c20 7820 2e2e 7820 5d2d 6d6f  ..s , x ..x ]-mo
│ │ │ │ -0005f210: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ -0005f220: 6620 286b 6b5b 7320 2e2e 7320 2c20 7820  f (kk[s ..s , x 
│ │ │ │ -0005f230: 2e2e 7820 5d29 2020 2020 2020 2020 2020  ..x ])          
│ │ │ │ -0005f240: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ -0005f250: 2020 2032 2020 2030 2020 2032 2020 2020     2   0   2    
│ │ │ │ -0005f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f270: 2020 2020 2020 2030 2020 2032 2020 2030         0   2   0
│ │ │ │ -0005f280: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0005f290: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +0005f1d0: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +0005f1e0: 2020 2020 2020 207c 0a7c 6f32 3020 3a20         |.|o20 : 
│ │ │ │ +0005f1f0: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78  kk[s ..s , x ..x
│ │ │ │ +0005f200: 205d 2d6d 6f64 756c 652c 2071 756f 7469   ]-module, quoti
│ │ │ │ +0005f210: 656e 7420 6f66 2028 6b6b 5b73 202e 2e73  ent of (kk[s ..s
│ │ │ │ +0005f220: 202c 2078 202e 2e78 205d 2920 2020 2020   , x ..x ])     
│ │ │ │ +0005f230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005f240: 2020 2020 3020 2020 3220 2020 3020 2020      0   2   0   
│ │ │ │ +0005f250: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0005f260: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0005f270: 3220 2020 3020 2020 3220 2020 2020 2020  2   0   2       
│ │ │ │ +0005f280: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +0005f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f2e0: 2d2d 7c0a 7c73 5f31 2073 5f30 2030 2020  --|.|s_1 s_0 0  
│ │ │ │ -0005f2f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f300: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f310: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f320: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f330: 2020 7c0a 7c30 2020 2030 2020 2073 5f32    |.|0   0   s_2
│ │ │ │ -0005f340: 2073 5f31 2073 5f30 2030 2020 2030 2020   s_1 s_0 0   0  
│ │ │ │ -0005f350: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f360: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f370: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f380: 2020 7c0a 7c30 2020 2030 2020 2030 2020    |.|0   0   0  
│ │ │ │ -0005f390: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -0005f3a0: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ -0005f3b0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f3c0: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f3d0: 2020 7c0a 7c30 2020 2030 2020 2030 2020    |.|0   0   0  
│ │ │ │ -0005f3e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f3f0: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ -0005f400: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f410: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f420: 2020 7c0a 7c30 2020 2030 2020 2030 2020    |.|0   0   0  
│ │ │ │ -0005f430: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f440: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f450: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -0005f460: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f470: 2020 7c0a 7c30 2020 2030 2020 2030 2020    |.|0   0   0  
│ │ │ │ -0005f480: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f490: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f4a0: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -0005f4b0: 2078 5f31 2078 5f30 2020 2020 2020 2020   x_1 x_0        
│ │ │ │ -0005f4c0: 2020 7c0a 7c30 2020 2030 2020 2030 2020    |.|0   0   0  
│ │ │ │ -0005f4d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f4e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f4f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f500: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f510: 2020 7c0a 7c30 2020 2030 2020 2030 2020    |.|0   0   0  
│ │ │ │ -0005f520: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f530: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f540: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005f550: 2030 2020 2030 2020 2020 2020 2020 2020   0   0          
│ │ │ │ -0005f560: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +0005f2d0: 2d2d 2d2d 2d2d 2d7c 0a7c 735f 3120 735f  -------|.|s_1 s_
│ │ │ │ +0005f2e0: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ +0005f2f0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f300: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f310: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ +0005f320: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f330: 2020 735f 3220 735f 3120 735f 3020 3020    s_2 s_1 s_0 0 
│ │ │ │ +0005f340: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f350: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f360: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ +0005f370: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f380: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ +0005f390: 3220 785f 3120 785f 3020 3020 2020 3020  2 x_1 x_0 0   0 
│ │ │ │ +0005f3a0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f3b0: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ +0005f3c0: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f3d0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f3e0: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ +0005f3f0: 3120 785f 3020 3020 2020 3020 2020 3020  1 x_0 0   0   0 
│ │ │ │ +0005f400: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ +0005f410: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f420: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f430: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f440: 2020 3020 2020 785f 3220 785f 3120 785f    0   x_2 x_1 x_
│ │ │ │ +0005f450: 3020 3020 2020 3020 2020 3020 2020 2020  0 0   0   0     
│ │ │ │ +0005f460: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f470: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f480: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f490: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f4a0: 2020 785f 3220 785f 3120 785f 3020 2020    x_2 x_1 x_0   
│ │ │ │ +0005f4b0: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f4c0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f4d0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f4e0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f4f0: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ +0005f500: 2020 2020 2020 207c 0a7c 3020 2020 3020         |.|0   0 
│ │ │ │ +0005f510: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f520: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f530: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +0005f540: 2020 3020 2020 3020 2020 3020 2020 2020    0   0   0     
│ │ │ │ +0005f550: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +0005f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f5b0: 2d2d 7c0a 7c30 2020 2020 2020 2020 2020  --|.|0          
│ │ │ │ -0005f5c0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f5d0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f5e0: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ -0005f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f600: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ -0005f610: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f620: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f630: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ -0005f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f650: 2020 7c0a 7c73 5f30 2d31 3173 5f31 2d34    |.|s_0-11s_1-4
│ │ │ │ -0005f660: 3073 5f32 202d 735f 3120 2020 2020 2020  0s_2 -s_1       
│ │ │ │ -0005f670: 2020 2020 2039 735f 312d 3233 735f 3220       9s_1-23s_2 
│ │ │ │ -0005f680: 2020 2020 3137 735f 3120 2020 2020 2020      17s_1       
│ │ │ │ -0005f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f6a0: 2020 7c0a 7c34 3573 5f31 2d33 3573 5f32    |.|45s_1-35s_2
│ │ │ │ -0005f6b0: 2020 2020 2033 3873 5f31 2020 2020 2020       38s_1      
│ │ │ │ -0005f6c0: 2020 2020 2073 5f30 2d37 735f 312b 3135       s_0-7s_1+15
│ │ │ │ -0005f6d0: 735f 3220 735f 3120 2020 2020 2020 2020  s_2 s_1         
│ │ │ │ -0005f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f6f0: 2020 7c0a 7c2d 3130 735f 312d 3236 735f    |.|-10s_1-26s_
│ │ │ │ -0005f700: 3220 2020 2073 5f30 2b34 3973 5f31 2d34  2    s_0+49s_1-4
│ │ │ │ -0005f710: 3073 5f32 2033 3473 5f31 2b34 735f 3220  0s_2 34s_1+4s_2 
│ │ │ │ -0005f720: 2020 2020 3973 5f31 2d32 3373 5f32 2020      9s_1-23s_2  
│ │ │ │ -0005f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f740: 2020 7c0a 7c35 3073 5f31 2d73 5f32 2020    |.|50s_1-s_2  
│ │ │ │ -0005f750: 2020 2020 2034 3573 5f31 2d33 3573 5f32       45s_1-35s_2
│ │ │ │ -0005f760: 2020 2020 2031 3073 5f31 2b32 3673 5f32       10s_1+26s_2
│ │ │ │ -0005f770: 2020 2020 735f 302d 3438 735f 312b 3135      s_0-48s_1+15
│ │ │ │ -0005f780: 735f 3220 2020 2020 2020 2020 2020 2020  s_2             
│ │ │ │ -0005f790: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ -0005f7a0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f7b0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f7c0: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ -0005f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7e0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ -0005f7f0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f800: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0005f810: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ -0005f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f830: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +0005f5a0: 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020 2020  -------|.|0     
│ │ │ │ +0005f5b0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f5c0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f5d0: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0005f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f5f0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f600: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f610: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f620: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0005f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f640: 2020 2020 2020 207c 0a7c 735f 302d 3131         |.|s_0-11
│ │ │ │ +0005f650: 735f 312d 3430 735f 3220 2d73 5f31 2020  s_1-40s_2 -s_1  
│ │ │ │ +0005f660: 2020 2020 2020 2020 2020 3973 5f31 2d32            9s_1-2
│ │ │ │ +0005f670: 3373 5f32 2020 2020 2031 3773 5f31 2020  3s_2     17s_1  
│ │ │ │ +0005f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f690: 2020 2020 2020 207c 0a7c 3435 735f 312d         |.|45s_1-
│ │ │ │ +0005f6a0: 3335 735f 3220 2020 2020 3338 735f 3120  35s_2     38s_1 
│ │ │ │ +0005f6b0: 2020 2020 2020 2020 2020 735f 302d 3773            s_0-7s
│ │ │ │ +0005f6c0: 5f31 2b31 3573 5f32 2073 5f31 2020 2020  _1+15s_2 s_1    
│ │ │ │ +0005f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f6e0: 2020 2020 2020 207c 0a7c 2d31 3073 5f31         |.|-10s_1
│ │ │ │ +0005f6f0: 2d32 3673 5f32 2020 2020 735f 302b 3439  -26s_2    s_0+49
│ │ │ │ +0005f700: 735f 312d 3430 735f 3220 3334 735f 312b  s_1-40s_2 34s_1+
│ │ │ │ +0005f710: 3473 5f32 2020 2020 2039 735f 312d 3233  4s_2     9s_1-23
│ │ │ │ +0005f720: 735f 3220 2020 2020 2020 2020 2020 2020  s_2             
│ │ │ │ +0005f730: 2020 2020 2020 207c 0a7c 3530 735f 312d         |.|50s_1-
│ │ │ │ +0005f740: 735f 3220 2020 2020 2020 3435 735f 312d  s_2       45s_1-
│ │ │ │ +0005f750: 3335 735f 3220 2020 2020 3130 735f 312b  35s_2     10s_1+
│ │ │ │ +0005f760: 3236 735f 3220 2020 2073 5f30 2d34 3873  26s_2    s_0-48s
│ │ │ │ +0005f770: 5f31 2b31 3573 5f32 2020 2020 2020 2020  _1+15s_2        
│ │ │ │ +0005f780: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f790: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f7a0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f7b0: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0005f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f7d0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f7e0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f7f0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f800: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0005f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f820: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +0005f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f880: 2d2d 7c0a 7c30 2020 2020 2020 2020 2020  --|.|0          
│ │ │ │ +0005f870: 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020 2020  -------|.|0     
│ │ │ │ +0005f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8d0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005f8c0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f920: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005f910: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f970: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005f960: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9c0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005f9b0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa10: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fa00: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa60: 2020 7c0a 7c73 5f30 5e32 2b34 3273 5f30    |.|s_0^2+42s_0
│ │ │ │ -0005fa70: 735f 312d 3330 735f 315e 322d 3235 735f  s_1-30s_1^2-25s_
│ │ │ │ -0005fa80: 3073 5f32 2d33 3573 5f31 735f 322b 3973  0s_2-35s_1s_2+9s
│ │ │ │ -0005fa90: 5f32 5e32 2020 2020 2020 2020 2020 2020  _2^2            
│ │ │ │ -0005faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fab0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fa50: 2020 2020 2020 207c 0a7c 735f 305e 322b         |.|s_0^2+
│ │ │ │ +0005fa60: 3432 735f 3073 5f31 2d33 3073 5f31 5e32  42s_0s_1-30s_1^2
│ │ │ │ +0005fa70: 2d32 3573 5f30 735f 322d 3335 735f 3173  -25s_0s_2-35s_1s
│ │ │ │ +0005fa80: 5f32 2b39 735f 325e 3220 2020 2020 2020  _2+9s_2^2       
│ │ │ │ +0005fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005faa0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb00: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │ +0005faf0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +0005fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb50: 2d2d 7c0a 7c30 2020 2020 2020 2020 2020  --|.|0          
│ │ │ │ +0005fb40: 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020 2020  -------|.|0     
│ │ │ │ +0005fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb80: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fba0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fb70: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fb90: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbd0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbf0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fbc0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fbe0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc20: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc40: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fc10: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fc30: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc70: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc90: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fc60: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fc80: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcc0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fce0: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fcb0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fcd0: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd10: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd30: 2020 7c0a 7c30 2020 2020 2020 2020 2020    |.|0          
│ │ │ │ +0005fd00: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fd20: 2020 2020 2020 207c 0a7c 3020 2020 2020         |.|0     
│ │ │ │ +0005fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd60: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -0005fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd80: 2020 7c0a 7c73 5f30 5e32 2b34 3273 5f30    |.|s_0^2+42s_0
│ │ │ │ -0005fd90: 735f 312d 3330 735f 315e 322d 3235 735f  s_1-30s_1^2-25s_
│ │ │ │ -0005fda0: 3073 5f32 2d33 3573 5f31 735f 322b 3973  0s_2-35s_1s_2+9s
│ │ │ │ -0005fdb0: 5f32 5e32 207c 2020 2020 2020 2020 2020  _2^2 |          
│ │ │ │ -0005fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fdd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005fd50: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +0005fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fd70: 2020 2020 2020 207c 0a7c 735f 305e 322b         |.|s_0^2+
│ │ │ │ +0005fd80: 3432 735f 3073 5f31 2d33 3073 5f31 5e32  42s_0s_1-30s_1^2
│ │ │ │ +0005fd90: 2d32 3573 5f30 735f 322d 3335 735f 3173  -25s_0s_2-35s_1s
│ │ │ │ +0005fda0: 5f32 2b39 735f 325e 3220 7c20 2020 2020  _2+9s_2^2 |     
│ │ │ │ +0005fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fdc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fe00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fe10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fe20: 2d2d 2b0a 7c69 3231 203a 2053 203d 2072  --+.|i21 : S = r
│ │ │ │ -0005fe30: 696e 6720 4553 2020 2020 2020 2020 2020  ing ES          
│ │ │ │ +0005fe10: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20  -------+.|i21 : 
│ │ │ │ +0005fe20: 5320 3d20 7269 6e67 2045 5320 2020 2020  S = ring ES     
│ │ │ │ +0005fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005fe60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fec0: 2020 7c0a 7c6f 3231 203d 2053 2020 2020    |.|o21 = S    
│ │ │ │ +0005feb0: 2020 2020 2020 207c 0a7c 6f32 3120 3d20         |.|o21 = 
│ │ │ │ +0005fec0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  0005fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0005ff00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff60: 2020 7c0a 7c6f 3231 203a 2050 6f6c 796e    |.|o21 : Polyn
│ │ │ │ -0005ff70: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +0005ff50: 2020 2020 2020 207c 0a7c 6f32 3120 3a20         |.|o21 : 
│ │ │ │ +0005ff60: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0005ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005ffa0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ffd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ffe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060000: 2d2d 2b0a 0a63 6f6d 7061 7265 2077 6974  --+..compare wit
│ │ │ │ -00060010: 6820 7468 6520 6275 696c 742d 696e 2045  h the built-in E
│ │ │ │ -00060020: 7874 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  xt..+-----------
│ │ │ │ +0005fff0: 2d2d 2d2d 2d2d 2d2b 0a0a 636f 6d70 6172  -------+..compar
│ │ │ │ +00060000: 6520 7769 7468 2074 6865 2062 7569 6c74  e with the built
│ │ │ │ +00060010: 2d69 6e20 4578 740a 0a2b 2d2d 2d2d 2d2d  -in Ext..+------
│ │ │ │ +00060020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00060060: 7c69 3232 203a 2045 4520 3d20 4578 7428  |i22 : EE = Ext(
│ │ │ │ -00060070: 4d62 6172 2c4b 293b 2020 2020 2020 2020  Mbar,K);        
│ │ │ │ -00060080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00060050: 2d2d 2d2b 0a7c 6932 3220 3a20 4545 203d  ---+.|i22 : EE =
│ │ │ │ +00060060: 2045 7874 284d 6261 722c 4b29 3b20 2020   Ext(Mbar,K);   
│ │ │ │ +00060070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00060090: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000600a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000600b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000600c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000600d0: 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a 2053  ------+.|i23 : S
│ │ │ │ -000600e0: 2720 3d20 7269 6e67 2045 4520 2d2d 206e  ' = ring EE -- n
│ │ │ │ -000600f0: 6f74 6520 7468 6174 2053 2720 6973 2074  ote that S' is t
│ │ │ │ -00060100: 6865 2070 6f6c 796e 6f6d 6961 6c20 7269  he polynomial ri
│ │ │ │ -00060110: 6e67 7c0a 7c20 2020 2020 2020 2020 2020  ng|.|           
│ │ │ │ +000600c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +000600d0: 3320 3a20 5327 203d 2072 696e 6720 4545  3 : S' = ring EE
│ │ │ │ +000600e0: 202d 2d20 6e6f 7465 2074 6861 7420 5327   -- note that S'
│ │ │ │ +000600f0: 2069 7320 7468 6520 706f 6c79 6e6f 6d69   is the polynomi
│ │ │ │ +00060100: 616c 2072 696e 677c 0a7c 2020 2020 2020  al ring|.|      
│ │ │ │ +00060110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00060150: 7c6f 3233 203d 2053 2720 2020 2020 2020  |o23 = S'       
│ │ │ │ +00060140: 2020 207c 0a7c 6f32 3320 3d20 5327 2020     |.|o23 = S'  
│ │ │ │ +00060150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00060170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00060180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00060190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000601a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000601b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000601c0: 2020 2020 2020 7c0a 7c6f 3233 203a 2050        |.|o23 : P
│ │ │ │ -000601d0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +000601b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000601c0: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +000601d0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  000601e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000601f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060200: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000601f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00060200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00060240: 0a54 6865 2074 776f 2076 6572 7369 6f6e  .The two version
│ │ │ │ -00060250: 7320 6f66 2045 7874 2061 7070 6561 7220  s of Ext appear 
│ │ │ │ -00060260: 746f 2062 6520 7468 6520 7361 6d65 2075  to be the same u
│ │ │ │ -00060270: 7020 746f 2063 6861 6e67 6520 6f66 2076  p to change of v
│ │ │ │ -00060280: 6172 6961 626c 6573 3a0a 0a2b 2d2d 2d2d  ariables:..+----
│ │ │ │ +00060230: 2d2d 2d2b 0a0a 5468 6520 7477 6f20 7665  ---+..The two ve
│ │ │ │ +00060240: 7273 696f 6e73 206f 6620 4578 7420 6170  rsions of Ext ap
│ │ │ │ +00060250: 7065 6172 2074 6f20 6265 2074 6865 2073  pear to be the s
│ │ │ │ +00060260: 616d 6520 7570 2074 6f20 6368 616e 6765  ame up to change
│ │ │ │ +00060270: 206f 6620 7661 7269 6162 6c65 733a 0a0a   of variables:..
│ │ │ │ +00060280: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00060290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000602a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000602b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000602c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000602d0: 2b0a 7c69 3234 203a 2041 203d 2072 6573  +.|i24 : A = res
│ │ │ │ -000602e0: 2045 5320 2020 2020 2020 2020 2020 2020   ES             
│ │ │ │ +000602c0: 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20 4120  -----+.|i24 : A 
│ │ │ │ +000602d0: 3d20 7265 7320 4553 2020 2020 2020 2020  = res ES        
│ │ │ │ +000602e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000602f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00060300: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00060310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00060360: 7c20 2020 2020 2020 3820 2020 2020 2033  |       8      3
│ │ │ │ -00060370: 3620 2020 2020 2036 3620 2020 2020 2036  6      66      6
│ │ │ │ -00060380: 3420 2020 2020 2033 3620 2020 2020 2031  4      36      1
│ │ │ │ -00060390: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ -000603a0: 2020 2020 207c 0a7c 6f32 3420 3d20 5320       |.|o24 = S 
│ │ │ │ -000603b0: 203c 2d2d 2053 2020 203c 2d2d 2053 2020   <-- S   <-- S  
│ │ │ │ -000603c0: 203c 2d2d 2053 2020 203c 2d2d 2053 2020   <-- S   <-- S  
│ │ │ │ -000603d0: 203c 2d2d 2053 2020 203c 2d2d 2053 2020   <-- S   <-- S  
│ │ │ │ -000603e0: 3c2d 2d20 3020 2020 2020 2020 7c0a 7c20  <-- 0       |.| 
│ │ │ │ +00060350: 2020 207c 0a7c 2020 2020 2020 2038 2020     |.|       8  
│ │ │ │ +00060360: 2020 2020 3336 2020 2020 2020 3636 2020      36      66  
│ │ │ │ +00060370: 2020 2020 3634 2020 2020 2020 3336 2020      64      36  
│ │ │ │ +00060380: 2020 2020 3132 2020 2020 2020 3220 2020      12      2   
│ │ │ │ +00060390: 2020 2020 2020 2020 2020 7c0a 7c6f 3234            |.|o24
│ │ │ │ +000603a0: 203d 2053 2020 3c2d 2d20 5320 2020 3c2d   = S  <-- S   <-
│ │ │ │ +000603b0: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d  - S   <-- S   <-
│ │ │ │ +000603c0: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d  - S   <-- S   <-
│ │ │ │ +000603d0: 2d20 5320 203c 2d2d 2030 2020 2020 2020  - S  <-- 0      
│ │ │ │ +000603e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000603f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060430: 2020 207c 0a7c 2020 2020 2020 3020 2020     |.|      0   
│ │ │ │ -00060440: 2020 2031 2020 2020 2020 2032 2020 2020     1       2    
│ │ │ │ -00060450: 2020 2033 2020 2020 2020 2034 2020 2020     3       4    
│ │ │ │ -00060460: 2020 2035 2020 2020 2020 2036 2020 2020     5       6    
│ │ │ │ -00060470: 2020 3720 2020 2020 2020 7c0a 7c20 2020    7       |.|   
│ │ │ │ +00060420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00060430: 2030 2020 2020 2020 3120 2020 2020 2020   0      1       
│ │ │ │ +00060440: 3220 2020 2020 2020 3320 2020 2020 2020  2       3       
│ │ │ │ +00060450: 3420 2020 2020 2020 3520 2020 2020 2020  4       5       
│ │ │ │ +00060460: 3620 2020 2020 2037 2020 2020 2020 207c  6      7       |
│ │ │ │ +00060470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00060480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000604a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000604b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000604c0: 207c 0a7c 6f32 3420 3a20 4368 6169 6e43   |.|o24 : ChainC
│ │ │ │ -000604d0: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +000604b0: 2020 2020 2020 7c0a 7c6f 3234 203a 2043        |.|o24 : C
│ │ │ │ +000604c0: 6861 696e 436f 6d70 6c65 7820 2020 2020  hainComplex     
│ │ │ │ +000604d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000604e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000604f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060500: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000604f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00060500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00060550: 0a7c 6932 3520 3a20 4220 3d20 7265 7320  .|i25 : B = res 
│ │ │ │ -00060560: 4545 2020 2020 2020 2020 2020 2020 2020  EE              
│ │ │ │ +00060540: 2d2d 2d2d 2b0a 7c69 3235 203a 2042 203d  ----+.|i25 : B =
│ │ │ │ +00060550: 2072 6573 2045 4520 2020 2020 2020 2020   res EE         
│ │ │ │ +00060560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00060580: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00060590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000605a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000605b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000605c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000605d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000605e0: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ -000605f0: 3336 2020 2020 2020 2036 3620 2020 2020  36       66     
│ │ │ │ -00060600: 2020 3634 2020 2020 2020 2033 3620 2020    64       36   
│ │ │ │ -00060610: 2020 2020 3132 2020 2020 2020 2032 2020      12       2  
│ │ │ │ -00060620: 2020 2020 7c0a 7c6f 3235 203d 2053 2720      |.|o25 = S' 
│ │ │ │ -00060630: 203c 2d2d 2053 2720 2020 3c2d 2d20 5327   <-- S'   <-- S'
│ │ │ │ -00060640: 2020 203c 2d2d 2053 2720 2020 3c2d 2d20     <-- S'   <-- 
│ │ │ │ -00060650: 5327 2020 203c 2d2d 2053 2720 2020 3c2d  S'   <-- S'   <-
│ │ │ │ -00060660: 2d20 5327 2020 3c2d 2d20 307c 0a7c 2020  - S'  <-- 0|.|  
│ │ │ │ +000605d0: 2020 7c0a 7c20 2020 2020 2020 2038 2020    |.|        8  
│ │ │ │ +000605e0: 2020 2020 2033 3620 2020 2020 2020 3636       36       66
│ │ │ │ +000605f0: 2020 2020 2020 2036 3420 2020 2020 2020         64       
│ │ │ │ +00060600: 3336 2020 2020 2020 2031 3220 2020 2020  36       12     
│ │ │ │ +00060610: 2020 3220 2020 2020 207c 0a7c 6f32 3520    2      |.|o25 
│ │ │ │ +00060620: 3d20 5327 2020 3c2d 2d20 5327 2020 203c  = S'  <-- S'   <
│ │ │ │ +00060630: 2d2d 2053 2720 2020 3c2d 2d20 5327 2020  -- S'   <-- S'  
│ │ │ │ +00060640: 203c 2d2d 2053 2720 2020 3c2d 2d20 5327   <-- S'   <-- S'
│ │ │ │ +00060650: 2020 203c 2d2d 2053 2720 203c 2d2d 2030     <-- S'  <-- 0
│ │ │ │ +00060660: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00060670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000606a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000606b0: 2020 7c0a 7c20 2020 2020 2030 2020 2020    |.|      0    
│ │ │ │ -000606c0: 2020 2031 2020 2020 2020 2020 3220 2020     1        2   
│ │ │ │ -000606d0: 2020 2020 2033 2020 2020 2020 2020 3420       3        4 
│ │ │ │ -000606e0: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ -000606f0: 3620 2020 2020 2020 377c 0a7c 2020 2020  6       7|.|    
│ │ │ │ +000606a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000606b0: 3020 2020 2020 2020 3120 2020 2020 2020  0       1       
│ │ │ │ +000606c0: 2032 2020 2020 2020 2020 3320 2020 2020   2        3     
│ │ │ │ +000606d0: 2020 2034 2020 2020 2020 2020 3520 2020     4        5   
│ │ │ │ +000606e0: 2020 2020 2036 2020 2020 2020 2037 7c0a       6       7|.
│ │ │ │ +000606f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00060700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060740: 7c0a 7c6f 3235 203a 2043 6861 696e 436f  |.|o25 : ChainCo
│ │ │ │ -00060750: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ +00060730: 2020 2020 207c 0a7c 6f32 3520 3a20 4368       |.|o25 : Ch
│ │ │ │ +00060740: 6169 6e43 6f6d 706c 6578 2020 2020 2020  ainComplex      
│ │ │ │ +00060750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060780: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00060770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00060780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000607c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000607d0: 7c69 3236 203a 2061 6c6c 286c 656e 6774  |i26 : all(lengt
│ │ │ │ -000607e0: 6820 412b 312c 2069 2d3e 2073 6f72 7420  h A+1, i-> sort 
│ │ │ │ -000607f0: 6465 6772 6565 7320 415f 6920 3d3d 2073  degrees A_i == s
│ │ │ │ -00060800: 6f72 7420 6465 6772 6565 7320 425f 6929  ort degrees B_i)
│ │ │ │ -00060810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000607c0: 2d2d 2d2b 0a7c 6932 3620 3a20 616c 6c28  ---+.|i26 : all(
│ │ │ │ +000607d0: 6c65 6e67 7468 2041 2b31 2c20 692d 3e20  length A+1, i-> 
│ │ │ │ +000607e0: 736f 7274 2064 6567 7265 6573 2041 5f69  sort degrees A_i
│ │ │ │ +000607f0: 203d 3d20 736f 7274 2064 6567 7265 6573   == sort degrees
│ │ │ │ +00060800: 2042 5f69 2920 2020 2020 7c0a 7c20 2020   B_i)     |.|   
│ │ │ │ +00060810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060850: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00060860: 3236 203d 2074 7275 6520 2020 2020 2020  26 = true       
│ │ │ │ +00060850: 207c 0a7c 6f32 3620 3d20 7472 7565 2020   |.|o26 = true  
│ │ │ │ +00060860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000608a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00060890: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000608a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000608b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000608c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000608d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000608e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a62 7574  ----------+..but
│ │ │ │ -000608f0: 2074 6865 7920 6861 7665 2061 7070 6172   they have appar
│ │ │ │ -00060900: 656e 746c 7920 6469 6666 6572 656e 7420  ently different 
│ │ │ │ -00060910: 616e 6e69 6869 6c61 746f 7273 0a0a 2b2d  annihilators..+-
│ │ │ │ +000608d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000608e0: 0a0a 6275 7420 7468 6579 2068 6176 6520  ..but they have 
│ │ │ │ +000608f0: 6170 7061 7265 6e74 6c79 2064 6966 6665  apparently diffe
│ │ │ │ +00060900: 7265 6e74 2061 6e6e 6968 696c 6174 6f72  rent annihilator
│ │ │ │ +00060910: 730a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  s..+------------
│ │ │ │  00060920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060960: 2d2d 2d2d 2b0a 7c69 3237 203a 2061 6e6e  ----+.|i27 : ann
│ │ │ │ -00060970: 2045 4520 2020 2020 2020 2020 2020 2020   EE             
│ │ │ │ +00060950: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3720  ---------+.|i27 
│ │ │ │ +00060960: 3a20 616e 6e20 4545 2020 2020 2020 2020  : ann EE        
│ │ │ │ +00060970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000609a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000609b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000609c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000609d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00060a20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00060a30: 2020 2020 2020 2020 2020 3220 7c0a 7c6f            2 |.|o
│ │ │ │ -00060a40: 3237 203d 2069 6465 616c 2028 7820 2c20  27 = ideal (x , 
│ │ │ │ -00060a50: 7820 2c20 7820 2c20 5820 202b 2034 3158  x , x , X  + 41X
│ │ │ │ -00060a60: 2058 2020 2d20 3337 5820 202d 2031 3458   X  - 37X  - 14X
│ │ │ │ -00060a70: 2058 2020 2d20 3239 5820 5820 202b 2034   X  - 29X X  + 4
│ │ │ │ -00060a80: 3558 2029 7c0a 7c20 2020 2020 2020 2020  5X )|.|         
│ │ │ │ -00060a90: 2020 2020 2032 2020 2031 2020 2030 2020       2   1   0  
│ │ │ │ -00060aa0: 2031 2020 2020 2020 3120 3220 2020 2020   1      1 2     
│ │ │ │ -00060ab0: 2032 2020 2020 2020 3120 3320 2020 2020   2      1 3     
│ │ │ │ -00060ac0: 2032 2033 2020 2020 2020 3320 7c0a 7c20   2 3      3 |.| 
│ │ │ │ +000609e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000609f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060a00: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00060a10: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00060a20: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00060a30: 207c 0a7c 6f32 3720 3d20 6964 6561 6c20   |.|o27 = ideal 
│ │ │ │ +00060a40: 2878 202c 2078 202c 2078 202c 2058 2020  (x , x , x , X  
│ │ │ │ +00060a50: 2b20 3431 5820 5820 202d 2033 3758 2020  + 41X X  - 37X  
│ │ │ │ +00060a60: 2d20 3134 5820 5820 202d 2032 3958 2058  - 14X X  - 29X X
│ │ │ │ +00060a70: 2020 2b20 3435 5820 297c 0a7c 2020 2020    + 45X )|.|    
│ │ │ │ +00060a80: 2020 2020 2020 2020 2020 3220 2020 3120            2   1 
│ │ │ │ +00060a90: 2020 3020 2020 3120 2020 2020 2031 2032    0   1      1 2
│ │ │ │ +00060aa0: 2020 2020 2020 3220 2020 2020 2031 2033        2      1 3
│ │ │ │ +00060ab0: 2020 2020 2020 3220 3320 2020 2020 2033        2 3      3
│ │ │ │ +00060ac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00060ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b10: 2020 2020 7c0a 7c6f 3237 203a 2049 6465      |.|o27 : Ide
│ │ │ │ -00060b20: 616c 206f 6620 5327 2020 2020 2020 2020  al of S'        
│ │ │ │ +00060b00: 2020 2020 2020 2020 207c 0a7c 6f32 3720           |.|o27 
│ │ │ │ +00060b10: 3a20 4964 6561 6c20 6f66 2053 2720 2020  : Ideal of S'   
│ │ │ │ +00060b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00060b50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00060b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060ba0: 2d2d 2d2d 2b0a 7c69 3238 203a 2061 6e6e  ----+.|i28 : ann
│ │ │ │ -00060bb0: 2045 5320 2020 2020 2020 2020 2020 2020   ES             
│ │ │ │ +00060b90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3820  ---------+.|i28 
│ │ │ │ +00060ba0: 3a20 616e 6e20 4553 2020 2020 2020 2020  : ann ES        
│ │ │ │ +00060bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00060be0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00060bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060c30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060c50: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00060c60: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00060c70: 2020 2020 2020 2020 2032 2020 7c0a 7c6f           2  |.|o
│ │ │ │ -00060c80: 3238 203d 2069 6465 616c 2028 7820 2c20  28 = ideal (x , 
│ │ │ │ -00060c90: 7820 2c20 7820 2c20 7320 202b 2034 3273  x , x , s  + 42s
│ │ │ │ -00060ca0: 2073 2020 2d20 3330 7320 202d 2032 3573   s  - 30s  - 25s
│ │ │ │ -00060cb0: 2073 2020 2d20 3335 7320 7320 202b 2039   s  - 35s s  + 9
│ │ │ │ -00060cc0: 7320 2920 7c0a 7c20 2020 2020 2020 2020  s ) |.|         
│ │ │ │ -00060cd0: 2020 2020 2032 2020 2031 2020 2030 2020       2   1   0  
│ │ │ │ -00060ce0: 2030 2020 2020 2020 3020 3120 2020 2020   0      0 1     
│ │ │ │ -00060cf0: 2031 2020 2020 2020 3020 3220 2020 2020   1      0 2     
│ │ │ │ -00060d00: 2031 2032 2020 2020 2032 2020 7c0a 7c20   1 2     2  |.| 
│ │ │ │ +00060c20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00060c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060c40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00060c50: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00060c60: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00060c70: 207c 0a7c 6f32 3820 3d20 6964 6561 6c20   |.|o28 = ideal 
│ │ │ │ +00060c80: 2878 202c 2078 202c 2078 202c 2073 2020  (x , x , x , s  
│ │ │ │ +00060c90: 2b20 3432 7320 7320 202d 2033 3073 2020  + 42s s  - 30s  
│ │ │ │ +00060ca0: 2d20 3235 7320 7320 202d 2033 3573 2073  - 25s s  - 35s s
│ │ │ │ +00060cb0: 2020 2b20 3973 2029 207c 0a7c 2020 2020    + 9s ) |.|    
│ │ │ │ +00060cc0: 2020 2020 2020 2020 2020 3220 2020 3120            2   1 
│ │ │ │ +00060cd0: 2020 3020 2020 3020 2020 2020 2030 2031    0   0      0 1
│ │ │ │ +00060ce0: 2020 2020 2020 3120 2020 2020 2030 2032        1      0 2
│ │ │ │ +00060cf0: 2020 2020 2020 3120 3220 2020 2020 3220        1 2     2 
│ │ │ │ +00060d00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00060d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060d50: 2020 2020 7c0a 7c6f 3238 203a 2049 6465      |.|o28 : Ide
│ │ │ │ -00060d60: 616c 206f 6620 5320 2020 2020 2020 2020  al of S         
│ │ │ │ +00060d40: 2020 2020 2020 2020 207c 0a7c 6f32 3820           |.|o28 
│ │ │ │ +00060d50: 3a20 4964 6561 6c20 6f66 2053 2020 2020  : Ideal of S    
│ │ │ │ +00060d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060d90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00060d90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00060da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060de0: 2d2d 2d2d 2b0a 0a61 6e64 2069 6e20 6661  ----+..and in fa
│ │ │ │ -00060df0: 6374 2074 6865 7920 6172 6520 6e6f 7420  ct they are not 
│ │ │ │ -00060e00: 6973 6f6d 6f72 7068 6963 3a0a 0a2b 2d2d  isomorphic:..+--
│ │ │ │ +00060dd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 616e 6420  ---------+..and 
│ │ │ │ +00060de0: 696e 2066 6163 7420 7468 6579 2061 7265  in fact they are
│ │ │ │ +00060df0: 206e 6f74 2069 736f 6d6f 7270 6869 633a   not isomorphic:
│ │ │ │ +00060e00: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00060e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00060e60: 3920 3a20 4545 746f 4553 203d 206d 6170  9 : EEtoES = map
│ │ │ │ -00060e70: 2872 696e 6720 4553 2c72 696e 6720 4545  (ring ES,ring EE
│ │ │ │ -00060e80: 2c20 6765 6e73 2072 696e 6720 4553 2920  , gens ring ES) 
│ │ │ │ +00060e50: 2b0a 7c69 3239 203a 2045 4574 6f45 5320  +.|i29 : EEtoES 
│ │ │ │ +00060e60: 3d20 6d61 7028 7269 6e67 2045 532c 7269  = map(ring ES,ri
│ │ │ │ +00060e70: 6e67 2045 452c 2067 656e 7320 7269 6e67  ng EE, gens ring
│ │ │ │ +00060e80: 2045 5329 2020 2020 2020 2020 2020 2020   ES)            
│ │ │ │  00060e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00060ea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00060eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ef0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00060f00: 3920 3d20 6d61 7020 2853 2c20 5327 2c20  9 = map (S, S', 
│ │ │ │ -00060f10: 7b73 202c 2073 202c 2073 202c 2078 202c  {s , s , s , x ,
│ │ │ │ -00060f20: 2078 202c 2078 207d 2920 2020 2020 2020   x , x })       
│ │ │ │ +00060ef0: 7c0a 7c6f 3239 203d 206d 6170 2028 532c  |.|o29 = map (S,
│ │ │ │ +00060f00: 2053 272c 207b 7320 2c20 7320 2c20 7320   S', {s , s , s 
│ │ │ │ +00060f10: 2c20 7820 2c20 7820 2c20 7820 7d29 2020  , x , x , x })  
│ │ │ │ +00060f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060f40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00060f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060f60: 2020 3020 2020 3120 2020 3220 2020 3020    0   1   2   0 
│ │ │ │ -00060f70: 2020 3120 2020 3220 2020 2020 2020 2020    1   2         
│ │ │ │ +00060f40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00060f50: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ +00060f60: 2020 2030 2020 2031 2020 2032 2020 2020     0   1   2    
│ │ │ │ +00060f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060f90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00060f90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00060fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060fe0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00060ff0: 3920 3a20 5269 6e67 4d61 7020 5320 3c2d  9 : RingMap S <-
│ │ │ │ -00061000: 2d20 5327 2020 2020 2020 2020 2020 2020  - S'            
│ │ │ │ +00060fe0: 7c0a 7c6f 3239 203a 2052 696e 674d 6170  |.|o29 : RingMap
│ │ │ │ +00060ff0: 2053 203c 2d2d 2053 2720 2020 2020 2020   S <-- S'       
│ │ │ │ +00061000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061030: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00061030: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00061040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00061090: 3020 3a20 4545 2720 3d20 636f 6b65 7220  0 : EE' = coker 
│ │ │ │ -000610a0: 4545 746f 4553 2070 7265 7365 6e74 6174  EEtoES presentat
│ │ │ │ -000610b0: 696f 6e20 4545 2020 2020 2020 2020 2020  ion EE          
│ │ │ │ +00061080: 2b0a 7c69 3330 203a 2045 4527 203d 2063  +.|i30 : EE' = c
│ │ │ │ +00061090: 6f6b 6572 2045 4574 6f45 5320 7072 6573  oker EEtoES pres
│ │ │ │ +000610a0: 656e 7461 7469 6f6e 2045 4520 2020 2020  entation EE     
│ │ │ │ +000610b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000610c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000610d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000610d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000610e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000610f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061120: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00061130: 3020 3d20 636f 6b65 726e 656c 207b 302c  0 = cokernel {0,
│ │ │ │ -00061140: 2030 7d20 2020 7c20 785f 3220 785f 3120   0}   | x_2 x_1 
│ │ │ │ -00061150: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ -00061160: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061170: 3020 2020 3020 2020 735f 327c 0a7c 2020  0   0   s_2|.|  
│ │ │ │ -00061180: 2020 2020 2020 2020 2020 2020 207b 302c               {0,
│ │ │ │ -00061190: 2030 7d20 2020 7c20 3020 2020 3020 2020   0}   | 0   0   
│ │ │ │ -000611a0: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ -000611b0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -000611c0: 3020 2020 3020 2020 3020 207c 0a7c 2020  0   0   0  |.|  
│ │ │ │ -000611d0: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ -000611e0: 2c20 2d32 7d20 7c20 3020 2020 3020 2020  , -2} | 0   0   
│ │ │ │ -000611f0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061200: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ -00061210: 3020 2020 3020 2020 3020 207c 0a7c 2020  0   0   0  |.|  
│ │ │ │ -00061220: 2020 2020 2020 2020 2020 2020 207b 2d31               {-1
│ │ │ │ -00061230: 2c20 2d32 7d20 7c20 3020 2020 3020 2020  , -2} | 0   0   
│ │ │ │ -00061240: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061250: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ -00061260: 785f 3120 785f 3020 3020 207c 0a7c 2020  x_1 x_0 0  |.|  
│ │ │ │ -00061270: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ -00061280: 2c20 2d33 7d20 7c20 3020 2020 3020 2020  , -3} | 0   0   
│ │ │ │ -00061290: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -000612a0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -000612b0: 3020 2020 3020 2020 3020 207c 0a7c 2020  0   0   0  |.|  
│ │ │ │ -000612c0: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ -000612d0: 2c20 2d33 7d20 7c20 3020 2020 3020 2020  , -3} | 0   0   
│ │ │ │ -000612e0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -000612f0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061300: 3020 2020 3020 2020 3020 207c 0a7c 2020  0   0   0  |.|  
│ │ │ │ -00061310: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ -00061320: 2c20 2d33 7d20 7c20 3020 2020 3020 2020  , -3} | 0   0   
│ │ │ │ -00061330: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061340: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061350: 3020 2020 3020 2020 3020 207c 0a7c 2020  0   0   0  |.|  
│ │ │ │ -00061360: 2020 2020 2020 2020 2020 2020 207b 2d32               {-2
│ │ │ │ -00061370: 2c20 2d33 7d20 7c20 3020 2020 3020 2020  , -3} | 0   0   
│ │ │ │ -00061380: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -00061390: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -000613a0: 3020 2020 3020 2020 3020 207c 0a7c 2020  0   0   0  |.|  
│ │ │ │ +00061120: 7c0a 7c6f 3330 203d 2063 6f6b 6572 6e65  |.|o30 = cokerne
│ │ │ │ +00061130: 6c20 7b30 2c20 307d 2020 207c 2078 5f32  l {0, 0}   | x_2
│ │ │ │ +00061140: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ +00061150: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061160: 2030 2020 2030 2020 2030 2020 2073 5f32   0   0   0   s_2
│ │ │ │ +00061170: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00061180: 2020 7b30 2c20 307d 2020 207c 2030 2020    {0, 0}   | 0  
│ │ │ │ +00061190: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ +000611a0: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ +000611b0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000611c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000611d0: 2020 7b2d 312c 202d 327d 207c 2030 2020    {-1, -2} | 0  
│ │ │ │ +000611e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000611f0: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ +00061200: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061210: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00061220: 2020 7b2d 312c 202d 327d 207c 2030 2020    {-1, -2} | 0  
│ │ │ │ +00061230: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061240: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061250: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ +00061260: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00061270: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +00061280: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061290: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000612a0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000612b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000612c0: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +000612d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000612e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000612f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00061310: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +00061320: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061330: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061340: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061350: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00061360: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +00061370: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061380: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +00061390: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ +000613a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000613b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000613c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000613d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000613e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000613f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000613f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00061400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061410: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +00061410: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │  00061420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061440: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00061450: 3020 3a20 532d 6d6f 6475 6c65 2c20 7175  0 : S-module, qu
│ │ │ │ -00061460: 6f74 6965 6e74 206f 6620 5320 2020 2020  otient of S     
│ │ │ │ +00061440: 7c0a 7c6f 3330 203a 2053 2d6d 6f64 756c  |.|o30 : S-modul
│ │ │ │ +00061450: 652c 2071 756f 7469 656e 7420 6f66 2053  e, quotient of S
│ │ │ │ +00061460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061490: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +00061490: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  000614a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000614b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000614c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000614d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000614e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 735f  -----------|.|s_
│ │ │ │ -000614f0: 3120 735f 3020 3020 2020 3020 2020 3020  1 s_0 0   0   0 
│ │ │ │ -00061500: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061510: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061520: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061530: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061540: 2020 3020 2020 735f 3220 735f 3120 735f    0   s_2 s_1 s_
│ │ │ │ -00061550: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ -00061560: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061570: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061580: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061590: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000615a0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000615b0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000615c0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000615d0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -000615e0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000615f0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061600: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061610: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061620: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061630: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061640: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ -00061650: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061660: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061670: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061680: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061690: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ -000616a0: 3220 785f 3120 785f 3020 3020 2020 3020  2 x_1 x_0 0   0 
│ │ │ │ -000616b0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000616c0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -000616d0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000616e0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -000616f0: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ -00061700: 3120 785f 3020 3020 2020 3020 2020 3020  1 x_0 0   0   0 
│ │ │ │ -00061710: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061720: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061730: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061740: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ -00061750: 2020 3020 2020 785f 3220 785f 3120 785f    0   x_2 x_1 x_
│ │ │ │ -00061760: 3020 2020 2020 2020 2020 207c 0a7c 2d2d  0          |.|--
│ │ │ │ +000614e0: 7c0a 7c73 5f31 2073 5f30 2030 2020 2030  |.|s_1 s_0 0   0
│ │ │ │ +000614f0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061500: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061510: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061520: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061530: 7c0a 7c30 2020 2030 2020 2073 5f32 2073  |.|0   0   s_2 s
│ │ │ │ +00061540: 5f31 2073 5f30 2030 2020 2030 2020 2030  _1 s_0 0   0   0
│ │ │ │ +00061550: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061560: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061570: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061580: 7c0a 7c30 2020 2030 2020 2030 2020 2030  |.|0   0   0   0
│ │ │ │ +00061590: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +000615a0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +000615b0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +000615c0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +000615d0: 7c0a 7c30 2020 2030 2020 2030 2020 2030  |.|0   0   0   0
│ │ │ │ +000615e0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +000615f0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061600: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061610: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061620: 7c0a 7c30 2020 2030 2020 2030 2020 2030  |.|0   0   0   0
│ │ │ │ +00061630: 2020 2030 2020 2078 5f32 2078 5f31 2078     0   x_2 x_1 x
│ │ │ │ +00061640: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ +00061650: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061660: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061670: 7c0a 7c30 2020 2030 2020 2030 2020 2030  |.|0   0   0   0
│ │ │ │ +00061680: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061690: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ +000616a0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +000616b0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +000616c0: 7c0a 7c30 2020 2030 2020 2030 2020 2030  |.|0   0   0   0
│ │ │ │ +000616d0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +000616e0: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ +000616f0: 5f32 2078 5f31 2078 5f30 2030 2020 2030  _2 x_1 x_0 0   0
│ │ │ │ +00061700: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061710: 7c0a 7c30 2020 2030 2020 2030 2020 2030  |.|0   0   0   0
│ │ │ │ +00061720: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061730: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +00061740: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ +00061750: 5f31 2078 5f30 2020 2020 2020 2020 2020  _1 x_0          
│ │ │ │ +00061760: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00061770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000617a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000617b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020  -----------|.|0 
│ │ │ │ -000617c0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000617d0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000617e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +000617b0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │ +000617c0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +000617d0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +000617e0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  000617f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061800: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061810: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00061820: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00061830: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00061800: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │ +00061810: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061820: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061830: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  00061840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061850: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -00061860: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00061870: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00061880: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00061850: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │ +00061860: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061870: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +00061880: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  00061890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000618a0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ -000618b0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000618c0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000618d0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +000618a0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │ +000618b0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +000618c0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +000618d0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │  000618e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000618f0: 2020 2020 2020 2020 2020 207c 0a7c 735f             |.|s_
│ │ │ │ -00061900: 302d 3138 735f 312d 3332 735f 3220 2d32  0-18s_1-32s_2 -2
│ │ │ │ -00061910: 3773 5f31 2b32 3573 5f32 2020 2020 3432  7s_1+25s_2    42
│ │ │ │ -00061920: 735f 3120 2020 2020 2020 2020 2020 2d32  s_1           -2
│ │ │ │ -00061930: 3273 5f31 2020 2020 2020 2020 2020 2020  2s_1            
│ │ │ │ -00061940: 2020 2020 2020 2020 2020 207c 0a7c 3233             |.|23
│ │ │ │ -00061950: 735f 312d 3431 735f 3220 2020 2020 735f  s_1-41s_2     s_
│ │ │ │ -00061960: 302d 3432 735f 312b 3138 735f 3220 2d34  0-42s_1+18s_2 -4
│ │ │ │ -00061970: 3573 5f31 2020 2020 2020 2020 2020 2d34  5s_1          -4
│ │ │ │ -00061980: 3273 5f31 2020 2020 2020 2020 2020 2020  2s_1            
│ │ │ │ -00061990: 2020 2020 2020 2020 2020 207c 0a7c 2d34             |.|-4
│ │ │ │ -000619a0: 3273 5f32 2020 2020 2020 2020 2020 3232  2s_2          22
│ │ │ │ -000619b0: 735f 3220 2020 2020 2020 2020 2020 735f  s_2           s_
│ │ │ │ -000619c0: 302d 3138 735f 312d 3332 735f 3220 2d32  0-18s_1-32s_2 -2
│ │ │ │ -000619d0: 3773 5f31 2b32 3573 5f32 2020 2020 2020  7s_1+25s_2      
│ │ │ │ -000619e0: 2020 2020 2020 2020 2020 207c 0a7c 3435             |.|45
│ │ │ │ -000619f0: 735f 3220 2020 2020 2020 2020 2020 3432  s_2           42
│ │ │ │ -00061a00: 735f 3220 2020 2020 2020 2020 2020 3233  s_2           23
│ │ │ │ -00061a10: 735f 312d 3431 735f 3220 2020 2020 735f  s_1-41s_2     s_
│ │ │ │ -00061a20: 302d 3432 735f 312b 3138 735f 3220 2020  0-42s_1+18s_2   
│ │ │ │ -00061a30: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000618f0: 7c0a 7c73 5f30 2d31 3873 5f31 2d33 3273  |.|s_0-18s_1-32s
│ │ │ │ +00061900: 5f32 202d 3237 735f 312b 3235 735f 3220  _2 -27s_1+25s_2 
│ │ │ │ +00061910: 2020 2034 3273 5f31 2020 2020 2020 2020     42s_1        
│ │ │ │ +00061920: 2020 202d 3232 735f 3120 2020 2020 2020     -22s_1       
│ │ │ │ +00061930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061940: 7c0a 7c32 3373 5f31 2d34 3173 5f32 2020  |.|23s_1-41s_2  
│ │ │ │ +00061950: 2020 2073 5f30 2d34 3273 5f31 2b31 3873     s_0-42s_1+18s
│ │ │ │ +00061960: 5f32 202d 3435 735f 3120 2020 2020 2020  _2 -45s_1       
│ │ │ │ +00061970: 2020 202d 3432 735f 3120 2020 2020 2020     -42s_1       
│ │ │ │ +00061980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061990: 7c0a 7c2d 3432 735f 3220 2020 2020 2020  |.|-42s_2       
│ │ │ │ +000619a0: 2020 2032 3273 5f32 2020 2020 2020 2020     22s_2        
│ │ │ │ +000619b0: 2020 2073 5f30 2d31 3873 5f31 2d33 3273     s_0-18s_1-32s
│ │ │ │ +000619c0: 5f32 202d 3237 735f 312b 3235 735f 3220  _2 -27s_1+25s_2 
│ │ │ │ +000619d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000619e0: 7c0a 7c34 3573 5f32 2020 2020 2020 2020  |.|45s_2        
│ │ │ │ +000619f0: 2020 2034 3273 5f32 2020 2020 2020 2020     42s_2        
│ │ │ │ +00061a00: 2020 2032 3373 5f31 2d34 3173 5f32 2020     23s_1-41s_2  
│ │ │ │ +00061a10: 2020 2073 5f30 2d34 3273 5f31 2b31 3873     s_0-42s_1+18s
│ │ │ │ +00061a20: 5f32 2020 2020 2020 2020 2020 2020 2020  _2              
│ │ │ │ +00061a30: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00061a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020  -----------|.|0 
│ │ │ │ +00061a80: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ad0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061ad0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b20: 2020 2020 2020 2020 2020 207c 0a7c 735f             |.|s_
│ │ │ │ -00061b30: 305e 322b 3431 735f 3073 5f31 2d33 3773  0^2+41s_0s_1-37s
│ │ │ │ -00061b40: 5f31 5e32 2d31 3473 5f30 735f 322d 3239  _1^2-14s_0s_2-29
│ │ │ │ -00061b50: 735f 3173 5f32 2b34 3573 5f32 5e32 2020  s_1s_2+45s_2^2  
│ │ │ │ +00061b20: 7c0a 7c73 5f30 5e32 2b34 3173 5f30 735f  |.|s_0^2+41s_0s_
│ │ │ │ +00061b30: 312d 3337 735f 315e 322d 3134 735f 3073  1-37s_1^2-14s_0s
│ │ │ │ +00061b40: 5f32 2d32 3973 5f31 735f 322b 3435 735f  _2-29s_1s_2+45s_
│ │ │ │ +00061b50: 325e 3220 2020 2020 2020 2020 2020 2020  2^2             
│ │ │ │  00061b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b70: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061b70: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bc0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061bc0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c10: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061c10: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c60: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061c60: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061cb0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061cb0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d00: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +00061d00: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020  -----------|.|0 
│ │ │ │ +00061d50: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061d80: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061da0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061da0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061dd0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061df0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061df0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061e20: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e40: 2020 2020 2020 2020 2020 207c 0a7c 735f             |.|s_
│ │ │ │ -00061e50: 305e 322b 3431 735f 3073 5f31 2d33 3773  0^2+41s_0s_1-37s
│ │ │ │ -00061e60: 5f31 5e32 2d31 3473 5f30 735f 322d 3239  _1^2-14s_0s_2-29
│ │ │ │ -00061e70: 735f 3173 5f32 2b34 3573 5f32 5e32 207c  s_1s_2+45s_2^2 |
│ │ │ │ +00061e40: 7c0a 7c73 5f30 5e32 2b34 3173 5f30 735f  |.|s_0^2+41s_0s_
│ │ │ │ +00061e50: 312d 3337 735f 315e 322d 3134 735f 3073  1-37s_1^2-14s_0s
│ │ │ │ +00061e60: 5f32 2d32 3973 5f31 735f 322b 3435 735f  _2-29s_1s_2+45s_
│ │ │ │ +00061e70: 325e 3220 7c20 2020 2020 2020 2020 2020  2^2 |           
│ │ │ │  00061e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e90: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061e90: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061ec0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ee0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061ee0: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061f10: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f30: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061f30: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061f60: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f80: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00061f80: 7c0a 7c30 2020 2020 2020 2020 2020 2020  |.|0            
│ │ │ │  00061f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061fb0: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │  00061fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061fd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00061fd0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00061fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00062030: 3120 3a20 4820 3d20 486f 6d28 4545 272c  1 : H = Hom(EE',
│ │ │ │ -00062040: 4553 293b 2020 2020 2020 2020 2020 2020  ES);            
│ │ │ │ +00062020: 2b0a 7c69 3331 203a 2048 203d 2048 6f6d  +.|i31 : H = Hom
│ │ │ │ +00062030: 2845 4527 2c45 5329 3b20 2020 2020 2020  (EE',ES);       
│ │ │ │ +00062040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062070: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00062070: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00062080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000620a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000620b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000620c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -000620d0: 3220 3a20 5120 3d20 706f 7369 7469 6f6e  2 : Q = position
│ │ │ │ -000620e0: 7328 6465 6772 6565 7320 7461 7267 6574  s(degrees target
│ │ │ │ -000620f0: 2070 7265 7365 6e74 6174 696f 6e20 482c   presentation H,
│ │ │ │ -00062100: 2069 2d3e 2069 203d 3d20 7b30 2c30 7d29   i-> i == {0,0})
│ │ │ │ -00062110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000620c0: 2b0a 7c69 3332 203a 2051 203d 2070 6f73  +.|i32 : Q = pos
│ │ │ │ +000620d0: 6974 696f 6e73 2864 6567 7265 6573 2074  itions(degrees t
│ │ │ │ +000620e0: 6172 6765 7420 7072 6573 656e 7461 7469  arget presentati
│ │ │ │ +000620f0: 6f6e 2048 2c20 692d 3e20 6920 3d3d 207b  on H, i-> i == {
│ │ │ │ +00062100: 302c 307d 2920 2020 2020 2020 2020 2020  0,0})           
│ │ │ │ +00062110: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00062120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062160: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00062170: 3220 3d20 7b38 2c20 392c 2031 302c 2031  2 = {8, 9, 10, 1
│ │ │ │ -00062180: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ +00062160: 7c0a 7c6f 3332 203d 207b 382c 2039 2c20  |.|o32 = {8, 9, 
│ │ │ │ +00062170: 3130 2c20 3131 7d20 2020 2020 2020 2020  10, 11}         
│ │ │ │ +00062180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000621b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000621c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062200: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00062210: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ +00062200: 7c0a 7c6f 3332 203a 204c 6973 7420 2020  |.|o32 : List   
│ │ │ │ +00062210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062250: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00062250: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
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│ │ │ │ -00062af0: 300a 2020 2a20 4f75 7470 7574 733a 0a20  0.  * Outputs:. 
│ │ │ │ -00062b00: 2020 2020 202a 2045 2c20 6120 2a6e 6f74       * E, a *not
│ │ │ │ -00062b10: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ -00062b20: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ -00062b30: 206f 7665 7220 6120 706f 6c79 6e6f 6d69   over a polynomi
│ │ │ │ -00062b40: 616c 2072 696e 6720 7769 7468 0a20 2020  al ring with.   
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│ │ │ │ -00062b60: 7265 6520 310a 0a44 6573 6372 6970 7469  ree 1..Descripti
│ │ │ │ -00062b70: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00062b80: 4578 7472 6163 7473 2074 6865 206f 6464  Extracts the odd
│ │ │ │ -00062b90: 2064 6567 7265 6520 7061 7274 2066 726f   degree part fro
│ │ │ │ -00062ba0: 6d20 4578 744d 6f64 756c 6520 4d2e 2049  m ExtModule M. I
│ │ │ │ -00062bb0: 6620 7468 6520 6f70 7469 6f6e 616c 2061  f the optional a
│ │ │ │ -00062bc0: 7267 756d 656e 7420 4f75 7452 696e 670a  rgument OutRing.
│ │ │ │ -00062bd0: 3d3e 2054 2069 7320 6769 7665 6e2c 2061  => T is given, a
│ │ │ │ -00062be0: 6e64 2063 6c61 7373 2054 203d 3d3d 2050  nd class T === P
│ │ │ │ -00062bf0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2074  olynomialRing, t
│ │ │ │ -00062c00: 6865 6e20 7468 6520 6f75 7470 7574 2077  hen the output w
│ │ │ │ -00062c10: 696c 6c20 6265 2061 206d 6f64 756c 650a  ill be a module.
│ │ │ │ -00062c20: 6f76 6572 2054 2e0a 0a2b 2d2d 2d2d 2d2d  over T...+------
│ │ │ │ +000629c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ +000629d0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ +000629e0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +000629f0: 2020 2020 4520 3d20 6f64 6445 7874 4d6f      E = oddExtMo
│ │ │ │ +00062a00: 6475 6c65 204d 0a20 202a 2049 6e70 7574  dule M.  * Input
│ │ │ │ +00062a10: 733a 0a20 2020 2020 202a 204d 2c20 6120  s:.      * M, a 
│ │ │ │ +00062a20: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +00062a30: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +00062a40: 6c65 2c2c 206f 7665 7220 6120 636f 6d70  le,, over a comp
│ │ │ │ +00062a50: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +00062a60: 6e0a 2020 2020 2020 2020 7269 6e67 0a20  n.        ring. 
│ │ │ │ +00062a70: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +00062a80: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +00062a90: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +00062aa0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00062ab0: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +00062ac0: 2020 2020 202a 204f 7574 5269 6e67 203d       * OutRing =
│ │ │ │ +00062ad0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00062ae0: 616c 7565 2030 0a20 202a 204f 7574 7075  alue 0.  * Outpu
│ │ │ │ +00062af0: 7473 3a0a 2020 2020 2020 2a20 452c 2061  ts:.      * E, a
│ │ │ │ +00062b00: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ +00062b10: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ +00062b20: 756c 652c 2c20 6f76 6572 2061 2070 6f6c  ule,, over a pol
│ │ │ │ +00062b30: 796e 6f6d 6961 6c20 7269 6e67 2077 6974  ynomial ring wit
│ │ │ │ +00062b40: 680a 2020 2020 2020 2020 6765 6e73 2069  h.        gens i
│ │ │ │ +00062b50: 6e20 6465 6772 6565 2031 0a0a 4465 7363  n degree 1..Desc
│ │ │ │ +00062b60: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00062b70: 3d3d 3d0a 0a45 7874 7261 6374 7320 7468  ===..Extracts th
│ │ │ │ +00062b80: 6520 6f64 6420 6465 6772 6565 2070 6172  e odd degree par
│ │ │ │ +00062b90: 7420 6672 6f6d 2045 7874 4d6f 6475 6c65  t from ExtModule
│ │ │ │ +00062ba0: 204d 2e20 4966 2074 6865 206f 7074 696f   M. If the optio
│ │ │ │ +00062bb0: 6e61 6c20 6172 6775 6d65 6e74 204f 7574  nal argument Out
│ │ │ │ +00062bc0: 5269 6e67 0a3d 3e20 5420 6973 2067 6976  Ring.=> T is giv
│ │ │ │ +00062bd0: 656e 2c20 616e 6420 636c 6173 7320 5420  en, and class T 
│ │ │ │ +00062be0: 3d3d 3d20 506f 6c79 6e6f 6d69 616c 5269  === PolynomialRi
│ │ │ │ +00062bf0: 6e67 2c20 7468 656e 2074 6865 206f 7574  ng, then the out
│ │ │ │ +00062c00: 7075 7420 7769 6c6c 2062 6520 6120 6d6f  put will be a mo
│ │ │ │ +00062c10: 6475 6c65 0a6f 7665 7220 542e 0a0a 2b2d  dule.over T...+-
│ │ │ │ +00062c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c60: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a  ---+.|i1 : kk= Z
│ │ │ │ -00062c70: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ +00062c50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00062c60: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020  kk= ZZ/101      
│ │ │ │ +00062c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062c90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00062ca0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062c90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00062ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062cd0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00062ce0: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │ +00062cd0: 7c0a 7c6f 3120 3d20 6b6b 2020 2020 2020  |.|o1 = kk      
│ │ │ │ +00062ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00062d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00062d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d50: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ -00062d60: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00062d40: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00062d50: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00062d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00062d90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00062d80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00062d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00062dd0: 203a 2053 203d 206b 6b5b 782c 792c 7a5d   : S = kk[x,y,z]
│ │ │ │ +00062dc0: 2b0a 7c69 3220 3a20 5320 3d20 6b6b 5b78  +.|i2 : S = kk[x
│ │ │ │ +00062dd0: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │  00062de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062e00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00062df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00062e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062e40: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ +00062e30: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +00062e40: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  00062e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00062e80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062e70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00062e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062eb0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00062ec0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -00062ed0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00062ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ef0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00062eb0: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69  |.|o2 : Polynomi
│ │ │ │ +00062ec0: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +00062ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00062ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062f30: 2d2d 2d2b 0a7c 6933 203a 2049 3220 3d20  ---+.|i3 : I2 = 
│ │ │ │ -00062f40: 6964 6561 6c22 7833 2c79 7a22 2020 2020  ideal"x3,yz"    
│ │ │ │ +00062f20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00062f30: 4932 203d 2069 6465 616c 2278 332c 797a  I2 = ideal"x3,yz
│ │ │ │ +00062f40: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  00062f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062f60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00062f70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062f60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00062f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062fa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00062fb0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00062fa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00062fb0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  00062fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062fe0: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -00062ff0: 6465 616c 2028 7820 2c20 792a 7a29 2020  deal (x , y*z)  
│ │ │ │ +00062fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00062fe0: 3320 3d20 6964 6561 6c20 2878 202c 2079  3 = ideal (x , y
│ │ │ │ +00062ff0: 2a7a 2920 2020 2020 2020 2020 2020 2020  *z)             
│ │ │ │  00063000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00063010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00063020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063060: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -00063070: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00063050: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561      |.|o3 : Idea
│ │ │ │ +00063060: 6c20 6f66 2053 2020 2020 2020 2020 2020  l of S          
│ │ │ │ +00063070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063090: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00063090: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000630a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000630b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ -000630e0: 3220 3d20 532f 4932 2020 2020 2020 2020  2 = S/I2        
│ │ │ │ +000630c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000630d0: 3420 3a20 5232 203d 2053 2f49 3220 2020  4 : R2 = S/I2   
│ │ │ │ +000630e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000630f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063110: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00063100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00063110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063150: 0a7c 6f34 203d 2052 3220 2020 2020 2020  .|o4 = R2       
│ │ │ │ +00063140: 2020 2020 7c0a 7c6f 3420 3d20 5232 2020      |.|o4 = R2  
│ │ │ │ +00063150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063180: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063180: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00063190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000631a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000631b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000631c0: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ -000631d0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +000631b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000631c0: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +000631d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000631e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000631f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063200: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000631f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00063200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00063240: 0a7c 6935 203a 204d 3220 3d20 5232 5e31  .|i5 : M2 = R2^1
│ │ │ │ -00063250: 2f69 6465 616c 2278 322c 792c 7a22 2020  /ideal"x2,y,z"  
│ │ │ │ +00063230: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d32 203d  ----+.|i5 : M2 =
│ │ │ │ +00063240: 2052 325e 312f 6964 6561 6c22 7832 2c79   R2^1/ideal"x2,y
│ │ │ │ +00063250: 2c7a 2220 2020 2020 2020 2020 2020 2020  ,z"             
│ │ │ │  00063260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063270: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063270: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00063280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000632a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000632b0: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ -000632c0: 6f6b 6572 6e65 6c20 7c20 7832 2079 207a  okernel | x2 y z
│ │ │ │ -000632d0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -000632e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000632f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000632a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000632b0: 3520 3d20 636f 6b65 726e 656c 207c 2078  5 = cokernel | x
│ │ │ │ +000632c0: 3220 7920 7a20 7c20 2020 2020 2020 2020  2 y z |         
│ │ │ │ +000632d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000632e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000632f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00063340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063350: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00063360: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00063370: 203a 2052 322d 6d6f 6475 6c65 2c20 7175   : R2-module, qu
│ │ │ │ -00063380: 6f74 6965 6e74 206f 6620 5232 2020 2020  otient of R2    
│ │ │ │ -00063390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000633a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00063320: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00063330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063340: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00063350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063360: 7c0a 7c6f 3520 3a20 5232 2d6d 6f64 756c  |.|o5 : R2-modul
│ │ │ │ +00063370: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +00063380: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00063390: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000633a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000633b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000633c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000633d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000633e0: 2d2d 2d2b 0a7c 6936 203a 2062 6574 7469  ---+.|i6 : betti
│ │ │ │ -000633f0: 2072 6573 2028 4d32 2c20 4c65 6e67 7468   res (M2, Length
│ │ │ │ -00063400: 4c69 6d69 7420 3d3e 3130 2920 2020 2020  Limit =>10)     
│ │ │ │ -00063410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000633d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +000633e0: 6265 7474 6920 7265 7320 284d 322c 204c  betti res (M2, L
│ │ │ │ +000633f0: 656e 6774 684c 696d 6974 203d 3e31 3029  engthLimit =>10)
│ │ │ │ +00063400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063410: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00063420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063450: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00063460: 2020 2020 2020 2020 2020 3020 3120 3220            0 1 2 
│ │ │ │ -00063470: 3320 3420 2035 2020 3620 2037 2020 3820  3 4  5  6  7  8 
│ │ │ │ -00063480: 2039 2031 3020 2020 2020 2020 2020 2020   9 10           
│ │ │ │ -00063490: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ -000634a0: 6f74 616c 3a20 3120 3320 3520 3720 3920  otal: 1 3 5 7 9 
│ │ │ │ -000634b0: 3131 2031 3320 3135 2031 3720 3139 2032  11 13 15 17 19 2
│ │ │ │ -000634c0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -000634d0: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ -000634e0: 3a20 3120 3220 3220 3220 3220 2032 2020  : 1 2 2 2 2  2  
│ │ │ │ -000634f0: 3220 2032 2020 3220 2032 2020 3220 2020  2  2  2  2  2   
│ │ │ │ -00063500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063510: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -00063520: 3120 3320 3420 3420 2034 2020 3420 2034  1 3 4 4  4  4  4
│ │ │ │ -00063530: 2020 3420 2034 2020 3420 2020 2020 2020    4  4  4       
│ │ │ │ -00063540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00063550: 2020 2020 2020 2032 3a20 2e20 2e20 2e20         2: . . . 
│ │ │ │ -00063560: 3120 3320 2034 2020 3420 2034 2020 3420  1 3  4  4  4  4 
│ │ │ │ -00063570: 2034 2020 3420 2020 2020 2020 2020 2020   4  4           
│ │ │ │ -00063580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00063590: 2020 2033 3a20 2e20 2e20 2e20 2e20 2e20     3: . . . . . 
│ │ │ │ -000635a0: 2031 2020 3320 2034 2020 3420 2034 2020   1  3  4  4  4  
│ │ │ │ -000635b0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -000635c0: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ -000635d0: 3a20 2e20 2e20 2e20 2e20 2e20 202e 2020  : . . . . .  .  
│ │ │ │ -000635e0: 2e20 2031 2020 3320 2034 2020 3420 2020  .  1  3  4  4   
│ │ │ │ -000635f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063600: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ -00063610: 2e20 2e20 2e20 2e20 202e 2020 2e20 202e  . . . .  .  .  .
│ │ │ │ -00063620: 2020 2e20 2031 2020 3320 2020 2020 2020    .  1  3       
│ │ │ │ -00063630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063450: 7c0a 7c20 2020 2020 2020 2020 2020 2030  |.|            0
│ │ │ │ +00063460: 2031 2032 2033 2034 2020 3520 2036 2020   1 2 3 4  5  6  
│ │ │ │ +00063470: 3720 2038 2020 3920 3130 2020 2020 2020  7  8  9 10      
│ │ │ │ +00063480: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00063490: 3620 3d20 746f 7461 6c3a 2031 2033 2035  6 = total: 1 3 5
│ │ │ │ +000634a0: 2037 2039 2031 3120 3133 2031 3520 3137   7 9 11 13 15 17
│ │ │ │ +000634b0: 2031 3920 3231 2020 2020 2020 2020 2020   19 21          
│ │ │ │ +000634c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000634d0: 2020 2020 303a 2031 2032 2032 2032 2032      0: 1 2 2 2 2
│ │ │ │ +000634e0: 2020 3220 2032 2020 3220 2032 2020 3220    2  2  2  2  2 
│ │ │ │ +000634f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00063500: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00063510: 313a 202e 2031 2033 2034 2034 2020 3420  1: . 1 3 4 4  4 
│ │ │ │ +00063520: 2034 2020 3420 2034 2020 3420 2034 2020   4  4  4  4  4  
│ │ │ │ +00063530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063540: 7c0a 7c20 2020 2020 2020 2020 323a 202e  |.|         2: .
│ │ │ │ +00063550: 202e 202e 2031 2033 2020 3420 2034 2020   . . 1 3  4  4  
│ │ │ │ +00063560: 3420 2034 2020 3420 2034 2020 2020 2020  4  4  4  4      
│ │ │ │ +00063570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00063580: 2020 2020 2020 2020 333a 202e 202e 202e          3: . . .
│ │ │ │ +00063590: 202e 202e 2020 3120 2033 2020 3420 2034   . .  1  3  4  4
│ │ │ │ +000635a0: 2020 3420 2034 2020 2020 2020 2020 2020    4  4          
│ │ │ │ +000635b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000635c0: 2020 2020 343a 202e 202e 202e 202e 202e      4: . . . . .
│ │ │ │ +000635d0: 2020 2e20 202e 2020 3120 2033 2020 3420    .  .  1  3  4 
│ │ │ │ +000635e0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +000635f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00063600: 353a 202e 202e 202e 202e 202e 2020 2e20  5: . . . . .  . 
│ │ │ │ +00063610: 202e 2020 2e20 202e 2020 3120 2033 2020   .  .  .  1  3  
│ │ │ │ +00063620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063630: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00063640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063670: 2020 2020 2020 207c 0a7c 6f36 203a 2042         |.|o6 : B
│ │ │ │ -00063680: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00063660: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00063670: 3620 3a20 4265 7474 6954 616c 6c79 2020  6 : BettiTally  
│ │ │ │ +00063680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000636a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000636b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000636c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000636d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000636e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000636f0: 0a7c 6937 203a 2045 203d 2045 7874 4d6f  .|i7 : E = ExtMo
│ │ │ │ -00063700: 6475 6c65 204d 3220 2020 2020 2020 2020  dule M2         
│ │ │ │ +000636e0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4520 3d20  ----+.|i7 : E = 
│ │ │ │ +000636f0: 4578 744d 6f64 756c 6520 4d32 2020 2020  ExtModule M2    
│ │ │ │ +00063700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063720: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00063730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063760: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00063770: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +00063750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00063760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063770: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │  00063780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637a0: 2020 207c 0a7c 6f37 203d 2028 6b6b 5b58     |.|o7 = (kk[X
│ │ │ │ -000637b0: 202e 2e58 205d 2920 2020 2020 2020 2020   ..X ])         
│ │ │ │ +00063790: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +000637a0: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020  (kk[X ..X ])    
│ │ │ │ +000637b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000637c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000637e0: 0a7c 2020 2020 2020 2020 2020 3020 2020  .|          0   
│ │ │ │ -000637f0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000637d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000637e0: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │ +000637f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063810: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00063820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063850: 2020 2020 2020 207c 0a7c 6f37 203a 206b         |.|o7 : k
│ │ │ │ -00063860: 6b5b 5820 2e2e 5820 5d2d 6d6f 6475 6c65  k[X ..X ]-module
│ │ │ │ -00063870: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ -00063880: 7b30 2e2e 312c 2032 3a31 2c20 333a 322c  {0..1, 2:1, 3:2,
│ │ │ │ -00063890: 2033 7d7c 0a7c 2020 2020 2020 2020 2030   3}|.|         0
│ │ │ │ -000638a0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00063840: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00063850: 3720 3a20 6b6b 5b58 202e 2e58 205d 2d6d  7 : kk[X ..X ]-m
│ │ │ │ +00063860: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ +00063870: 7265 6573 207b 302e 2e31 2c20 323a 312c  rees {0..1, 2:1,
│ │ │ │ +00063880: 2033 3a32 2c20 337d 7c0a 7c20 2020 2020   3:2, 3}|.|     
│ │ │ │ +00063890: 2020 2020 3020 2020 3120 2020 2020 2020      0   1       
│ │ │ │ +000638a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000638b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000638d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000638c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000638d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000638e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000638f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -00063910: 203a 2061 7070 6c79 2874 6f4c 6973 7428   : apply(toList(
│ │ │ │ -00063920: 302e 2e31 3029 2c20 692d 3e68 696c 6265  0..10), i->hilbe
│ │ │ │ -00063930: 7274 4675 6e63 7469 6f6e 2869 2c20 4529  rtFunction(i, E)
│ │ │ │ -00063940: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │ +00063900: 2b0a 7c69 3820 3a20 6170 706c 7928 746f  +.|i8 : apply(to
│ │ │ │ +00063910: 4c69 7374 2830 2e2e 3130 292c 2069 2d3e  List(0..10), i->
│ │ │ │ +00063920: 6869 6c62 6572 7446 756e 6374 696f 6e28  hilbertFunction(
│ │ │ │ +00063930: 692c 2045 2929 2020 2020 2020 7c0a 7c20  i, E))      |.| 
│ │ │ │ +00063940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063980: 2020 207c 0a7c 6f38 203d 207b 312c 2033     |.|o8 = {1, 3
│ │ │ │ -00063990: 2c20 352c 2037 2c20 392c 2031 312c 2031  , 5, 7, 9, 11, 1
│ │ │ │ -000639a0: 332c 2031 352c 2031 372c 2031 392c 2032  3, 15, 17, 19, 2
│ │ │ │ -000639b0: 317d 2020 2020 2020 2020 2020 2020 207c  1}             |
│ │ │ │ -000639c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00063970: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ +00063980: 7b31 2c20 332c 2035 2c20 372c 2039 2c20  {1, 3, 5, 7, 9, 
│ │ │ │ +00063990: 3131 2c20 3133 2c20 3135 2c20 3137 2c20  11, 13, 15, 17, 
│ │ │ │ +000639a0: 3139 2c20 3231 7d20 2020 2020 2020 2020  19, 21}         
│ │ │ │ +000639b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000639c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000639d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000639e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000639f0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -00063a00: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +000639f0: 7c0a 7c6f 3820 3a20 4c69 7374 2020 2020  |.|o8 : List    
│ │ │ │ +00063a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00063a20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00063a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063a70: 2d2d 2d2b 0a7c 6939 203a 2045 6f64 6420  ---+.|i9 : Eodd 
│ │ │ │ -00063a80: 3d20 6f64 6445 7874 4d6f 6475 6c65 204d  = oddExtModule M
│ │ │ │ -00063a90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00063aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063ab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00063a60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ +00063a70: 456f 6464 203d 206f 6464 4578 744d 6f64  Eodd = oddExtMod
│ │ │ │ +00063a80: 756c 6520 4d32 2020 2020 2020 2020 2020  ule M2          
│ │ │ │ +00063a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063aa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00063ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00063af0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +00063ae0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00063af0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │  00063b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b20: 2020 2020 2020 207c 0a7c 6f39 203d 2028         |.|o9 = (
│ │ │ │ -00063b30: 6b6b 5b58 202e 2e58 205d 2920 2020 2020  kk[X ..X ])     
│ │ │ │ +00063b10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00063b20: 3920 3d20 286b 6b5b 5820 2e2e 5820 5d29  9 = (kk[X ..X ])
│ │ │ │ +00063b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00063b70: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +00063b50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00063b60: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +00063b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00063b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00063ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063bd0: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -00063be0: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f   : kk[X ..X ]-mo
│ │ │ │ -00063bf0: 6475 6c65 2c20 6672 6565 2c20 6465 6772  dule, free, degr
│ │ │ │ -00063c00: 6565 7320 7b33 3a30 2c20 317d 2020 2020  ees {3:0, 1}    
│ │ │ │ -00063c10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00063c20: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ +00063bd0: 7c0a 7c6f 3920 3a20 6b6b 5b58 202e 2e58  |.|o9 : kk[X ..X
│ │ │ │ +00063be0: 205d 2d6d 6f64 756c 652c 2066 7265 652c   ]-module, free,
│ │ │ │ +00063bf0: 2064 6567 7265 6573 207b 333a 302c 2031   degrees {3:0, 1
│ │ │ │ +00063c00: 7d20 2020 2020 2020 2020 2020 7c0a 7c20  }           |.| 
│ │ │ │ +00063c10: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00063c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00063c40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00063c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00063c90: 0a7c 6931 3020 3a20 6170 706c 7928 746f  .|i10 : apply(to
│ │ │ │ -00063ca0: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ -00063cb0: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ -00063cc0: 2c20 456f 6464 2929 2020 207c 0a7c 2020  , Eodd))   |.|  
│ │ │ │ +00063c80: 2d2d 2d2d 2b0a 7c69 3130 203a 2061 7070  ----+.|i10 : app
│ │ │ │ +00063c90: 6c79 2874 6f4c 6973 7428 302e 2e35 292c  ly(toList(0..5),
│ │ │ │ +00063ca0: 2069 2d3e 6869 6c62 6572 7446 756e 6374   i->hilbertFunct
│ │ │ │ +00063cb0: 696f 6e28 692c 2045 6f64 6429 2920 2020  ion(i, Eodd))   
│ │ │ │ +00063cc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00063cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d00: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ -00063d10: 7b33 2c20 372c 2031 312c 2031 352c 2031  {3, 7, 11, 15, 1
│ │ │ │ -00063d20: 392c 2032 337d 2020 2020 2020 2020 2020  9, 23}          
│ │ │ │ -00063d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00063cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00063d00: 3130 203d 207b 332c 2037 2c20 3131 2c20  10 = {3, 7, 11, 
│ │ │ │ +00063d10: 3135 2c20 3139 2c20 3233 7d20 2020 2020  15, 19, 23}     
│ │ │ │ +00063d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00063d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00063d80: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020  .|o10 : List    
│ │ │ │ +00063d70: 2020 2020 7c0a 7c6f 3130 203a 204c 6973      |.|o10 : Lis
│ │ │ │ +00063d80: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00063d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063db0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00063db0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00063dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063df0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -00063e00: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -00063e10: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65   *note ExtModule
│ │ │ │ -00063e20: 3a20 4578 744d 6f64 756c 652c 202d 2d20  : ExtModule, -- 
│ │ │ │ -00063e30: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ -00063e40: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -00063e50: 7365 6374 696f 6e20 6173 0a20 2020 206d  section as.    m
│ │ │ │ -00063e60: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ -00063e70: 6572 6174 6f72 2072 696e 670a 2020 2a20  erator ring.  * 
│ │ │ │ -00063e80: 2a6e 6f74 6520 6576 656e 4578 744d 6f64  *note evenExtMod
│ │ │ │ -00063e90: 756c 653a 2065 7665 6e45 7874 4d6f 6475  ule: evenExtModu
│ │ │ │ -00063ea0: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274  le, -- even part
│ │ │ │ -00063eb0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ -00063ec0: 7665 7220 610a 2020 2020 636f 6d70 6c65  ver a.    comple
│ │ │ │ -00063ed0: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ -00063ee0: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ -00063ef0: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ -00063f00: 2020 2a20 2a6e 6f74 6520 4f75 7452 696e    * *note OutRin
│ │ │ │ -00063f10: 673a 204f 7574 5269 6e67 2c20 2d2d 204f  g: OutRing, -- O
│ │ │ │ -00063f20: 7074 696f 6e20 616c 6c6f 7769 6e67 2073  ption allowing s
│ │ │ │ -00063f30: 7065 6369 6669 6361 7469 6f6e 206f 6620  pecification of 
│ │ │ │ -00063f40: 7468 6520 7269 6e67 206f 7665 720a 2020  the ring over.  
│ │ │ │ -00063f50: 2020 7768 6963 6820 7468 6520 6f75 7470    which the outp
│ │ │ │ -00063f60: 7574 2069 7320 6465 6669 6e65 640a 0a57  ut is defined..W
│ │ │ │ -00063f70: 6179 7320 746f 2075 7365 206f 6464 4578  ays to use oddEx
│ │ │ │ -00063f80: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d  tModule:.=======
│ │ │ │ -00063f90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00063fa0: 3d3d 0a0a 2020 2a20 226f 6464 4578 744d  ==..  * "oddExtM
│ │ │ │ -00063fb0: 6f64 756c 6528 4d6f 6475 6c65 2922 0a0a  odule(Module)"..
│ │ │ │ -00063fc0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00063fd0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00063fe0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00063ff0: 7420 2a6e 6f74 6520 6f64 6445 7874 4d6f  t *note oddExtMo
│ │ │ │ -00064000: 6475 6c65 3a20 6f64 6445 7874 4d6f 6475  dule: oddExtModu
│ │ │ │ -00064010: 6c65 2c20 6973 2061 202a 6e6f 7465 206d  le, is a *note m
│ │ │ │ -00064020: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ -00064030: 6974 680a 6f70 7469 6f6e 733a 2028 4d61  ith.options: (Ma
│ │ │ │ -00064040: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00064050: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ -00064060: 696f 6e73 2c2e 0a1f 0a46 696c 653a 2043  ions,....File: C
│ │ │ │ -00064070: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00064080: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -00064090: 6e66 6f2c 204e 6f64 653a 204f 7074 696d  nfo, Node: Optim
│ │ │ │ -000640a0: 6973 6d2c 204e 6578 743a 204f 7574 5269  ism, Next: OutRi
│ │ │ │ -000640b0: 6e67 2c20 5072 6576 3a20 6f64 6445 7874  ng, Prev: oddExt
│ │ │ │ -000640c0: 4d6f 6475 6c65 2c20 5570 3a20 546f 700a  Module, Up: Top.
│ │ │ │ -000640d0: 0a4f 7074 696d 6973 6d20 2d2d 204f 7074  .Optimism -- Opt
│ │ │ │ -000640e0: 696f 6e20 746f 2068 6967 6853 797a 7967  ion to highSyzyg
│ │ │ │ -000640f0: 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  y.**************
│ │ │ │ -00064100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064110: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ -00064120: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ -00064130: 3a20 0a20 2020 2020 2020 2068 6967 6853  : .        highS
│ │ │ │ -00064140: 797a 7967 7928 4d2c 204f 7074 696d 6973  yzygy(M, Optimis
│ │ │ │ -00064150: 6d20 3d3e 2031 290a 2020 2a20 496e 7075  m => 1).  * Inpu
│ │ │ │ -00064160: 7473 3a0a 2020 2020 2020 2a20 4f70 7469  ts:.      * Opti
│ │ │ │ -00064170: 6d69 736d 2c20 616e 202a 6e6f 7465 2069  mism, an *note i
│ │ │ │ -00064180: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ -00064190: 7932 446f 6329 5a5a 2c2c 200a 0a44 6573  y2Doc)ZZ,, ..Des
│ │ │ │ -000641a0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -000641b0: 3d3d 3d3d 0a0a 4966 2068 6967 6853 797a  ====..If highSyz
│ │ │ │ -000641c0: 7967 7928 4d29 2063 686f 6f73 6573 2074  ygy(M) chooses t
│ │ │ │ -000641d0: 6865 2070 2d74 6820 7379 7a79 6779 2c20  he p-th syzygy, 
│ │ │ │ -000641e0: 7468 656e 2068 6967 6853 797a 7967 7928  then highSyzygy(
│ │ │ │ -000641f0: 4d2c 4f70 7469 6d69 736d 3d3e 7229 0a63  M,Optimism=>r).c
│ │ │ │ -00064200: 686f 6f73 6573 2074 6865 2028 702d 7229  hooses the (p-r)
│ │ │ │ -00064210: 2d74 6820 7379 7a79 6779 2e20 2850 6f73  -th syzygy. (Pos
│ │ │ │ -00064220: 6974 6976 6520 4f70 7469 6d69 736d 2063  itive Optimism c
│ │ │ │ -00064230: 686f 6f73 6573 2061 206c 6f77 6572 2022  hooses a lower "
│ │ │ │ -00064240: 6869 6768 2220 7379 7a79 6779 2c0a 6e65  high" syzygy,.ne
│ │ │ │ -00064250: 6761 7469 7665 204f 7074 696d 6973 6d20  gative Optimism 
│ │ │ │ -00064260: 6120 6869 6768 6572 2022 6869 6768 2220  a higher "high" 
│ │ │ │ -00064270: 7379 7a79 6779 2e0a 0a43 6176 6561 740a  syzygy...Caveat.
│ │ │ │ -00064280: 3d3d 3d3d 3d3d 0a0a 4172 6520 7468 6572  ======..Are ther
│ │ │ │ -00064290: 6520 6361 7365 7320 7768 656e 2070 6f73  e cases when pos
│ │ │ │ -000642a0: 6974 6976 6520 4f70 7469 6d69 736d 2069  itive Optimism i
│ │ │ │ -000642b0: 7320 6a75 7374 6966 6965 643f 0a0a 5365  s justified?..Se
│ │ │ │ -000642c0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -000642d0: 0a20 202a 202a 6e6f 7465 206d 6642 6f75  .  * *note mfBou
│ │ │ │ -000642e0: 6e64 3a20 6d66 426f 756e 642c 202d 2d20  nd: mfBound, -- 
│ │ │ │ -000642f0: 6465 7465 726d 696e 6573 2068 6f77 2068  determines how h
│ │ │ │ -00064300: 6967 6820 6120 7379 7a79 6779 2074 6f20  igh a syzygy to 
│ │ │ │ -00064310: 7461 6b65 2066 6f72 0a20 2020 2022 6d61  take for.    "ma
│ │ │ │ -00064320: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -00064330: 6e22 0a20 202a 202a 6e6f 7465 2068 6967  n".  * *note hig
│ │ │ │ -00064340: 6853 797a 7967 793a 2068 6967 6853 797a  hSyzygy: highSyz
│ │ │ │ -00064350: 7967 792c 202d 2d20 5265 7475 726e 7320  ygy, -- Returns 
│ │ │ │ -00064360: 6120 7379 7a79 6779 206d 6f64 756c 6520  a syzygy module 
│ │ │ │ -00064370: 6f6e 6520 6265 796f 6e64 2074 6865 0a20  one beyond the. 
│ │ │ │ -00064380: 2020 2072 6567 756c 6172 6974 7920 6f66     regularity of
│ │ │ │ -00064390: 2045 7874 284d 2c6b 290a 0a46 756e 6374   Ext(M,k)..Funct
│ │ │ │ -000643a0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ -000643b0: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ -000643c0: 6420 4f70 7469 6d69 736d 3a0a 3d3d 3d3d  d Optimism:.====
│ │ │ │ +00063de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00063df0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00063e00: 0a0a 2020 2a20 2a6e 6f74 6520 4578 744d  ..  * *note ExtM
│ │ │ │ +00063e10: 6f64 756c 653a 2045 7874 4d6f 6475 6c65  odule: ExtModule
│ │ │ │ +00063e20: 2c20 2d2d 2045 7874 5e2a 284d 2c6b 2920  , -- Ext^*(M,k) 
│ │ │ │ +00063e30: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ +00063e40: 696e 7465 7273 6563 7469 6f6e 2061 730a  intersection as.
│ │ │ │ +00063e50: 2020 2020 6d6f 6475 6c65 206f 7665 7220      module over 
│ │ │ │ +00063e60: 4349 206f 7065 7261 746f 7220 7269 6e67  CI operator ring
│ │ │ │ +00063e70: 0a20 202a 202a 6e6f 7465 2065 7665 6e45  .  * *note evenE
│ │ │ │ +00063e80: 7874 4d6f 6475 6c65 3a20 6576 656e 4578  xtModule: evenEx
│ │ │ │ +00063e90: 744d 6f64 756c 652c 202d 2d20 6576 656e  tModule, -- even
│ │ │ │ +00063ea0: 2070 6172 7420 6f66 2045 7874 5e2a 284d   part of Ext^*(M
│ │ │ │ +00063eb0: 2c6b 2920 6f76 6572 2061 0a20 2020 2063  ,k) over a.    c
│ │ │ │ +00063ec0: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ +00063ed0: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ +00063ee0: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ +00063ef0: 7269 6e67 0a20 202a 202a 6e6f 7465 204f  ring.  * *note O
│ │ │ │ +00063f00: 7574 5269 6e67 3a20 4f75 7452 696e 672c  utRing: OutRing,
│ │ │ │ +00063f10: 202d 2d20 4f70 7469 6f6e 2061 6c6c 6f77   -- Option allow
│ │ │ │ +00063f20: 696e 6720 7370 6563 6966 6963 6174 696f  ing specificatio
│ │ │ │ +00063f30: 6e20 6f66 2074 6865 2072 696e 6720 6f76  n of the ring ov
│ │ │ │ +00063f40: 6572 0a20 2020 2077 6869 6368 2074 6865  er.    which the
│ │ │ │ +00063f50: 206f 7574 7075 7420 6973 2064 6566 696e   output is defin
│ │ │ │ +00063f60: 6564 0a0a 5761 7973 2074 6f20 7573 6520  ed..Ways to use 
│ │ │ │ +00063f70: 6f64 6445 7874 4d6f 6475 6c65 3a0a 3d3d  oddExtModule:.==
│ │ │ │ +00063f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00063f90: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6f64  =======..  * "od
│ │ │ │ +00063fa0: 6445 7874 4d6f 6475 6c65 284d 6f64 756c  dExtModule(Modul
│ │ │ │ +00063fb0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ +00063fc0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +00063fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00063fe0: 6f62 6a65 6374 202a 6e6f 7465 206f 6464  object *note odd
│ │ │ │ +00063ff0: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ +00064000: 744d 6f64 756c 652c 2069 7320 6120 2a6e  tModule, is a *n
│ │ │ │ +00064010: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ +00064020: 696f 6e20 7769 7468 0a6f 7074 696f 6e73  ion with.options
│ │ │ │ +00064030: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00064040: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ +00064050: 7468 4f70 7469 6f6e 732c 2e0a 1f0a 4669  thOptions,....Fi
│ │ │ │ +00064060: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +00064070: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +00064080: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +00064090: 4f70 7469 6d69 736d 2c20 4e65 7874 3a20  Optimism, Next: 
│ │ │ │ +000640a0: 4f75 7452 696e 672c 2050 7265 763a 206f  OutRing, Prev: o
│ │ │ │ +000640b0: 6464 4578 744d 6f64 756c 652c 2055 703a  ddExtModule, Up:
│ │ │ │ +000640c0: 2054 6f70 0a0a 4f70 7469 6d69 736d 202d   Top..Optimism -
│ │ │ │ +000640d0: 2d20 4f70 7469 6f6e 2074 6f20 6869 6768  - Option to high
│ │ │ │ +000640e0: 5379 7a79 6779 0a2a 2a2a 2a2a 2a2a 2a2a  Syzygy.*********
│ │ │ │ +000640f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00064100: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ +00064110: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ +00064120: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00064130: 6869 6768 5379 7a79 6779 284d 2c20 4f70  highSyzygy(M, Op
│ │ │ │ +00064140: 7469 6d69 736d 203d 3e20 3129 0a20 202a  timism => 1).  *
│ │ │ │ +00064150: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00064160: 204f 7074 696d 6973 6d2c 2061 6e20 2a6e   Optimism, an *n
│ │ │ │ +00064170: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +00064180: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +00064190: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +000641a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 6869  =========..If hi
│ │ │ │ +000641b0: 6768 5379 7a79 6779 284d 2920 6368 6f6f  ghSyzygy(M) choo
│ │ │ │ +000641c0: 7365 7320 7468 6520 702d 7468 2073 797a  ses the p-th syz
│ │ │ │ +000641d0: 7967 792c 2074 6865 6e20 6869 6768 5379  ygy, then highSy
│ │ │ │ +000641e0: 7a79 6779 284d 2c4f 7074 696d 6973 6d3d  zygy(M,Optimism=
│ │ │ │ +000641f0: 3e72 290a 6368 6f6f 7365 7320 7468 6520  >r).chooses the 
│ │ │ │ +00064200: 2870 2d72 292d 7468 2073 797a 7967 792e  (p-r)-th syzygy.
│ │ │ │ +00064210: 2028 506f 7369 7469 7665 204f 7074 696d   (Positive Optim
│ │ │ │ +00064220: 6973 6d20 6368 6f6f 7365 7320 6120 6c6f  ism chooses a lo
│ │ │ │ +00064230: 7765 7220 2268 6967 6822 2073 797a 7967  wer "high" syzyg
│ │ │ │ +00064240: 792c 0a6e 6567 6174 6976 6520 4f70 7469  y,.negative Opti
│ │ │ │ +00064250: 6d69 736d 2061 2068 6967 6865 7220 2268  mism a higher "h
│ │ │ │ +00064260: 6967 6822 2073 797a 7967 792e 0a0a 4361  igh" syzygy...Ca
│ │ │ │ +00064270: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a41 7265  veat.======..Are
│ │ │ │ +00064280: 2074 6865 7265 2063 6173 6573 2077 6865   there cases whe
│ │ │ │ +00064290: 6e20 706f 7369 7469 7665 204f 7074 696d  n positive Optim
│ │ │ │ +000642a0: 6973 6d20 6973 206a 7573 7469 6669 6564  ism is justified
│ │ │ │ +000642b0: 3f0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ?..See also.====
│ │ │ │ +000642c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +000642d0: 6d66 426f 756e 643a 206d 6642 6f75 6e64  mfBound: mfBound
│ │ │ │ +000642e0: 2c20 2d2d 2064 6574 6572 6d69 6e65 7320  , -- determines 
│ │ │ │ +000642f0: 686f 7720 6869 6768 2061 2073 797a 7967  how high a syzyg
│ │ │ │ +00064300: 7920 746f 2074 616b 6520 666f 720a 2020  y to take for.  
│ │ │ │ +00064310: 2020 226d 6174 7269 7846 6163 746f 7269    "matrixFactori
│ │ │ │ +00064320: 7a61 7469 6f6e 220a 2020 2a20 2a6e 6f74  zation".  * *not
│ │ │ │ +00064330: 6520 6869 6768 5379 7a79 6779 3a20 6869  e highSyzygy: hi
│ │ │ │ +00064340: 6768 5379 7a79 6779 2c20 2d2d 2052 6574  ghSyzygy, -- Ret
│ │ │ │ +00064350: 7572 6e73 2061 2073 797a 7967 7920 6d6f  urns a syzygy mo
│ │ │ │ +00064360: 6475 6c65 206f 6e65 2062 6579 6f6e 6420  dule one beyond 
│ │ │ │ +00064370: 7468 650a 2020 2020 7265 6775 6c61 7269  the.    regulari
│ │ │ │ +00064380: 7479 206f 6620 4578 7428 4d2c 6b29 0a0a  ty of Ext(M,k)..
│ │ │ │ +00064390: 4675 6e63 7469 6f6e 7320 7769 7468 206f  Functions with o
│ │ │ │ +000643a0: 7074 696f 6e61 6c20 6172 6775 6d65 6e74  ptional argument
│ │ │ │ +000643b0: 206e 616d 6564 204f 7074 696d 6973 6d3a   named Optimism:
│ │ │ │ +000643c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │  000643d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000643e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000643f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -00064400: 2a20 2268 6967 6853 797a 7967 7928 2e2e  * "highSyzygy(..
│ │ │ │ -00064410: 2e2c 4f70 7469 6d69 736d 3d3e 2e2e 2e29  .,Optimism=>...)
│ │ │ │ -00064420: 2220 2d2d 2073 6565 202a 6e6f 7465 2068  " -- see *note h
│ │ │ │ -00064430: 6967 6853 797a 7967 793a 2068 6967 6853  ighSyzygy: highS
│ │ │ │ -00064440: 797a 7967 792c 202d 2d0a 2020 2020 5265  yzygy, --.    Re
│ │ │ │ -00064450: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ -00064460: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ -00064470: 2074 6865 2072 6567 756c 6172 6974 7920   the regularity 
│ │ │ │ -00064480: 6f66 2045 7874 284d 2c6b 290a 2020 2a20  of Ext(M,k).  * 
│ │ │ │ -00064490: 2274 776f 4d6f 6e6f 6d69 616c 7328 2e2e  "twoMonomials(..
│ │ │ │ -000644a0: 2e2c 4f70 7469 6d69 736d 3d3e 2e2e 2e29  .,Optimism=>...)
│ │ │ │ -000644b0: 2220 2d2d 2073 6565 202a 6e6f 7465 2074  " -- see *note t
│ │ │ │ -000644c0: 776f 4d6f 6e6f 6d69 616c 733a 2074 776f  woMonomials: two
│ │ │ │ -000644d0: 4d6f 6e6f 6d69 616c 732c 0a20 2020 202d  Monomials,.    -
│ │ │ │ -000644e0: 2d20 7461 6c6c 7920 7468 6520 7365 7175  - tally the sequ
│ │ │ │ -000644f0: 656e 6365 7320 6f66 2042 5261 6e6b 7320  ences of BRanks 
│ │ │ │ -00064500: 666f 7220 6365 7274 6169 6e20 6578 616d  for certain exam
│ │ │ │ -00064510: 706c 6573 0a0a 466f 7220 7468 6520 7072  ples..For the pr
│ │ │ │ -00064520: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -00064530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -00064540: 206f 626a 6563 7420 2a6e 6f74 6520 4f70   object *note Op
│ │ │ │ -00064550: 7469 6d69 736d 3a20 4f70 7469 6d69 736d  timism: Optimism
│ │ │ │ -00064560: 2c20 6973 2061 202a 6e6f 7465 2073 796d  , is a *note sym
│ │ │ │ -00064570: 626f 6c3a 2028 4d61 6361 756c 6179 3244  bol: (Macaulay2D
│ │ │ │ -00064580: 6f63 2953 796d 626f 6c2c 2e0a 1f0a 4669  oc)Symbol,....Fi
│ │ │ │ -00064590: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -000645a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -000645b0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ -000645c0: 4f75 7452 696e 672c 204e 6578 743a 2070  OutRing, Next: p
│ │ │ │ -000645d0: 7369 4d61 7073 2c20 5072 6576 3a20 4f70  siMaps, Prev: Op
│ │ │ │ -000645e0: 7469 6d69 736d 2c20 5570 3a20 546f 700a  timism, Up: Top.
│ │ │ │ -000645f0: 0a4f 7574 5269 6e67 202d 2d20 4f70 7469  .OutRing -- Opti
│ │ │ │ -00064600: 6f6e 2061 6c6c 6f77 696e 6720 7370 6563  on allowing spec
│ │ │ │ -00064610: 6966 6963 6174 696f 6e20 6f66 2074 6865  ification of the
│ │ │ │ -00064620: 2072 696e 6720 6f76 6572 2077 6869 6368   ring over which
│ │ │ │ -00064630: 2074 6865 206f 7574 7075 7420 6973 2064   the output is d
│ │ │ │ -00064640: 6566 696e 6564 0a2a 2a2a 2a2a 2a2a 2a2a  efined.*********
│ │ │ │ +000643f0: 3d0a 0a20 202a 2022 6869 6768 5379 7a79  =..  * "highSyzy
│ │ │ │ +00064400: 6779 282e 2e2e 2c4f 7074 696d 6973 6d3d  gy(...,Optimism=
│ │ │ │ +00064410: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e  >...)" -- see *n
│ │ │ │ +00064420: 6f74 6520 6869 6768 5379 7a79 6779 3a20  ote highSyzygy: 
│ │ │ │ +00064430: 6869 6768 5379 7a79 6779 2c20 2d2d 0a20  highSyzygy, --. 
│ │ │ │ +00064440: 2020 2052 6574 7572 6e73 2061 2073 797a     Returns a syz
│ │ │ │ +00064450: 7967 7920 6d6f 6475 6c65 206f 6e65 2062  ygy module one b
│ │ │ │ +00064460: 6579 6f6e 6420 7468 6520 7265 6775 6c61  eyond the regula
│ │ │ │ +00064470: 7269 7479 206f 6620 4578 7428 4d2c 6b29  rity of Ext(M,k)
│ │ │ │ +00064480: 0a20 202a 2022 7477 6f4d 6f6e 6f6d 6961  .  * "twoMonomia
│ │ │ │ +00064490: 6c73 282e 2e2e 2c4f 7074 696d 6973 6d3d  ls(...,Optimism=
│ │ │ │ +000644a0: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e  >...)" -- see *n
│ │ │ │ +000644b0: 6f74 6520 7477 6f4d 6f6e 6f6d 6961 6c73  ote twoMonomials
│ │ │ │ +000644c0: 3a20 7477 6f4d 6f6e 6f6d 6961 6c73 2c0a  : twoMonomials,.
│ │ │ │ +000644d0: 2020 2020 2d2d 2074 616c 6c79 2074 6865      -- tally the
│ │ │ │ +000644e0: 2073 6571 7565 6e63 6573 206f 6620 4252   sequences of BR
│ │ │ │ +000644f0: 616e 6b73 2066 6f72 2063 6572 7461 696e  anks for certain
│ │ │ │ +00064500: 2065 7861 6d70 6c65 730a 0a46 6f72 2074   examples..For t
│ │ │ │ +00064510: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +00064520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00064530: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +00064540: 7465 204f 7074 696d 6973 6d3a 204f 7074  te Optimism: Opt
│ │ │ │ +00064550: 696d 6973 6d2c 2069 7320 6120 2a6e 6f74  imism, is a *not
│ │ │ │ +00064560: 6520 7379 6d62 6f6c 3a20 284d 6163 6175  e symbol: (Macau
│ │ │ │ +00064570: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e  lay2Doc)Symbol,.
│ │ │ │ +00064580: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00064590: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +000645a0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +000645b0: 6f64 653a 204f 7574 5269 6e67 2c20 4e65  ode: OutRing, Ne
│ │ │ │ +000645c0: 7874 3a20 7073 694d 6170 732c 2050 7265  xt: psiMaps, Pre
│ │ │ │ +000645d0: 763a 204f 7074 696d 6973 6d2c 2055 703a  v: Optimism, Up:
│ │ │ │ +000645e0: 2054 6f70 0a0a 4f75 7452 696e 6720 2d2d   Top..OutRing --
│ │ │ │ +000645f0: 204f 7074 696f 6e20 616c 6c6f 7769 6e67   Option allowing
│ │ │ │ +00064600: 2073 7065 6369 6669 6361 7469 6f6e 206f   specification o
│ │ │ │ +00064610: 6620 7468 6520 7269 6e67 206f 7665 7220  f the ring over 
│ │ │ │ +00064620: 7768 6963 6820 7468 6520 6f75 7470 7574  which the output
│ │ │ │ +00064630: 2069 7320 6465 6669 6e65 640a 2a2a 2a2a   is defined.****
│ │ │ │ +00064640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5365  ************..Se
│ │ │ │ -000646a0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -000646b0: 0a20 202a 202a 6e6f 7465 2065 7665 6e45  .  * *note evenE
│ │ │ │ -000646c0: 7874 4d6f 6475 6c65 3a20 6576 656e 4578  xtModule: evenEx
│ │ │ │ -000646d0: 744d 6f64 756c 652c 202d 2d20 6576 656e  tModule, -- even
│ │ │ │ -000646e0: 2070 6172 7420 6f66 2045 7874 5e2a 284d   part of Ext^*(M
│ │ │ │ -000646f0: 2c6b 2920 6f76 6572 2061 0a20 2020 2063  ,k) over a.    c
│ │ │ │ -00064700: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ -00064710: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ -00064720: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ -00064730: 7269 6e67 0a20 202a 202a 6e6f 7465 206f  ring.  * *note o
│ │ │ │ -00064740: 6464 4578 744d 6f64 756c 653a 206f 6464  ddExtModule: odd
│ │ │ │ -00064750: 4578 744d 6f64 756c 652c 202d 2d20 6f64  ExtModule, -- od
│ │ │ │ -00064760: 6420 7061 7274 206f 6620 4578 745e 2a28  d part of Ext^*(
│ │ │ │ -00064770: 4d2c 6b29 206f 7665 7220 6120 636f 6d70  M,k) over a comp
│ │ │ │ -00064780: 6c65 7465 0a20 2020 2069 6e74 6572 7365  lete.    interse
│ │ │ │ -00064790: 6374 696f 6e20 6173 206d 6f64 756c 6520  ction as module 
│ │ │ │ -000647a0: 6f76 6572 2043 4920 6f70 6572 6174 6f72  over CI operator
│ │ │ │ -000647b0: 2072 696e 670a 0a46 756e 6374 696f 6e73   ring..Functions
│ │ │ │ -000647c0: 2077 6974 6820 6f70 7469 6f6e 616c 2061   with optional a
│ │ │ │ -000647d0: 7267 756d 656e 7420 6e61 6d65 6420 4f75  rgument named Ou
│ │ │ │ -000647e0: 7452 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d  tRing:.=========
│ │ │ │ +00064690: 2a0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  *..See also.====
│ │ │ │ +000646a0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +000646b0: 6576 656e 4578 744d 6f64 756c 653a 2065  evenExtModule: e
│ │ │ │ +000646c0: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d  venExtModule, --
│ │ │ │ +000646d0: 2065 7665 6e20 7061 7274 206f 6620 4578   even part of Ex
│ │ │ │ +000646e0: 745e 2a28 4d2c 6b29 206f 7665 7220 610a  t^*(M,k) over a.
│ │ │ │ +000646f0: 2020 2020 636f 6d70 6c65 7465 2069 6e74      complete int
│ │ │ │ +00064700: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ +00064710: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +00064720: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ +00064730: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65  ote oddExtModule
│ │ │ │ +00064740: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20  : oddExtModule, 
│ │ │ │ +00064750: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ +00064760: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ +00064770: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ +00064780: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ +00064790: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ +000647a0: 7261 746f 7220 7269 6e67 0a0a 4675 6e63  rator ring..Func
│ │ │ │ +000647b0: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +000647c0: 6e61 6c20 6172 6775 6d65 6e74 206e 616d  nal argument nam
│ │ │ │ +000647d0: 6564 204f 7574 5269 6e67 3a0a 3d3d 3d3d  ed OutRing:.====
│ │ │ │ +000647e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000647f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00064800: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00064810: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7665  ======..  * "eve
│ │ │ │ -00064820: 6e45 7874 4d6f 6475 6c65 282e 2e2e 2c4f  nExtModule(...,O
│ │ │ │ -00064830: 7574 5269 6e67 3d3e 2e2e 2e29 2220 2d2d  utRing=>...)" --
│ │ │ │ -00064840: 2073 6565 202a 6e6f 7465 2065 7665 6e45   see *note evenE
│ │ │ │ -00064850: 7874 4d6f 6475 6c65 3a0a 2020 2020 6576  xtModule:.    ev
│ │ │ │ -00064860: 656e 4578 744d 6f64 756c 652c 202d 2d20  enExtModule, -- 
│ │ │ │ -00064870: 6576 656e 2070 6172 7420 6f66 2045 7874  even part of Ext
│ │ │ │ -00064880: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063  ^*(M,k) over a c
│ │ │ │ -00064890: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ -000648a0: 7469 6f6e 2061 730a 2020 2020 6d6f 6475  tion as.    modu
│ │ │ │ -000648b0: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ -000648c0: 746f 7220 7269 6e67 0a20 202a 2022 6f64  tor ring.  * "od
│ │ │ │ -000648d0: 6445 7874 4d6f 6475 6c65 282e 2e2e 2c4f  dExtModule(...,O
│ │ │ │ -000648e0: 7574 5269 6e67 3d3e 2e2e 2e29 2220 2d2d  utRing=>...)" --
│ │ │ │ -000648f0: 2073 6565 202a 6e6f 7465 206f 6464 4578   see *note oddEx
│ │ │ │ -00064900: 744d 6f64 756c 653a 206f 6464 4578 744d  tModule: oddExtM
│ │ │ │ -00064910: 6f64 756c 652c 0a20 2020 202d 2d20 6f64  odule,.    -- od
│ │ │ │ -00064920: 6420 7061 7274 206f 6620 4578 745e 2a28  d part of Ext^*(
│ │ │ │ -00064930: 4d2c 6b29 206f 7665 7220 6120 636f 6d70  M,k) over a comp
│ │ │ │ -00064940: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ -00064950: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ -00064960: 2043 490a 2020 2020 6f70 6572 6174 6f72   CI.    operator
│ │ │ │ -00064970: 2072 696e 670a 0a46 6f72 2074 6865 2070   ring..For the p
│ │ │ │ -00064980: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00064990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -000649a0: 6520 6f62 6a65 6374 202a 6e6f 7465 204f  e object *note O
│ │ │ │ -000649b0: 7574 5269 6e67 3a20 4f75 7452 696e 672c  utRing: OutRing,
│ │ │ │ -000649c0: 2069 7320 6120 2a6e 6f74 6520 7379 6d62   is a *note symb
│ │ │ │ -000649d0: 6f6c 3a20 284d 6163 6175 6c61 7932 446f  ol: (Macaulay2Do
│ │ │ │ -000649e0: 6329 5379 6d62 6f6c 2c2e 0a1f 0a46 696c  c)Symbol,....Fil
│ │ │ │ -000649f0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00064a00: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00064a10: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2070  ns.info, Node: p
│ │ │ │ -00064a20: 7369 4d61 7073 2c20 4e65 7874 3a20 7265  siMaps, Next: re
│ │ │ │ -00064a30: 6775 6c61 7269 7479 5365 7175 656e 6365  gularitySequence
│ │ │ │ -00064a40: 2c20 5072 6576 3a20 4f75 7452 696e 672c  , Prev: OutRing,
│ │ │ │ -00064a50: 2055 703a 2054 6f70 0a0a 7073 694d 6170   Up: Top..psiMap
│ │ │ │ -00064a60: 7320 2d2d 206c 6973 7420 7468 6520 6d61  s -- list the ma
│ │ │ │ -00064a70: 7073 2020 7073 6928 7029 3a20 425f 3128  ps  psi(p): B_1(
│ │ │ │ -00064a80: 7029 202d 2d3e 2041 5f30 2870 2d31 2920  p) --> A_0(p-1) 
│ │ │ │ -00064a90: 696e 2061 206d 6174 7269 7846 6163 746f  in a matrixFacto
│ │ │ │ -00064aa0: 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a  rization.*******
│ │ │ │ +00064800: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00064810: 2022 6576 656e 4578 744d 6f64 756c 6528   "evenExtModule(
│ │ │ │ +00064820: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e  ...,OutRing=>...
│ │ │ │ +00064830: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ +00064840: 6576 656e 4578 744d 6f64 756c 653a 0a20  evenExtModule:. 
│ │ │ │ +00064850: 2020 2065 7665 6e45 7874 4d6f 6475 6c65     evenExtModule
│ │ │ │ +00064860: 2c20 2d2d 2065 7665 6e20 7061 7274 206f  , -- even part o
│ │ │ │ +00064870: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ +00064880: 7220 6120 636f 6d70 6c65 7465 2069 6e74  r a complete int
│ │ │ │ +00064890: 6572 7365 6374 696f 6e20 6173 0a20 2020  ersection as.   
│ │ │ │ +000648a0: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ +000648b0: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ +000648c0: 2a20 226f 6464 4578 744d 6f64 756c 6528  * "oddExtModule(
│ │ │ │ +000648d0: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e  ...,OutRing=>...
│ │ │ │ +000648e0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ +000648f0: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64  oddExtModule: od
│ │ │ │ +00064900: 6445 7874 4d6f 6475 6c65 2c0a 2020 2020  dExtModule,.    
│ │ │ │ +00064910: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ +00064920: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ +00064930: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +00064940: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65  ection as module
│ │ │ │ +00064950: 206f 7665 7220 4349 0a20 2020 206f 7065   over CI.    ope
│ │ │ │ +00064960: 7261 746f 7220 7269 6e67 0a0a 466f 7220  rator ring..For 
│ │ │ │ +00064970: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00064980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00064990: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +000649a0: 6f74 6520 4f75 7452 696e 673a 204f 7574  ote OutRing: Out
│ │ │ │ +000649b0: 5269 6e67 2c20 6973 2061 202a 6e6f 7465  Ring, is a *note
│ │ │ │ +000649c0: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
│ │ │ │ +000649d0: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a  ay2Doc)Symbol,..
│ │ │ │ +000649e0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +000649f0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +00064a00: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +00064a10: 6465 3a20 7073 694d 6170 732c 204e 6578  de: psiMaps, Nex
│ │ │ │ +00064a20: 743a 2072 6567 756c 6172 6974 7953 6571  t: regularitySeq
│ │ │ │ +00064a30: 7565 6e63 652c 2050 7265 763a 204f 7574  uence, Prev: Out
│ │ │ │ +00064a40: 5269 6e67 2c20 5570 3a20 546f 700a 0a70  Ring, Up: Top..p
│ │ │ │ +00064a50: 7369 4d61 7073 202d 2d20 6c69 7374 2074  siMaps -- list t
│ │ │ │ +00064a60: 6865 206d 6170 7320 2070 7369 2870 293a  he maps  psi(p):
│ │ │ │ +00064a70: 2042 5f31 2870 2920 2d2d 3e20 415f 3028   B_1(p) --> A_0(
│ │ │ │ +00064a80: 702d 3129 2069 6e20 6120 6d61 7472 6978  p-1) in a matrix
│ │ │ │ +00064a90: 4661 6374 6f72 697a 6174 696f 6e0a 2a2a  Factorization.**
│ │ │ │ +00064aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064af0: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ -00064b00: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ -00064b10: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00064b20: 7073 6d61 7073 203d 2070 7369 4d61 7073  psmaps = psiMaps
│ │ │ │ -00064b30: 206d 660a 2020 2a20 496e 7075 7473 3a0a   mf.  * Inputs:.
│ │ │ │ -00064b40: 2020 2020 2020 2a20 6d66 2c20 6120 2a6e        * mf, a *n
│ │ │ │ -00064b50: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ -00064b60: 6c61 7932 446f 6329 4c69 7374 2c2c 206f  lay2Doc)List,, o
│ │ │ │ -00064b70: 7574 7075 7420 6f66 2061 206d 6174 7269  utput of a matri
│ │ │ │ -00064b80: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ -00064b90: 2020 2020 2020 2063 6f6d 7075 7461 7469         computati
│ │ │ │ -00064ba0: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a  on.  * Outputs:.
│ │ │ │ -00064bb0: 2020 2020 2020 2a20 7073 6d61 7073 2c20        * psmaps, 
│ │ │ │ -00064bc0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -00064bd0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -00064be0: 2c2c 206c 6973 7420 6d61 7472 6963 6573  ,, list matrices
│ │ │ │ -00064bf0: 2024 645f 703a 0a20 2020 2020 2020 2042   $d_p:.        B
│ │ │ │ -00064c00: 5f31 2870 295c 746f 2041 5f30 2870 2d31  _1(p)\to A_0(p-1
│ │ │ │ -00064c10: 2924 0a0a 4465 7363 7269 7074 696f 6e0a  )$..Description.
│ │ │ │ -00064c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 6565  ===========..See
│ │ │ │ -00064c30: 2074 6865 2064 6f63 756d 656e 7461 7469   the documentati
│ │ │ │ -00064c40: 6f6e 2066 6f72 206d 6174 7269 7846 6163  on for matrixFac
│ │ │ │ -00064c50: 746f 7269 7a61 7469 6f6e 2066 6f72 2061  torization for a
│ │ │ │ -00064c60: 6e20 6578 616d 706c 652e 0a0a 5365 6520  n example...See 
│ │ │ │ -00064c70: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -00064c80: 202a 202a 6e6f 7465 206d 6174 7269 7846   * *note matrixF
│ │ │ │ -00064c90: 6163 746f 7269 7a61 7469 6f6e 3a20 6d61  actorization: ma
│ │ │ │ -00064ca0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -00064cb0: 6e2c 202d 2d20 4d61 7073 2069 6e20 6120  n, -- Maps in a 
│ │ │ │ -00064cc0: 6869 6768 6572 0a20 2020 2063 6f64 696d  higher.    codim
│ │ │ │ -00064cd0: 656e 7369 6f6e 206d 6174 7269 7820 6661  ension matrix fa
│ │ │ │ -00064ce0: 6374 6f72 697a 6174 696f 6e0a 2020 2a20  ctorization.  * 
│ │ │ │ -00064cf0: 2a6e 6f74 6520 4252 616e 6b73 3a20 4252  *note BRanks: BR
│ │ │ │ -00064d00: 616e 6b73 2c20 2d2d 2072 616e 6b73 206f  anks, -- ranks o
│ │ │ │ -00064d10: 6620 7468 6520 6d6f 6475 6c65 7320 425f  f the modules B_
│ │ │ │ -00064d20: 6928 6429 2069 6e20 610a 2020 2020 6d61  i(d) in a.    ma
│ │ │ │ -00064d30: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -00064d40: 6e0a 2020 2a20 2a6e 6f74 6520 624d 6170  n.  * *note bMap
│ │ │ │ -00064d50: 733a 2062 4d61 7073 2c20 2d2d 206c 6973  s: bMaps, -- lis
│ │ │ │ -00064d60: 7420 7468 6520 6d61 7073 2020 645f 703a  t the maps  d_p:
│ │ │ │ -00064d70: 425f 3128 7029 2d2d 3e42 5f30 2870 2920  B_1(p)-->B_0(p) 
│ │ │ │ -00064d80: 696e 2061 0a20 2020 206d 6174 7269 7846  in a.    matrixF
│ │ │ │ -00064d90: 6163 746f 7269 7a61 7469 6f6e 0a20 202a  actorization.  *
│ │ │ │ -00064da0: 202a 6e6f 7465 2064 4d61 7073 3a20 644d   *note dMaps: dM
│ │ │ │ -00064db0: 6170 732c 202d 2d20 6c69 7374 2074 6865  aps, -- list the
│ │ │ │ -00064dc0: 206d 6170 7320 2064 2870 293a 415f 3128   maps  d(p):A_1(
│ │ │ │ -00064dd0: 7029 2d2d 3e20 415f 3028 7029 2069 6e20  p)--> A_0(p) in 
│ │ │ │ -00064de0: 610a 2020 2020 6d61 7472 6978 4661 6374  a.    matrixFact
│ │ │ │ -00064df0: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e  orization.  * *n
│ │ │ │ -00064e00: 6f74 6520 684d 6170 733a 2068 4d61 7073  ote hMaps: hMaps
│ │ │ │ -00064e10: 2c20 2d2d 206c 6973 7420 7468 6520 6d61  , -- list the ma
│ │ │ │ -00064e20: 7073 2020 6828 7029 3a20 415f 3028 7029  ps  h(p): A_0(p)
│ │ │ │ -00064e30: 2d2d 3e20 415f 3128 7029 2069 6e20 610a  --> A_1(p) in a.
│ │ │ │ -00064e40: 2020 2020 6d61 7472 6978 4661 6374 6f72      matrixFactor
│ │ │ │ -00064e50: 697a 6174 696f 6e0a 0a57 6179 7320 746f  ization..Ways to
│ │ │ │ -00064e60: 2075 7365 2070 7369 4d61 7073 3a0a 3d3d   use psiMaps:.==
│ │ │ │ -00064e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00064e80: 3d3d 0a0a 2020 2a20 2270 7369 4d61 7073  ==..  * "psiMaps
│ │ │ │ -00064e90: 284c 6973 7429 220a 0a46 6f72 2074 6865  (List)"..For the
│ │ │ │ -00064ea0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00064eb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00064ec0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00064ed0: 2070 7369 4d61 7073 3a20 7073 694d 6170   psiMaps: psiMap
│ │ │ │ -00064ee0: 732c 2069 7320 6120 2a6e 6f74 6520 6d65  s, is a *note me
│ │ │ │ -00064ef0: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ -00064f00: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -00064f10: 686f 6446 756e 6374 696f 6e2c 2e0a 1f0a  hodFunction,....
│ │ │ │ -00064f20: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ -00064f30: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -00064f40: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ -00064f50: 3a20 7265 6775 6c61 7269 7479 5365 7175  : regularitySequ
│ │ │ │ -00064f60: 656e 6365 2c20 4e65 7874 3a20 5332 2c20  ence, Next: S2, 
│ │ │ │ -00064f70: 5072 6576 3a20 7073 694d 6170 732c 2055  Prev: psiMaps, U
│ │ │ │ -00064f80: 703a 2054 6f70 0a0a 7265 6775 6c61 7269  p: Top..regulari
│ │ │ │ -00064f90: 7479 5365 7175 656e 6365 202d 2d20 7265  tySequence -- re
│ │ │ │ -00064fa0: 6775 6c61 7269 7479 206f 6620 4578 7420  gularity of Ext 
│ │ │ │ -00064fb0: 6d6f 6475 6c65 7320 666f 7220 6120 7365  modules for a se
│ │ │ │ -00064fc0: 7175 656e 6365 206f 6620 4d43 4d20 6170  quence of MCM ap
│ │ │ │ -00064fd0: 7072 6f78 696d 6174 696f 6e73 0a2a 2a2a  proximations.***
│ │ │ │ +00064ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ +00064af0: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ +00064b00: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00064b10: 2020 2020 2070 736d 6170 7320 3d20 7073       psmaps = ps
│ │ │ │ +00064b20: 694d 6170 7320 6d66 0a20 202a 2049 6e70  iMaps mf.  * Inp
│ │ │ │ +00064b30: 7574 733a 0a20 2020 2020 202a 206d 662c  uts:.      * mf,
│ │ │ │ +00064b40: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +00064b50: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +00064b60: 742c 2c20 6f75 7470 7574 206f 6620 6120  t,, output of a 
│ │ │ │ +00064b70: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +00064b80: 696f 6e0a 2020 2020 2020 2020 636f 6d70  ion.        comp
│ │ │ │ +00064b90: 7574 6174 696f 6e0a 2020 2a20 4f75 7470  utation.  * Outp
│ │ │ │ +00064ba0: 7574 733a 0a20 2020 2020 202a 2070 736d  uts:.      * psm
│ │ │ │ +00064bb0: 6170 732c 2061 202a 6e6f 7465 206c 6973  aps, a *note lis
│ │ │ │ +00064bc0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +00064bd0: 294c 6973 742c 2c20 6c69 7374 206d 6174  )List,, list mat
│ │ │ │ +00064be0: 7269 6365 7320 2464 5f70 3a0a 2020 2020  rices $d_p:.    
│ │ │ │ +00064bf0: 2020 2020 425f 3128 7029 5c74 6f20 415f      B_1(p)\to A_
│ │ │ │ +00064c00: 3028 702d 3129 240a 0a44 6573 6372 6970  0(p-1)$..Descrip
│ │ │ │ +00064c10: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00064c20: 0a0a 5365 6520 7468 6520 646f 6375 6d65  ..See the docume
│ │ │ │ +00064c30: 6e74 6174 696f 6e20 666f 7220 6d61 7472  ntation for matr
│ │ │ │ +00064c40: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ +00064c50: 666f 7220 616e 2065 7861 6d70 6c65 2e0a  for an example..
│ │ │ │ +00064c60: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00064c70: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ +00064c80: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00064c90: 6e3a 206d 6174 7269 7846 6163 746f 7269  n: matrixFactori
│ │ │ │ +00064ca0: 7a61 7469 6f6e 2c20 2d2d 204d 6170 7320  zation, -- Maps 
│ │ │ │ +00064cb0: 696e 2061 2068 6967 6865 720a 2020 2020  in a higher.    
│ │ │ │ +00064cc0: 636f 6469 6d65 6e73 696f 6e20 6d61 7472  codimension matr
│ │ │ │ +00064cd0: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ +00064ce0: 0a20 202a 202a 6e6f 7465 2042 5261 6e6b  .  * *note BRank
│ │ │ │ +00064cf0: 733a 2042 5261 6e6b 732c 202d 2d20 7261  s: BRanks, -- ra
│ │ │ │ +00064d00: 6e6b 7320 6f66 2074 6865 206d 6f64 756c  nks of the modul
│ │ │ │ +00064d10: 6573 2042 5f69 2864 2920 696e 2061 0a20  es B_i(d) in a. 
│ │ │ │ +00064d20: 2020 206d 6174 7269 7846 6163 746f 7269     matrixFactori
│ │ │ │ +00064d30: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465  zation.  * *note
│ │ │ │ +00064d40: 2062 4d61 7073 3a20 624d 6170 732c 202d   bMaps: bMaps, -
│ │ │ │ +00064d50: 2d20 6c69 7374 2074 6865 206d 6170 7320  - list the maps 
│ │ │ │ +00064d60: 2064 5f70 3a42 5f31 2870 292d 2d3e 425f   d_p:B_1(p)-->B_
│ │ │ │ +00064d70: 3028 7029 2069 6e20 610a 2020 2020 6d61  0(p) in a.    ma
│ │ │ │ +00064d80: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00064d90: 6e0a 2020 2a20 2a6e 6f74 6520 644d 6170  n.  * *note dMap
│ │ │ │ +00064da0: 733a 2064 4d61 7073 2c20 2d2d 206c 6973  s: dMaps, -- lis
│ │ │ │ +00064db0: 7420 7468 6520 6d61 7073 2020 6428 7029  t the maps  d(p)
│ │ │ │ +00064dc0: 3a41 5f31 2870 292d 2d3e 2041 5f30 2870  :A_1(p)--> A_0(p
│ │ │ │ +00064dd0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ +00064de0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ +00064df0: 202a 202a 6e6f 7465 2068 4d61 7073 3a20   * *note hMaps: 
│ │ │ │ +00064e00: 684d 6170 732c 202d 2d20 6c69 7374 2074  hMaps, -- list t
│ │ │ │ +00064e10: 6865 206d 6170 7320 2068 2870 293a 2041  he maps  h(p): A
│ │ │ │ +00064e20: 5f30 2870 292d 2d3e 2041 5f31 2870 2920  _0(p)--> A_1(p) 
│ │ │ │ +00064e30: 696e 2061 0a20 2020 206d 6174 7269 7846  in a.    matrixF
│ │ │ │ +00064e40: 6163 746f 7269 7a61 7469 6f6e 0a0a 5761  actorization..Wa
│ │ │ │ +00064e50: 7973 2074 6f20 7573 6520 7073 694d 6170  ys to use psiMap
│ │ │ │ +00064e60: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
│ │ │ │ +00064e70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7073  =======..  * "ps
│ │ │ │ +00064e80: 694d 6170 7328 4c69 7374 2922 0a0a 466f  iMaps(List)"..Fo
│ │ │ │ +00064e90: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00064ea0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00064eb0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00064ec0: 2a6e 6f74 6520 7073 694d 6170 733a 2070  *note psiMaps: p
│ │ │ │ +00064ed0: 7369 4d61 7073 2c20 6973 2061 202a 6e6f  siMaps, is a *no
│ │ │ │ +00064ee0: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +00064ef0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ +00064f00: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00064f10: 2c2e 0a1f 0a46 696c 653a 2043 6f6d 706c  ,....File: Compl
│ │ │ │ +00064f20: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +00064f30: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ +00064f40: 204e 6f64 653a 2072 6567 756c 6172 6974   Node: regularit
│ │ │ │ +00064f50: 7953 6571 7565 6e63 652c 204e 6578 743a  ySequence, Next:
│ │ │ │ +00064f60: 2053 322c 2050 7265 763a 2070 7369 4d61   S2, Prev: psiMa
│ │ │ │ +00064f70: 7073 2c20 5570 3a20 546f 700a 0a72 6567  ps, Up: Top..reg
│ │ │ │ +00064f80: 756c 6172 6974 7953 6571 7565 6e63 6520  ularitySequence 
│ │ │ │ +00064f90: 2d2d 2072 6567 756c 6172 6974 7920 6f66  -- regularity of
│ │ │ │ +00064fa0: 2045 7874 206d 6f64 756c 6573 2066 6f72   Ext modules for
│ │ │ │ +00064fb0: 2061 2073 6571 7565 6e63 6520 6f66 204d   a sequence of M
│ │ │ │ +00064fc0: 434d 2061 7070 726f 7869 6d61 7469 6f6e  CM approximation
│ │ │ │ +00064fd0: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │  00064fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00065000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00065010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00065020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00065030: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -00065040: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -00065050: 200a 2020 2020 2020 2020 4c20 3d20 7265   .        L = re
│ │ │ │ -00065060: 6775 6c61 7269 7479 5365 7175 656e 6365  gularitySequence
│ │ │ │ -00065070: 2028 522c 4d29 0a20 202a 2049 6e70 7574   (R,M).  * Input
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│ │ │ │ -00065090: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ -000650a0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ -000650b0: 206c 6973 7420 6f66 2072 696e 6773 2052   list of rings R
│ │ │ │ -000650c0: 5f69 203d 0a20 2020 2020 2020 2053 2f28  _i =.        S/(
│ │ │ │ -000650d0: 665f 302e 2e66 5f7b 2869 2d31 297d 292c  f_0..f_{(i-1)}),
│ │ │ │ -000650e0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -000650f0: 6563 7469 6f6e 730a 2020 2020 2020 2a20  ections.      * 
│ │ │ │ -00065100: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -00065110: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00065120: 294d 6f64 756c 652c 2c20 6d6f 6475 6c65  )Module,, module
│ │ │ │ -00065130: 206f 7665 7220 525f 6320 7768 6572 6520   over R_c where 
│ │ │ │ -00065140: 6320 3d0a 2020 2020 2020 2020 6c65 6e67  c =.        leng
│ │ │ │ -00065150: 7468 2052 202d 2031 2e0a 2020 2a20 4f75  th R - 1..  * Ou
│ │ │ │ -00065160: 7470 7574 733a 0a20 2020 2020 202a 204c  tputs:.      * L
│ │ │ │ -00065170: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -00065180: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -00065190: 7374 2c2c 204c 6973 7420 6f66 2070 6169  st,, List of pai
│ │ │ │ -000651a0: 7273 207b 7265 6775 6c61 7269 7479 0a20  rs {regularity. 
│ │ │ │ -000651b0: 2020 2020 2020 2065 7665 6e45 7874 4d6f         evenExtMo
│ │ │ │ -000651c0: 6475 6c65 204d 5f69 2c20 7265 6775 6c61  dule M_i, regula
│ │ │ │ -000651d0: 7269 7479 206f 6464 4578 744d 6f64 756c  rity oddExtModul
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│ │ │ │ -000651f0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00065200: 0a43 6f6d 7075 7465 7320 7468 6520 6e6f  .Computes the no
│ │ │ │ -00065210: 6e2d 6672 6565 2070 6172 7473 204d 5f69  n-free parts M_i
│ │ │ │ -00065220: 206f 6620 7468 6520 4d43 4d20 6170 7072   of the MCM appr
│ │ │ │ -00065230: 6f78 696d 6174 696f 6e20 746f 204d 206f  oximation to M o
│ │ │ │ -00065240: 7665 7220 525f 692c 0a73 746f 7070 696e  ver R_i,.stoppin
│ │ │ │ -00065250: 6720 7768 656e 204d 5f69 2062 6563 6f6d  g when M_i becom
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│ │ │ │ -000652a0: 6775 6c61 7269 7469 6573 2c20 7374 6172  gularities, star
│ │ │ │ -000652b0: 7469 6e67 2077 6974 6820 4d5f 7b28 632d  ting with M_{(c-
│ │ │ │ -000652c0: 3129 7d20 4e6f 7465 2074 6861 7420 7468  1)} Note that th
│ │ │ │ -000652d0: 6520 6669 7273 7420 7061 6972 2069 7320  e first pair is 
│ │ │ │ -000652e0: 666f 720a 7468 650a 0a2b 2d2d 2d2d 2d2d  for.the..+------
│ │ │ │ +00065020: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00065030: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +00065040: 7361 6765 3a20 0a20 2020 2020 2020 204c  sage: .        L
│ │ │ │ +00065050: 203d 2072 6567 756c 6172 6974 7953 6571   = regularitySeq
│ │ │ │ +00065060: 7565 6e63 6520 2852 2c4d 290a 2020 2a20  uence (R,M).  * 
│ │ │ │ +00065070: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00065080: 522c 2061 202a 6e6f 7465 206c 6973 743a  R, a *note list:
│ │ │ │ +00065090: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ +000650a0: 6973 742c 2c20 6c69 7374 206f 6620 7269  ist,, list of ri
│ │ │ │ +000650b0: 6e67 7320 525f 6920 3d0a 2020 2020 2020  ngs R_i =.      
│ │ │ │ +000650c0: 2020 532f 2866 5f30 2e2e 665f 7b28 692d    S/(f_0..f_{(i-
│ │ │ │ +000650d0: 3129 7d29 2c20 636f 6d70 6c65 7465 2069  1)}), complete i
│ │ │ │ +000650e0: 6e74 6572 7365 6374 696f 6e73 0a20 2020  ntersections.   
│ │ │ │ +000650f0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +00065100: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +00065110: 7932 446f 6329 4d6f 6475 6c65 2c2c 206d  y2Doc)Module,, m
│ │ │ │ +00065120: 6f64 756c 6520 6f76 6572 2052 5f63 2077  odule over R_c w
│ │ │ │ +00065130: 6865 7265 2063 203d 0a20 2020 2020 2020  here c =.       
│ │ │ │ +00065140: 206c 656e 6774 6820 5220 2d20 312e 0a20   length R - 1.. 
│ │ │ │ +00065150: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00065160: 2020 2a20 4c2c 2061 202a 6e6f 7465 206c    * L, a *note l
│ │ │ │ +00065170: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
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│ │ │ │ +000651b0: 4578 744d 6f64 756c 6520 4d5f 692c 2072  ExtModule M_i, r
│ │ │ │ +000651c0: 6567 756c 6172 6974 7920 6f64 6445 7874  egularity oddExt
│ │ │ │ +000651d0: 4d6f 6475 6c65 204d 5f69 290a 0a44 6573  Module M_i)..Des
│ │ │ │ +000651e0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +000651f0: 3d3d 3d3d 0a0a 436f 6d70 7574 6573 2074  ====..Computes t
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│ │ │ │ +00065210: 7320 4d5f 6920 6f66 2074 6865 204d 434d  s M_i of the MCM
│ │ │ │ +00065220: 2061 7070 726f 7869 6d61 7469 6f6e 2074   approximation t
│ │ │ │ +00065230: 6f20 4d20 6f76 6572 2052 5f69 2c0a 7374  o M over R_i,.st
│ │ │ │ +00065240: 6f70 7069 6e67 2077 6865 6e20 4d5f 6920  opping when M_i 
│ │ │ │ +00065250: 6265 636f 6d65 7320 6672 6565 2c20 616e  becomes free, an
│ │ │ │ +00065260: 6420 7265 7475 726e 7320 7468 6520 6c69  d returns the li
│ │ │ │ +00065270: 7374 2077 686f 7365 2065 6c65 6d65 6e74  st whose element
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│ │ │ │ +00065290: 6f66 2072 6567 756c 6172 6974 6965 732c  of regularities,
│ │ │ │ +000652a0: 2073 7461 7274 696e 6720 7769 7468 204d   starting with M
│ │ │ │ +000652b0: 5f7b 2863 2d31 297d 204e 6f74 6520 7468  _{(c-1)} Note th
│ │ │ │ +000652c0: 6174 2074 6865 2066 6972 7374 2070 6169  at the first pai
│ │ │ │ +000652d0: 7220 6973 2066 6f72 0a74 6865 0a0a 2b2d  r is for.the..+-
│ │ │ │ +000652e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000652f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065320: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00065330: 6320 3d20 333b 643d 3220 2020 2020 2020  c = 3;d=2       
│ │ │ │ +00065310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00065320: 6931 203a 2063 203d 2033 3b64 3d32 2020  i1 : c = 3;d=2  
│ │ │ │ +00065330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065360: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00065350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065360: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00065370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000653a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -000653b0: 3d20 3220 2020 2020 2020 2020 2020 2020  = 2             
│ │ │ │ +00065390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000653a0: 0a7c 6f32 203d 2032 2020 2020 2020 2020  .|o2 = 2        
│ │ │ │ +000653b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000653c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000653d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000653e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000653e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000653f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00065430: 3320 3a20 5220 3d20 7365 7475 7052 696e  3 : R = setupRin
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│ │ │ │ +00065420: 2d2b 0a7c 6933 203a 2052 203d 2073 6574  -+.|i3 : R = set
│ │ │ │ +00065430: 7570 5269 6e67 7328 632c 6429 3b20 2020  upRings(c,d);   
│ │ │ │ +00065440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065460: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00065460: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00065470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000654a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000654b0: 7c69 3420 3a20 5263 203d 2052 5f63 2020  |i4 : Rc = R_c  
│ │ │ │ +000654a0: 2d2d 2d2b 0a7c 6934 203a 2052 6320 3d20  ---+.|i4 : Rc = 
│ │ │ │ +000654b0: 525f 6320 2020 2020 2020 2020 2020 2020  R_c             
│ │ │ │  000654c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000654d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000654e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000654f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000654e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000654f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065530: 7c0a 7c6f 3420 3d20 5263 2020 2020 2020  |.|o4 = Rc      
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│ │ │ │ +00065530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00065560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00065570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000655a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000655b0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +000655c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000655d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000655e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000655f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000655e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000655f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065630: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d20 3d20  ----+.|i5 : M = 
│ │ │ │ -00065640: 636f 6b65 7220 6d61 7472 6978 7b7b 5263  coker matrix{{Rc
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│ │ │ │ -00065660: 635f 312c 5263 5f32 2c52 635f 307d 7d20  c_1,Rc_2,Rc_0}} 
│ │ │ │ -00065670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00065620: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
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│ │ │ │ +00065660: 5f30 7d7d 2020 2020 2020 7c0a 7c20 2020  _0}}      |.|   
│ │ │ │ +00065670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656b0: 2020 2020 2020 7c0a 7c6f 3520 3d20 636f        |.|o5 = co
│ │ │ │ -000656c0: 6b65 726e 656c 207c 2078 5f30 2078 5f31  kernel | x_0 x_1
│ │ │ │ -000656d0: 2078 5f32 207c 2020 2020 2020 2020 2020   x_2 |          
│ │ │ │ -000656e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00065700: 2020 2020 2020 2020 7c20 785f 3120 785f          | x_1 x_
│ │ │ │ -00065710: 3220 785f 3020 7c20 2020 2020 2020 2020  2 x_0 |         
│ │ │ │ -00065720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000656a0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +000656b0: 203d 2063 6f6b 6572 6e65 6c20 7c20 785f   = cokernel | x_
│ │ │ │ +000656c0: 3020 785f 3120 785f 3220 7c20 2020 2020  0 x_1 x_2 |     
│ │ │ │ +000656d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000656e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000656f0: 2020 2020 2020 2020 2020 2020 207c 2078               | x
│ │ │ │ +00065700: 5f31 2078 5f32 2078 5f30 207c 2020 2020  _1 x_2 x_0 |    
│ │ │ │ +00065710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00065730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00065780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065790: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000657a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000657b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -000657c0: 3a20 5263 2d6d 6f64 756c 652c 2071 756f  : Rc-module, quo
│ │ │ │ -000657d0: 7469 656e 7420 6f66 2052 6320 2020 2020  tient of Rc     
│ │ │ │ +00065760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065770: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00065780: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00065790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000657a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000657b0: 0a7c 6f35 203a 2052 632d 6d6f 6475 6c65  .|o5 : Rc-module
│ │ │ │ +000657c0: 2c20 7175 6f74 6965 6e74 206f 6620 5263  , quotient of Rc
│ │ │ │ +000657d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000657e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000657f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000657f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00065800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00065840: 3620 3a20 7265 6775 6c61 7269 7479 5365  6 : regularitySe
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│ │ │ │ +00065830: 2d2b 0a7c 6936 203a 2072 6567 756c 6172  -+.|i6 : regular
│ │ │ │ +00065840: 6974 7953 6571 7565 6e63 6528 522c 4d29  itySequence(R,M)
│ │ │ │ +00065850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00065880: 7265 6720 6576 656e 2065 7874 2c20 736f  reg even ext, so
│ │ │ │ -00065890: 6320 6465 6773 2065 7665 6e20 6578 742c  c degs even ext,
│ │ │ │ -000658a0: 2072 6567 206f 6464 2065 7874 2c20 736f   reg odd ext, so
│ │ │ │ -000658b0: 6320 6465 6773 206f 6464 2065 7874 7c0a  c degs odd ext|.
│ │ │ │ -000658c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +00065a90: 0a20 202a 202a 6e6f 7465 2061 7573 6c61  .  * *note ausla
│ │ │ │ +00065aa0: 6e64 6572 496e 7661 7269 616e 743a 2028  nderInvariant: (
│ │ │ │ +00065ab0: 4d43 4d41 7070 726f 7869 6d61 7469 6f6e  MCMApproximation
│ │ │ │ +00065ac0: 7329 6175 736c 616e 6465 7249 6e76 6172  s)auslanderInvar
│ │ │ │ +00065ad0: 6961 6e74 2c20 2d2d 0a20 2020 206d 6561  iant, --.    mea
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│ │ │ │ +00065b00: 2074 6865 2065 7373 656e 7469 616c 204d   the essential M
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│ │ │ │  00065b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │  00065ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00065cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00065d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00065d10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
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│ │ │ │ -00065d30: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -00065d40: 2020 6620 3d20 5332 2862 2c4d 290a 2020    f = S2(b,M).  
│ │ │ │ -00065d50: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00065d60: 2a20 622c 2061 6e20 2a6e 6f74 6520 696e  * b, an *note in
│ │ │ │ -00065d70: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ -00065d80: 3244 6f63 295a 5a2c 2c20 6465 6772 6565  2Doc)ZZ,, degree
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│ │ │ │ -00065db0: 2074 6865 2063 6f6d 7075 7461 7469 6f6e   the computation
│ │ │ │ -00065dc0: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
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│ │ │ │ -00065de0: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
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│ │ │ │ -00065e50: 2020 2020 2020 6167 7265 6573 2077 6974        agrees wit
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│ │ │ │ -00065e80: 6573 2024 5c67 6571 2062 240a 0a44 6573  es $\geq b$..Des
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│ │ │ │ -00065ea0: 3d3d 3d3d 0a0a 4966 204d 2069 7320 6120  ====..If M is a 
│ │ │ │ -00065eb0: 6772 6164 6564 206d 6f64 756c 6520 6f76  graded module ov
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│ │ │ │ -00065f50: 616e 7920 7365 7175 656e 6365 206f 6620  any sequence of 
│ │ │ │ -00065f60: 6964 6561 6c73 2063 6f6e 7461 696e 6564  ideals contained
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│ │ │ │ -00065f90: 535f 2b2e 2054 6865 7265 2069 7320 6120  S_+. There is a 
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│ │ │ │ -00065fd0: 495f 642c 4d29 2e20 5765 0a63 6f6d 7075  I_d,M). We.compu
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│ │ │ │ -00066120: 6f20 636f 6d70 7574 6520 486f 6d28 492c  o compute Hom(I,
│ │ │ │ -00066130: 4d29 2077 6865 7265 2049 205c 7375 6273  M) where I \subs
│ │ │ │ -00066140: 6574 2028 535f 2b29 5e7b 722d 627d 2e0a  et (S_+)^{r-b}..
│ │ │ │ -00066150: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00065d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00065d10: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ +00065d20: 3d0a 0a20 202a 2055 7361 6765 3a20 0a20  =..  * Usage: . 
│ │ │ │ +00065d30: 2020 2020 2020 2066 203d 2053 3228 622c         f = S2(b,
│ │ │ │ +00065d40: 4d29 0a20 202a 2049 6e70 7574 733a 0a20  M).  * Inputs:. 
│ │ │ │ +00065d50: 2020 2020 202a 2062 2c20 616e 202a 6e6f       * b, an *no
│ │ │ │ +00065d60: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00065d70: 6175 6c61 7932 446f 6329 5a5a 2c2c 2064  aulay2Doc)ZZ,, d
│ │ │ │ +00065d80: 6567 7265 6520 626f 756e 6420 746f 2077  egree bound to w
│ │ │ │ +00065d90: 6869 6368 2074 6f20 6361 7272 790a 2020  hich to carry.  
│ │ │ │ +00065da0: 2020 2020 2020 7468 6520 636f 6d70 7574        the comput
│ │ │ │ +00065db0: 6174 696f 6e0a 2020 2020 2020 2a20 4d2c  ation.      * M,
│ │ │ │ +00065dc0: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ +00065dd0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00065de0: 6f64 756c 652c 2c20 0a20 202a 204f 7574  odule,, .  * Out
│ │ │ │ +00065df0: 7075 7473 3a0a 2020 2020 2020 2a20 662c  puts:.      * f,
│ │ │ │ +00065e00: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +00065e10: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00065e20: 6174 7269 782c 2c20 6465 6669 6e69 6e67  atrix,, defining
│ │ │ │ +00065e30: 2061 206d 6170 204d 2d2d 3e4d 2720 7468   a map M-->M' th
│ │ │ │ +00065e40: 6174 0a20 2020 2020 2020 2061 6772 6565  at.        agree
│ │ │ │ +00065e50: 7320 7769 7468 2074 6865 2053 322d 6966  s with the S2-if
│ │ │ │ +00065e60: 6963 6174 696f 6e20 6f66 204d 2069 6e20  ication of M in 
│ │ │ │ +00065e70: 6465 6772 6565 7320 245c 6765 7120 6224  degrees $\geq b$
│ │ │ │ +00065e80: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00065e90: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 4d20  =========..If M 
│ │ │ │ +00065ea0: 6973 2061 2067 7261 6465 6420 6d6f 6475  is a graded modu
│ │ │ │ +00065eb0: 6c65 206f 7665 7220 6120 7269 6e67 2053  le over a ring S
│ │ │ │ +00065ec0: 2c20 7468 656e 2074 6865 2053 322d 6966  , then the S2-if
│ │ │ │ +00065ed0: 6963 6174 696f 6e20 6f66 204d 2069 7320  ication of M is 
│ │ │ │ +00065ee0: 5c73 756d 5f7b 640a 5c69 6e20 5a5a 7d20  \sum_{d.\in ZZ} 
│ │ │ │ +00065ef0: 485e 3028 2873 6865 6166 204d 2928 6429  H^0((sheaf M)(d)
│ │ │ │ +00065f00: 292c 2077 6869 6368 206d 6179 2062 6520  ), which may be 
│ │ │ │ +00065f10: 636f 6d70 7574 6564 2061 7320 6c69 6d5f  computed as lim_
│ │ │ │ +00065f20: 7b64 2d3e 5c69 6e66 7479 7d20 486f 6d28  {d->\infty} Hom(
│ │ │ │ +00065f30: 495f 642c 4d29 2c0a 7768 6572 6520 495f  I_d,M),.where I_
│ │ │ │ +00065f40: 6420 6973 2061 6e79 2073 6571 7565 6e63  d is any sequenc
│ │ │ │ +00065f50: 6520 6f66 2069 6465 616c 7320 636f 6e74  e of ideals cont
│ │ │ │ +00065f60: 6169 6e65 6420 696e 2068 6967 6865 7220  ained in higher 
│ │ │ │ +00065f70: 616e 6420 6869 6768 6572 2070 6f77 6572  and higher power
│ │ │ │ +00065f80: 7320 6f66 0a53 5f2b 2e20 5468 6572 6520  s of.S_+. There 
│ │ │ │ +00065f90: 6973 2061 206e 6174 7572 616c 2072 6573  is a natural res
│ │ │ │ +00065fa0: 7472 6963 7469 6f6e 206d 6170 2066 3a20  triction map f: 
│ │ │ │ +00065fb0: 4d20 3d20 486f 6d28 532c 4d29 205c 746f  M = Hom(S,M) \to
│ │ │ │ +00065fc0: 2048 6f6d 2849 5f64 2c4d 292e 2057 650a   Hom(I_d,M). We.
│ │ │ │ +00065fd0: 636f 6d70 7574 6520 616c 6c20 7468 6973  compute all this
│ │ │ │ +00065fe0: 2075 7369 6e67 2074 6865 2069 6465 616c   using the ideal
│ │ │ │ +00065ff0: 7320 495f 6420 6765 6e65 7261 7465 6420  s I_d generated 
│ │ │ │ +00066000: 6279 2074 6865 2064 2d74 6820 706f 7765  by the d-th powe
│ │ │ │ +00066010: 7273 206f 6620 7468 650a 7661 7269 6162  rs of the.variab
│ │ │ │ +00066020: 6c65 7320 696e 2053 2e0a 0a53 696e 6365  les in S...Since
│ │ │ │ +00066030: 2074 6865 2072 6573 756c 7420 6d61 7920   the result may 
│ │ │ │ +00066040: 6e6f 7420 6265 2066 696e 6974 656c 7920  not be finitely 
│ │ │ │ +00066050: 6765 6e65 7261 7465 6420 2874 6869 7320  generated (this 
│ │ │ │ +00066060: 6861 7070 656e 7320 6966 2061 6e64 206f  happens if and o
│ │ │ │ +00066070: 6e6c 7920 6966 204d 0a68 6173 2061 6e20  nly if M.has an 
│ │ │ │ +00066080: 6173 736f 6369 6174 6564 2070 7269 6d65  associated prime
│ │ │ │ +00066090: 206f 6620 6469 6d65 6e73 696f 6e20 3129   of dimension 1)
│ │ │ │ +000660a0: 2c20 7765 2063 6f6d 7075 7465 206f 6e6c  , we compute onl
│ │ │ │ +000660b0: 7920 7570 2074 6f20 6120 7370 6563 6966  y up to a specif
│ │ │ │ +000660c0: 6965 640a 6465 6772 6565 2062 6f75 6e64  ied.degree bound
│ │ │ │ +000660d0: 2062 2e20 466f 7220 7468 6520 7265 7375   b. For the resu
│ │ │ │ +000660e0: 6c74 2074 6f20 6265 2063 6f72 7265 6374  lt to be correct
│ │ │ │ +000660f0: 2064 6f77 6e20 746f 2064 6567 7265 6520   down to degree 
│ │ │ │ +00066100: 622c 2069 7420 6973 2073 7566 6669 6369  b, it is suffici
│ │ │ │ +00066110: 656e 740a 746f 2063 6f6d 7075 7465 2048  ent.to compute H
│ │ │ │ +00066120: 6f6d 2849 2c4d 2920 7768 6572 6520 4920  om(I,M) where I 
│ │ │ │ +00066130: 5c73 7562 7365 7420 2853 5f2b 295e 7b72  \subset (S_+)^{r
│ │ │ │ +00066140: 2d62 7d2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  -b}...+---------
│ │ │ │ +00066150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000661a0: 0a7c 6931 203a 206b 6b3d 5a5a 2f31 3031  .|i1 : kk=ZZ/101
│ │ │ │ +00066190: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d5a  ----+.|i1 : kk=Z
│ │ │ │ +000661a0: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │  000661b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000661c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000661d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000661e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000661f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000661e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000661f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066240: 0a7c 6f31 203d 206b 6b20 2020 2020 2020  .|o1 = kk       
│ │ │ │ +00066230: 2020 2020 7c0a 7c6f 3120 3d20 6b6b 2020      |.|o1 = kk  
│ │ │ │ +00066240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066280: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000662a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000662b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000662c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000662d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000662e0: 0a7c 6f31 203a 2051 756f 7469 656e 7452  .|o1 : QuotientR
│ │ │ │ -000662f0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +000662d0: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74      |.|o1 : Quot
│ │ │ │ +000662e0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +000662f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00066320: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00066330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00066380: 0a7c 6932 203a 2053 203d 206b 6b5b 612c  .|i2 : S = kk[a,
│ │ │ │ -00066390: 622c 632c 645d 2020 2020 2020 2020 2020  b,c,d]          
│ │ │ │ +00066370: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20  ----+.|i2 : S = 
│ │ │ │ +00066380: 6b6b 5b61 2c62 2c63 2c64 5d20 2020 2020  kk[a,b,c,d]     
│ │ │ │ +00066390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000663a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000663b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000663c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000663d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000663c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000663d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000663e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000663f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066420: 0a7c 6f32 203d 2053 2020 2020 2020 2020  .|o2 = S        
│ │ │ │ +00066410: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020      |.|o2 = S   
│ │ │ │ +00066420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066460: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000664a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000664b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000664c0: 0a7c 6f32 203a 2050 6f6c 796e 6f6d 6961  .|o2 : Polynomia
│ │ │ │ -000664d0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +000664b0: 2020 2020 7c0a 7c6f 3220 3a20 506f 6c79      |.|o2 : Poly
│ │ │ │ +000664c0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +000664d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000664e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000664f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066510: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00066500: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00066510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00066560: 0a7c 6933 203a 204d 203d 2074 7275 6e63  .|i3 : M = trunc
│ │ │ │ -00066570: 6174 6528 332c 535e 3129 2020 2020 2020  ate(3,S^1)      
│ │ │ │ +00066550: 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20 3d20  ----+.|i3 : M = 
│ │ │ │ +00066560: 7472 756e 6361 7465 2833 2c53 5e31 2920  truncate(3,S^1) 
│ │ │ │ +00066570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000665a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000665b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000665a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000665b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000665c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000665d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000665e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000665f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066600: 0a7c 6f33 203d 2069 6d61 6765 207c 2064  .|o3 = image | d
│ │ │ │ -00066610: 3320 6364 3220 6264 3220 6164 3220 6332  3 cd2 bd2 ad2 c2
│ │ │ │ -00066620: 6420 6263 6420 6163 6420 6232 6420 6162  d bcd acd b2d ab
│ │ │ │ -00066630: 6420 6132 6420 6333 2062 6332 2061 6332  d a2d c3 bc2 ac2
│ │ │ │ -00066640: 2062 3263 2061 6263 2061 3263 2062 337c   b2c abc a2c b3|
│ │ │ │ -00066650: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000665f0: 2020 2020 7c0a 7c6f 3320 3d20 696d 6167      |.|o3 = imag
│ │ │ │ +00066600: 6520 7c20 6433 2063 6432 2062 6432 2061  e | d3 cd2 bd2 a
│ │ │ │ +00066610: 6432 2063 3264 2062 6364 2061 6364 2062  d2 c2d bcd acd b
│ │ │ │ +00066620: 3264 2061 6264 2061 3264 2063 3320 6263  2d abd a2d c3 bc
│ │ │ │ +00066630: 3220 6163 3220 6232 6320 6162 6320 6132  2 ac2 b2c abc a2
│ │ │ │ +00066640: 6320 6233 7c0a 7c20 2020 2020 2020 2020  c b3|.|         
│ │ │ │ +00066650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000666a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000666b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00066690: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000666a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000666b0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  000666c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000666d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000666e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000666f0: 0a7c 6f33 203a 2053 2d6d 6f64 756c 652c  .|o3 : S-module,
│ │ │ │ -00066700: 2073 7562 6d6f 6475 6c65 206f 6620 5320   submodule of S 
│ │ │ │ +000666e0: 2020 2020 7c0a 7c6f 3320 3a20 532d 6d6f      |.|o3 : S-mo
│ │ │ │ +000666f0: 6475 6c65 2c20 7375 626d 6f64 756c 6520  dule, submodule 
│ │ │ │ +00066700: 6f66 2053 2020 2020 2020 2020 2020 2020  of S            
│ │ │ │  00066710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066740: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00066730: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00066740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00066790: 0a7c 6162 3220 6132 6220 6133 207c 2020  .|ab2 a2b a3 |  
│ │ │ │ +00066780: 2d2d 2d2d 7c0a 7c61 6232 2061 3262 2061  ----|.|ab2 a2b a
│ │ │ │ +00066790: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │  000667a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000667b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000667c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000667d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000667e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000667d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000667e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000667f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00066830: 0a7c 6934 203a 2062 6574 7469 206d 6174  .|i4 : betti mat
│ │ │ │ -00066840: 7269 7820 5332 2830 2c4d 2920 2020 2020  rix S2(0,M)     
│ │ │ │ +00066820: 2d2d 2d2d 2b0a 7c69 3420 3a20 6265 7474  ----+.|i4 : bett
│ │ │ │ +00066830: 6920 6d61 7472 6978 2053 3228 302c 4d29  i matrix S2(0,M)
│ │ │ │ +00066840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066870: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000668a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000668b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000668c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000668d0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ -000668e0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +000668c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000668d0: 2020 2030 2020 3120 2020 2020 2020 2020     0  1         
│ │ │ │ +000668e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000668f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066920: 0a7c 6f34 203d 2074 6f74 616c 3a20 3120  .|o4 = total: 1 
│ │ │ │ -00066930: 3230 2020 2020 2020 2020 2020 2020 2020  20              
│ │ │ │ +00066910: 2020 2020 7c0a 7c6f 3420 3d20 746f 7461      |.|o4 = tota
│ │ │ │ +00066920: 6c3a 2031 2032 3020 2020 2020 2020 2020  l: 1 20         
│ │ │ │ +00066930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066970: 0a7c 2020 2020 2020 2020 2030 3a20 3120  .|         0: 1 
│ │ │ │ -00066980: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00066960: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066970: 303a 2031 2020 2e20 2020 2020 2020 2020  0: 1  .         
│ │ │ │ +00066980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000669a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000669b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000669c0: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -000669d0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +000669b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000669c0: 313a 202e 2020 2e20 2020 2020 2020 2020  1: .  .         
│ │ │ │ +000669d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000669e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000669f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066a10: 0a7c 2020 2020 2020 2020 2032 3a20 2e20  .|         2: . 
│ │ │ │ -00066a20: 3230 2020 2020 2020 2020 2020 2020 2020  20              
│ │ │ │ +00066a00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066a10: 323a 202e 2032 3020 2020 2020 2020 2020  2: . 20         
│ │ │ │ +00066a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066a60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066a50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066ab0: 0a7c 6f34 203a 2042 6574 7469 5461 6c6c  .|o4 : BettiTall
│ │ │ │ -00066ac0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +00066aa0: 2020 2020 7c0a 7c6f 3420 3a20 4265 7474      |.|o4 : Bett
│ │ │ │ +00066ab0: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +00066ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066b00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00066af0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00066b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00066b50: 0a7c 6935 203a 2062 6574 7469 206d 6174  .|i5 : betti mat
│ │ │ │ -00066b60: 7269 7820 5332 2831 2c4d 2920 2020 2020  rix S2(1,M)     
│ │ │ │ +00066b40: 2d2d 2d2d 2b0a 7c69 3520 3a20 6265 7474  ----+.|i5 : bett
│ │ │ │ +00066b50: 6920 6d61 7472 6978 2053 3228 312c 4d29  i matrix S2(1,M)
│ │ │ │ +00066b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066bf0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ -00066c00: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00066be0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066bf0: 2020 2030 2020 3120 2020 2020 2020 2020     0  1         
│ │ │ │ +00066c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066c40: 0a7c 6f35 203d 2074 6f74 616c 3a20 3120  .|o5 = total: 1 
│ │ │ │ -00066c50: 3230 2020 2020 2020 2020 2020 2020 2020  20              
│ │ │ │ +00066c30: 2020 2020 7c0a 7c6f 3520 3d20 746f 7461      |.|o5 = tota
│ │ │ │ +00066c40: 6c3a 2031 2032 3020 2020 2020 2020 2020  l: 1 20         
│ │ │ │ +00066c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066c90: 0a7c 2020 2020 2020 2020 2030 3a20 3120  .|         0: 1 
│ │ │ │ -00066ca0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00066c80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066c90: 303a 2031 2020 2e20 2020 2020 2020 2020  0: 1  .         
│ │ │ │ +00066ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066ce0: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -00066cf0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00066cd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066ce0: 313a 202e 2020 2e20 2020 2020 2020 2020  1: .  .         
│ │ │ │ +00066cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066d30: 0a7c 2020 2020 2020 2020 2032 3a20 2e20  .|         2: . 
│ │ │ │ -00066d40: 3230 2020 2020 2020 2020 2020 2020 2020  20              
│ │ │ │ +00066d20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066d30: 323a 202e 2032 3020 2020 2020 2020 2020  2: . 20         
│ │ │ │ +00066d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066d80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066d70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066dd0: 0a7c 6f35 203a 2042 6574 7469 5461 6c6c  .|o5 : BettiTall
│ │ │ │ -00066de0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +00066dc0: 2020 2020 7c0a 7c6f 3520 3a20 4265 7474      |.|o5 : Bett
│ │ │ │ +00066dd0: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +00066de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066e20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00066e10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00066e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00066e70: 0a7c 6936 203a 204d 203d 2053 5e31 2f69  .|i6 : M = S^1/i
│ │ │ │ -00066e80: 6e74 6572 7365 6374 2869 6465 616c 2261  ntersect(ideal"a
│ │ │ │ -00066e90: 2c62 2c63 222c 2069 6465 616c 2262 2c63  ,b,c", ideal"b,c
│ │ │ │ -00066ea0: 2c64 222c 6964 6561 6c22 632c 642c 6122  ,d",ideal"c,d,a"
│ │ │ │ -00066eb0: 2c69 6465 616c 2264 2c61 2c62 2229 207c  ,ideal"d,a,b") |
│ │ │ │ -00066ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066e60: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d20  ----+.|i6 : M = 
│ │ │ │ +00066e70: 535e 312f 696e 7465 7273 6563 7428 6964  S^1/intersect(id
│ │ │ │ +00066e80: 6561 6c22 612c 622c 6322 2c20 6964 6561  eal"a,b,c", idea
│ │ │ │ +00066e90: 6c22 622c 632c 6422 2c69 6465 616c 2263  l"b,c,d",ideal"c
│ │ │ │ +00066ea0: 2c64 2c61 222c 6964 6561 6c22 642c 612c  ,d,a",ideal"d,a,
│ │ │ │ +00066eb0: 6222 2920 7c0a 7c20 2020 2020 2020 2020  b") |.|         
│ │ │ │ +00066ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066f10: 0a7c 6f36 203d 2063 6f6b 6572 6e65 6c20  .|o6 = cokernel 
│ │ │ │ -00066f20: 7c20 6364 2062 6420 6164 2062 6320 6163  | cd bd ad bc ac
│ │ │ │ -00066f30: 2061 6220 7c20 2020 2020 2020 2020 2020   ab |           
│ │ │ │ +00066f00: 2020 2020 7c0a 7c6f 3620 3d20 636f 6b65      |.|o6 = coke
│ │ │ │ +00066f10: 726e 656c 207c 2063 6420 6264 2061 6420  rnel | cd bd ad 
│ │ │ │ +00066f20: 6263 2061 6320 6162 207c 2020 2020 2020  bc ac ab |      
│ │ │ │ +00066f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00066f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00066fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00066fc0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00066fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00066fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066fc0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  00066fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067000: 0a7c 6f36 203a 2053 2d6d 6f64 756c 652c  .|o6 : S-module,
│ │ │ │ -00067010: 2071 756f 7469 656e 7420 6f66 2053 2020   quotient of S  
│ │ │ │ +00066ff0: 2020 2020 7c0a 7c6f 3620 3a20 532d 6d6f      |.|o6 : S-mo
│ │ │ │ +00067000: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +00067010: 6620 5320 2020 2020 2020 2020 2020 2020  f S             
│ │ │ │  00067020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067050: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00067040: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00067050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000670a0: 0a7c 6937 203a 2070 7275 6e65 2073 6f75  .|i7 : prune sou
│ │ │ │ -000670b0: 7263 6520 5332 2830 2c4d 2920 2020 2020  rce S2(0,M)     
│ │ │ │ +00067090: 2d2d 2d2d 2b0a 7c69 3720 3a20 7072 756e  ----+.|i7 : prun
│ │ │ │ +000670a0: 6520 736f 7572 6365 2053 3228 302c 4d29  e source S2(0,M)
│ │ │ │ +000670b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000670c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000670d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000670e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000670f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000670e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000670f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067140: 0a7c 6f37 203d 2063 6f6b 6572 6e65 6c20  .|o7 = cokernel 
│ │ │ │ -00067150: 7c20 6364 2062 6420 6164 2062 6320 6163  | cd bd ad bc ac
│ │ │ │ -00067160: 2061 6220 7c20 2020 2020 2020 2020 2020   ab |           
│ │ │ │ +00067130: 2020 2020 7c0a 7c6f 3720 3d20 636f 6b65      |.|o7 = coke
│ │ │ │ +00067140: 726e 656c 207c 2063 6420 6264 2061 6420  rnel | cd bd ad 
│ │ │ │ +00067150: 6263 2061 6320 6162 207c 2020 2020 2020  bc ac ab |      
│ │ │ │ +00067160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00067180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000671a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000671b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000671c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000671d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000671e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000671f0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000671d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000671e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000671f0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  00067200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067230: 0a7c 6f37 203a 2053 2d6d 6f64 756c 652c  .|o7 : S-module,
│ │ │ │ -00067240: 2071 756f 7469 656e 7420 6f66 2053 2020   quotient of S  
│ │ │ │ +00067220: 2020 2020 7c0a 7c6f 3720 3a20 532d 6d6f      |.|o7 : S-mo
│ │ │ │ +00067230: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +00067240: 6620 5320 2020 2020 2020 2020 2020 2020  f S             
│ │ │ │  00067250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067280: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00067270: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00067280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000672a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000672b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000672c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000672d0: 0a7c 6938 203a 2070 7275 6e65 2074 6172  .|i8 : prune tar
│ │ │ │ -000672e0: 6765 7420 5332 2830 2c4d 2920 2020 2020  get S2(0,M)     
│ │ │ │ +000672c0: 2d2d 2d2d 2b0a 7c69 3820 3a20 7072 756e  ----+.|i8 : prun
│ │ │ │ +000672d0: 6520 7461 7267 6574 2053 3228 302c 4d29  e target S2(0,M)
│ │ │ │ +000672e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000672f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067320: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00067310: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067370: 0a7c 6f38 203d 2063 6f6b 6572 6e65 6c20  .|o8 = cokernel 
│ │ │ │ -00067380: 7b2d 317d 207c 2064 2063 2062 2030 2030  {-1} | d c b 0 0
│ │ │ │ -00067390: 2030 2030 2030 2030 2030 2030 2030 207c   0 0 0 0 0 0 0 |
│ │ │ │ +00067360: 2020 2020 7c0a 7c6f 3820 3d20 636f 6b65      |.|o8 = coke
│ │ │ │ +00067370: 726e 656c 207b 2d31 7d20 7c20 6420 6320  rnel {-1} | d c 
│ │ │ │ +00067380: 6220 3020 3020 3020 3020 3020 3020 3020  b 0 0 0 0 0 0 0 
│ │ │ │ +00067390: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │  000673a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000673c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000673d0: 7b2d 317d 207c 2030 2030 2030 2064 2063  {-1} | 0 0 0 d c
│ │ │ │ -000673e0: 2061 2030 2030 2030 2030 2030 2030 207c   a 0 0 0 0 0 0 |
│ │ │ │ +000673b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000673c0: 2020 2020 207b 2d31 7d20 7c20 3020 3020       {-1} | 0 0 
│ │ │ │ +000673d0: 3020 6420 6320 6120 3020 3020 3020 3020  0 d c a 0 0 0 0 
│ │ │ │ +000673e0: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │  000673f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067410: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00067420: 7b2d 317d 207c 2030 2030 2030 2030 2030  {-1} | 0 0 0 0 0
│ │ │ │ -00067430: 2030 2064 2062 2061 2030 2030 2030 207c   0 d b a 0 0 0 |
│ │ │ │ +00067400: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067410: 2020 2020 207b 2d31 7d20 7c20 3020 3020       {-1} | 0 0 
│ │ │ │ +00067420: 3020 3020 3020 3020 6420 6220 6120 3020  0 0 0 0 d b a 0 
│ │ │ │ +00067430: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │  00067440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00067470: 7b2d 317d 207c 2030 2030 2030 2030 2030  {-1} | 0 0 0 0 0
│ │ │ │ -00067480: 2030 2030 2030 2030 2063 2062 2061 207c   0 0 0 0 c b a |
│ │ │ │ +00067450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067460: 2020 2020 207b 2d31 7d20 7c20 3020 3020       {-1} | 0 0 
│ │ │ │ +00067470: 3020 3020 3020 3020 3020 3020 3020 6320  0 0 0 0 0 0 0 c 
│ │ │ │ +00067480: 6220 6120 7c20 2020 2020 2020 2020 2020  b a |           
│ │ │ │  00067490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000674a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000674b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000674a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000674b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000674c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000674d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000674e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000674f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00067510: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +000674f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067510: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │  00067520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067550: 0a7c 6f38 203a 2053 2d6d 6f64 756c 652c  .|o8 : S-module,
│ │ │ │ -00067560: 2071 756f 7469 656e 7420 6f66 2053 2020   quotient of S  
│ │ │ │ +00067540: 2020 2020 7c0a 7c6f 3820 3a20 532d 6d6f      |.|o8 : S-mo
│ │ │ │ +00067550: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +00067560: 6620 5320 2020 2020 2020 2020 2020 2020  f S             
│ │ │ │  00067570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000675a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00067590: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000675a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000675b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000675c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000675d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000675e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000675f0: 0a0a 4174 206f 6e65 2074 696d 6520 4445  ..At one time DE
│ │ │ │ -00067600: 2068 6f70 6564 2074 6861 742c 2069 6620   hoped that, if 
│ │ │ │ -00067610: 4d20 7765 7265 2061 206d 6f64 756c 6520  M were a module 
│ │ │ │ -00067620: 6f76 6572 2074 6865 2063 6f6d 706c 6574  over the complet
│ │ │ │ -00067630: 6520 696e 7465 7273 6563 7469 6f6e 2052  e intersection R
│ │ │ │ -00067640: 0a77 6974 6820 7265 7369 6475 6520 6669  .with residue fi
│ │ │ │ -00067650: 656c 6420 6b2c 2074 6865 6e20 7468 6520  eld k, then the 
│ │ │ │ -00067660: 6e61 7475 7261 6c20 6d61 7020 6672 6f6d  natural map from
│ │ │ │ -00067670: 2022 636f 6d70 6c65 7465 2220 4578 7420   "complete" Ext 
│ │ │ │ -00067680: 6d6f 6475 6c65 2022 2877 6964 6568 6174  module "(widehat
│ │ │ │ -00067690: 0a45 7874 295f 5228 4d2c 6b29 2220 746f  .Ext)_R(M,k)" to
│ │ │ │ -000676a0: 2074 6865 2053 322d 6966 6963 6174 696f   the S2-ificatio
│ │ │ │ -000676b0: 6e20 6f66 2045 7874 5f52 284d 2c6b 2920  n of Ext_R(M,k) 
│ │ │ │ -000676c0: 776f 756c 6420 6265 2073 7572 6a65 6374  would be surject
│ │ │ │ -000676d0: 6976 653b 0a65 7175 6976 616c 656e 746c  ive;.equivalentl
│ │ │ │ -000676e0: 792c 2069 6620 4e20 7765 7265 2061 2073  y, if N were a s
│ │ │ │ -000676f0: 7566 6669 6369 656e 746c 7920 6e65 6761  ufficiently nega
│ │ │ │ -00067700: 7469 7665 2073 797a 7967 7920 6f66 204d  tive syzygy of M
│ │ │ │ -00067710: 2c20 7468 656e 2074 6865 2066 6972 7374  , then the first
│ │ │ │ -00067720: 0a6c 6f63 616c 2063 6f68 6f6d 6f6c 6f67  .local cohomolog
│ │ │ │ -00067730: 7920 6d6f 6475 6c65 206f 6620 4578 745f  y module of Ext_
│ │ │ │ -00067740: 5228 4d2c 6b29 2077 6f75 6c64 2062 6520  R(M,k) would be 
│ │ │ │ -00067750: 7a65 726f 2e20 5468 6973 2069 7320 6661  zero. This is fa
│ │ │ │ -00067760: 6c73 652c 2061 7320 7368 6f77 6e20 6279  lse, as shown by
│ │ │ │ -00067770: 0a74 6865 2066 6f6c 6c6f 7769 6e67 2065  .the following e
│ │ │ │ -00067780: 7861 6d70 6c65 3a0a 0a2b 2d2d 2d2d 2d2d  xample:..+------
│ │ │ │ +000675e0: 2d2d 2d2d 2b0a 0a41 7420 6f6e 6520 7469  ----+..At one ti
│ │ │ │ +000675f0: 6d65 2044 4520 686f 7065 6420 7468 6174  me DE hoped that
│ │ │ │ +00067600: 2c20 6966 204d 2077 6572 6520 6120 6d6f  , if M were a mo
│ │ │ │ +00067610: 6475 6c65 206f 7665 7220 7468 6520 636f  dule over the co
│ │ │ │ +00067620: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ +00067630: 696f 6e20 520a 7769 7468 2072 6573 6964  ion R.with resid
│ │ │ │ +00067640: 7565 2066 6965 6c64 206b 2c20 7468 656e  ue field k, then
│ │ │ │ +00067650: 2074 6865 206e 6174 7572 616c 206d 6170   the natural map
│ │ │ │ +00067660: 2066 726f 6d20 2263 6f6d 706c 6574 6522   from "complete"
│ │ │ │ +00067670: 2045 7874 206d 6f64 756c 6520 2228 7769   Ext module "(wi
│ │ │ │ +00067680: 6465 6861 740a 4578 7429 5f52 284d 2c6b  dehat.Ext)_R(M,k
│ │ │ │ +00067690: 2922 2074 6f20 7468 6520 5332 2d69 6669  )" to the S2-ifi
│ │ │ │ +000676a0: 6361 7469 6f6e 206f 6620 4578 745f 5228  cation of Ext_R(
│ │ │ │ +000676b0: 4d2c 6b29 2077 6f75 6c64 2062 6520 7375  M,k) would be su
│ │ │ │ +000676c0: 726a 6563 7469 7665 3b0a 6571 7569 7661  rjective;.equiva
│ │ │ │ +000676d0: 6c65 6e74 6c79 2c20 6966 204e 2077 6572  lently, if N wer
│ │ │ │ +000676e0: 6520 6120 7375 6666 6963 6965 6e74 6c79  e a sufficiently
│ │ │ │ +000676f0: 206e 6567 6174 6976 6520 7379 7a79 6779   negative syzygy
│ │ │ │ +00067700: 206f 6620 4d2c 2074 6865 6e20 7468 6520   of M, then the 
│ │ │ │ +00067710: 6669 7273 740a 6c6f 6361 6c20 636f 686f  first.local coho
│ │ │ │ +00067720: 6d6f 6c6f 6779 206d 6f64 756c 6520 6f66  mology module of
│ │ │ │ +00067730: 2045 7874 5f52 284d 2c6b 2920 776f 756c   Ext_R(M,k) woul
│ │ │ │ +00067740: 6420 6265 207a 6572 6f2e 2054 6869 7320  d be zero. This 
│ │ │ │ +00067750: 6973 2066 616c 7365 2c20 6173 2073 686f  is false, as sho
│ │ │ │ +00067760: 776e 2062 790a 7468 6520 666f 6c6c 6f77  wn by.the follow
│ │ │ │ +00067770: 696e 6720 6578 616d 706c 653a 0a0a 2b2d  ing example:..+-
│ │ │ │ +00067780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000677a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000677b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ -000677c0: 5320 3d20 5a5a 2f31 3031 5b78 5f30 2e2e  S = ZZ/101[x_0..
│ │ │ │ -000677d0: 785f 325d 3b20 2020 2020 2020 2020 2020  x_2];           
│ │ │ │ -000677e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000677a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000677b0: 6939 203a 2053 203d 205a 5a2f 3130 315b  i9 : S = ZZ/101[
│ │ │ │ +000677c0: 785f 302e 2e78 5f32 5d3b 2020 2020 2020  x_0..x_2];      
│ │ │ │ +000677d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000677e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000677f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067810: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ -00067820: 203a 2066 6620 3d20 6170 706c 7928 332c   : ff = apply(3,
│ │ │ │ -00067830: 2069 2d3e 785f 695e 3229 3b20 2020 2020   i->x_i^2);     
│ │ │ │ -00067840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00067800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00067810: 0a7c 6931 3020 3a20 6666 203d 2061 7070  .|i10 : ff = app
│ │ │ │ +00067820: 6c79 2833 2c20 692d 3e78 5f69 5e32 293b  ly(3, i->x_i^2);
│ │ │ │ +00067830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067840: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00067850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00067880: 3131 203a 2052 203d 2053 2f69 6465 616c  11 : R = S/ideal
│ │ │ │ -00067890: 2066 663b 2020 2020 2020 2020 2020 2020   ff;            
│ │ │ │ -000678a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00067870: 2d2b 0a7c 6931 3120 3a20 5220 3d20 532f  -+.|i11 : R = S/
│ │ │ │ +00067880: 6964 6561 6c20 6666 3b20 2020 2020 2020  ideal ff;       
│ │ │ │ +00067890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000678a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000678b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000678c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000678d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000678e0: 7c69 3132 203a 204d 203d 2063 6f6b 6572  |i12 : M = coker
│ │ │ │ -000678f0: 6e65 6c20 6d61 7472 6978 207b 7b78 5f30  nel matrix {{x_0
│ │ │ │ -00067900: 2c20 785f 312a 785f 327d 7d3b 2020 207c  , x_1*x_2}};   |
│ │ │ │ -00067910: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000678d0: 2d2d 2d2b 0a7c 6931 3220 3a20 4d20 3d20  ---+.|i12 : M = 
│ │ │ │ +000678e0: 636f 6b65 726e 656c 206d 6174 7269 7820  cokernel matrix 
│ │ │ │ +000678f0: 7b7b 785f 302c 2078 5f31 2a78 5f32 7d7d  {{x_0, x_1*x_2}}
│ │ │ │ +00067900: 3b20 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  ;   |.+---------
│ │ │ │ +00067910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067940: 2b0a 7c69 3133 203a 2062 203d 2035 3b20  +.|i13 : b = 5; 
│ │ │ │ +00067930: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6220  -----+.|i13 : b 
│ │ │ │ +00067940: 3d20 353b 2020 2020 2020 2020 2020 2020  = 5;            
│ │ │ │  00067950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067970: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00067960: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00067970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000679a0: 2d2d 2b0a 7c69 3134 203a 204d 6220 3d20  --+.|i14 : Mb = 
│ │ │ │ -000679b0: 7072 756e 6520 7379 7a79 6779 4d6f 6475  prune syzygyModu
│ │ │ │ -000679c0: 6c65 282d 622c 4d29 3b20 2020 2020 2020  le(-b,M);       
│ │ │ │ -000679d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00067990: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +000679a0: 4d62 203d 2070 7275 6e65 2073 797a 7967  Mb = prune syzyg
│ │ │ │ +000679b0: 794d 6f64 756c 6528 2d62 2c4d 293b 2020  yModule(-b,M);  
│ │ │ │ +000679c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000679d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000679e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067a00: 2d2d 2d2d 2b0a 7c69 3135 203a 2045 203d  ----+.|i15 : E =
│ │ │ │ -00067a10: 2070 7275 6e65 2065 7665 6e45 7874 4d6f   prune evenExtMo
│ │ │ │ -00067a20: 6475 6c65 204d 623b 2020 2020 2020 2020  dule Mb;        
│ │ │ │ -00067a30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000679f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520  ---------+.|i15 
│ │ │ │ +00067a00: 3a20 4520 3d20 7072 756e 6520 6576 656e  : E = prune even
│ │ │ │ +00067a10: 4578 744d 6f64 756c 6520 4d62 3b20 2020  ExtModule Mb;   
│ │ │ │ +00067a20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00067a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067a60: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2053  ------+.|i16 : S
│ │ │ │ -00067a70: 326d 6170 203d 2053 3228 302c 4529 3b20  2map = S2(0,E); 
│ │ │ │ -00067a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00067a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00067a60: 3620 3a20 5332 6d61 7020 3d20 5332 2830  6 : S2map = S2(0
│ │ │ │ +00067a70: 2c45 293b 2020 2020 2020 2020 2020 2020  ,E);            
│ │ │ │ +00067a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00067a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067ac0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a          |.|o16 :
│ │ │ │ -00067ad0: 204d 6174 7269 7820 2020 2020 2020 2020   Matrix         
│ │ │ │ -00067ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067af0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00067ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00067ac0: 6f31 3620 3a20 4d61 7472 6978 2020 2020  o16 : Matrix    
│ │ │ │ +00067ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067ae0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00067af0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00067b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │ -00067b30: 203a 2053 4520 3d20 7072 756e 6520 7461   : SE = prune ta
│ │ │ │ -00067b40: 7267 6574 2053 326d 6170 3b20 2020 2020  rget S2map;     
│ │ │ │ -00067b50: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00067b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00067b20: 0a7c 6931 3720 3a20 5345 203d 2070 7275  .|i17 : SE = pru
│ │ │ │ +00067b30: 6e65 2074 6172 6765 7420 5332 6d61 703b  ne target S2map;
│ │ │ │ +00067b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067b50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00067b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00067b90: 3138 203a 2065 7874 7261 203d 2070 7275  18 : extra = pru
│ │ │ │ -00067ba0: 6e65 2063 6f6b 6572 2053 326d 6170 3b20  ne coker S2map; 
│ │ │ │ -00067bb0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00067b80: 2d2b 0a7c 6931 3820 3a20 6578 7472 6120  -+.|i18 : extra 
│ │ │ │ +00067b90: 3d20 7072 756e 6520 636f 6b65 7220 5332  = prune coker S2
│ │ │ │ +00067ba0: 6d61 703b 2020 2020 2020 2020 2020 2020  map;            
│ │ │ │ +00067bb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00067bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00067bf0: 7c69 3139 203a 204b 4520 3d20 7072 756e  |i19 : KE = prun
│ │ │ │ -00067c00: 6520 6b65 7220 5332 6d61 703b 2020 2020  e ker S2map;    
│ │ │ │ -00067c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067c20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00067be0: 2d2d 2d2b 0a7c 6931 3920 3a20 4b45 203d  ---+.|i19 : KE =
│ │ │ │ +00067bf0: 2070 7275 6e65 206b 6572 2053 326d 6170   prune ker S2map
│ │ │ │ +00067c00: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00067c10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00067c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c50: 2b0a 7c69 3230 203a 2062 6574 7469 2072  +.|i20 : betti r
│ │ │ │ -00067c60: 6573 284d 622c 204c 656e 6774 684c 696d  es(Mb, LengthLim
│ │ │ │ -00067c70: 6974 203d 3e20 3130 2920 2020 2020 2020  it => 10)       
│ │ │ │ -00067c80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00067c40: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 6265  -----+.|i20 : be
│ │ │ │ +00067c50: 7474 6920 7265 7328 4d62 2c20 4c65 6e67  tti res(Mb, Leng
│ │ │ │ +00067c60: 7468 4c69 6d69 7420 3d3e 2031 3029 2020  thLimit => 10)  
│ │ │ │ +00067c70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00067c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00067cc0: 2020 2030 2020 3120 3220 3320 3420 3520     0  1 2 3 4 5 
│ │ │ │ -00067cd0: 3620 3720 3820 2039 2031 3020 2020 2020  6 7 8  9 10     
│ │ │ │ -00067ce0: 2020 207c 0a7c 6f32 3020 3d20 746f 7461     |.|o20 = tota
│ │ │ │ -00067cf0: 6c3a 2032 3020 3134 2039 2035 2032 2031  l: 20 14 9 5 2 1
│ │ │ │ -00067d00: 2032 2034 2037 2031 3120 3136 2020 2020   2 4 7 11 16    
│ │ │ │ -00067d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00067d20: 2d36 3a20 3230 2031 3420 3920 3520 3220  -6: 20 14 9 5 2 
│ │ │ │ -00067d30: 2e20 2e20 2e20 2e20 202e 2020 2e20 2020  . . . .  .  .   
│ │ │ │ -00067d40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00067d50: 202d 353a 2020 2e20 202e 202e 202e 202e   -5:  .  . . . .
│ │ │ │ -00067d60: 2031 2031 2031 2031 2020 3120 2031 2020   1 1 1 1  1  1  
│ │ │ │ -00067d70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00067d80: 2020 2d34 3a20 202e 2020 2e20 2e20 2e20    -4:  .  . . . 
│ │ │ │ -00067d90: 2e20 2e20 3120 3320 3620 3130 2031 3520  . . 1 3 6 10 15 
│ │ │ │ -00067da0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00067ca0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00067cb0: 2020 2020 2020 2020 3020 2031 2032 2033          0  1 2 3
│ │ │ │ +00067cc0: 2034 2035 2036 2037 2038 2020 3920 3130   4 5 6 7 8  9 10
│ │ │ │ +00067cd0: 2020 2020 2020 2020 7c0a 7c6f 3230 203d          |.|o20 =
│ │ │ │ +00067ce0: 2074 6f74 616c 3a20 3230 2031 3420 3920   total: 20 14 9 
│ │ │ │ +00067cf0: 3520 3220 3120 3220 3420 3720 3131 2031  5 2 1 2 4 7 11 1
│ │ │ │ +00067d00: 3620 2020 2020 2020 207c 0a7c 2020 2020  6        |.|    
│ │ │ │ +00067d10: 2020 2020 202d 363a 2032 3020 3134 2039       -6: 20 14 9
│ │ │ │ +00067d20: 2035 2032 202e 202e 202e 202e 2020 2e20   5 2 . . . .  . 
│ │ │ │ +00067d30: 202e 2020 2020 2020 2020 7c0a 7c20 2020   .        |.|   
│ │ │ │ +00067d40: 2020 2020 2020 2d35 3a20 202e 2020 2e20        -5:  .  . 
│ │ │ │ +00067d50: 2e20 2e20 2e20 3120 3120 3120 3120 2031  . . . 1 1 1 1  1
│ │ │ │ +00067d60: 2020 3120 2020 2020 2020 207c 0a7c 2020    1        |.|  
│ │ │ │ +00067d70: 2020 2020 2020 202d 343a 2020 2e20 202e         -4:  .  .
│ │ │ │ +00067d80: 202e 202e 202e 202e 2031 2033 2036 2031   . . . . 1 3 6 1
│ │ │ │ +00067d90: 3020 3135 2020 2020 2020 2020 7c0a 7c20  0 15        |.| 
│ │ │ │ +00067da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067dd0: 2020 2020 2020 2020 7c0a 7c6f 3230 203a          |.|o20 :
│ │ │ │ -00067de0: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ -00067df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067e00: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00067dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00067dd0: 6f32 3020 3a20 4265 7474 6954 616c 6c79  o20 : BettiTally
│ │ │ │ +00067de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00067e00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00067e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │ -00067e40: 203a 2061 7070 6c79 2028 352c 2069 2d3e   : apply (5, i->
│ │ │ │ -00067e50: 2068 696c 6265 7274 4675 6e63 7469 6f6e   hilbertFunction
│ │ │ │ -00067e60: 2869 2c20 4b45 2929 2020 207c 0a7c 2020  (i, KE))   |.|  
│ │ │ │ +00067e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00067e30: 0a7c 6932 3120 3a20 6170 706c 7920 2835  .|i21 : apply (5
│ │ │ │ +00067e40: 2c20 692d 3e20 6869 6c62 6572 7446 756e  , i-> hilbertFun
│ │ │ │ +00067e50: 6374 696f 6e28 692c 204b 4529 2920 2020  ction(i, KE))   
│ │ │ │ +00067e60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00067e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067e90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00067ea0: 3231 203d 207b 3230 2c20 392c 2032 2c20  21 = {20, 9, 2, 
│ │ │ │ -00067eb0: 302c 2030 7d20 2020 2020 2020 2020 2020  0, 0}           
│ │ │ │ -00067ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00067e90: 207c 0a7c 6f32 3120 3d20 7b32 302c 2039   |.|o21 = {20, 9
│ │ │ │ +00067ea0: 2c20 322c 2030 2c20 307d 2020 2020 2020  , 2, 0, 0}      
│ │ │ │ +00067eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067ec0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00067ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00067f00: 7c6f 3231 203a 204c 6973 7420 2020 2020  |o21 : List     
│ │ │ │ +00067ef0: 2020 207c 0a7c 6f32 3120 3a20 4c69 7374     |.|o21 : List
│ │ │ │ +00067f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067f30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00067f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00067f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067f60: 2b0a 7c69 3232 203a 2061 7070 6c79 2028  +.|i22 : apply (
│ │ │ │ -00067f70: 352c 2069 2d3e 2068 696c 6265 7274 4675  5, i-> hilbertFu
│ │ │ │ -00067f80: 6e63 7469 6f6e 2869 2c20 4529 2920 2020  nction(i, E))   
│ │ │ │ -00067f90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00067f50: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 6170  -----+.|i22 : ap
│ │ │ │ +00067f60: 706c 7920 2835 2c20 692d 3e20 6869 6c62  ply (5, i-> hilb
│ │ │ │ +00067f70: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ +00067f80: 2929 2020 2020 7c0a 7c20 2020 2020 2020  ))    |.|       
│ │ │ │ +00067f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067fc0: 2020 7c0a 7c6f 3232 203d 207b 3230 2c20    |.|o22 = {20, 
│ │ │ │ -00067fd0: 392c 2032 2c20 322c 2037 7d20 2020 2020  9, 2, 2, 7}     
│ │ │ │ -00067fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067ff0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00067fb0: 2020 2020 2020 207c 0a7c 6f32 3220 3d20         |.|o22 = 
│ │ │ │ +00067fc0: 7b32 302c 2039 2c20 322c 2032 2c20 377d  {20, 9, 2, 2, 7}
│ │ │ │ +00067fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00067ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068020: 2020 2020 7c0a 7c6f 3232 203a 204c 6973      |.|o22 : Lis
│ │ │ │ -00068030: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00068040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068050: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00068010: 2020 2020 2020 2020 207c 0a7c 6f32 3220           |.|o22 
│ │ │ │ +00068020: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00068030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068040: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00068050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068080: 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a 2061  ------+.|i23 : a
│ │ │ │ -00068090: 7070 6c79 2028 352c 2069 2d3e 2068 696c  pply (5, i-> hil
│ │ │ │ -000680a0: 6265 7274 4675 6e63 7469 6f6e 2869 2c20  bertFunction(i, 
│ │ │ │ -000680b0: 5345 2929 2020 207c 0a7c 2020 2020 2020  SE))   |.|      
│ │ │ │ +00068070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00068080: 3320 3a20 6170 706c 7920 2835 2c20 692d  3 : apply (5, i-
│ │ │ │ +00068090: 3e20 6869 6c62 6572 7446 756e 6374 696f  > hilbertFunctio
│ │ │ │ +000680a0: 6e28 692c 2053 4529 2920 2020 7c0a 7c20  n(i, SE))   |.| 
│ │ │ │ +000680b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000680c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000680d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000680e0: 2020 2020 2020 2020 7c0a 7c6f 3233 203d          |.|o23 =
│ │ │ │ -000680f0: 207b 312c 2031 2c20 312c 2032 2c20 377d   {1, 1, 1, 2, 7}
│ │ │ │ -00068100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068110: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000680d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000680e0: 6f32 3320 3d20 7b31 2c20 312c 2031 2c20  o23 = {1, 1, 1, 
│ │ │ │ +000680f0: 322c 2037 7d20 2020 2020 2020 2020 2020  2, 7}           
│ │ │ │ +00068100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00068110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00068120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068140: 2020 2020 2020 2020 2020 7c0a 7c6f 3233            |.|o23
│ │ │ │ -00068150: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00068130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00068140: 0a7c 6f32 3320 3a20 4c69 7374 2020 2020  .|o23 : List    
│ │ │ │ +00068150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068170: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00068170: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00068180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000681a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000681b0: 3234 203a 2061 7070 6c79 2028 352c 2069  24 : apply (5, i
│ │ │ │ -000681c0: 2d3e 2068 696c 6265 7274 4675 6e63 7469  -> hilbertFuncti
│ │ │ │ -000681d0: 6f6e 2869 2c20 6578 7472 6129 297c 0a7c  on(i, extra))|.|
│ │ │ │ +000681a0: 2d2b 0a7c 6932 3420 3a20 6170 706c 7920  -+.|i24 : apply 
│ │ │ │ +000681b0: 2835 2c20 692d 3e20 6869 6c62 6572 7446  (5, i-> hilbertF
│ │ │ │ +000681c0: 756e 6374 696f 6e28 692c 2065 7874 7261  unction(i, extra
│ │ │ │ +000681d0: 2929 7c0a 7c20 2020 2020 2020 2020 2020  ))|.|           
│ │ │ │  000681e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000681f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00068210: 7c6f 3234 203d 207b 312c 2031 2c20 312c  |o24 = {1, 1, 1,
│ │ │ │ -00068220: 2030 2c20 307d 2020 2020 2020 2020 2020   0, 0}          
│ │ │ │ -00068230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00068240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00068200: 2020 207c 0a7c 6f32 3420 3d20 7b31 2c20     |.|o24 = {1, 
│ │ │ │ +00068210: 312c 2031 2c20 302c 2030 7d20 2020 2020  1, 1, 0, 0}     
│ │ │ │ +00068220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068230: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00068240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068270: 7c0a 7c6f 3234 203a 204c 6973 7420 2020  |.|o24 : List   
│ │ │ │ +00068260: 2020 2020 207c 0a7c 6f32 3420 3a20 4c69       |.|o24 : Li
│ │ │ │ +00068270: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00068280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000682a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00068290: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000682a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000682b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000682c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000682d0: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d  --+..Caveat.====
│ │ │ │ -000682e0: 3d3d 0a0a 5465 7874 2053 322d 6966 6963  ==..Text S2-ific
│ │ │ │ -000682f0: 6174 696f 6e20 6973 2072 656c 6174 6564  ation is related
│ │ │ │ -00068300: 2074 6f20 636f 6d70 7574 696e 6720 636f   to computing co
│ │ │ │ -00068310: 686f 6d6f 6c6f 6779 2061 6e64 2074 6f20  homology and to 
│ │ │ │ -00068320: 636f 6d70 7574 696e 6720 696e 7465 6772  computing integr
│ │ │ │ -00068330: 616c 0a63 6c6f 7375 7265 3b20 7468 6572  al.closure; ther
│ │ │ │ -00068340: 6520 6172 6520 7363 7269 7074 7320 696e  e are scripts in
│ │ │ │ -00068350: 2074 686f 7365 2070 6163 6b61 6765 7320   those packages 
│ │ │ │ -00068360: 7468 6174 2070 726f 6475 6365 2061 6e20  that produce an 
│ │ │ │ -00068370: 5332 2d69 6669 6361 7469 6f6e 2c20 6275  S2-ification, bu
│ │ │ │ -00068380: 740a 6f6e 6520 7461 6b65 7320 6120 7269  t.one takes a ri
│ │ │ │ -00068390: 6e67 2061 7320 6172 6775 6d65 6e74 2061  ng as argument a
│ │ │ │ -000683a0: 6e64 2074 6865 206f 7468 6572 2064 6f65  nd the other doe
│ │ │ │ -000683b0: 736e 2774 2070 726f 6475 6365 2074 6865  sn't produce the
│ │ │ │ -000683c0: 2063 6f6d 7061 7269 736f 6e20 6d61 702e   comparison map.
│ │ │ │ -000683d0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -000683e0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2049  ===..  * *note I
│ │ │ │ -000683f0: 6e74 6567 7261 6c43 6c6f 7375 7265 3a20  ntegralClosure: 
│ │ │ │ -00068400: 2849 6e74 6567 7261 6c43 6c6f 7375 7265  (IntegralClosure
│ │ │ │ -00068410: 2954 6f70 2c20 2d2d 2072 6f75 7469 6e65  )Top, -- routine
│ │ │ │ -00068420: 7320 666f 7220 696e 7465 6772 616c 0a20  s for integral. 
│ │ │ │ -00068430: 2020 2063 6c6f 7375 7265 206f 6620 6166     closure of af
│ │ │ │ -00068440: 6669 6e65 2064 6f6d 6169 6e73 2061 6e64  fine domains and
│ │ │ │ -00068450: 2069 6465 616c 730a 2020 2a20 2a6e 6f74   ideals.  * *not
│ │ │ │ -00068460: 6520 6d61 6b65 5332 3a20 2849 6e74 6567  e makeS2: (Integ
│ │ │ │ -00068470: 7261 6c43 6c6f 7375 7265 296d 616b 6553  ralClosure)makeS
│ │ │ │ -00068480: 322c 202d 2d20 636f 6d70 7574 6520 7468  2, -- compute th
│ │ │ │ -00068490: 6520 5332 6966 6963 6174 696f 6e20 6f66  e S2ification of
│ │ │ │ -000684a0: 2061 0a20 2020 2072 6564 7563 6564 2072   a.    reduced r
│ │ │ │ -000684b0: 696e 670a 2020 2a20 2a6e 6f74 6520 4247  ing.  * *note BG
│ │ │ │ -000684c0: 473a 2028 4247 4729 546f 702c 202d 2d20  G: (BGG)Top, -- 
│ │ │ │ -000684d0: 4265 726e 7374 6569 6e2d 4765 6c27 6661  Bernstein-Gel'fa
│ │ │ │ -000684e0: 6e64 2d47 656c 2766 616e 6420 636f 7272  nd-Gel'fand corr
│ │ │ │ -000684f0: 6573 706f 6e64 656e 6365 0a20 202a 202a  espondence.  * *
│ │ │ │ -00068500: 6e6f 7465 2063 6f68 6f6d 6f6c 6f67 793a  note cohomology:
│ │ │ │ -00068510: 2028 4d61 6361 756c 6179 3244 6f63 2963   (Macaulay2Doc)c
│ │ │ │ -00068520: 6f68 6f6d 6f6c 6f67 792c 202d 2d20 6765  ohomology, -- ge
│ │ │ │ -00068530: 6e65 7261 6c20 636f 686f 6d6f 6c6f 6779  neral cohomology
│ │ │ │ -00068540: 2066 756e 6374 6f72 0a20 202a 2048 485e   functor.  * HH^
│ │ │ │ -00068550: 5a5a 2053 756d 4f66 5477 6973 7473 2028  ZZ SumOfTwists (
│ │ │ │ -00068560: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ -00068570: 6174 696f 6e29 0a0a 5761 7973 2074 6f20  ation)..Ways to 
│ │ │ │ -00068580: 7573 6520 5332 3a0a 3d3d 3d3d 3d3d 3d3d  use S2:.========
│ │ │ │ -00068590: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 5332  =======..  * "S2
│ │ │ │ -000685a0: 285a 5a2c 4d6f 6475 6c65 2922 0a0a 466f  (ZZ,Module)"..Fo
│ │ │ │ -000685b0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -000685c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -000685d0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -000685e0: 2a6e 6f74 6520 5332 3a20 5332 2c20 6973  *note S2: S2, is
│ │ │ │ -000685f0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -00068600: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ -00068610: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -00068620: 6e63 7469 6f6e 2c2e 0a1f 0a46 696c 653a  nction,....File:
│ │ │ │ -00068630: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ -00068640: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ -00068650: 2e69 6e66 6f2c 204e 6f64 653a 2053 6861  .info, Node: Sha
│ │ │ │ -00068660: 6d61 7368 2c20 4e65 7874 3a20 7370 6c69  mash, Next: spli
│ │ │ │ -00068670: 7474 696e 6773 2c20 5072 6576 3a20 5332  ttings, Prev: S2
│ │ │ │ -00068680: 2c20 5570 3a20 546f 700a 0a53 6861 6d61  , Up: Top..Shama
│ │ │ │ -00068690: 7368 202d 2d20 436f 6d70 7574 6573 2074  sh -- Computes t
│ │ │ │ -000686a0: 6865 2053 6861 6d61 7368 2043 6f6d 706c  he Shamash Compl
│ │ │ │ -000686b0: 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ex.*************
│ │ │ │ -000686c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000686d0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -000686e0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -000686f0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -00068700: 2020 2046 4620 3d20 5368 616d 6173 6828     FF = Shamash(
│ │ │ │ -00068710: 6666 2c46 2c6c 656e 290a 2020 2020 2020  ff,F,len).      
│ │ │ │ -00068720: 2020 4646 203d 2053 6861 6d61 7368 2852    FF = Shamash(R
│ │ │ │ -00068730: 6261 722c 462c 6c65 6e29 0a20 202a 2049  bar,F,len).  * I
│ │ │ │ -00068740: 6e70 7574 733a 0a20 2020 2020 202a 2066  nputs:.      * f
│ │ │ │ -00068750: 662c 2061 202a 6e6f 7465 206d 6174 7269  f, a *note matri
│ │ │ │ -00068760: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -00068770: 294d 6174 7269 782c 2c20 3120 7820 3120  )Matrix,, 1 x 1 
│ │ │ │ -00068780: 4d61 7472 6978 206f 7665 7220 7269 6e67  Matrix over ring
│ │ │ │ -00068790: 2046 2e0a 2020 2020 2020 2a20 5262 6172   F..      * Rbar
│ │ │ │ -000687a0: 2c20 6120 2a6e 6f74 6520 7269 6e67 3a20  , a *note ring: 
│ │ │ │ -000687b0: 284d 6163 6175 6c61 7932 446f 6329 5269  (Macaulay2Doc)Ri
│ │ │ │ -000687c0: 6e67 2c2c 2072 696e 6720 4620 6d6f 6420  ng,, ring F mod 
│ │ │ │ -000687d0: 6964 6561 6c20 6666 0a20 2020 2020 202a  ideal ff.      *
│ │ │ │ -000687e0: 2046 2c20 6120 2a6e 6f74 6520 6368 6169   F, a *note chai
│ │ │ │ -000687f0: 6e20 636f 6d70 6c65 783a 2028 4d61 6361  n complex: (Maca
│ │ │ │ -00068800: 756c 6179 3244 6f63 2943 6861 696e 436f  ulay2Doc)ChainCo
│ │ │ │ -00068810: 6d70 6c65 782c 2c20 6465 6669 6e65 6420  mplex,, defined 
│ │ │ │ -00068820: 6f76 6572 0a20 2020 2020 2020 2072 696e  over.        rin
│ │ │ │ -00068830: 6720 6666 0a20 2020 2020 202a 206c 656e  g ff.      * len
│ │ │ │ -00068840: 2c20 616e 202a 6e6f 7465 2069 6e74 6567  , an *note integ
│ │ │ │ -00068850: 6572 3a20 284d 6163 6175 6c61 7932 446f  er: (Macaulay2Do
│ │ │ │ -00068860: 6329 5a5a 2c2c 200a 2020 2a20 4f75 7470  c)ZZ,, .  * Outp
│ │ │ │ -00068870: 7574 733a 0a20 2020 2020 202a 2046 462c  uts:.      * FF,
│ │ │ │ -00068880: 2061 202a 6e6f 7465 2063 6861 696e 2063   a *note chain c
│ │ │ │ -00068890: 6f6d 706c 6578 3a20 284d 6163 6175 6c61  omplex: (Macaula
│ │ │ │ -000688a0: 7932 446f 6329 4368 6169 6e43 6f6d 706c  y2Doc)ChainCompl
│ │ │ │ -000688b0: 6578 2c2c 2063 6861 696e 2063 6f6d 706c  ex,, chain compl
│ │ │ │ -000688c0: 6578 0a20 2020 2020 2020 206f 7665 7220  ex.        over 
│ │ │ │ -000688d0: 2872 696e 6720 4629 2f28 6964 6561 6c20  (ring F)/(ideal 
│ │ │ │ -000688e0: 6666 290a 0a44 6573 6372 6970 7469 6f6e  ff)..Description
│ │ │ │ -000688f0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4c65  .===========..Le
│ │ │ │ -00068900: 7420 5220 3d20 7269 6e67 2046 203d 2072  t R = ring F = r
│ │ │ │ -00068910: 696e 6720 6666 2c20 616e 6420 5262 6172  ing ff, and Rbar
│ │ │ │ -00068920: 203d 2052 2f28 6964 6561 6c20 6629 2c20   = R/(ideal f), 
│ │ │ │ -00068930: 7768 6572 6520 6666 203d 206d 6174 7269  where ff = matri
│ │ │ │ -00068940: 787b 7b66 7d7d 2069 7320 610a 3178 3120  x{{f}} is a.1x1 
│ │ │ │ -00068950: 6d61 7472 6978 2077 686f 7365 2065 6e74  matrix whose ent
│ │ │ │ -00068960: 7279 2069 7320 6120 6e6f 6e7a 6572 6f64  ry is a nonzerod
│ │ │ │ -00068970: 6976 6973 6f72 2069 6e20 522e 2054 6865  ivisor in R. The
│ │ │ │ -00068980: 2063 6f6d 706c 6578 2046 2073 686f 756c   complex F shoul
│ │ │ │ -00068990: 6420 6164 6d69 7420 610a 7379 7374 656d  d admit a.system
│ │ │ │ -000689a0: 206f 6620 6869 6768 6572 2068 6f6d 6f74   of higher homot
│ │ │ │ -000689b0: 6f70 6965 7320 666f 7220 7468 6520 656e  opies for the en
│ │ │ │ -000689c0: 7472 7920 6f66 2066 662c 2072 6574 7572  try of ff, retur
│ │ │ │ -000689d0: 6e65 6420 6279 2074 6865 2063 616c 6c0a  ned by the call.
│ │ │ │ -000689e0: 6d61 6b65 486f 6d6f 746f 7069 6573 2866  makeHomotopies(f
│ │ │ │ -000689f0: 662c 4629 2e0a 0a54 6865 2063 6f6d 706c  f,F)...The compl
│ │ │ │ -00068a00: 6578 2046 4620 6861 7320 7465 726d 730a  ex FF has terms.
│ │ │ │ -00068a10: 0a46 465f 7b32 2a69 7d20 3d20 5262 6172  .FF_{2*i} = Rbar
│ │ │ │ -00068a20: 2a2a 2846 5f30 202b 2b20 465f 3220 2b2b  **(F_0 ++ F_2 ++
│ │ │ │ -00068a30: 202e 2e20 2b2b 2046 5f69 290a 0a46 465f   .. ++ F_i)..FF_
│ │ │ │ -00068a40: 7b32 2a69 2b31 7d20 3d20 5262 6172 2a2a  {2*i+1} = Rbar**
│ │ │ │ -00068a50: 2846 5f31 202b 2b20 465f 3320 2b2b 2e2e  (F_1 ++ F_3 ++..
│ │ │ │ -00068a60: 2b2b 465f 7b32 2a69 2b31 7d29 0a0a 616e  ++F_{2*i+1})..an
│ │ │ │ -00068a70: 6420 6d61 7073 206d 6164 6520 6672 6f6d  d maps made from
│ │ │ │ -00068a80: 2074 6865 2068 6967 6865 7220 686f 6d6f   the higher homo
│ │ │ │ -00068a90: 746f 7069 6573 2e0a 0a46 6f72 2074 6865  topies...For the
│ │ │ │ -00068aa0: 2063 6173 6520 6f66 2061 2063 6f6d 706c   case of a compl
│ │ │ │ -00068ab0: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ -00068ac0: 206f 6620 6869 6768 6572 2063 6f64 696d   of higher codim
│ │ │ │ -00068ad0: 656e 7369 6f6e 2c20 6f72 2074 6f20 7365  ension, or to se
│ │ │ │ -00068ae0: 6520 7468 650a 636f 6d70 6f6e 656e 7473  e the.components
│ │ │ │ -00068af0: 206f 6620 7468 6520 7265 736f 6c75 7469   of the resoluti
│ │ │ │ -00068b00: 6f6e 2061 7320 7375 6d6d 616e 6473 206f  on as summands o
│ │ │ │ -00068b10: 6620 4646 5f6a 2c20 7573 6520 7468 6520  f FF_j, use the 
│ │ │ │ -00068b20: 726f 7574 696e 650a 4569 7365 6e62 7564  routine.Eisenbud
│ │ │ │ -00068b30: 5368 616d 6173 6820 696e 7374 6561 642e  Shamash instead.
│ │ │ │ -00068b40: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +000682c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174  -------+..Caveat
│ │ │ │ +000682d0: 0a3d 3d3d 3d3d 3d0a 0a54 6578 7420 5332  .======..Text S2
│ │ │ │ +000682e0: 2d69 6669 6361 7469 6f6e 2069 7320 7265  -ification is re
│ │ │ │ +000682f0: 6c61 7465 6420 746f 2063 6f6d 7075 7469  lated to computi
│ │ │ │ +00068300: 6e67 2063 6f68 6f6d 6f6c 6f67 7920 616e  ng cohomology an
│ │ │ │ +00068310: 6420 746f 2063 6f6d 7075 7469 6e67 2069  d to computing i
│ │ │ │ +00068320: 6e74 6567 7261 6c0a 636c 6f73 7572 653b  ntegral.closure;
│ │ │ │ +00068330: 2074 6865 7265 2061 7265 2073 6372 6970   there are scrip
│ │ │ │ +00068340: 7473 2069 6e20 7468 6f73 6520 7061 636b  ts in those pack
│ │ │ │ +00068350: 6167 6573 2074 6861 7420 7072 6f64 7563  ages that produc
│ │ │ │ +00068360: 6520 616e 2053 322d 6966 6963 6174 696f  e an S2-ificatio
│ │ │ │ +00068370: 6e2c 2062 7574 0a6f 6e65 2074 616b 6573  n, but.one takes
│ │ │ │ +00068380: 2061 2072 696e 6720 6173 2061 7267 756d   a ring as argum
│ │ │ │ +00068390: 656e 7420 616e 6420 7468 6520 6f74 6865  ent and the othe
│ │ │ │ +000683a0: 7220 646f 6573 6e27 7420 7072 6f64 7563  r doesn't produc
│ │ │ │ +000683b0: 6520 7468 6520 636f 6d70 6172 6973 6f6e  e the comparison
│ │ │ │ +000683c0: 206d 6170 2e0a 0a53 6565 2061 6c73 6f0a   map...See also.
│ │ │ │ +000683d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000683e0: 6f74 6520 496e 7465 6772 616c 436c 6f73  ote IntegralClos
│ │ │ │ +000683f0: 7572 653a 2028 496e 7465 6772 616c 436c  ure: (IntegralCl
│ │ │ │ +00068400: 6f73 7572 6529 546f 702c 202d 2d20 726f  osure)Top, -- ro
│ │ │ │ +00068410: 7574 696e 6573 2066 6f72 2069 6e74 6567  utines for integ
│ │ │ │ +00068420: 7261 6c0a 2020 2020 636c 6f73 7572 6520  ral.    closure 
│ │ │ │ +00068430: 6f66 2061 6666 696e 6520 646f 6d61 696e  of affine domain
│ │ │ │ +00068440: 7320 616e 6420 6964 6561 6c73 0a20 202a  s and ideals.  *
│ │ │ │ +00068450: 202a 6e6f 7465 206d 616b 6553 323a 2028   *note makeS2: (
│ │ │ │ +00068460: 496e 7465 6772 616c 436c 6f73 7572 6529  IntegralClosure)
│ │ │ │ +00068470: 6d61 6b65 5332 2c20 2d2d 2063 6f6d 7075  makeS2, -- compu
│ │ │ │ +00068480: 7465 2074 6865 2053 3269 6669 6361 7469  te the S2ificati
│ │ │ │ +00068490: 6f6e 206f 6620 610a 2020 2020 7265 6475  on of a.    redu
│ │ │ │ +000684a0: 6365 6420 7269 6e67 0a20 202a 202a 6e6f  ced ring.  * *no
│ │ │ │ +000684b0: 7465 2042 4747 3a20 2842 4747 2954 6f70  te BGG: (BGG)Top
│ │ │ │ +000684c0: 2c20 2d2d 2042 6572 6e73 7465 696e 2d47  , -- Bernstein-G
│ │ │ │ +000684d0: 656c 2766 616e 642d 4765 6c27 6661 6e64  el'fand-Gel'fand
│ │ │ │ +000684e0: 2063 6f72 7265 7370 6f6e 6465 6e63 650a   correspondence.
│ │ │ │ +000684f0: 2020 2a20 2a6e 6f74 6520 636f 686f 6d6f    * *note cohomo
│ │ │ │ +00068500: 6c6f 6779 3a20 284d 6163 6175 6c61 7932  logy: (Macaulay2
│ │ │ │ +00068510: 446f 6329 636f 686f 6d6f 6c6f 6779 2c20  Doc)cohomology, 
│ │ │ │ +00068520: 2d2d 2067 656e 6572 616c 2063 6f68 6f6d  -- general cohom
│ │ │ │ +00068530: 6f6c 6f67 7920 6675 6e63 746f 720a 2020  ology functor.  
│ │ │ │ +00068540: 2a20 4848 5e5a 5a20 5375 6d4f 6654 7769  * HH^ZZ SumOfTwi
│ │ │ │ +00068550: 7374 7320 286d 6973 7369 6e67 2064 6f63  sts (missing doc
│ │ │ │ +00068560: 756d 656e 7461 7469 6f6e 290a 0a57 6179  umentation)..Way
│ │ │ │ +00068570: 7320 746f 2075 7365 2053 323a 0a3d 3d3d  s to use S2:.===
│ │ │ │ +00068580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00068590: 2a20 2253 3228 5a5a 2c4d 6f64 756c 6529  * "S2(ZZ,Module)
│ │ │ │ +000685a0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +000685b0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +000685c0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +000685d0: 6a65 6374 202a 6e6f 7465 2053 323a 2053  ject *note S2: S
│ │ │ │ +000685e0: 322c 2069 7320 6120 2a6e 6f74 6520 6d65  2, is a *note me
│ │ │ │ +000685f0: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ +00068600: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +00068610: 686f 6446 756e 6374 696f 6e2c 2e0a 1f0a  hodFunction,....
│ │ │ │ +00068620: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +00068630: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00068640: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +00068650: 3a20 5368 616d 6173 682c 204e 6578 743a  : Shamash, Next:
│ │ │ │ +00068660: 2073 706c 6974 7469 6e67 732c 2050 7265   splittings, Pre
│ │ │ │ +00068670: 763a 2053 322c 2055 703a 2054 6f70 0a0a  v: S2, Up: Top..
│ │ │ │ +00068680: 5368 616d 6173 6820 2d2d 2043 6f6d 7075  Shamash -- Compu
│ │ │ │ +00068690: 7465 7320 7468 6520 5368 616d 6173 6820  tes the Shamash 
│ │ │ │ +000686a0: 436f 6d70 6c65 780a 2a2a 2a2a 2a2a 2a2a  Complex.********
│ │ │ │ +000686b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000686c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +000686d0: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +000686e0: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +000686f0: 2020 2020 2020 2020 4646 203d 2053 6861          FF = Sha
│ │ │ │ +00068700: 6d61 7368 2866 662c 462c 6c65 6e29 0a20  mash(ff,F,len). 
│ │ │ │ +00068710: 2020 2020 2020 2046 4620 3d20 5368 616d         FF = Sham
│ │ │ │ +00068720: 6173 6828 5262 6172 2c46 2c6c 656e 290a  ash(Rbar,F,len).
│ │ │ │ +00068730: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00068740: 2020 2a20 6666 2c20 6120 2a6e 6f74 6520    * ff, a *note 
│ │ │ │ +00068750: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ +00068760: 7932 446f 6329 4d61 7472 6978 2c2c 2031  y2Doc)Matrix,, 1
│ │ │ │ +00068770: 2078 2031 204d 6174 7269 7820 6f76 6572   x 1 Matrix over
│ │ │ │ +00068780: 2072 696e 6720 462e 0a20 2020 2020 202a   ring F..      *
│ │ │ │ +00068790: 2052 6261 722c 2061 202a 6e6f 7465 2072   Rbar, a *note r
│ │ │ │ +000687a0: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ +000687b0: 6f63 2952 696e 672c 2c20 7269 6e67 2046  oc)Ring,, ring F
│ │ │ │ +000687c0: 206d 6f64 2069 6465 616c 2066 660a 2020   mod ideal ff.  
│ │ │ │ +000687d0: 2020 2020 2a20 462c 2061 202a 6e6f 7465      * F, a *note
│ │ │ │ +000687e0: 2063 6861 696e 2063 6f6d 706c 6578 3a20   chain complex: 
│ │ │ │ +000687f0: 284d 6163 6175 6c61 7932 446f 6329 4368  (Macaulay2Doc)Ch
│ │ │ │ +00068800: 6169 6e43 6f6d 706c 6578 2c2c 2064 6566  ainComplex,, def
│ │ │ │ +00068810: 696e 6564 206f 7665 720a 2020 2020 2020  ined over.      
│ │ │ │ +00068820: 2020 7269 6e67 2066 660a 2020 2020 2020    ring ff.      
│ │ │ │ +00068830: 2a20 6c65 6e2c 2061 6e20 2a6e 6f74 6520  * len, an *note 
│ │ │ │ +00068840: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +00068850: 6179 3244 6f63 295a 5a2c 2c20 0a20 202a  ay2Doc)ZZ,, .  *
│ │ │ │ +00068860: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +00068870: 2a20 4646 2c20 6120 2a6e 6f74 6520 6368  * FF, a *note ch
│ │ │ │ +00068880: 6169 6e20 636f 6d70 6c65 783a 2028 4d61  ain complex: (Ma
│ │ │ │ +00068890: 6361 756c 6179 3244 6f63 2943 6861 696e  caulay2Doc)Chain
│ │ │ │ +000688a0: 436f 6d70 6c65 782c 2c20 6368 6169 6e20  Complex,, chain 
│ │ │ │ +000688b0: 636f 6d70 6c65 780a 2020 2020 2020 2020  complex.        
│ │ │ │ +000688c0: 6f76 6572 2028 7269 6e67 2046 292f 2869  over (ring F)/(i
│ │ │ │ +000688d0: 6465 616c 2066 6629 0a0a 4465 7363 7269  deal ff)..Descri
│ │ │ │ +000688e0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +000688f0: 3d0a 0a4c 6574 2052 203d 2072 696e 6720  =..Let R = ring 
│ │ │ │ +00068900: 4620 3d20 7269 6e67 2066 662c 2061 6e64  F = ring ff, and
│ │ │ │ +00068910: 2052 6261 7220 3d20 522f 2869 6465 616c   Rbar = R/(ideal
│ │ │ │ +00068920: 2066 292c 2077 6865 7265 2066 6620 3d20   f), where ff = 
│ │ │ │ +00068930: 6d61 7472 6978 7b7b 667d 7d20 6973 2061  matrix{{f}} is a
│ │ │ │ +00068940: 0a31 7831 206d 6174 7269 7820 7768 6f73  .1x1 matrix whos
│ │ │ │ +00068950: 6520 656e 7472 7920 6973 2061 206e 6f6e  e entry is a non
│ │ │ │ +00068960: 7a65 726f 6469 7669 736f 7220 696e 2052  zerodivisor in R
│ │ │ │ +00068970: 2e20 5468 6520 636f 6d70 6c65 7820 4620  . The complex F 
│ │ │ │ +00068980: 7368 6f75 6c64 2061 646d 6974 2061 0a73  should admit a.s
│ │ │ │ +00068990: 7973 7465 6d20 6f66 2068 6967 6865 7220  ystem of higher 
│ │ │ │ +000689a0: 686f 6d6f 746f 7069 6573 2066 6f72 2074  homotopies for t
│ │ │ │ +000689b0: 6865 2065 6e74 7279 206f 6620 6666 2c20  he entry of ff, 
│ │ │ │ +000689c0: 7265 7475 726e 6564 2062 7920 7468 6520  returned by the 
│ │ │ │ +000689d0: 6361 6c6c 0a6d 616b 6548 6f6d 6f74 6f70  call.makeHomotop
│ │ │ │ +000689e0: 6965 7328 6666 2c46 292e 0a0a 5468 6520  ies(ff,F)...The 
│ │ │ │ +000689f0: 636f 6d70 6c65 7820 4646 2068 6173 2074  complex FF has t
│ │ │ │ +00068a00: 6572 6d73 0a0a 4646 5f7b 322a 697d 203d  erms..FF_{2*i} =
│ │ │ │ +00068a10: 2052 6261 722a 2a28 465f 3020 2b2b 2046   Rbar**(F_0 ++ F
│ │ │ │ +00068a20: 5f32 202b 2b20 2e2e 202b 2b20 465f 6929  _2 ++ .. ++ F_i)
│ │ │ │ +00068a30: 0a0a 4646 5f7b 322a 692b 317d 203d 2052  ..FF_{2*i+1} = R
│ │ │ │ +00068a40: 6261 722a 2a28 465f 3120 2b2b 2046 5f33  bar**(F_1 ++ F_3
│ │ │ │ +00068a50: 202b 2b2e 2e2b 2b46 5f7b 322a 692b 317d   ++..++F_{2*i+1}
│ │ │ │ +00068a60: 290a 0a61 6e64 206d 6170 7320 6d61 6465  )..and maps made
│ │ │ │ +00068a70: 2066 726f 6d20 7468 6520 6869 6768 6572   from the higher
│ │ │ │ +00068a80: 2068 6f6d 6f74 6f70 6965 732e 0a0a 466f   homotopies...Fo
│ │ │ │ +00068a90: 7220 7468 6520 6361 7365 206f 6620 6120  r the case of a 
│ │ │ │ +00068aa0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +00068ab0: 6374 696f 6e20 6f66 2068 6967 6865 7220  ction of higher 
│ │ │ │ +00068ac0: 636f 6469 6d65 6e73 696f 6e2c 206f 7220  codimension, or 
│ │ │ │ +00068ad0: 746f 2073 6565 2074 6865 0a63 6f6d 706f  to see the.compo
│ │ │ │ +00068ae0: 6e65 6e74 7320 6f66 2074 6865 2072 6573  nents of the res
│ │ │ │ +00068af0: 6f6c 7574 696f 6e20 6173 2073 756d 6d61  olution as summa
│ │ │ │ +00068b00: 6e64 7320 6f66 2046 465f 6a2c 2075 7365  nds of FF_j, use
│ │ │ │ +00068b10: 2074 6865 2072 6f75 7469 6e65 0a45 6973   the routine.Eis
│ │ │ │ +00068b20: 656e 6275 6453 6861 6d61 7368 2069 6e73  enbudShamash ins
│ │ │ │ +00068b30: 7465 6164 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  tead...+--------
│ │ │ │ +00068b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068b70: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ -00068b80: 203d 205a 5a2f 3130 315b 782c 792c 7a5d   = ZZ/101[x,y,z]
│ │ │ │ +00068b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00068b70: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b78  1 : S = ZZ/101[x
│ │ │ │ +00068b80: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │  00068b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00068bb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00068ba0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00068bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068be0: 2020 2020 207c 0a7c 6f31 203d 2053 2020       |.|o1 = S  
│ │ │ │ +00068bd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00068be0: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │  00068bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068c10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00068c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c50: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e     |.|o1 : Polyn
│ │ │ │ -00068c60: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ -00068c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c80: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00068c40: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00068c50: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +00068c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00068c80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00068c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068cc0: 2d2b 0a7c 6932 203a 2052 203d 2053 2f69  -+.|i2 : R = S/i
│ │ │ │ -00068cd0: 6465 616c 2278 332c 7933 2220 2020 2020  deal"x3,y3"     
│ │ │ │ -00068ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068cf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00068cb0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5220  ------+.|i2 : R 
│ │ │ │ +00068cc0: 3d20 532f 6964 6561 6c22 7833 2c79 3322  = S/ideal"x3,y3"
│ │ │ │ +00068cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00068cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00068d30: 0a7c 6f32 203d 2052 2020 2020 2020 2020  .|o2 = R        
│ │ │ │ +00068d20: 2020 2020 7c0a 7c6f 3220 3d20 5220 2020      |.|o2 = R   
│ │ │ │ +00068d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068d50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00068d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00068da0: 6f32 203a 2051 756f 7469 656e 7452 696e  o2 : QuotientRin
│ │ │ │ -00068db0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00068dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068dd0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00068d90: 2020 7c0a 7c6f 3220 3a20 5175 6f74 6965    |.|o2 : Quotie
│ │ │ │ +00068da0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +00068db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068dc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00068dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00068e10: 203a 204d 203d 2052 5e31 2f69 6465 616c   : M = R^1/ideal
│ │ │ │ -00068e20: 2878 2c79 2c7a 2920 2020 2020 2020 2020  (x,y,z)         
│ │ │ │ -00068e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00068e00: 2b0a 7c69 3320 3a20 4d20 3d20 525e 312f  +.|i3 : M = R^1/
│ │ │ │ +00068e10: 6964 6561 6c28 782c 792c 7a29 2020 2020  ideal(x,y,z)    
│ │ │ │ +00068e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068e30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e70: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ -00068e80: 2063 6f6b 6572 6e65 6c20 7c20 7820 7920   cokernel | x y 
│ │ │ │ -00068e90: 7a20 7c20 2020 2020 2020 2020 2020 2020  z |             
│ │ │ │ -00068ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068eb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00068e60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00068e70: 7c6f 3320 3d20 636f 6b65 726e 656c 207c  |o3 = cokernel |
│ │ │ │ +00068e80: 2078 2079 207a 207c 2020 2020 2020 2020   x y z |        
│ │ │ │ +00068e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00068eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ee0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068f00: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -00068f10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00068f20: 7c6f 3320 3a20 522d 6d6f 6475 6c65 2c20  |o3 : R-module, 
│ │ │ │ -00068f30: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ -00068f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068f50: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00068ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00068ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068ef0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +00068f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068f10: 2020 207c 0a7c 6f33 203a 2052 2d6d 6f64     |.|o3 : R-mod
│ │ │ │ +00068f20: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ +00068f30: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00068f40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00068f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00068f90: 3420 3a20 4620 3d20 7265 7320 4d20 2020  4 : F = res M   
│ │ │ │ +00068f80: 2d2b 0a7c 6934 203a 2046 203d 2072 6573  -+.|i4 : F = res
│ │ │ │ +00068f90: 204d 2020 2020 2020 2020 2020 2020 2020   M              
│ │ │ │  00068fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00068fb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00068fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00069000: 2020 2031 2020 2020 2020 3320 2020 2020     1      3     
│ │ │ │ -00069010: 2035 2020 2020 2020 3720 2020 2020 2039   5      7      9
│ │ │ │ -00069020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069030: 207c 0a7c 6f34 203d 2052 2020 3c2d 2d20   |.|o4 = R  <-- 
│ │ │ │ -00069040: 5220 203c 2d2d 2052 2020 3c2d 2d20 5220  R  <-- R  <-- R 
│ │ │ │ -00069050: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ -00069060: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00068fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00068ff0: 0a7c 2020 2020 2020 3120 2020 2020 2033  .|      1      3
│ │ │ │ +00069000: 2020 2020 2020 3520 2020 2020 2037 2020        5      7  
│ │ │ │ +00069010: 2020 2020 3920 2020 2020 2020 2020 2020      9           
│ │ │ │ +00069020: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220        |.|o4 = R 
│ │ │ │ +00069030: 203c 2d2d 2052 2020 3c2d 2d20 5220 203c   <-- R  <-- R  <
│ │ │ │ +00069040: 2d2d 2052 2020 3c2d 2d20 5220 2020 2020  -- R  <-- R     
│ │ │ │ +00069050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00069060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000690a0: 0a7c 2020 2020 2030 2020 2020 2020 3120  .|     0      1 
│ │ │ │ -000690b0: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -000690c0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -000690d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00069090: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ +000690a0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ +000690b0: 2033 2020 2020 2020 3420 2020 2020 2020   3      4       
│ │ │ │ +000690c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000690d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000690e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000690f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00069110: 6f34 203a 2043 6861 696e 436f 6d70 6c65  o4 : ChainComple
│ │ │ │ -00069120: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00069130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069140: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00069100: 2020 7c0a 7c6f 3420 3a20 4368 6169 6e43    |.|o4 : ChainC
│ │ │ │ +00069110: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00069120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069130: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00069140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00069180: 203a 2066 6620 3d20 6d61 7472 6978 7b7b   : ff = matrix{{
│ │ │ │ -00069190: 7a5e 337d 7d20 2020 2020 2020 2020 2020  z^3}}           
│ │ │ │ -000691a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000691b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00069170: 2b0a 7c69 3520 3a20 6666 203d 206d 6174  +.|i5 : ff = mat
│ │ │ │ +00069180: 7269 787b 7b7a 5e33 7d7d 2020 2020 2020  rix{{z^3}}      
│ │ │ │ +00069190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000691a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000691b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000691d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000691e0: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ -000691f0: 207c 207a 3320 7c20 2020 2020 2020 2020   | z3 |         
│ │ │ │ +000691d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000691e0: 7c6f 3520 3d20 7c20 7a33 207c 2020 2020  |o5 = | z3 |    
│ │ │ │ +000691f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069220: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00069210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069250: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069260: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00069240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069250: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00069260: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  00069270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00069290: 7c6f 3520 3a20 4d61 7472 6978 2052 2020  |o5 : Matrix R  
│ │ │ │ -000692a0: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ -000692b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00069280: 2020 207c 0a7c 6f35 203a 204d 6174 7269     |.|o5 : Matri
│ │ │ │ +00069290: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ +000692a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000692b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000692c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000692d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000692e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000692f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00069300: 3620 3a20 5231 203d 2052 2f69 6465 616c  6 : R1 = R/ideal
│ │ │ │ -00069310: 2066 6620 2020 2020 2020 2020 2020 2020   ff             
│ │ │ │ -00069320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069330: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000692f0: 2d2b 0a7c 6936 203a 2052 3120 3d20 522f  -+.|i6 : R1 = R/
│ │ │ │ +00069300: 6964 6561 6c20 6666 2020 2020 2020 2020  ideal ff        
│ │ │ │ +00069310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00069330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069360: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00069370: 3d20 5231 2020 2020 2020 2020 2020 2020  = R1            
│ │ │ │ +00069350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00069360: 0a7c 6f36 203d 2052 3120 2020 2020 2020  .|o6 = R1       
│ │ │ │ +00069370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00069390: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000693a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000693b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693d0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -000693e0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000693d0: 6f36 203a 2051 756f 7469 656e 7452 696e  o6 : QuotientRin
│ │ │ │ +000693e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  000693f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069410: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00069400: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00069410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069440: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6265  ------+.|i7 : be
│ │ │ │ -00069450: 7474 6920 4620 2020 2020 2020 2020 2020  tti F           
│ │ │ │ +00069430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00069440: 203a 2062 6574 7469 2046 2020 2020 2020   : betti F      
│ │ │ │ +00069450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00069470: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00069480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000694c0: 2020 2030 2031 2032 2033 2034 2020 2020     0 1 2 3 4    
│ │ │ │ +000694a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000694b0: 2020 2020 2020 2020 3020 3120 3220 3320          0 1 2 3 
│ │ │ │ +000694c0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  000694d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694e0: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -000694f0: 203d 2074 6f74 616c 3a20 3120 3320 3520   = total: 1 3 5 
│ │ │ │ -00069500: 3720 3920 2020 2020 2020 2020 2020 2020  7 9             
│ │ │ │ -00069510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069520: 2020 7c0a 7c20 2020 2020 2020 2020 303a    |.|         0:
│ │ │ │ -00069530: 2031 2033 2033 2031 202e 2020 2020 2020   1 3 3 1 .      
│ │ │ │ -00069540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069550: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00069560: 2020 2020 2031 3a20 2e20 2e20 3220 3620       1: . . 2 6 
│ │ │ │ -00069570: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ -00069580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069590: 7c0a 7c20 2020 2020 2020 2020 323a 202e  |.|         2: .
│ │ │ │ -000695a0: 202e 202e 202e 2033 2020 2020 2020 2020   . . . 3        
│ │ │ │ -000695b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000694e0: 7c0a 7c6f 3720 3d20 746f 7461 6c3a 2031  |.|o7 = total: 1
│ │ │ │ +000694f0: 2033 2035 2037 2039 2020 2020 2020 2020   3 5 7 9        
│ │ │ │ +00069500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069520: 2020 2030 3a20 3120 3320 3320 3120 2e20     0: 1 3 3 1 . 
│ │ │ │ +00069530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069550: 7c20 2020 2020 2020 2020 313a 202e 202e  |         1: . .
│ │ │ │ +00069560: 2032 2036 2036 2020 2020 2020 2020 2020   2 6 6          
│ │ │ │ +00069570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069580: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069590: 2032 3a20 2e20 2e20 2e20 2e20 3320 2020   2: . . . . 3   
│ │ │ │ +000695a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000695b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000695c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000695d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000695e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00069600: 7c6f 3720 3a20 4265 7474 6954 616c 6c79  |o7 : BettiTally
│ │ │ │ +000695f0: 2020 207c 0a7c 6f37 203a 2042 6574 7469     |.|o7 : Betti
│ │ │ │ +00069600: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │  00069610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069630: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00069620: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00069630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00069670: 3820 3a20 4646 203d 2053 6861 6d61 7368  8 : FF = Shamash
│ │ │ │ -00069680: 2866 662c 462c 3429 2020 2020 2020 2020  (ff,F,4)        
│ │ │ │ -00069690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000696a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00069660: 2d2b 0a7c 6938 203a 2046 4620 3d20 5368  -+.|i8 : FF = Sh
│ │ │ │ +00069670: 616d 6173 6828 6666 2c46 2c34 2920 2020  amash(ff,F,4)   
│ │ │ │ +00069680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000696a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000696b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000696c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000696d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000696e0: 2020 2f20 525c 3120 2020 2020 2f20 525c    / R\1     / R\
│ │ │ │ -000696f0: 3320 2020 2020 2f20 525c 3620 2020 2020  3     / R\6     
│ │ │ │ -00069700: 2f20 525c 3130 2020 2020 202f 2052 5c31  / R\10     / R\1
│ │ │ │ -00069710: 357c 0a7c 6f38 203d 207c 2d2d 7c20 203c  5|.|o8 = |--|  <
│ │ │ │ -00069720: 2d2d 207c 2d2d 7c20 203c 2d2d 207c 2d2d  -- |--|  <-- |--
│ │ │ │ -00069730: 7c20 203c 2d2d 207c 2d2d 7c20 2020 3c2d  |  <-- |--|   <-
│ │ │ │ -00069740: 2d20 7c2d 2d7c 2020 7c0a 7c20 2020 2020  - |--|  |.|     
│ │ │ │ -00069750: 7c20 337c 2020 2020 2020 7c20 337c 2020  | 3|      | 3|  
│ │ │ │ -00069760: 2020 2020 7c20 337c 2020 2020 2020 7c20      | 3|      | 
│ │ │ │ -00069770: 337c 2020 2020 2020 207c 2033 7c20 207c  3|       | 3|  |
│ │ │ │ -00069780: 0a7c 2020 2020 205c 7a20 2f20 2020 2020  .|     \z /     
│ │ │ │ -00069790: 205c 7a20 2f20 2020 2020 205c 7a20 2f20   \z /      \z / 
│ │ │ │ -000697a0: 2020 2020 205c 7a20 2f20 2020 2020 2020       \z /       
│ │ │ │ -000697b0: 5c7a 202f 2020 7c0a 7c20 2020 2020 2020  \z /  |.|       
│ │ │ │ +000696c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000696d0: 0a7c 2020 2020 202f 2052 5c31 2020 2020  .|     / R\1    
│ │ │ │ +000696e0: 202f 2052 5c33 2020 2020 202f 2052 5c36   / R\3     / R\6
│ │ │ │ +000696f0: 2020 2020 202f 2052 5c31 3020 2020 2020       / R\10     
│ │ │ │ +00069700: 2f20 525c 3135 7c0a 7c6f 3820 3d20 7c2d  / R\15|.|o8 = |-
│ │ │ │ +00069710: 2d7c 2020 3c2d 2d20 7c2d 2d7c 2020 3c2d  -|  <-- |--|  <-
│ │ │ │ +00069720: 2d20 7c2d 2d7c 2020 3c2d 2d20 7c2d 2d7c  - |--|  <-- |--|
│ │ │ │ +00069730: 2020 203c 2d2d 207c 2d2d 7c20 207c 0a7c     <-- |--|  |.|
│ │ │ │ +00069740: 2020 2020 207c 2033 7c20 2020 2020 207c       | 3|      |
│ │ │ │ +00069750: 2033 7c20 2020 2020 207c 2033 7c20 2020   3|      | 3|   
│ │ │ │ +00069760: 2020 207c 2033 7c20 2020 2020 2020 7c20     | 3|       | 
│ │ │ │ +00069770: 337c 2020 7c0a 7c20 2020 2020 5c7a 202f  3|  |.|     \z /
│ │ │ │ +00069780: 2020 2020 2020 5c7a 202f 2020 2020 2020        \z /      
│ │ │ │ +00069790: 5c7a 202f 2020 2020 2020 5c7a 202f 2020  \z /      \z /  
│ │ │ │ +000697a0: 2020 2020 205c 7a20 2f20 207c 0a7c 2020       \z /  |.|  
│ │ │ │ +000697b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000697c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000697d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000697e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000697f0: 2020 2020 2030 2020 2020 2020 2020 2031       0         1
│ │ │ │ -00069800: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00069810: 2020 2033 2020 2020 2020 2020 2020 3420     3          4 
│ │ │ │ -00069820: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000697e0: 2020 7c0a 7c20 2020 2020 3020 2020 2020    |.|     0     
│ │ │ │ +000697f0: 2020 2020 3120 2020 2020 2020 2020 3220      1         2 
│ │ │ │ +00069800: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +00069810: 2020 2034 2020 2020 207c 0a7c 2020 2020     4     |.|    
│ │ │ │ +00069820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069850: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -00069860: 203a 2043 6861 696e 436f 6d70 6c65 7820   : ChainComplex 
│ │ │ │ +00069850: 7c0a 7c6f 3820 3a20 4368 6169 6e43 6f6d  |.|o8 : ChainCom
│ │ │ │ +00069860: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │  00069870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069890: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00069880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000698a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000698b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000698c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ -000698d0: 2047 4720 3d20 5368 616d 6173 6828 5231   GG = Shamash(R1
│ │ │ │ -000698e0: 2c46 2c34 2920 2020 2020 2020 2020 2020  ,F,4)           
│ │ │ │ -000698f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069900: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000698b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000698c0: 7c69 3920 3a20 4747 203d 2053 6861 6d61  |i9 : GG = Shama
│ │ │ │ +000698d0: 7368 2852 312c 462c 3429 2020 2020 2020  sh(R1,F,4)      
│ │ │ │ +000698e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000698f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069930: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069940: 2031 2020 2020 2020 2033 2020 2020 2020   1       3      
│ │ │ │ -00069950: 2036 2020 2020 2020 2031 3020 2020 2020   6       10     
│ │ │ │ -00069960: 2020 3135 2020 2020 2020 2020 2020 7c0a    15          |.
│ │ │ │ -00069970: 7c6f 3920 3d20 5231 2020 3c2d 2d20 5231  |o9 = R1  <-- R1
│ │ │ │ -00069980: 2020 3c2d 2d20 5231 2020 3c2d 2d20 5231    <-- R1  <-- R1
│ │ │ │ -00069990: 2020 203c 2d2d 2052 3120 2020 2020 2020     <-- R1       
│ │ │ │ -000699a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00069930: 2020 2020 2020 3120 2020 2020 2020 3320        1       3 
│ │ │ │ +00069940: 2020 2020 2020 3620 2020 2020 2020 3130        6       10
│ │ │ │ +00069950: 2020 2020 2020 2031 3520 2020 2020 2020         15       
│ │ │ │ +00069960: 2020 207c 0a7c 6f39 203d 2052 3120 203c     |.|o9 = R1  <
│ │ │ │ +00069970: 2d2d 2052 3120 203c 2d2d 2052 3120 203c  -- R1  <-- R1  <
│ │ │ │ +00069980: 2d2d 2052 3120 2020 3c2d 2d20 5231 2020  -- R1   <-- R1  
│ │ │ │ +00069990: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000699a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000699b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000699c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000699d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000699e0: 2020 2020 3020 2020 2020 2020 3120 2020      0       1   
│ │ │ │ -000699f0: 2020 2020 3220 2020 2020 2020 3320 2020      2       3   
│ │ │ │ -00069a00: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -00069a10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000699d0: 207c 0a7c 2020 2020 2030 2020 2020 2020   |.|     0      
│ │ │ │ +000699e0: 2031 2020 2020 2020 2032 2020 2020 2020   1       2      
│ │ │ │ +000699f0: 2033 2020 2020 2020 2020 3420 2020 2020   3        4     
│ │ │ │ +00069a00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00069a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069a40: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -00069a50: 3a20 4368 6169 6e43 6f6d 706c 6578 2020  : ChainComplex  
│ │ │ │ +00069a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00069a40: 0a7c 6f39 203a 2043 6861 696e 436f 6d70  .|o9 : ChainComp
│ │ │ │ +00069a50: 6c65 7820 2020 2020 2020 2020 2020 2020  lex             
│ │ │ │  00069a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069a80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00069a70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069ab0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ -00069ac0: 2062 6574 7469 2046 4620 2020 2020 2020   betti FF       
│ │ │ │ +00069aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00069ab0: 6931 3020 3a20 6265 7474 6920 4646 2020  i10 : betti FF  
│ │ │ │ +00069ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069b20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00069b30: 2020 2020 2020 3020 3120 3220 2033 2020        0 1 2  3  
│ │ │ │ -00069b40: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00069b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00069b60: 6f31 3020 3d20 746f 7461 6c3a 2031 2033  o10 = total: 1 3
│ │ │ │ -00069b70: 2036 2031 3020 3135 2020 2020 2020 2020   6 10 15        
│ │ │ │ -00069b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069ba0: 2030 3a20 3120 3320 3320 2031 2020 2e20   0: 1 3 3  1  . 
│ │ │ │ +00069b10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00069b20: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +00069b30: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +00069b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069b50: 2020 7c0a 7c6f 3130 203d 2074 6f74 616c    |.|o10 = total
│ │ │ │ +00069b60: 3a20 3120 3320 3620 3130 2031 3520 2020  : 1 3 6 10 15   
│ │ │ │ +00069b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069b80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00069b90: 2020 2020 2020 303a 2031 2033 2033 2020        0: 1 3 3  
│ │ │ │ +00069ba0: 3120 202e 2020 2020 2020 2020 2020 2020  1  .            
│ │ │ │  00069bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069bc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00069bd0: 2020 2020 2020 2020 313a 202e 202e 2033          1: . . 3
│ │ │ │ -00069be0: 2020 3920 2039 2020 2020 2020 2020 2020    9  9          
│ │ │ │ -00069bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069c00: 2020 7c0a 7c20 2020 2020 2020 2020 2032    |.|          2
│ │ │ │ -00069c10: 3a20 2e20 2e20 2e20 202e 2020 3620 2020  : . . .  .  6   
│ │ │ │ -00069c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00069bc0: 7c0a 7c20 2020 2020 2020 2020 2031 3a20  |.|          1: 
│ │ │ │ +00069bd0: 2e20 2e20 3320 2039 2020 3920 2020 2020  . . 3  9  9     
│ │ │ │ +00069be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069bf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069c00: 2020 2020 323a 202e 202e 202e 2020 2e20      2: . . .  . 
│ │ │ │ +00069c10: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00069c20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069c30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00069c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069c70: 7c0a 7c6f 3130 203a 2042 6574 7469 5461  |.|o10 : BettiTa
│ │ │ │ -00069c80: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ -00069c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069ca0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069c60: 2020 2020 207c 0a7c 6f31 3020 3a20 4265       |.|o10 : Be
│ │ │ │ +00069c70: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +00069c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069c90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00069ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00069ce0: 7c69 3131 203a 2062 6574 7469 2047 4720  |i11 : betti GG 
│ │ │ │ +00069cd0: 2d2d 2d2b 0a7c 6931 3120 3a20 6265 7474  ---+.|i11 : bett
│ │ │ │ +00069ce0: 6920 4747 2020 2020 2020 2020 2020 2020  i GG            
│ │ │ │  00069cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069d00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00069d50: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -00069d60: 3220 2033 2020 3420 2020 2020 2020 2020  2  3  4         
│ │ │ │ -00069d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069d80: 2020 207c 0a7c 6f31 3120 3d20 746f 7461     |.|o11 = tota
│ │ │ │ -00069d90: 6c3a 2031 2033 2036 2031 3020 3135 2020  l: 1 3 6 10 15  
│ │ │ │ -00069da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069db0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00069dc0: 2020 2020 2020 2030 3a20 3120 3320 3320         0: 1 3 3 
│ │ │ │ -00069dd0: 2031 2020 2e20 2020 2020 2020 2020 2020   1  .           
│ │ │ │ -00069de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069df0: 207c 0a7c 2020 2020 2020 2020 2020 313a   |.|          1:
│ │ │ │ -00069e00: 202e 202e 2033 2020 3920 2039 2020 2020   . . 3  9  9    
│ │ │ │ -00069e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069e20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00069e30: 2020 2020 2032 3a20 2e20 2e20 2e20 202e       2: . . .  .
│ │ │ │ -00069e40: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ -00069e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069e60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069d40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00069d50: 2030 2031 2032 2020 3320 2034 2020 2020   0 1 2  3  4    
│ │ │ │ +00069d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069d70: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +00069d80: 2074 6f74 616c 3a20 3120 3320 3620 3130   total: 1 3 6 10
│ │ │ │ +00069d90: 2031 3520 2020 2020 2020 2020 2020 2020   15             
│ │ │ │ +00069da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00069db0: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ +00069dc0: 2033 2033 2020 3120 202e 2020 2020 2020   3 3  1  .      
│ │ │ │ +00069dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069de0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00069df0: 2020 2031 3a20 2e20 2e20 3320 2039 2020     1: . . 3  9  
│ │ │ │ +00069e00: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +00069e10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00069e20: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ +00069e30: 202e 2020 2e20 2036 2020 2020 2020 2020   .  .  6        
│ │ │ │ +00069e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069e90: 2020 2020 2020 7c0a 7c6f 3131 203a 2042        |.|o11 : B
│ │ │ │ -00069ea0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00069e80: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00069e90: 3120 3a20 4265 7474 6954 616c 6c79 2020  1 : BettiTally  
│ │ │ │ +00069ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069ec0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00069ec0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00069ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069f00: 2d2d 2d2d 2b0a 7c69 3132 203a 2072 696e  ----+.|i12 : rin
│ │ │ │ -00069f10: 6720 4747 2020 2020 2020 2020 2020 2020  g GG            
│ │ │ │ +00069ef0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220  ---------+.|i12 
│ │ │ │ +00069f00: 3a20 7269 6e67 2047 4720 2020 2020 2020  : ring GG       
│ │ │ │ +00069f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00069f30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00069f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f70: 2020 7c0a 7c6f 3132 203d 2052 3120 2020    |.|o12 = R1   
│ │ │ │ +00069f60: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +00069f70: 5231 2020 2020 2020 2020 2020 2020 2020  R1              
│ │ │ │  00069f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00069f90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069fa0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00069fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fe0: 7c0a 7c6f 3132 203a 2051 756f 7469 656e  |.|o12 : Quotien
│ │ │ │ -00069ff0: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ -0006a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a010: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00069fd0: 2020 2020 207c 0a7c 6f31 3220 3a20 5175       |.|o12 : Qu
│ │ │ │ +00069fe0: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00069ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a000: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0006a050: 7c69 3133 203a 2061 7070 6c79 286c 656e  |i13 : apply(len
│ │ │ │ -0006a060: 6774 6820 4747 2c20 692d 3e70 7275 6e65  gth GG, i->prune
│ │ │ │ -0006a070: 2048 485f 6920 4646 2920 2020 2020 2020   HH_i FF)       
│ │ │ │ -0006a080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a040: 2d2d 2d2b 0a7c 6931 3320 3a20 6170 706c  ---+.|i13 : appl
│ │ │ │ +0006a050: 7928 6c65 6e67 7468 2047 472c 2069 2d3e  y(length GG, i->
│ │ │ │ +0006a060: 7072 756e 6520 4848 5f69 2046 4629 2020  prune HH_i FF)  
│ │ │ │ +0006a070: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006a0c0: 3133 203d 207b 636f 6b65 726e 656c 207c  13 = {cokernel |
│ │ │ │ -0006a0d0: 207a 2079 2078 207c 2c20 302c 2030 2c20   z y x |, 0, 0, 
│ │ │ │ -0006a0e0: 307d 2020 2020 2020 2020 2020 2020 2020  0}              
│ │ │ │ -0006a0f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006a0b0: 207c 0a7c 6f31 3320 3d20 7b63 6f6b 6572   |.|o13 = {coker
│ │ │ │ +0006a0c0: 6e65 6c20 7c20 7a20 7920 7820 7c2c 2030  nel | z y x |, 0
│ │ │ │ +0006a0d0: 2c20 302c 2030 7d20 2020 2020 2020 2020  , 0, 0}         
│ │ │ │ +0006a0e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a120: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -0006a130: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0006a110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a120: 0a7c 6f31 3320 3a20 4c69 7374 2020 2020  .|o13 : List    
│ │ │ │ +0006a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a160: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0006a150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a190: 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561  --------+..Cavea
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│ │ │ │ -0006a1c0: 6f6d 6f6c 6f67 6963 616c 2063 6f6d 706c  omological compl
│ │ │ │ -0006a1d0: 6578 2073 7461 7274 696e 6720 6672 6f6d  ex starting from
│ │ │ │ -0006a1e0: 2046 5f30 2e20 5468 6520 6d61 7472 6978   F_0. The matrix
│ │ │ │ -0006a1f0: 2066 6620 6d75 7374 0a62 6520 3178 312e   ff must.be 1x1.
│ │ │ │ -0006a200: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0006a210: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2045  ===..  * *note E
│ │ │ │ -0006a220: 6973 656e 6275 6453 6861 6d61 7368 3a20  isenbudShamash: 
│ │ │ │ -0006a230: 4569 7365 6e62 7564 5368 616d 6173 682c  EisenbudShamash,
│ │ │ │ -0006a240: 202d 2d20 436f 6d70 7574 6573 2074 6865   -- Computes the
│ │ │ │ -0006a250: 2045 6973 656e 6275 642d 5368 616d 6173   Eisenbud-Shamas
│ │ │ │ -0006a260: 680a 2020 2020 436f 6d70 6c65 780a 2020  h.    Complex.  
│ │ │ │ -0006a270: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f  * *note makeHomo
│ │ │ │ -0006a280: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f  topies: makeHomo
│ │ │ │ -0006a290: 746f 7069 6573 2c20 2d2d 2072 6574 7572  topies, -- retur
│ │ │ │ -0006a2a0: 6e73 2061 2073 7973 7465 6d20 6f66 2068  ns a system of h
│ │ │ │ -0006a2b0: 6967 6865 720a 2020 2020 686f 6d6f 746f  igher.    homoto
│ │ │ │ -0006a2c0: 7069 6573 0a0a 5761 7973 2074 6f20 7573  pies..Ways to us
│ │ │ │ -0006a2d0: 6520 5368 616d 6173 683a 0a3d 3d3d 3d3d  e Shamash:.=====
│ │ │ │ -0006a2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0006a2f0: 0a20 202a 2022 5368 616d 6173 6828 4d61  .  * "Shamash(Ma
│ │ │ │ -0006a300: 7472 6978 2c43 6861 696e 436f 6d70 6c65  trix,ChainComple
│ │ │ │ -0006a310: 782c 5a5a 2922 0a20 202a 2022 5368 616d  x,ZZ)".  * "Sham
│ │ │ │ -0006a320: 6173 6828 5269 6e67 2c43 6861 696e 436f  ash(Ring,ChainCo
│ │ │ │ -0006a330: 6d70 6c65 782c 5a5a 2922 0a0a 466f 7220  mplex,ZZ)"..For 
│ │ │ │ -0006a340: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0006a350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006a360: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0006a370: 6f74 6520 5368 616d 6173 683a 2053 6861  ote Shamash: Sha
│ │ │ │ -0006a380: 6d61 7368 2c20 6973 2061 202a 6e6f 7465  mash, is a *note
│ │ │ │ -0006a390: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ -0006a3a0: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ -0006a3b0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ -0006a3c0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -0006a3d0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -0006a3e0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -0006a3f0: 6f64 653a 2073 706c 6974 7469 6e67 732c  ode: splittings,
│ │ │ │ -0006a400: 204e 6578 743a 2073 7461 626c 6548 6f6d   Next: stableHom
│ │ │ │ -0006a410: 2c20 5072 6576 3a20 5368 616d 6173 682c  , Prev: Shamash,
│ │ │ │ -0006a420: 2055 703a 2054 6f70 0a0a 7370 6c69 7474   Up: Top..splitt
│ │ │ │ -0006a430: 696e 6773 202d 2d20 636f 6d70 7574 6520  ings -- compute 
│ │ │ │ -0006a440: 7468 6520 7370 6c69 7474 696e 6773 206f  the splittings o
│ │ │ │ -0006a450: 6620 6120 7370 6c69 7420 7269 6768 7420  f a split right 
│ │ │ │ -0006a460: 6578 6163 7420 7365 7175 656e 6365 0a2a  exact sequence.*
│ │ │ │ +0006a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0006a190: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a46  Caveat.======..F
│ │ │ │ +0006a1a0: 2069 7320 6173 7375 6d65 6420 746f 2062   is assumed to b
│ │ │ │ +0006a1b0: 6520 6120 686f 6d6f 6c6f 6769 6361 6c20  e a homological 
│ │ │ │ +0006a1c0: 636f 6d70 6c65 7820 7374 6172 7469 6e67  complex starting
│ │ │ │ +0006a1d0: 2066 726f 6d20 465f 302e 2054 6865 206d   from F_0. The m
│ │ │ │ +0006a1e0: 6174 7269 7820 6666 206d 7573 740a 6265  atrix ff must.be
│ │ │ │ +0006a1f0: 2031 7831 2e0a 0a53 6565 2061 6c73 6f0a   1x1...See also.
│ │ │ │ +0006a200: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0006a210: 6f74 6520 4569 7365 6e62 7564 5368 616d  ote EisenbudSham
│ │ │ │ +0006a220: 6173 683a 2045 6973 656e 6275 6453 6861  ash: EisenbudSha
│ │ │ │ +0006a230: 6d61 7368 2c20 2d2d 2043 6f6d 7075 7465  mash, -- Compute
│ │ │ │ +0006a240: 7320 7468 6520 4569 7365 6e62 7564 2d53  s the Eisenbud-S
│ │ │ │ +0006a250: 6861 6d61 7368 0a20 2020 2043 6f6d 706c  hamash.    Compl
│ │ │ │ +0006a260: 6578 0a20 202a 202a 6e6f 7465 206d 616b  ex.  * *note mak
│ │ │ │ +0006a270: 6548 6f6d 6f74 6f70 6965 733a 206d 616b  eHomotopies: mak
│ │ │ │ +0006a280: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20  eHomotopies, -- 
│ │ │ │ +0006a290: 7265 7475 726e 7320 6120 7379 7374 656d  returns a system
│ │ │ │ +0006a2a0: 206f 6620 6869 6768 6572 0a20 2020 2068   of higher.    h
│ │ │ │ +0006a2b0: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320  omotopies..Ways 
│ │ │ │ +0006a2c0: 746f 2075 7365 2053 6861 6d61 7368 3a0a  to use Shamash:.
│ │ │ │ +0006a2d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006a2e0: 3d3d 3d3d 0a0a 2020 2a20 2253 6861 6d61  ====..  * "Shama
│ │ │ │ +0006a2f0: 7368 284d 6174 7269 782c 4368 6169 6e43  sh(Matrix,ChainC
│ │ │ │ +0006a300: 6f6d 706c 6578 2c5a 5a29 220a 2020 2a20  omplex,ZZ)".  * 
│ │ │ │ +0006a310: 2253 6861 6d61 7368 2852 696e 672c 4368  "Shamash(Ring,Ch
│ │ │ │ +0006a320: 6169 6e43 6f6d 706c 6578 2c5a 5a29 220a  ainComplex,ZZ)".
│ │ │ │ +0006a330: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0006a340: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0006a350: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +0006a360: 6374 202a 6e6f 7465 2053 6861 6d61 7368  ct *note Shamash
│ │ │ │ +0006a370: 3a20 5368 616d 6173 682c 2069 7320 6120  : Shamash, is a 
│ │ │ │ +0006a380: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +0006a390: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ +0006a3a0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +0006a3b0: 696f 6e2c 2e0a 1f0a 4669 6c65 3a20 436f  ion,....File: Co
│ │ │ │ +0006a3c0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +0006a3d0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +0006a3e0: 666f 2c20 4e6f 6465 3a20 7370 6c69 7474  fo, Node: splitt
│ │ │ │ +0006a3f0: 696e 6773 2c20 4e65 7874 3a20 7374 6162  ings, Next: stab
│ │ │ │ +0006a400: 6c65 486f 6d2c 2050 7265 763a 2053 6861  leHom, Prev: Sha
│ │ │ │ +0006a410: 6d61 7368 2c20 5570 3a20 546f 700a 0a73  mash, Up: Top..s
│ │ │ │ +0006a420: 706c 6974 7469 6e67 7320 2d2d 2063 6f6d  plittings -- com
│ │ │ │ +0006a430: 7075 7465 2074 6865 2073 706c 6974 7469  pute the splitti
│ │ │ │ +0006a440: 6e67 7320 6f66 2061 2073 706c 6974 2072  ngs of a split r
│ │ │ │ +0006a450: 6967 6874 2065 7861 6374 2073 6571 7565  ight exact seque
│ │ │ │ +0006a460: 6e63 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  nce.************
│ │ │ │  0006a470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006a480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006a490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006a4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006a4b0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -0006a4c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573 6167  ======..  * Usag
│ │ │ │ -0006a4d0: 653a 200a 2020 2020 2020 2020 7820 3d20  e: .        x = 
│ │ │ │ -0006a4e0: 7370 6c69 7474 696e 6773 2861 2c62 290a  splittings(a,b).
│ │ │ │ -0006a4f0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -0006a500: 2020 2a20 612c 2061 202a 6e6f 7465 206d    * a, a *note m
│ │ │ │ -0006a510: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ -0006a520: 3244 6f63 294d 6174 7269 782c 2c20 6d61  2Doc)Matrix,, ma
│ │ │ │ -0006a530: 7073 2069 6e74 6f20 7468 6520 6b65 726e  ps into the kern
│ │ │ │ -0006a540: 656c 206f 6620 620a 2020 2020 2020 2a20  el of b.      * 
│ │ │ │ -0006a550: 622c 2061 202a 6e6f 7465 206d 6174 7269  b, a *note matri
│ │ │ │ -0006a560: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -0006a570: 294d 6174 7269 782c 2c20 7265 7072 6573  )Matrix,, repres
│ │ │ │ -0006a580: 656e 7469 6e67 2061 2073 7572 6a65 6374  enting a surject
│ │ │ │ -0006a590: 696f 6e0a 2020 2020 2020 2020 6672 6f6d  ion.        from
│ │ │ │ -0006a5a0: 2074 6172 6765 7420 6120 746f 2061 2066   target a to a f
│ │ │ │ -0006a5b0: 7265 6520 6d6f 6475 6c65 0a20 202a 204f  ree module.  * O
│ │ │ │ -0006a5c0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0006a5d0: 4c2c 2061 202a 6e6f 7465 206c 6973 743a  L, a *note list:
│ │ │ │ -0006a5e0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0006a5f0: 6973 742c 2c20 4c20 3d20 5c7b 7369 676d  ist,, L = \{sigm
│ │ │ │ -0006a600: 612c 7461 755c 7d2c 2073 706c 6974 7469  a,tau\}, splitti
│ │ │ │ -0006a610: 6e67 7320 6f66 0a20 2020 2020 2020 2061  ngs of.        a
│ │ │ │ -0006a620: 2c62 2072 6573 7065 6374 6976 656c 790a  ,b respectively.
│ │ │ │ -0006a630: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ -0006a640: 3d3d 3d3d 3d3d 3d3d 0a0a 4173 7375 6d69  ========..Assumi
│ │ │ │ -0006a650: 6e67 2074 6861 7420 2861 2c62 2920 6172  ng that (a,b) ar
│ │ │ │ -0006a660: 6520 7468 6520 6d61 7073 206f 6620 6120  e the maps of a 
│ │ │ │ -0006a670: 7269 6768 7420 6578 6163 7420 7365 7175  right exact sequ
│ │ │ │ -0006a680: 656e 6365 0a0a 2430 5c74 6f20 415c 746f  ence..$0\to A\to
│ │ │ │ -0006a690: 2042 5c74 6f20 4320 5c74 6f20 3024 0a0a   B\to C \to 0$..
│ │ │ │ -0006a6a0: 7769 7468 2042 2c20 4320 6672 6565 2c20  with B, C free, 
│ │ │ │ -0006a6b0: 7468 6520 7363 7269 7074 2070 726f 6475  the script produ
│ │ │ │ -0006a6c0: 6365 7320 6120 7061 6972 206f 6620 6d61  ces a pair of ma
│ │ │ │ -0006a6d0: 7073 2073 6967 6d61 2c20 7461 7520 7769  ps sigma, tau wi
│ │ │ │ -0006a6e0: 7468 2024 7461 753a 2043 205c 746f 0a42  th $tau: C \to.B
│ │ │ │ -0006a6f0: 2420 6120 7370 6c69 7474 696e 6720 6f66  $ a splitting of
│ │ │ │ -0006a700: 2062 2061 6e64 2024 7369 676d 613a 2042   b and $sigma: B
│ │ │ │ -0006a710: 205c 746f 2041 2420 6120 7370 6c69 7474   \to A$ a splitt
│ │ │ │ -0006a720: 696e 6720 6f66 2061 3b20 7468 6174 2069  ing of a; that i
│ │ │ │ -0006a730: 732c 0a0a 2461 2a73 6967 6d61 2b74 6175  s,..$a*sigma+tau
│ │ │ │ -0006a740: 2a62 203d 2031 5f42 240a 0a24 7369 676d  *b = 1_B$..$sigm
│ │ │ │ -0006a750: 612a 6120 3d20 315f 4124 0a0a 2462 2a74  a*a = 1_A$..$b*t
│ │ │ │ -0006a760: 6175 203d 2031 5f43 240a 0a2b 2d2d 2d2d  au = 1_C$..+----
│ │ │ │ +0006a4a0: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ +0006a4b0: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ +0006a4c0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0006a4d0: 2078 203d 2073 706c 6974 7469 6e67 7328   x = splittings(
│ │ │ │ +0006a4e0: 612c 6229 0a20 202a 2049 6e70 7574 733a  a,b).  * Inputs:
│ │ │ │ +0006a4f0: 0a20 2020 2020 202a 2061 2c20 6120 2a6e  .      * a, a *n
│ │ │ │ +0006a500: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ +0006a510: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ +0006a520: 2c2c 206d 6170 7320 696e 746f 2074 6865  ,, maps into the
│ │ │ │ +0006a530: 206b 6572 6e65 6c20 6f66 2062 0a20 2020   kernel of b.   
│ │ │ │ +0006a540: 2020 202a 2062 2c20 6120 2a6e 6f74 6520     * b, a *note 
│ │ │ │ +0006a550: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ +0006a560: 7932 446f 6329 4d61 7472 6978 2c2c 2072  y2Doc)Matrix,, r
│ │ │ │ +0006a570: 6570 7265 7365 6e74 696e 6720 6120 7375  epresenting a su
│ │ │ │ +0006a580: 726a 6563 7469 6f6e 0a20 2020 2020 2020  rjection.       
│ │ │ │ +0006a590: 2066 726f 6d20 7461 7267 6574 2061 2074   from target a t
│ │ │ │ +0006a5a0: 6f20 6120 6672 6565 206d 6f64 756c 650a  o a free module.
│ │ │ │ +0006a5b0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0006a5c0: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ +0006a5d0: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +0006a5e0: 446f 6329 4c69 7374 2c2c 204c 203d 205c  Doc)List,, L = \
│ │ │ │ +0006a5f0: 7b73 6967 6d61 2c74 6175 5c7d 2c20 7370  {sigma,tau\}, sp
│ │ │ │ +0006a600: 6c69 7474 696e 6773 206f 660a 2020 2020  littings of.    
│ │ │ │ +0006a610: 2020 2020 612c 6220 7265 7370 6563 7469      a,b respecti
│ │ │ │ +0006a620: 7665 6c79 0a0a 4465 7363 7269 7074 696f  vely..Descriptio
│ │ │ │ +0006a630: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41  n.===========..A
│ │ │ │ +0006a640: 7373 756d 696e 6720 7468 6174 2028 612c  ssuming that (a,
│ │ │ │ +0006a650: 6229 2061 7265 2074 6865 206d 6170 7320  b) are the maps 
│ │ │ │ +0006a660: 6f66 2061 2072 6967 6874 2065 7861 6374  of a right exact
│ │ │ │ +0006a670: 2073 6571 7565 6e63 650a 0a24 305c 746f   sequence..$0\to
│ │ │ │ +0006a680: 2041 5c74 6f20 425c 746f 2043 205c 746f   A\to B\to C \to
│ │ │ │ +0006a690: 2030 240a 0a77 6974 6820 422c 2043 2066   0$..with B, C f
│ │ │ │ +0006a6a0: 7265 652c 2074 6865 2073 6372 6970 7420  ree, the script 
│ │ │ │ +0006a6b0: 7072 6f64 7563 6573 2061 2070 6169 7220  produces a pair 
│ │ │ │ +0006a6c0: 6f66 206d 6170 7320 7369 676d 612c 2074  of maps sigma, t
│ │ │ │ +0006a6d0: 6175 2077 6974 6820 2474 6175 3a20 4320  au with $tau: C 
│ │ │ │ +0006a6e0: 5c74 6f0a 4224 2061 2073 706c 6974 7469  \to.B$ a splitti
│ │ │ │ +0006a6f0: 6e67 206f 6620 6220 616e 6420 2473 6967  ng of b and $sig
│ │ │ │ +0006a700: 6d61 3a20 4220 5c74 6f20 4124 2061 2073  ma: B \to A$ a s
│ │ │ │ +0006a710: 706c 6974 7469 6e67 206f 6620 613b 2074  plitting of a; t
│ │ │ │ +0006a720: 6861 7420 6973 2c0a 0a24 612a 7369 676d  hat is,..$a*sigm
│ │ │ │ +0006a730: 612b 7461 752a 6220 3d20 315f 4224 0a0a  a+tau*b = 1_B$..
│ │ │ │ +0006a740: 2473 6967 6d61 2a61 203d 2031 5f41 240a  $sigma*a = 1_A$.
│ │ │ │ +0006a750: 0a24 622a 7461 7520 3d20 315f 4324 0a0a  .$b*tau = 1_C$..
│ │ │ │ +0006a760: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0006a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a7b0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b  ------+.|i1 : kk
│ │ │ │ -0006a7c0: 3d20 5a5a 2f31 3031 2020 2020 2020 2020  = ZZ/101        
│ │ │ │ +0006a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0006a7b0: 203a 206b 6b3d 205a 5a2f 3130 3120 2020   : kk= ZZ/101   
│ │ │ │ +0006a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006a7f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a850: 7c0a 7c6f 3120 3d20 6b6b 2020 2020 2020  |.|o1 = kk      
│ │ │ │ +0006a840: 2020 2020 207c 0a7c 6f31 203d 206b 6b20       |.|o1 = kk 
│ │ │ │ +0006a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006a890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a8e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0006a8f0: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +0006a8d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a8e0: 0a7c 6f31 203a 2051 756f 7469 656e 7452  .|o1 : QuotientR
│ │ │ │ +0006a8f0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  0006a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a930: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006a920: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a980: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20  ----+.|i2 : S = 
│ │ │ │ -0006a990: 6b6b 5b78 2c79 2c7a 5d20 2020 2020 2020  kk[x,y,z]       
│ │ │ │ +0006a970: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +0006a980: 2053 203d 206b 6b5b 782c 792c 7a5d 2020   S = kk[x,y,z]  
│ │ │ │ +0006a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a9d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0006a9c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aa10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006aa20: 7c6f 3220 3d20 5320 2020 2020 2020 2020  |o2 = S         
│ │ │ │ +0006aa10: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ +0006aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aa60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006aa60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0006aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aab0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -0006aac0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0006aaa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006aab0: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52  o2 : PolynomialR
│ │ │ │ +0006aac0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  0006aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ab00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006aaf0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ab50: 2d2d 2b0a 7c69 3320 3a20 7365 7452 616e  --+.|i3 : setRan
│ │ │ │ -0006ab60: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ +0006ab40: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2073  -------+.|i3 : s
│ │ │ │ +0006ab50: 6574 5261 6e64 6f6d 5365 6564 2030 2020  etRandomSeed 0  
│ │ │ │ +0006ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ab90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006aba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006ab90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006abe0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006abf0: 3320 3d20 3020 2020 2020 2020 2020 2020  3 = 0           
│ │ │ │ +0006abe0: 207c 0a7c 6f33 203d 2030 2020 2020 2020   |.|o3 = 0      
│ │ │ │ +0006abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ac30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0006ac20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006ac30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0006ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ac80: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7420  ------+.|i4 : t 
│ │ │ │ -0006ac90: 3d20 7261 6e64 6f6d 2853 5e7b 323a 2d31  = random(S^{2:-1
│ │ │ │ -0006aca0: 2c32 3a2d 327d 2c20 535e 7b33 3a2d 312c  ,2:-2}, S^{3:-1,
│ │ │ │ -0006acb0: 343a 2d32 7d29 2020 2020 2020 2020 2020  4:-2})          
│ │ │ │ -0006acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006acd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +0006ac80: 203a 2074 203d 2072 616e 646f 6d28 535e   : t = random(S^
│ │ │ │ +0006ac90: 7b32 3a2d 312c 323a 2d32 7d2c 2053 5e7b  {2:-1,2:-2}, S^{
│ │ │ │ +0006aca0: 333a 2d31 2c34 3a2d 327d 2920 2020 2020  3:-1,4:-2})     
│ │ │ │ +0006acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006acc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ad20: 7c0a 7c6f 3420 3d20 7b31 7d20 7c20 3234  |.|o4 = {1} | 24
│ │ │ │ -0006ad30: 2020 2d33 3620 2d33 3020 3339 782d 3433    -36 -30 39x-43
│ │ │ │ -0006ad40: 792b 3435 7a20 2032 3178 2d31 3579 2d33  y+45z  21x-15y-3
│ │ │ │ -0006ad50: 347a 2033 3478 2d32 3879 2d34 387a 2020  4z 34x-28y-48z  
│ │ │ │ -0006ad60: 3139 782d 3437 792d 3437 7a20 7c7c 0a7c  19x-47y-47z ||.|
│ │ │ │ -0006ad70: 2020 2020 207b 317d 207c 202d 3239 2031       {1} | -29 1
│ │ │ │ -0006ad80: 3920 2031 3920 202d 3437 782b 3338 792b  9  19  -47x+38y+
│ │ │ │ -0006ad90: 3437 7a20 2d33 3978 2b32 792b 3139 7a20  47z -39x+2y+19z 
│ │ │ │ -0006ada0: 2d31 3878 2b31 3679 2d31 367a 202d 3133  -18x+16y-16z -13
│ │ │ │ -0006adb0: 782b 3232 792b 377a 207c 7c0a 7c20 2020  x+22y+7z ||.|   
│ │ │ │ -0006adc0: 2020 7b32 7d20 7c20 3020 2020 3020 2020    {2} | 0   0   
│ │ │ │ -0006add0: 3020 2020 2d31 3020 2020 2020 2020 2020  0   -10         
│ │ │ │ -0006ade0: 202d 3239 2020 2020 2020 2020 202d 3820   -29         -8 
│ │ │ │ -0006adf0: 2020 2020 2020 2020 2020 2d32 3220 2020            -22   
│ │ │ │ -0006ae00: 2020 2020 2020 7c7c 0a7c 2020 2020 207b        ||.|     {
│ │ │ │ -0006ae10: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0006ae20: 202d 3239 2020 2020 2020 2020 2020 2d32   -29          -2
│ │ │ │ -0006ae30: 3420 2020 2020 2020 2020 2d33 3820 2020  4         -38   
│ │ │ │ -0006ae40: 2020 2020 2020 202d 3136 2020 2020 2020         -16      
│ │ │ │ -0006ae50: 2020 207c 7c0a 7c20 2020 2020 2020 2020     ||.|         
│ │ │ │ +0006ad10: 2020 2020 207c 0a7c 6f34 203d 207b 317d       |.|o4 = {1}
│ │ │ │ +0006ad20: 207c 2032 3420 202d 3336 202d 3330 2033   | 24  -36 -30 3
│ │ │ │ +0006ad30: 3978 2d34 3379 2b34 357a 2020 3231 782d  9x-43y+45z  21x-
│ │ │ │ +0006ad40: 3135 792d 3334 7a20 3334 782d 3238 792d  15y-34z 34x-28y-
│ │ │ │ +0006ad50: 3438 7a20 2031 3978 2d34 3779 2d34 377a  48z  19x-47y-47z
│ │ │ │ +0006ad60: 207c 7c0a 7c20 2020 2020 7b31 7d20 7c20   ||.|     {1} | 
│ │ │ │ +0006ad70: 2d32 3920 3139 2020 3139 2020 2d34 3778  -29 19  19  -47x
│ │ │ │ +0006ad80: 2b33 3879 2b34 377a 202d 3339 782b 3279  +38y+47z -39x+2y
│ │ │ │ +0006ad90: 2b31 397a 202d 3138 782b 3136 792d 3136  +19z -18x+16y-16
│ │ │ │ +0006ada0: 7a20 2d31 3378 2b32 3279 2b37 7a20 7c7c  z -13x+22y+7z ||
│ │ │ │ +0006adb0: 0a7c 2020 2020 207b 327d 207c 2030 2020  .|     {2} | 0  
│ │ │ │ +0006adc0: 2030 2020 2030 2020 202d 3130 2020 2020   0   0   -10    
│ │ │ │ +0006add0: 2020 2020 2020 2d32 3920 2020 2020 2020        -29       
│ │ │ │ +0006ade0: 2020 2d38 2020 2020 2020 2020 2020 202d    -8           -
│ │ │ │ +0006adf0: 3232 2020 2020 2020 2020 207c 7c0a 7c20  22         ||.| 
│ │ │ │ +0006ae00: 2020 2020 7b32 7d20 7c20 3020 2020 3020      {2} | 0   0 
│ │ │ │ +0006ae10: 2020 3020 2020 2d32 3920 2020 2020 2020    0   -29       
│ │ │ │ +0006ae20: 2020 202d 3234 2020 2020 2020 2020 202d     -24         -
│ │ │ │ +0006ae30: 3338 2020 2020 2020 2020 2020 2d31 3620  38          -16 
│ │ │ │ +0006ae40: 2020 2020 2020 2020 7c7c 0a7c 2020 2020          ||.|    
│ │ │ │ +0006ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aea0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0006aeb0: 2034 2020 2020 2020 3720 2020 2020 2020   4      7       
│ │ │ │ +0006ae90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006aea0: 2020 2020 2020 3420 2020 2020 2037 2020        4      7  
│ │ │ │ +0006aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006aef0: 7c6f 3420 3a20 4d61 7472 6978 2053 2020  |o4 : Matrix S  
│ │ │ │ -0006af00: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ +0006aee0: 2020 207c 0a7c 6f34 203a 204d 6174 7269     |.|o4 : Matri
│ │ │ │ +0006aef0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +0006af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006af30: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0006af40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006af80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -0006af90: 7373 203d 2073 706c 6974 7469 6e67 7328  ss = splittings(
│ │ │ │ -0006afa0: 7379 7a20 742c 2074 2920 2020 2020 2020  syz t, t)       
│ │ │ │ +0006af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0006af80: 6935 203a 2073 7320 3d20 7370 6c69 7474  i5 : ss = splitt
│ │ │ │ +0006af90: 696e 6773 2873 797a 2074 2c20 7429 2020  ings(syz t, t)  
│ │ │ │ +0006afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006afc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b020: 2020 7c0a 7c6f 3520 3d20 7b7b 317d 207c    |.|o5 = {{1} |
│ │ │ │ -0006b030: 2030 2030 2031 2030 2030 2030 2020 2030   0 0 1 0 0 0   0
│ │ │ │ -0006b040: 2020 7c2c 207b 317d 207c 202d 3237 2032    |, {1} | -27 2
│ │ │ │ -0006b050: 2020 3133 782d 3130 792b 3433 7a20 3530    13x-10y+43z 50
│ │ │ │ -0006b060: 782d 3334 792d 3530 7a20 7c7d 2020 207c  x-34y-50z |}   |
│ │ │ │ -0006b070: 0a7c 2020 2020 2020 7b32 7d20 7c20 3020  .|      {2} | 0 
│ │ │ │ -0006b080: 3020 3020 3020 3020 2d33 3120 2d36 207c  0 0 0 0 -31 -6 |
│ │ │ │ -0006b090: 2020 7b31 7d20 7c20 2d34 2020 3335 2032    {1} | -4  35 2
│ │ │ │ -0006b0a0: 3278 2b33 3279 2b34 337a 202d 3778 2d38  2x+32y+43z -7x-8
│ │ │ │ -0006b0b0: 792d 3237 7a20 207c 2020 2020 7c0a 7c20  y-27z  |    |.| 
│ │ │ │ -0006b0c0: 2020 2020 207b 327d 207c 2030 2030 2030       {2} | 0 0 0
│ │ │ │ -0006b0d0: 2030 2030 2032 3920 2039 2020 7c20 207b   0 0 29  9  |  {
│ │ │ │ -0006b0e0: 317d 207c 2030 2020 2030 2020 3020 2020  1} | 0   0  0   
│ │ │ │ -0006b0f0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0006b100: 2020 2020 7c20 2020 207c 0a7c 2020 2020      |    |.|    
│ │ │ │ +0006b010: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ +0006b020: 7b31 7d20 7c20 3020 3020 3120 3020 3020  {1} | 0 0 1 0 0 
│ │ │ │ +0006b030: 3020 2020 3020 207c 2c20 7b31 7d20 7c20  0   0  |, {1} | 
│ │ │ │ +0006b040: 2d32 3720 3220 2031 3378 2d31 3079 2b34  -27 2  13x-10y+4
│ │ │ │ +0006b050: 337a 2035 3078 2d33 3479 2d35 307a 207c  3z 50x-34y-50z |
│ │ │ │ +0006b060: 7d20 2020 7c0a 7c20 2020 2020 207b 327d  }   |.|      {2}
│ │ │ │ +0006b070: 207c 2030 2030 2030 2030 2030 202d 3331   | 0 0 0 0 0 -31
│ │ │ │ +0006b080: 202d 3620 7c20 207b 317d 207c 202d 3420   -6 |  {1} | -4 
│ │ │ │ +0006b090: 2033 3520 3232 782b 3332 792b 3433 7a20   35 22x+32y+43z 
│ │ │ │ +0006b0a0: 2d37 782d 3879 2d32 377a 2020 7c20 2020  -7x-8y-27z  |   
│ │ │ │ +0006b0b0: 207c 0a7c 2020 2020 2020 7b32 7d20 7c20   |.|      {2} | 
│ │ │ │ +0006b0c0: 3020 3020 3020 3020 3020 3239 2020 3920  0 0 0 0 0 29  9 
│ │ │ │ +0006b0d0: 207c 2020 7b31 7d20 7c20 3020 2020 3020   |  {1} | 0   0 
│ │ │ │ +0006b0e0: 2030 2020 2020 2020 2020 2020 2030 2020   0           0  
│ │ │ │ +0006b0f0: 2020 2020 2020 2020 207c 2020 2020 7c0a           |    |.
│ │ │ │ +0006b100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0006b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b120: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -0006b130: 7c20 3020 2020 3020 202d 3235 2020 2020  | 0   0  -25    
│ │ │ │ -0006b140: 2020 2020 2032 3620 2020 2020 2020 2020       26         
│ │ │ │ -0006b150: 207c 2020 2020 7c0a 7c20 2020 2020 2020   |    |.|       
│ │ │ │ -0006b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b170: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
│ │ │ │ -0006b180: 2020 2030 2020 3236 2020 2020 2020 2020     0  26        
│ │ │ │ -0006b190: 2020 2d32 2020 2020 2020 2020 2020 7c20    -2          | 
│ │ │ │ -0006b1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0006b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b1c0: 2020 2020 2020 7b32 7d20 7c20 3020 2020        {2} | 0   
│ │ │ │ -0006b1d0: 3020 2030 2020 2020 2020 2020 2020 2030  0  0           0
│ │ │ │ -0006b1e0: 2020 2020 2020 2020 2020 207c 2020 2020             |    
│ │ │ │ -0006b1f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0006b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b210: 2020 207b 327d 207c 2030 2020 2030 2020     {2} | 0   0  
│ │ │ │ -0006b220: 3020 2020 2020 2020 2020 2020 3020 2020  0           0   
│ │ │ │ -0006b230: 2020 2020 2020 2020 7c20 2020 207c 0a7c          |    |.|
│ │ │ │ +0006b120: 207b 327d 207c 2030 2020 2030 2020 2d32   {2} | 0   0  -2
│ │ │ │ +0006b130: 3520 2020 2020 2020 2020 3236 2020 2020  5         26    
│ │ │ │ +0006b140: 2020 2020 2020 7c20 2020 207c 0a7c 2020        |    |.|  
│ │ │ │ +0006b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b160: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +0006b170: 7d20 7c20 3020 2020 3020 2032 3620 2020  } | 0   0  26   
│ │ │ │ +0006b180: 2020 2020 2020 202d 3220 2020 2020 2020         -2       
│ │ │ │ +0006b190: 2020 207c 2020 2020 7c0a 7c20 2020 2020     |    |.|     
│ │ │ │ +0006b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b1b0: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +0006b1c0: 2030 2020 2030 2020 3020 2020 2020 2020   0   0  0       
│ │ │ │ +0006b1d0: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ +0006b1e0: 7c20 2020 207c 0a7c 2020 2020 2020 2020  |    |.|        
│ │ │ │ +0006b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b200: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ +0006b210: 2020 3020 2030 2020 2020 2020 2020 2020    0  0          
│ │ │ │ +0006b220: 2030 2020 2020 2020 2020 2020 207c 2020   0           |  
│ │ │ │ +0006b230: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b280: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -0006b290: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +0006b270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b280: 0a7c 6f35 203a 204c 6973 7420 2020 2020  .|o5 : List     
│ │ │ │ +0006b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b2d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006b2c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b320: 2d2d 2d2d 2b0a 7c69 3620 3a20 7373 2f62  ----+.|i6 : ss/b
│ │ │ │ -0006b330: 6574 7469 2020 2020 2020 2020 2020 2020  etti            
│ │ │ │ +0006b310: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0006b320: 2073 732f 6265 7474 6920 2020 2020 2020   ss/betti       
│ │ │ │ +0006b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b370: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0006b360: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b3b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006b3c0: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ -0006b3d0: 3120 2020 2020 2020 2020 3020 3120 2020  1         0 1   
│ │ │ │ +0006b3b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b3c0: 2020 2030 2031 2020 2020 2020 2020 2030     0 1         0
│ │ │ │ +0006b3d0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  0006b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b400: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -0006b410: 203d 207b 746f 7461 6c3a 2033 2037 2c20   = {total: 3 7, 
│ │ │ │ -0006b420: 746f 7461 6c3a 2037 2034 7d20 2020 2020  total: 7 4}     
│ │ │ │ +0006b400: 7c0a 7c6f 3620 3d20 7b74 6f74 616c 3a20  |.|o6 = {total: 
│ │ │ │ +0006b410: 3320 372c 2074 6f74 616c 3a20 3720 347d  3 7, total: 7 4}
│ │ │ │ +0006b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b450: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006b460: 2020 2020 2030 3a20 2e20 3320 2020 2020       0: . 3     
│ │ │ │ -0006b470: 2030 3a20 2e20 3220 2020 2020 2020 2020   0: . 2         
│ │ │ │ +0006b440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b450: 2020 2020 2020 2020 2020 303a 202e 2033            0: . 3
│ │ │ │ +0006b460: 2020 2020 2020 303a 202e 2032 2020 2020        0: . 2    
│ │ │ │ +0006b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006b4b0: 2020 313a 2031 2034 2020 2020 2020 313a    1: 1 4      1:
│ │ │ │ -0006b4c0: 2033 2032 2020 2020 2020 2020 2020 2020   3 2            
│ │ │ │ +0006b490: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006b4a0: 2020 2020 2020 2031 3a20 3120 3420 2020         1: 1 4   
│ │ │ │ +0006b4b0: 2020 2031 3a20 3320 3220 2020 2020 2020     1: 3 2       
│ │ │ │ +0006b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4f0: 2020 7c0a 7c20 2020 2020 2020 2020 2032    |.|          2
│ │ │ │ -0006b500: 3a20 3220 2e20 2020 2020 2032 3a20 3420  : 2 .      2: 4 
│ │ │ │ -0006b510: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0006b4e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006b4f0: 2020 2020 323a 2032 202e 2020 2020 2020      2: 2 .      
│ │ │ │ +0006b500: 323a 2034 202e 2020 2020 2020 2020 2020  2: 4 .          
│ │ │ │ +0006b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006b530: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006b590: 3620 3a20 4c69 7374 2020 2020 2020 2020  6 : List        
│ │ │ │ +0006b580: 207c 0a7c 6f36 203a 204c 6973 7420 2020   |.|o6 : List   
│ │ │ │ +0006b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0006b5c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006b5d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0006b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b620: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f  ------+..Ways to
│ │ │ │ -0006b630: 2075 7365 2073 706c 6974 7469 6e67 733a   use splittings:
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│ │ │ │ -0006b650: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2273  ========..  * "s
│ │ │ │ -0006b660: 706c 6974 7469 6e67 7328 4d61 7472 6978  plittings(Matrix
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│ │ │ │ -0006b680: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -0006b690: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006b6a0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -0006b6b0: 7465 2073 706c 6974 7469 6e67 733a 2073  te splittings: s
│ │ │ │ -0006b6c0: 706c 6974 7469 6e67 732c 2069 7320 6120  plittings, is a 
│ │ │ │ -0006b6d0: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ -0006b6e0: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ -0006b6f0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -0006b700: 696f 6e2c 2e0a 1f0a 4669 6c65 3a20 436f  ion,....File: Co
│ │ │ │ -0006b710: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -0006b720: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ -0006b730: 666f 2c20 4e6f 6465 3a20 7374 6162 6c65  fo, Node: stable
│ │ │ │ -0006b740: 486f 6d2c 204e 6578 743a 2073 756d 5477  Hom, Next: sumTw
│ │ │ │ -0006b750: 6f4d 6f6e 6f6d 6961 6c73 2c20 5072 6576  oMonomials, Prev
│ │ │ │ -0006b760: 3a20 7370 6c69 7474 696e 6773 2c20 5570  : splittings, Up
│ │ │ │ -0006b770: 3a20 546f 700a 0a73 7461 626c 6548 6f6d  : Top..stableHom
│ │ │ │ -0006b780: 202d 2d20 6d61 7020 6672 6f6d 2048 6f6d   -- map from Hom
│ │ │ │ -0006b790: 284d 2c4e 2920 746f 2074 6865 2073 7461  (M,N) to the sta
│ │ │ │ -0006b7a0: 626c 6520 486f 6d20 6d6f 6475 6c65 0a2a  ble Hom module.*
│ │ │ │ +0006b610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761  -----------+..Wa
│ │ │ │ +0006b620: 7973 2074 6f20 7573 6520 7370 6c69 7474  ys to use splitt
│ │ │ │ +0006b630: 696e 6773 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  ings:.==========
│ │ │ │ +0006b640: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +0006b650: 202a 2022 7370 6c69 7474 696e 6773 284d   * "splittings(M
│ │ │ │ +0006b660: 6174 7269 782c 4d61 7472 6978 2922 0a0a  atrix,Matrix)"..
│ │ │ │ +0006b670: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +0006b680: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +0006b690: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +0006b6a0: 7420 2a6e 6f74 6520 7370 6c69 7474 696e  t *note splittin
│ │ │ │ +0006b6b0: 6773 3a20 7370 6c69 7474 696e 6773 2c20  gs: splittings, 
│ │ │ │ +0006b6c0: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +0006b6d0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ +0006b6e0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +0006b6f0: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ +0006b700: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ +0006b710: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +0006b720: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2073  ns.info, Node: s
│ │ │ │ +0006b730: 7461 626c 6548 6f6d 2c20 4e65 7874 3a20  tableHom, Next: 
│ │ │ │ +0006b740: 7375 6d54 776f 4d6f 6e6f 6d69 616c 732c  sumTwoMonomials,
│ │ │ │ +0006b750: 2050 7265 763a 2073 706c 6974 7469 6e67   Prev: splitting
│ │ │ │ +0006b760: 732c 2055 703a 2054 6f70 0a0a 7374 6162  s, Up: Top..stab
│ │ │ │ +0006b770: 6c65 486f 6d20 2d2d 206d 6170 2066 726f  leHom -- map fro
│ │ │ │ +0006b780: 6d20 486f 6d28 4d2c 4e29 2074 6f20 7468  m Hom(M,N) to th
│ │ │ │ +0006b790: 6520 7374 6162 6c65 2048 6f6d 206d 6f64  e stable Hom mod
│ │ │ │ +0006b7a0: 756c 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ule.************
│ │ │ │  0006b7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006b7c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006b7d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006b7e0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -0006b7f0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ -0006b800: 7361 6765 3a20 0a20 2020 2020 2020 2070  sage: .        p
│ │ │ │ -0006b810: 203d 2073 7461 626c 6548 6f6d 284d 2c4e   = stableHom(M,N
│ │ │ │ -0006b820: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -0006b830: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ -0006b840: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ -0006b850: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ -0006b860: 0a20 2020 2020 202a 204e 2c20 6120 2a6e  .      * N, a *n
│ │ │ │ -0006b870: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ -0006b880: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ -0006b890: 2c2c 200a 2020 2a20 4f75 7470 7574 733a  ,, .  * Outputs:
│ │ │ │ -0006b8a0: 0a20 2020 2020 202a 2070 2c20 6120 2a6e  .      * p, a *n
│ │ │ │ -0006b8b0: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -0006b8c0: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -0006b8d0: 2c2c 2070 726f 6a65 6374 696f 6e20 6672  ,, projection fr
│ │ │ │ -0006b8e0: 6f6d 2048 6f6d 284d 2c4e 2920 746f 0a20  om Hom(M,N) to. 
│ │ │ │ -0006b8f0: 2020 2020 2020 2074 6865 2073 7461 626c         the stabl
│ │ │ │ -0006b900: 6520 486f 6d0a 0a44 6573 6372 6970 7469  e Hom..Descripti
│ │ │ │ -0006b910: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -0006b920: 5468 6520 7374 6162 6c65 2048 6f6d 2069  The stable Hom i
│ │ │ │ -0006b930: 7320 486f 6d28 4d2c 4e29 2f54 2077 6865  s Hom(M,N)/T whe
│ │ │ │ -0006b940: 7265 2054 2069 7320 7468 6520 7375 626d  re T is the subm
│ │ │ │ -0006b950: 6f64 756c 6520 6f66 2068 6f6d 6f6d 6f72  odule of homomor
│ │ │ │ -0006b960: 7068 6973 6d73 2074 6861 740a 6661 6374  phisms that.fact
│ │ │ │ -0006b970: 6f72 2074 6872 6f75 6768 2061 2066 7265  or through a fre
│ │ │ │ -0006b980: 6520 636f 7665 7220 6f66 204e 2028 6f72  e cover of N (or
│ │ │ │ -0006b990: 2c20 6571 7569 7661 6c65 6e74 6c79 2c20  , equivalently, 
│ │ │ │ -0006b9a0: 7468 726f 7567 6820 616e 7920 7072 6f6a  through any proj
│ │ │ │ -0006b9b0: 6563 7469 7665 290a 0a53 6565 2061 6c73  ective)..See als
│ │ │ │ -0006b9c0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -0006b9d0: 2a6e 6f74 6520 6973 5374 6162 6c79 5472  *note isStablyTr
│ │ │ │ -0006b9e0: 6976 6961 6c3a 2069 7353 7461 626c 7954  ivial: isStablyT
│ │ │ │ -0006b9f0: 7269 7669 616c 2c20 2d2d 2072 6574 7572  rivial, -- retur
│ │ │ │ -0006ba00: 6e73 2074 7275 6520 6966 2074 6865 206d  ns true if the m
│ │ │ │ -0006ba10: 6170 2067 6f65 7320 746f 0a20 2020 2030  ap goes to.    0
│ │ │ │ -0006ba20: 2075 6e64 6572 2073 7461 626c 6548 6f6d   under stableHom
│ │ │ │ -0006ba30: 0a0a 5761 7973 2074 6f20 7573 6520 7374  ..Ways to use st
│ │ │ │ -0006ba40: 6162 6c65 486f 6d3a 0a3d 3d3d 3d3d 3d3d  ableHom:.=======
│ │ │ │ -0006ba50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0006ba60: 0a20 202a 2022 7374 6162 6c65 486f 6d28  .  * "stableHom(
│ │ │ │ -0006ba70: 4d6f 6475 6c65 2c4d 6f64 756c 6529 220a  Module,Module)".
│ │ │ │ -0006ba80: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -0006ba90: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -0006baa0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -0006bab0: 6374 202a 6e6f 7465 2073 7461 626c 6548  ct *note stableH
│ │ │ │ -0006bac0: 6f6d 3a20 7374 6162 6c65 486f 6d2c 2069  om: stableHom, i
│ │ │ │ -0006bad0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ -0006bae0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ -0006baf0: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -0006bb00: 756e 6374 696f 6e2c 2e0a 1f0a 4669 6c65  unction,....File
│ │ │ │ -0006bb10: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -0006bb20: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0006bb30: 732e 696e 666f 2c20 4e6f 6465 3a20 7375  s.info, Node: su
│ │ │ │ -0006bb40: 6d54 776f 4d6f 6e6f 6d69 616c 732c 204e  mTwoMonomials, N
│ │ │ │ -0006bb50: 6578 743a 2054 6174 6552 6573 6f6c 7574  ext: TateResolut
│ │ │ │ -0006bb60: 696f 6e2c 2050 7265 763a 2073 7461 626c  ion, Prev: stabl
│ │ │ │ -0006bb70: 6548 6f6d 2c20 5570 3a20 546f 700a 0a73  eHom, Up: Top..s
│ │ │ │ -0006bb80: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 202d  umTwoMonomials -
│ │ │ │ -0006bb90: 2d20 7461 6c6c 7920 7468 6520 7365 7175  - tally the sequ
│ │ │ │ -0006bba0: 656e 6365 7320 6f66 2042 5261 6e6b 7320  ences of BRanks 
│ │ │ │ -0006bbb0: 666f 7220 6365 7274 6169 6e20 6578 616d  for certain exam
│ │ │ │ -0006bbc0: 706c 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ples.***********
│ │ │ │ +0006b7d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ +0006b7e0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ +0006b7f0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +0006b800: 2020 2020 7020 3d20 7374 6162 6c65 486f      p = stableHo
│ │ │ │ +0006b810: 6d28 4d2c 4e29 0a20 202a 2049 6e70 7574  m(M,N).  * Input
│ │ │ │ +0006b820: 733a 0a20 2020 2020 202a 204d 2c20 6120  s:.      * M, a 
│ │ │ │ +0006b830: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +0006b840: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +0006b850: 6c65 2c2c 200a 2020 2020 2020 2a20 4e2c  le,, .      * N,
│ │ │ │ +0006b860: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ +0006b870: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0006b880: 6f64 756c 652c 2c20 0a20 202a 204f 7574  odule,, .  * Out
│ │ │ │ +0006b890: 7075 7473 3a0a 2020 2020 2020 2a20 702c  puts:.      * p,
│ │ │ │ +0006b8a0: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +0006b8b0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0006b8c0: 6174 7269 782c 2c20 7072 6f6a 6563 7469  atrix,, projecti
│ │ │ │ +0006b8d0: 6f6e 2066 726f 6d20 486f 6d28 4d2c 4e29  on from Hom(M,N)
│ │ │ │ +0006b8e0: 2074 6f0a 2020 2020 2020 2020 7468 6520   to.        the 
│ │ │ │ +0006b8f0: 7374 6162 6c65 2048 6f6d 0a0a 4465 7363  stable Hom..Desc
│ │ │ │ +0006b900: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0006b910: 3d3d 3d0a 0a54 6865 2073 7461 626c 6520  ===..The stable 
│ │ │ │ +0006b920: 486f 6d20 6973 2048 6f6d 284d 2c4e 292f  Hom is Hom(M,N)/
│ │ │ │ +0006b930: 5420 7768 6572 6520 5420 6973 2074 6865  T where T is the
│ │ │ │ +0006b940: 2073 7562 6d6f 6475 6c65 206f 6620 686f   submodule of ho
│ │ │ │ +0006b950: 6d6f 6d6f 7270 6869 736d 7320 7468 6174  momorphisms that
│ │ │ │ +0006b960: 0a66 6163 746f 7220 7468 726f 7567 6820  .factor through 
│ │ │ │ +0006b970: 6120 6672 6565 2063 6f76 6572 206f 6620  a free cover of 
│ │ │ │ +0006b980: 4e20 286f 722c 2065 7175 6976 616c 656e  N (or, equivalen
│ │ │ │ +0006b990: 746c 792c 2074 6872 6f75 6768 2061 6e79  tly, through any
│ │ │ │ +0006b9a0: 2070 726f 6a65 6374 6976 6529 0a0a 5365   projective)..Se
│ │ │ │ +0006b9b0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +0006b9c0: 0a20 202a 202a 6e6f 7465 2069 7353 7461  .  * *note isSta
│ │ │ │ +0006b9d0: 626c 7954 7269 7669 616c 3a20 6973 5374  blyTrivial: isSt
│ │ │ │ +0006b9e0: 6162 6c79 5472 6976 6961 6c2c 202d 2d20  ablyTrivial, -- 
│ │ │ │ +0006b9f0: 7265 7475 726e 7320 7472 7565 2069 6620  returns true if 
│ │ │ │ +0006ba00: 7468 6520 6d61 7020 676f 6573 2074 6f0a  the map goes to.
│ │ │ │ +0006ba10: 2020 2020 3020 756e 6465 7220 7374 6162      0 under stab
│ │ │ │ +0006ba20: 6c65 486f 6d0a 0a57 6179 7320 746f 2075  leHom..Ways to u
│ │ │ │ +0006ba30: 7365 2073 7461 626c 6548 6f6d 3a0a 3d3d  se stableHom:.==
│ │ │ │ +0006ba40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006ba50: 3d3d 3d3d 0a0a 2020 2a20 2273 7461 626c  ====..  * "stabl
│ │ │ │ +0006ba60: 6548 6f6d 284d 6f64 756c 652c 4d6f 6475  eHom(Module,Modu
│ │ │ │ +0006ba70: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ +0006ba80: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0006ba90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0006baa0: 206f 626a 6563 7420 2a6e 6f74 6520 7374   object *note st
│ │ │ │ +0006bab0: 6162 6c65 486f 6d3a 2073 7461 626c 6548  ableHom: stableH
│ │ │ │ +0006bac0: 6f6d 2c20 6973 2061 202a 6e6f 7465 206d  om, is a *note m
│ │ │ │ +0006bad0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ +0006bae0: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ +0006baf0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a1f  thodFunction,...
│ │ │ │ +0006bb00: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ +0006bb10: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0006bb20: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ +0006bb30: 653a 2073 756d 5477 6f4d 6f6e 6f6d 6961  e: sumTwoMonomia
│ │ │ │ +0006bb40: 6c73 2c20 4e65 7874 3a20 5461 7465 5265  ls, Next: TateRe
│ │ │ │ +0006bb50: 736f 6c75 7469 6f6e 2c20 5072 6576 3a20  solution, Prev: 
│ │ │ │ +0006bb60: 7374 6162 6c65 486f 6d2c 2055 703a 2054  stableHom, Up: T
│ │ │ │ +0006bb70: 6f70 0a0a 7375 6d54 776f 4d6f 6e6f 6d69  op..sumTwoMonomi
│ │ │ │ +0006bb80: 616c 7320 2d2d 2074 616c 6c79 2074 6865  als -- tally the
│ │ │ │ +0006bb90: 2073 6571 7565 6e63 6573 206f 6620 4252   sequences of BR
│ │ │ │ +0006bba0: 616e 6b73 2066 6f72 2063 6572 7461 696e  anks for certain
│ │ │ │ +0006bbb0: 2065 7861 6d70 6c65 730a 2a2a 2a2a 2a2a   examples.******
│ │ │ │ +0006bbc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006bbd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006bbe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006bbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006bc00: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -0006bc10: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -0006bc20: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -0006bc30: 2020 2073 756d 5477 6f4d 6f6e 6f6d 6961     sumTwoMonomia
│ │ │ │ -0006bc40: 6c73 2863 2c64 290a 2020 2a20 496e 7075  ls(c,d).  * Inpu
│ │ │ │ -0006bc50: 7473 3a0a 2020 2020 2020 2a20 632c 2061  ts:.      * c, a
│ │ │ │ -0006bc60: 6e20 2a6e 6f74 6520 696e 7465 6765 723a  n *note integer:
│ │ │ │ -0006bc70: 2028 4d61 6361 756c 6179 3244 6f63 295a   (Macaulay2Doc)Z
│ │ │ │ -0006bc80: 5a2c 2c20 636f 6469 6d65 6e73 696f 6e20  Z,, codimension 
│ │ │ │ -0006bc90: 696e 2077 6869 6368 2074 6f20 776f 726b  in which to work
│ │ │ │ -0006bca0: 0a20 2020 2020 202a 2064 2c20 616e 202a  .      * d, an *
│ │ │ │ -0006bcb0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ -0006bcc0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ -0006bcd0: 2064 6567 7265 6520 6f66 2074 6865 206d   degree of the m
│ │ │ │ -0006bce0: 6f6e 6f6d 6961 6c73 2074 6f20 7461 6b65  onomials to take
│ │ │ │ -0006bcf0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -0006bd00: 2020 2020 2a20 542c 2061 202a 6e6f 7465      * T, a *note
│ │ │ │ -0006bd10: 2074 616c 6c79 3a20 284d 6163 6175 6c61   tally: (Macaula
│ │ │ │ -0006bd20: 7932 446f 6329 5461 6c6c 792c 2c20 0a0a  y2Doc)Tally,, ..
│ │ │ │ -0006bd30: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -0006bd40: 3d3d 3d3d 3d3d 3d0a 0a74 616c 6c69 6573  =======..tallies
│ │ │ │ -0006bd50: 2074 6865 2073 6571 7565 6e63 6573 206f   the sequences o
│ │ │ │ -0006bd60: 6620 422d 7261 6e6b 7320 7468 6174 206f  f B-ranks that o
│ │ │ │ -0006bd70: 6363 7572 2066 6f72 2073 756d 7320 6f66  ccur for sums of
│ │ │ │ -0006bd80: 2070 6169 7273 206f 6620 6d6f 6e6f 6d69   pairs of monomi
│ │ │ │ -0006bd90: 616c 7320 696e 2052 0a3d 2053 2f28 642d  als in R.= S/(d-
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│ │ │ │ -0006bdd0: 7920 283d 6329 3b20 7468 6174 2069 732c  y (=c); that is,
│ │ │ │ -0006bde0: 2066 6f72 2061 6e0a 6170 7072 6f70 7269   for an.appropri
│ │ │ │ -0006bdf0: 6174 6520 7379 7a79 6779 204d 206f 6620  ate syzygy M of 
│ │ │ │ -0006be00: 4d30 203d 2052 2f28 6d31 2b6d 3229 2077  M0 = R/(m1+m2) w
│ │ │ │ -0006be10: 6865 7265 206d 3120 616e 6420 6d32 2061  here m1 and m2 a
│ │ │ │ -0006be20: 7265 206d 6f6e 6f6d 6961 6c73 206f 6620  re monomials of 
│ │ │ │ -0006be30: 7468 650a 7361 6d65 2064 6567 7265 652e  the.same degree.
│ │ │ │ -0006be40: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0006bbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +0006bc00: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +0006bc10: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +0006bc20: 2020 2020 2020 2020 7375 6d54 776f 4d6f          sumTwoMo
│ │ │ │ +0006bc30: 6e6f 6d69 616c 7328 632c 6429 0a20 202a  nomials(c,d).  *
│ │ │ │ +0006bc40: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +0006bc50: 2063 2c20 616e 202a 6e6f 7465 2069 6e74   c, an *note int
│ │ │ │ +0006bc60: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ +0006bc70: 446f 6329 5a5a 2c2c 2063 6f64 696d 656e  Doc)ZZ,, codimen
│ │ │ │ +0006bc80: 7369 6f6e 2069 6e20 7768 6963 6820 746f  sion in which to
│ │ │ │ +0006bc90: 2077 6f72 6b0a 2020 2020 2020 2a20 642c   work.      * d,
│ │ │ │ +0006bca0: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ +0006bcb0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +0006bcc0: 295a 5a2c 2c20 6465 6772 6565 206f 6620  )ZZ,, degree of 
│ │ │ │ +0006bcd0: 7468 6520 6d6f 6e6f 6d69 616c 7320 746f  the monomials to
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│ │ │ │ +0006bd00: 2a6e 6f74 6520 7461 6c6c 793a 2028 4d61  *note tally: (Ma
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│ │ │ │ +0006bd20: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e  ,, ..Description
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│ │ │ │ +0006bd70: 6d73 206f 6620 7061 6972 7320 6f66 206d  ms of pairs of m
│ │ │ │ +0006bd80: 6f6e 6f6d 6961 6c73 2069 6e20 520a 3d20  onomials in R.= 
│ │ │ │ +0006bd90: 532f 2864 2d74 6820 706f 7765 7273 206f  S/(d-th powers o
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│ │ │ │ +0006bdb0: 2c20 7769 7468 2066 756c 6c20 636f 6d70  , with full comp
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│ │ │ │ +0006bdd0: 7420 6973 2c20 666f 7220 616e 0a61 7070  t is, for an.app
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│ │ │ │ +0006bdf0: 4d20 6f66 204d 3020 3d20 522f 286d 312b  M of M0 = R/(m1+
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│ │ │ │ +0006be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
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│ │ │ │ -0006be90: 6420 3020 2020 2020 2020 2020 2020 2020  d 0             
│ │ │ │ -0006bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006beb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006be70: 2d2b 0a7c 6931 203a 2073 6574 5261 6e64  -+.|i1 : setRand
│ │ │ │ +0006be80: 6f6d 5365 6564 2030 2020 2020 2020 2020  omSeed 0        
│ │ │ │ +0006be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bef0: 2020 2020 7c0a 7c6f 3120 3d20 3020 2020      |.|o1 = 0   
│ │ │ │ +0006bee0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ +0006bef0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  0006bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf30: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0006bf20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006bf30: 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  ----------------
│ │ │ │ -0006bf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0006bf70: 3220 3a20 7375 6d54 776f 4d6f 6e6f 6d69  2 : sumTwoMonomi
│ │ │ │ -0006bf80: 616c 7328 322c 3329 2020 2020 2020 2020  als(2,3)        
│ │ │ │ -0006bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bfa0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -0006bfb0: 7365 6420 302e 3737 3332 3538 7320 2863  sed 0.773258s (c
│ │ │ │ -0006bfc0: 7075 293b 2030 2e34 3439 3738 3773 2028  pu); 0.449787s (
│ │ │ │ -0006bfd0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -0006bfe0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -0006bff0: 302e 3333 3435 3873 2028 6370 7529 3b20  0.33458s (cpu); 
│ │ │ │ -0006c000: 302e 3230 3233 3436 7320 2874 6872 6561  0.202346s (threa
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│ │ │ │ -0006c020: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030  |.| -- used 0.00
│ │ │ │ -0006c030: 3031 3430 3235 3373 2028 6370 7529 3b20  0140253s (cpu); 
│ │ │ │ -0006c040: 322e 3830 3665 2d30 3673 2028 7468 7265  2.806e-06s (thre
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│ │ │ │ +0006bf60: 2d2b 0a7c 6932 203a 2073 756d 5477 6f4d  -+.|i2 : sumTwoM
│ │ │ │ +0006bf70: 6f6e 6f6d 6961 6c73 2832 2c33 2920 2020  onomials(2,3)   
│ │ │ │ +0006bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bf90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006bfa0: 202d 2d20 7573 6564 2031 2e34 3638 3536   -- used 1.46856
│ │ │ │ +0006bfb0: 7320 2863 7075 293b 2030 2e35 3436 3630  s (cpu); 0.54660
│ │ │ │ +0006bfc0: 3773 2028 7468 7265 6164 293b 2030 7320  7s (thread); 0s 
│ │ │ │ +0006bfd0: 2867 6329 2020 2020 207c 0a7c 202d 2d20  (gc)     |.| -- 
│ │ │ │ +0006bfe0: 7573 6564 2030 2e35 3331 3832 3973 2028  used 0.531829s (
│ │ │ │ +0006bff0: 6370 7529 3b20 302e 3234 3138 3532 7320  cpu); 0.241852s 
│ │ │ │ +0006c000: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0006c010: 2920 2020 207c 0a7c 202d 2d20 7573 6564  )    |.| -- used
│ │ │ │ +0006c020: 2030 2e30 3030 3131 3838 3333 7320 2863   0.000118833s (c
│ │ │ │ +0006c030: 7075 293b 2031 2e37 3633 652d 3036 7320  pu); 1.763e-06s 
│ │ │ │ +0006c040: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0006c050: 297c 0a7c 3220 2020 2020 2020 2020 2020  )|.|2           
│ │ │ │  0006c060: 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                  
│ │ │ │ -0006c090: 2020 2020 2020 2020 7c0a 7c54 616c 6c79          |.|Tally
│ │ │ │ -0006c0a0: 7b7b 7b32 2c20 327d 2c20 7b31 2c20 327d  {{{2, 2}, {1, 2}
│ │ │ │ -0006c0b0: 7d20 3d3e 2033 7d20 2020 2020 2020 2020  } => 3}         
│ │ │ │ -0006c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c0d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006c080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006c090: 5461 6c6c 797b 7b7b 322c 2032 7d2c 207b  Tally{{{2, 2}, {
│ │ │ │ +0006c0a0: 312c 2032 7d7d 203d 3e20 337d 2020 2020  1, 2}} => 3}    
│ │ │ │ +0006c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c0c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c110: 7c0a 7c33 2020 2020 2020 2020 2020 2020  |.|3            
│ │ │ │ +0006c100: 2020 2020 207c 0a7c 3320 2020 2020 2020       |.|3       
│ │ │ │ +0006c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c140: 2020 2020 2020 2020 2020 2020 7c0a 7c54              |.|T
│ │ │ │ -0006c150: 616c 6c79 7b7b 7b32 2c20 327d 2c20 7b31  ally{{{2, 2}, {1
│ │ │ │ -0006c160: 2c20 327d 7d20 3d3e 2031 7d20 2020 2020  , 2}} => 1}     
│ │ │ │ -0006c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c180: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006c140: 207c 0a7c 5461 6c6c 797b 7b7b 322c 2032   |.|Tally{{{2, 2
│ │ │ │ +0006c150: 7d2c 207b 312c 2032 7d7d 203d 3e20 317d  }, {1, 2}} => 1}
│ │ │ │ +0006c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1c0: 2020 2020 7c0a 7c34 2020 2020 2020 2020      |.|4        
│ │ │ │ +0006c1b0: 2020 2020 2020 2020 207c 0a7c 3420 2020           |.|4   
│ │ │ │ +0006c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c200: 7c0a 7c54 616c 6c79 7b7d 2020 2020 2020  |.|Tally{}      
│ │ │ │ +0006c1f0: 2020 2020 207c 0a7c 5461 6c6c 797b 7d20       |.|Tally{} 
│ │ │ │ +0006c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c230: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006c230: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0006c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c270: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -0006c280: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0006c290: 2a20 2a6e 6f74 6520 7477 6f4d 6f6e 6f6d  * *note twoMonom
│ │ │ │ -0006c2a0: 6961 6c73 3a20 7477 6f4d 6f6e 6f6d 6961  ials: twoMonomia
│ │ │ │ -0006c2b0: 6c73 2c20 2d2d 2074 616c 6c79 2074 6865  ls, -- tally the
│ │ │ │ -0006c2c0: 2073 6571 7565 6e63 6573 206f 6620 4252   sequences of BR
│ │ │ │ -0006c2d0: 616e 6b73 2066 6f72 0a20 2020 2063 6572  anks for.    cer
│ │ │ │ -0006c2e0: 7461 696e 2065 7861 6d70 6c65 730a 0a57  tain examples..W
│ │ │ │ -0006c2f0: 6179 7320 746f 2075 7365 2073 756d 5477  ays to use sumTw
│ │ │ │ -0006c300: 6f4d 6f6e 6f6d 6961 6c73 3a0a 3d3d 3d3d  oMonomials:.====
│ │ │ │ -0006c310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006c320: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2273  ========..  * "s
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│ │ │ │ -0006c350: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -0006c360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
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│ │ │ │ -0006c380: 7375 6d54 776f 4d6f 6e6f 6d69 616c 733a  sumTwoMonomials:
│ │ │ │ -0006c390: 2073 756d 5477 6f4d 6f6e 6f6d 6961 6c73   sumTwoMonomials
│ │ │ │ -0006c3a0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0006c3b0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ -0006c3c0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0006c3d0: 6f64 4675 6e63 7469 6f6e 2c2e 0a1f 0a46  odFunction,....F
│ │ │ │ -0006c3e0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0006c3f0: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0006c400: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0006c410: 2054 6174 6552 6573 6f6c 7574 696f 6e2c   TateResolution,
│ │ │ │ -0006c420: 204e 6578 743a 2074 656e 736f 7257 6974   Next: tensorWit
│ │ │ │ -0006c430: 6843 6f6d 706f 6e65 6e74 732c 2050 7265  hComponents, Pre
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│ │ │ │ -0006c450: 6c73 2c20 5570 3a20 546f 700a 0a54 6174  ls, Up: Top..Tat
│ │ │ │ -0006c460: 6552 6573 6f6c 7574 696f 6e20 2d2d 2054  eResolution -- T
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│ │ │ │ +0006c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0006c270: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +0006c280: 3d0a 0a20 202a 202a 6e6f 7465 2074 776f  =..  * *note two
│ │ │ │ +0006c290: 4d6f 6e6f 6d69 616c 733a 2074 776f 4d6f  Monomials: twoMo
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│ │ │ │ +0006c2b0: 7920 7468 6520 7365 7175 656e 6365 7320  y the sequences 
│ │ │ │ +0006c2c0: 6f66 2042 5261 6e6b 7320 666f 720a 2020  of BRanks for.  
│ │ │ │ +0006c2d0: 2020 6365 7274 6169 6e20 6578 616d 706c    certain exampl
│ │ │ │ +0006c2e0: 6573 0a0a 5761 7973 2074 6f20 7573 6520  es..Ways to use 
│ │ │ │ +0006c2f0: 7375 6d54 776f 4d6f 6e6f 6d69 616c 733a  sumTwoMonomials:
│ │ │ │ +0006c300: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0006c310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +0006c320: 202a 2022 7375 6d54 776f 4d6f 6e6f 6d69   * "sumTwoMonomi
│ │ │ │ +0006c330: 616c 7328 5a5a 2c5a 5a29 220a 0a46 6f72  als(ZZ,ZZ)"..For
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│ │ │ │ +0006c350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006c360: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
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│ │ │ │ +0006c4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006c4b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006c4c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c4d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c4e0: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
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│ │ │ │ -0006c740: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0006c4d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ +0006c4e0: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ +0006c4f0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
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│ │ │ │ +0006c5a0: 2020 2a20 7570 7065 722c 2061 6e20 2a6e    * upper, an *n
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│ │ │ │ +0006c5c0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
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│ │ │ │ +0006c610: 2020 2020 202a 2046 2c20 6120 2a6e 6f74       * F, a *not
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│ │ │ │ +0006c630: 2028 4d61 6361 756c 6179 3244 6f63 2943   (Macaulay2Doc)C
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│ │ │ │ +0006c6d0: 2061 2047 6f72 656e 7374 6569 6e20 7269   a Gorenstein ri
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│ │ │ │ +0006c6f0: 6d6f 6475 6c65 206f 7665 7220 616e 0a65  module over an.e
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│ │ │ │ +0006c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0006c7a0: 3031 5b61 2c62 2c63 2c20 536b 6577 436f  01[a,b,c, SkewCo
│ │ │ │ -0006c7b0: 6d6d 7574 6174 6976 653d 3e74 7275 655d  mmutative=>true]
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│ │ │ │ +0006c7b0: 7472 7565 5d20 2020 2020 2020 2020 2020  true]           
│ │ │ │  0006c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c7e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006c7d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c830: 7c0a 7c6f 3120 3d20 4520 2020 2020 2020  |.|o1 = E       
│ │ │ │ +0006c820: 2020 2020 207c 0a7c 6f31 203d 2045 2020       |.|o1 = E  
│ │ │ │ +0006c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c880: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006c870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006c900: 626c 6528 7329 2020 2020 2020 2020 2020  ble(s)          
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│ │ │ │ +0006c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c970: 2b0a 7c69 3220 3a20 4d20 3d20 636f 6b65  +.|i2 : M = coke
│ │ │ │ -0006c980: 7220 6d61 7028 455e 322c 2045 5e7b 2d31  r map(E^2, E^{-1
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│ │ │ │ -0006c9a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0006c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c9c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006c960: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d  -----+.|i2 : M =
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│ │ │ │ +0006c980: 455e 7b2d 317d 2c20 6d61 7472 6978 2261  E^{-1}, matrix"a
│ │ │ │ +0006c990: 623b 6263 2229 2020 2020 2020 2020 2020  b;bc")          
│ │ │ │ +0006c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c9b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca10: 7c0a 7c6f 3220 3d20 636f 6b65 726e 656c  |.|o2 = cokernel
│ │ │ │ -0006ca20: 207c 2061 6220 7c20 2020 2020 2020 2020   | ab |         
│ │ │ │ +0006ca00: 2020 2020 207c 0a7c 6f32 203d 2063 6f6b       |.|o2 = cok
│ │ │ │ +0006ca10: 6572 6e65 6c20 7c20 6162 207c 2020 2020  ernel | ab |    
│ │ │ │ +0006ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0006ca70: 207c 2062 6320 7c20 2020 2020 2020 2020   | bc |         
│ │ │ │ +0006ca50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ca60: 2020 2020 2020 7c20 6263 207c 2020 2020        | bc |    
│ │ │ │ +0006ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cab0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006caa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0006cb10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0006caf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cb10: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0006cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006cb60: 2c20 7175 6f74 6965 6e74 206f 6620 4520  , quotient of E 
│ │ │ │ +0006cb40: 2020 2020 207c 0a7c 6f32 203a 2045 2d6d       |.|o2 : E-m
│ │ │ │ +0006cb50: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +0006cb60: 6f66 2045 2020 2020 2020 2020 2020 2020  of E            
│ │ │ │  0006cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cba0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0006cb90: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cbf0: 2b0a 7c69 3320 3a20 7072 6573 656e 7461  +.|i3 : presenta
│ │ │ │ -0006cc00: 7469 6f6e 204d 2020 2020 2020 2020 2020  tion M          
│ │ │ │ +0006cbe0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2070 7265  -----+.|i3 : pre
│ │ │ │ +0006cbf0: 7365 6e74 6174 696f 6e20 4d20 2020 2020  sentation M     
│ │ │ │ +0006cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006cc30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc90: 7c0a 7c6f 3320 3d20 7c20 6162 207c 2020  |.|o3 = | ab |  
│ │ │ │ +0006cc80: 2020 2020 207c 0a7c 6f33 203d 207c 2061       |.|o3 = | a
│ │ │ │ +0006cc90: 6220 7c20 2020 2020 2020 2020 2020 2020  b |             
│ │ │ │  0006cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cce0: 7c0a 7c20 2020 2020 7c20 6263 207c 2020  |.|     | bc |  
│ │ │ │ +0006ccd0: 2020 2020 207c 0a7c 2020 2020 207c 2062       |.|     | b
│ │ │ │ +0006cce0: 6320 7c20 2020 2020 2020 2020 2020 2020  c |             
│ │ │ │  0006ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006d7e0: 2020 6469 7265 6374 2073 756d 2069 6e66    direct sum inf
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│ │ │ │ +0006dbd0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +0006dbe0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ +0006dbf0: 6e66 6f2c 204e 6f64 653a 2074 776f 4d6f  nfo, Node: twoMo
│ │ │ │ +0006dc00: 6e6f 6d69 616c 732c 2050 7265 763a 2074  nomials, Prev: t
│ │ │ │ +0006dc10: 6f41 7272 6179 2c20 5570 3a20 546f 700a  oArray, Up: Top.
│ │ │ │ +0006dc20: 0a74 776f 4d6f 6e6f 6d69 616c 7320 2d2d  .twoMonomials --
│ │ │ │ +0006dc30: 2074 616c 6c79 2074 6865 2073 6571 7565   tally the seque
│ │ │ │ +0006dc40: 6e63 6573 206f 6620 4252 616e 6b73 2066  nces of BRanks f
│ │ │ │ +0006dc50: 6f72 2063 6572 7461 696e 2065 7861 6d70  or certain examp
│ │ │ │ +0006dc60: 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  les.************
│ │ │ │  0006dc70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006dc80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006dc90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dcb0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ -0006dcc0: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ -0006dcd0: 200a 2020 2020 2020 2020 5420 3d20 5477   .        T = Tw
│ │ │ │ -0006dce0: 6f4d 6f6e 6f6d 6961 6c73 2863 2c64 290a  oMonomials(c,d).
│ │ │ │ -0006dcf0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -0006dd00: 2020 2a20 632c 2061 6e20 2a6e 6f74 6520    * c, an *note 
│ │ │ │ -0006dd10: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -0006dd20: 6179 3244 6f63 295a 5a2c 2c20 636f 6469  ay2Doc)ZZ,, codi
│ │ │ │ -0006dd30: 6d65 6e73 696f 6e20 696e 2077 6869 6368  mension in which
│ │ │ │ -0006dd40: 2074 6f20 776f 726b 0a20 2020 2020 202a   to work.      *
│ │ │ │ -0006dd50: 2064 2c20 616e 202a 6e6f 7465 2069 6e74   d, an *note int
│ │ │ │ -0006dd60: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -0006dd70: 446f 6329 5a5a 2c2c 2064 6567 7265 6520  Doc)ZZ,, degree 
│ │ │ │ -0006dd80: 6f66 2074 6865 206d 6f6e 6f6d 6961 6c73  of the monomials
│ │ │ │ -0006dd90: 2074 6f20 7461 6b65 0a20 202a 202a 6e6f   to take.  * *no
│ │ │ │ -0006dda0: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ -0006ddb0: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ -0006ddc0: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ -0006ddd0: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ -0006dde0: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ -0006ddf0: 204f 7074 696d 6973 6d20 3d3e 202e 2e2e   Optimism => ...
│ │ │ │ -0006de00: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0006de10: 300a 2020 2a20 4f75 7470 7574 733a 0a20  0.  * Outputs:. 
│ │ │ │ -0006de20: 2020 2020 202a 2054 2c20 6120 2a6e 6f74       * T, a *not
│ │ │ │ -0006de30: 6520 7461 6c6c 793a 2028 4d61 6361 756c  e tally: (Macaul
│ │ │ │ -0006de40: 6179 3244 6f63 2954 616c 6c79 2c2c 200a  ay2Doc)Tally,, .
│ │ │ │ -0006de50: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ -0006de60: 3d3d 3d3d 3d3d 3d3d 0a0a 7461 6c6c 6965  ========..tallie
│ │ │ │ -0006de70: 7320 7468 6520 7365 7175 656e 6365 7320  s the sequences 
│ │ │ │ -0006de80: 6f66 2042 2d72 616e 6b73 2074 6861 7420  of B-ranks that 
│ │ │ │ -0006de90: 6f63 6375 7220 666f 7220 6964 6561 6c73  occur for ideals
│ │ │ │ -0006dea0: 2067 656e 6572 6174 6564 2062 7920 7061   generated by pa
│ │ │ │ -0006deb0: 6972 7320 6f66 0a6d 6f6e 6f6d 6961 6c73  irs of.monomials
│ │ │ │ -0006dec0: 2069 6e20 5220 3d20 532f 2864 2d74 6820   in R = S/(d-th 
│ │ │ │ -0006ded0: 706f 7765 7273 206f 6620 7468 6520 7661  powers of the va
│ │ │ │ -0006dee0: 7269 6162 6c65 7329 2c20 7769 7468 2066  riables), with f
│ │ │ │ -0006def0: 756c 6c20 636f 6d70 6c65 7869 7479 2028  ull complexity (
│ │ │ │ -0006df00: 3d63 293b 0a74 6861 7420 6973 2c20 666f  =c);.that is, fo
│ │ │ │ -0006df10: 7220 616e 2061 7070 726f 7072 6961 7465  r an appropriate
│ │ │ │ -0006df20: 2073 797a 7967 7920 4d20 6f66 204d 3020   syzygy M of M0 
│ │ │ │ -0006df30: 3d20 522f 286d 312c 206d 3229 2077 6865  = R/(m1, m2) whe
│ │ │ │ -0006df40: 7265 206d 3120 616e 6420 6d32 2061 7265  re m1 and m2 are
│ │ │ │ -0006df50: 0a6d 6f6e 6f6d 6961 6c73 206f 6620 7468  .monomials of th
│ │ │ │ -0006df60: 6520 7361 6d65 2064 6567 7265 652e 0a0a  e same degree...
│ │ │ │ -0006df70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0006dca0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +0006dcb0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ +0006dcc0: 7361 6765 3a20 0a20 2020 2020 2020 2054  sage: .        T
│ │ │ │ +0006dcd0: 203d 2054 776f 4d6f 6e6f 6d69 616c 7328   = TwoMonomials(
│ │ │ │ +0006dce0: 632c 6429 0a20 202a 2049 6e70 7574 733a  c,d).  * Inputs:
│ │ │ │ +0006dcf0: 0a20 2020 2020 202a 2063 2c20 616e 202a  .      * c, an *
│ │ │ │ +0006dd00: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ +0006dd10: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ +0006dd20: 2063 6f64 696d 656e 7369 6f6e 2069 6e20   codimension in 
│ │ │ │ +0006dd30: 7768 6963 6820 746f 2077 6f72 6b0a 2020  which to work.  
│ │ │ │ +0006dd40: 2020 2020 2a20 642c 2061 6e20 2a6e 6f74      * d, an *not
│ │ │ │ +0006dd50: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ +0006dd60: 756c 6179 3244 6f63 295a 5a2c 2c20 6465  ulay2Doc)ZZ,, de
│ │ │ │ +0006dd70: 6772 6565 206f 6620 7468 6520 6d6f 6e6f  gree of the mono
│ │ │ │ +0006dd80: 6d69 616c 7320 746f 2074 616b 650a 2020  mials to take.  
│ │ │ │ +0006dd90: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ +0006dda0: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ +0006ddb0: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ +0006ddc0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +0006ddd0: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ +0006dde0: 2020 2020 2a20 4f70 7469 6d69 736d 203d      * Optimism =
│ │ │ │ +0006ddf0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +0006de00: 616c 7565 2030 0a20 202a 204f 7574 7075  alue 0.  * Outpu
│ │ │ │ +0006de10: 7473 3a0a 2020 2020 2020 2a20 542c 2061  ts:.      * T, a
│ │ │ │ +0006de20: 202a 6e6f 7465 2074 616c 6c79 3a20 284d   *note tally: (M
│ │ │ │ +0006de30: 6163 6175 6c61 7932 446f 6329 5461 6c6c  acaulay2Doc)Tall
│ │ │ │ +0006de40: 792c 2c20 0a0a 4465 7363 7269 7074 696f  y,, ..Descriptio
│ │ │ │ +0006de50: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a74  n.===========..t
│ │ │ │ +0006de60: 616c 6c69 6573 2074 6865 2073 6571 7565  allies the seque
│ │ │ │ +0006de70: 6e63 6573 206f 6620 422d 7261 6e6b 7320  nces of B-ranks 
│ │ │ │ +0006de80: 7468 6174 206f 6363 7572 2066 6f72 2069  that occur for i
│ │ │ │ +0006de90: 6465 616c 7320 6765 6e65 7261 7465 6420  deals generated 
│ │ │ │ +0006dea0: 6279 2070 6169 7273 206f 660a 6d6f 6e6f  by pairs of.mono
│ │ │ │ +0006deb0: 6d69 616c 7320 696e 2052 203d 2053 2f28  mials in R = S/(
│ │ │ │ +0006dec0: 642d 7468 2070 6f77 6572 7320 6f66 2074  d-th powers of t
│ │ │ │ +0006ded0: 6865 2076 6172 6961 626c 6573 292c 2077  he variables), w
│ │ │ │ +0006dee0: 6974 6820 6675 6c6c 2063 6f6d 706c 6578  ith full complex
│ │ │ │ +0006def0: 6974 7920 283d 6329 3b0a 7468 6174 2069  ity (=c);.that i
│ │ │ │ +0006df00: 732c 2066 6f72 2061 6e20 6170 7072 6f70  s, for an approp
│ │ │ │ +0006df10: 7269 6174 6520 7379 7a79 6779 204d 206f  riate syzygy M o
│ │ │ │ +0006df20: 6620 4d30 203d 2052 2f28 6d31 2c20 6d32  f M0 = R/(m1, m2
│ │ │ │ +0006df30: 2920 7768 6572 6520 6d31 2061 6e64 206d  ) where m1 and m
│ │ │ │ +0006df40: 3220 6172 650a 6d6f 6e6f 6d69 616c 7320  2 are.monomials 
│ │ │ │ +0006df50: 6f66 2074 6865 2073 616d 6520 6465 6772  of the same degr
│ │ │ │ +0006df60: 6565 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ee...+----------
│ │ │ │ +0006df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006dfa0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365  ------+.|i1 : se
│ │ │ │ -0006dfb0: 7452 616e 646f 6d53 6565 6420 3020 2020  tRandomSeed 0   
│ │ │ │ +0006df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0006dfa0: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ +0006dfb0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  0006dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006dfd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006dfe0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006dfd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e010: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ +0006e000: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0006e010: 203d 2030 2020 2020 2020 2020 2020 2020   = 0            
│ │ │ │  0006e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e050: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0006e040: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0006e050: 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  ----------------
│ │ │ │ -0006e080: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7477  ------+.|i2 : tw
│ │ │ │ -0006e090: 6f4d 6f6e 6f6d 6961 6c73 2832 2c33 2920  oMonomials(2,3) 
│ │ │ │ +0006e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0006e080: 203a 2074 776f 4d6f 6e6f 6d69 616c 7328   : twoMonomials(
│ │ │ │ +0006e090: 322c 3329 2020 2020 2020 2020 2020 2020  2,3)            
│ │ │ │  0006e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e0c0: 7c20 2d2d 2075 7365 6420 312e 3235 3736  | -- used 1.2576
│ │ │ │ -0006e0d0: 3873 2028 6370 7529 3b20 302e 3636 3432  8s (cpu); 0.6642
│ │ │ │ -0006e0e0: 3737 7320 2874 6872 6561 6429 3b20 3073  77s (thread); 0s
│ │ │ │ -0006e0f0: 2028 6763 2920 7c0a 7c32 2020 2020 2020   (gc) |.|2      
│ │ │ │ +0006e0b0: 2020 207c 0a7c 202d 2d20 7573 6564 2032     |.| -- used 2
│ │ │ │ +0006e0c0: 2e36 3031 3439 7320 2863 7075 293b 2031  .60149s (cpu); 1
│ │ │ │ +0006e0d0: 2e31 3732 3937 7320 2874 6872 6561 6429  .17297s (thread)
│ │ │ │ +0006e0e0: 3b20 3073 2028 6763 2920 207c 0a7c 3220  ; 0s (gc)  |.|2 
│ │ │ │ +0006e0f0: 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                  
│ │ │ │ -0006e120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e130: 7c54 616c 6c79 7b7b 7b31 2c20 317d 7d20  |Tally{{{1, 1}} 
│ │ │ │ -0006e140: 3d3e 2032 2020 2020 2020 2020 7d20 2020  => 2        }   
│ │ │ │ -0006e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e160: 2020 2020 2020 7c0a 7c20 2020 2020 207b        |.|      {
│ │ │ │ -0006e170: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -0006e180: 3d3e 2034 2020 2020 2020 2020 2020 2020  => 4            
│ │ │ │ -0006e190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e1a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006e120: 2020 207c 0a7c 5461 6c6c 797b 7b7b 312c     |.|Tally{{{1,
│ │ │ │ +0006e130: 2031 7d7d 203d 3e20 3220 2020 2020 2020   1}} => 2       
│ │ │ │ +0006e140: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +0006e150: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006e160: 2020 2020 7b7b 322c 2032 7d2c 207b 312c      {{2, 2}, {1,
│ │ │ │ +0006e170: 2032 7d7d 203d 3e20 3420 2020 2020 2020   2}} => 4       
│ │ │ │ +0006e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e190: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006e1a0: 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                  
│ │ │ │ -0006e1d0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -0006e1e0: 6420 302e 3634 3832 3032 7320 2863 7075  d 0.648202s (cpu
│ │ │ │ -0006e1f0: 293b 2030 2e33 3732 3638 3973 2028 7468  ); 0.372689s (th
│ │ │ │ -0006e200: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │ -0006e210: 7c33 2020 2020 2020 2020 2020 2020 2020  |3              
│ │ │ │ +0006e1c0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0006e1d0: 2d20 7573 6564 2031 2e38 3131 3536 7320  - used 1.81156s 
│ │ │ │ +0006e1e0: 2863 7075 293b 2030 2e35 3139 3838 3473  (cpu); 0.519884s
│ │ │ │ +0006e1f0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +0006e200: 6329 207c 0a7c 3320 2020 2020 2020 2020  c) |.|3         
│ │ │ │ +0006e210: 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                  
│ │ │ │ -0006e240: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b        |.|Tally{{
│ │ │ │ -0006e250: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -0006e260: 3d3e 2032 7d20 2020 2020 2020 2020 2020  => 2}           
│ │ │ │ -0006e270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e280: 7c20 2020 2020 207b 7b33 2c20 337d 2c20  |      {{3, 3}, 
│ │ │ │ -0006e290: 7b32 2c20 337d 7d20 3d3e 2031 2020 2020  {2, 3}} => 1    
│ │ │ │ -0006e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e2b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006e230: 2020 2020 2020 2020 2020 207c 0a7c 5461             |.|Ta
│ │ │ │ +0006e240: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c  lly{{{2, 2}, {1,
│ │ │ │ +0006e250: 2032 7d7d 203d 3e20 327d 2020 2020 2020   2}} => 2}      
│ │ │ │ +0006e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e270: 2020 207c 0a7c 2020 2020 2020 7b7b 332c     |.|      {{3,
│ │ │ │ +0006e280: 2033 7d2c 207b 322c 2033 7d7d 203d 3e20   3}, {2, 3}} => 
│ │ │ │ +0006e290: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0006e2a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e2e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e2f0: 7c20 2d2d 2075 7365 6420 302e 3236 3531  | -- used 0.2651
│ │ │ │ -0006e300: 3573 2028 6370 7529 3b20 302e 3132 3631  5s (cpu); 0.1261
│ │ │ │ -0006e310: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │ -0006e320: 2867 6329 2020 7c0a 7c34 2020 2020 2020  (gc)  |.|4      
│ │ │ │ +0006e2e0: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ +0006e2f0: 2e34 3131 3637 3673 2028 6370 7529 3b20  .411676s (cpu); 
│ │ │ │ +0006e300: 302e 3235 3033 3439 7320 2874 6872 6561  0.250349s (threa
│ │ │ │ +0006e310: 6429 3b20 3073 2028 6763 297c 0a7c 3420  d); 0s (gc)|.|4 
│ │ │ │ +0006e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e360: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20  |Tally{{{2, 2}, 
│ │ │ │ -0006e370: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020  {1, 2}} => 1}   
│ │ │ │ -0006e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e390: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006e350: 2020 207c 0a7c 5461 6c6c 797b 7b7b 322c     |.|Tally{{{2,
│ │ │ │ +0006e360: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20   2}, {1, 2}} => 
│ │ │ │ +0006e370: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ +0006e380: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ +0006eac0: 6861 6d61 7368 7f34 3237 3535 300a 4e6f  hamash.427550.No
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│ │ │ │ +0006eb60: 4172 7261 797f 3434 3837 3636 0a4e 6f64  Array.448766.Nod
│ │ │ │ +0006eb70: 653a 2074 776f 4d6f 6e6f 6d69 616c 737f  e: twoMonomials.
│ │ │ │ +0006eb80: 3434 3934 3739 0a1f 0a45 6e64 2054 6167  449479...End Tag
│ │ │ │ +0006eb90: 2054 6162 6c65 0a                         Table.
│ │ ├── ./usr/share/info/Cremona.info.gz
│ │ │ ├── Cremona.info
│ │ │ │ @@ -147,16 +147,16 @@
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│ │ │ │ @@ -322,16 +322,16 @@
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│ │ │ │  00001460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │ +00001490: 7075 293b 2030 2e30 3334 3032 3535 7320  pu); 0.0340255s 
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│ │ │ │  000014b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000014d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00001500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -387,18 +387,18 @@
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│ │ │ │  00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
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│ │ │ │  00001850: 6920 2020 2020 2020 2020 2020 2020 2020  i               
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│ │ │ │  00001870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001880: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │ -000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -000018c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00001890: 6564 2030 2e31 3234 3836 3873 2028 6370  ed 0.124868s (cp
│ │ │ │ +000018a0: 7529 3b20 302e 3036 3734 3935 3673 2028  u); 0.0674956s (
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│ │ │ │ +000018c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000018d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000018e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000018f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00001910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001920: 2020 2020 2020 207c 0a7c 6f34 203d 2031         |.|o4 = 1
│ │ │ │  00001930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -412,17 +412,17 @@
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│ │ │ │  000019f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │ +00001a30: 293b 2030 2e33 3937 3037 3873 2028 7468  ); 0.397078s (th
│ │ │ │ +00001a40: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
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│ │ │ │  00001a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00001c20: 7265 6573 3d3e 3029 2020 2020 2020 2020  rees=>0)        
│ │ │ │  00001c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001c40: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │ +00001c60: 7075 293b 2030 2e30 3436 3136 3632 7320  pu); 0.0461662s 
│ │ │ │  00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  00001c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00001c90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00001e40: 7068 6920 2020 2020 2020 2020 2020 2020  phi             
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│ │ │ │  00001e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │ -00001e90: 2863 7075 2020 2020 2020 2020 2020 2020  (cpu            
│ │ │ │ +00001e80: 6564 2030 2e30 3031 3035 3731 3573 2028  ed 0.00105715s (
│ │ │ │ +00001e90: 6370 7520 2020 2020 2020 2020 2020 2020  cpu             
│ │ │ │  00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -548,130 +548,130 @@
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│ │ │ │  00002280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002290: 2020 2020 2020 2033 3030 3030 3720 2030         300007  0
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│ │ │ │  000023e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  000023f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00002590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000025a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2878 2078 2020  -------|.|(x x  
│ │ │ │ -000025b0: 2d20 7820 7820 202b 2078 2078 202c 2078  - x x  + x x , x
│ │ │ │ -000025c0: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ -000025d0: 202c 2078 2078 2020 2d20 7820 7820 202b   , x x  - x x  +
│ │ │ │ -000025e0: 2078 2078 202c 2078 2078 2020 2d20 7820   x x , x x  - x 
│ │ │ │ -000025f0: 7820 202b 2078 207c 0a7c 2020 3620 3720  x  + x |.|  6 7 
│ │ │ │ -00002600: 2020 2035 2038 2020 2020 3420 3920 2020     5 8    4 9   
│ │ │ │ -00002610: 3320 3720 2020 2032 2038 2020 2020 3120  3 7    2 8    1 
│ │ │ │ -00002620: 3920 2020 3320 3520 2020 2032 2036 2020  9   3 5    2 6  
│ │ │ │ -00002630: 2020 3020 3920 2020 3320 3420 2020 2031    0 9   3 4    1
│ │ │ │ -00002640: 2036 2020 2020 307c 0a7c 2020 2020 2020   6    0|.|      
│ │ │ │ +000025a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2028 7820 7820  -------|.| (x x 
│ │ │ │ +000025b0: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ +000025c0: 7820 7820 202d 2078 2078 2020 2b20 7820  x x  - x x  + x 
│ │ │ │ +000025d0: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ +000025e0: 2b20 7820 7820 2c20 7820 7820 202d 2078  + x x , x x  - x
│ │ │ │ +000025f0: 2078 2020 2b20 787c 0a7c 2020 2036 2037   x  + x|.|   6 7
│ │ │ │ +00002600: 2020 2020 3520 3820 2020 2034 2039 2020      5 8    4 9  
│ │ │ │ +00002610: 2033 2037 2020 2020 3220 3820 2020 2031   3 7    2 8    1
│ │ │ │ +00002620: 2039 2020 2033 2035 2020 2020 3220 3620   9   3 5    2 6 
│ │ │ │ +00002630: 2020 2030 2039 2020 2033 2034 2020 2020     0 9   3 4    
│ │ │ │ +00002640: 3120 3620 2020 207c 0a7c 2020 2020 2020  1 6    |.|      
│ │ │ │  00002650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002690: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000026a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000026c0: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ +000026c0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │  000026d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000026f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002710: 2020 2020 2020 2020 2020 202d 2d2d 2d2d             -----
│ │ │ │ -00002720: 2d5b 7820 2e2e 7820 5d20 2020 2020 2020  -[x ..x ]       
│ │ │ │ +00002710: 2020 2020 2020 2020 2020 2020 2d2d 2d2d              ----
│ │ │ │ +00002720: 2d2d 5b78 202e 2e78 205d 2020 2020 2020  --[x ..x ]      
│ │ │ │  00002730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002760: 2020 2020 2020 2020 2020 2033 3030 3030             30000
│ │ │ │ -00002770: 3720 2030 2020 2039 2020 2020 2020 2020  7  0   9        
│ │ │ │ -00002780: 2020 2020 2020 207c 0a7c 5d20 3c2d 2d20         |.|] <-- 
│ │ │ │ -00002790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002760: 2020 2020 2020 2020 2020 2020 3330 3030              3000
│ │ │ │ +00002770: 3037 2020 3020 2020 3920 2020 2020 2020  07  0   9       
│ │ │ │ +00002780: 2020 2020 2020 207c 0a7c 205d 203c 2d2d         |.| ] <--
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│ │ │ │  000027c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000027e0: 2878 2078 2020 2d20 7820 7820 202b 2078  (x x  - x x  + x
│ │ │ │ -000027f0: 2078 202c 2078 2078 2020 2d20 7820 7820   x , x x  - x x 
│ │ │ │ -00002800: 202b 2078 2078 202c 2078 2078 2020 2d20   + x x , x x  - 
│ │ │ │ -00002810: 7820 7820 202b 2078 2078 202c 2078 2078  x x  + x x , x x
│ │ │ │ -00002820: 2020 2d20 7820 787c 0a7c 2020 2020 2020    - x x|.|      
│ │ │ │ -00002830: 2020 3620 3720 2020 2035 2038 2020 2020    6 7    5 8    
│ │ │ │ -00002840: 3420 3920 2020 3320 3720 2020 2032 2038  4 9   3 7    2 8
│ │ │ │ -00002850: 2020 2020 3120 3920 2020 3320 3520 2020      1 9   3 5   
│ │ │ │ -00002860: 2032 2036 2020 2020 3020 3920 2020 3320   2 6    0 9   3 
│ │ │ │ -00002870: 3420 2020 2031 207c 0a7c 2d2d 2d2d 2d2d  4    1 |.|------
│ │ │ │ +000027d0: 2d2d 2d2d 2d2d 2d7c 0a7c 3620 2020 2020  -------|.|6     
│ │ │ │ +000027e0: 2028 7820 7820 202d 2078 2078 2020 2b20   (x x  - x x  + 
│ │ │ │ +000027f0: 7820 7820 2c20 7820 7820 202d 2078 2078  x x , x x  - x x
│ │ │ │ +00002800: 2020 2b20 7820 7820 2c20 7820 7820 202d    + x x , x x  -
│ │ │ │ +00002810: 2078 2078 2020 2b20 7820 7820 2c20 7820   x x  + x x , x 
│ │ │ │ +00002820: 7820 202d 2078 207c 0a7c 2020 2020 2020  x  - x |.|      
│ │ │ │ +00002830: 2020 2036 2037 2020 2020 3520 3820 2020     6 7    5 8   
│ │ │ │ +00002840: 2034 2039 2020 2033 2037 2020 2020 3220   4 9   3 7    2 
│ │ │ │ +00002850: 3820 2020 2031 2039 2020 2033 2035 2020  8    1 9   3 5  
│ │ │ │ +00002860: 2020 3220 3620 2020 2030 2039 2020 2033    2 6    0 9   3
│ │ │ │ +00002870: 2034 2020 2020 317c 0a7c 2d2d 2d2d 2d2d   4    1|.|------
│ │ │ │  00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028c0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │  000028d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000028e0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -000028f0: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ -00002900: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000028e0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +000028f0: 2020 2020 2020 2020 2032 2020 2020 3220           2    2 
│ │ │ │ +00002900: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00002910: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00002940: 7420 202d 2074 2074 2020 2d20 7420 7420  t  - t t  - t t 
│ │ │ │ -00002950: 202b 2074 2074 2074 202c 202d 2074 2074   + t t t , - t t
│ │ │ │ -00002960: 2020 2b20 2020 207c 0a7c 7820 2c20 7820    +    |.|x , x 
│ │ │ │ -00002970: 7820 202d 2078 2078 2020 2b20 7820 7820  x  - x x  + x x 
│ │ │ │ -00002980: 2920 2020 2020 2032 2020 2020 2031 2032  )      2     1 2
│ │ │ │ -00002990: 2033 2020 2020 3020 3320 2020 2031 2034   3    0 3    1 4
│ │ │ │ -000029a0: 2020 2020 3020 3220 3420 2020 2020 3220      0 2 4     2 
│ │ │ │ -000029b0: 3320 2020 2020 207c 0a7c 2038 2020 2032  3      |.| 8   2
│ │ │ │ -000029c0: 2034 2020 2020 3120 3520 2020 2030 2037   4    1 5    0 7
│ │ │ │ -000029d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002930: 2d2d 2c20 7b2d 2074 2020 2b20 3274 2074  --, {- t  + 2t t
│ │ │ │ +00002940: 2074 2020 2d20 7420 7420 202d 2074 2074   t  - t t  - t t
│ │ │ │ +00002950: 2020 2b20 7420 7420 7420 2c20 2d20 7420    + t t t , - t 
│ │ │ │ +00002960: 7420 202b 2020 207c 0a7c 2078 202c 2078  t  +   |.| x , x
│ │ │ │ +00002970: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ +00002980: 2029 2020 2020 2020 3220 2020 2020 3120   )      2     1 
│ │ │ │ +00002990: 3220 3320 2020 2030 2033 2020 2020 3120  2 3    0 3    1 
│ │ │ │ +000029a0: 3420 2020 2030 2032 2034 2020 2020 2032  4    0 2 4     2
│ │ │ │ +000029b0: 2033 2020 2020 207c 0a7c 3020 3820 2020   3     |.|0 8   
│ │ │ │ +000029c0: 3220 3420 2020 2031 2035 2020 2020 3020  2 4    1 5    0 
│ │ │ │ +000029d0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  000029e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000029f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -688,25 +688,25 @@
│ │ │ │  00002af0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b40: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002b60: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +00002b60: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │  00002b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002b90: 2020 2020 2020 207c 0a7c 2020 2b20 7820         |.|  + x 
│ │ │ │ -00002ba0: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ -00002bb0: 2b20 7820 7820 2920 2020 2020 2020 2020  + x x )         
│ │ │ │ +00002b90: 2020 2020 2020 207c 0a7c 7820 202b 2078         |.|x  + x
│ │ │ │ +00002ba0: 2078 202c 2078 2078 2020 2d20 7820 7820   x , x x  - x x 
│ │ │ │ +00002bb0: 202b 2078 2078 2029 2020 2020 2020 2020   + x x )        
│ │ │ │  00002bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002be0: 2020 2020 2020 207c 0a7c 3620 2020 2030         |.|6    0
│ │ │ │ -00002bf0: 2038 2020 2032 2034 2020 2020 3120 3520   8   2 4    1 5 
│ │ │ │ -00002c00: 2020 2030 2037 2020 2020 2020 2020 2020     0 7          
│ │ │ │ +00002be0: 2020 2020 2020 207c 0a7c 2036 2020 2020         |.| 6    
│ │ │ │ +00002bf0: 3020 3820 2020 3220 3420 2020 2031 2035  0 8   2 4    1 5
│ │ │ │ +00002c00: 2020 2020 3020 3720 2020 2020 2020 2020      0 7         
│ │ │ │  00002c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002c30: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -832,17 +832,17 @@
│ │ │ │  000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074  -------+.|i8 : t
│ │ │ │  00003410: 696d 6520 7073 6920 3d20 696e 7665 7273  ime psi = invers
│ │ │ │  00003420: 654d 6170 2070 6869 2020 2020 2020 2020  eMap phi        
│ │ │ │  00003430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00003460: 6564 2030 2e35 3636 3739 3573 2028 6370  ed 0.566795s (cp
│ │ │ │ -00003470: 7529 3b20 302e 3431 3735 3573 2028 7468  u); 0.41755s (th
│ │ │ │ -00003480: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00003460: 6564 2030 2e35 3939 3836 3873 2028 6370  ed 0.599868s (cp
│ │ │ │ +00003470: 7529 3b20 302e 3434 3534 7320 2874 6872  u); 0.4454s (thr
│ │ │ │ +00003480: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00003490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000034b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ @@ -1117,18 +1117,18 @@
│ │ │ │  000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074  -------+.|i9 : t
│ │ │ │  000045e0: 696d 6520 6973 496e 7665 7273 654d 6170  ime isInverseMap
│ │ │ │  000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020  (phi,psi)       
│ │ │ │  00004600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004630: 6564 2030 2e30 3130 3133 3233 7320 2863  ed 0.0101323s (c
│ │ │ │ -00004640: 7075 293b 2030 2e30 3039 3436 3335 3673  pu); 0.00946356s
│ │ │ │ -00004650: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00004660: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00004630: 6564 2030 2e30 3037 3938 3633 3273 2028  ed 0.00798632s (
│ │ │ │ +00004640: 6370 7529 3b20 302e 3030 3731 3237 3339  cpu); 0.00712739
│ │ │ │ +00004650: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00004660: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  00004670: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00004680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046c0: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │  000046d0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ @@ -1142,16 +1142,16 @@
│ │ │ │  00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │  00004770: 7469 6d65 2064 6567 7265 654d 6170 2070  time degreeMap p
│ │ │ │  00004780: 7369 2020 2020 2020 2020 2020 2020 2020  si              
│ │ │ │  00004790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -000047c0: 6564 2030 2e32 3937 3931 3773 2028 6370  ed 0.297917s (cp
│ │ │ │ -000047d0: 7529 3b20 302e 3233 3338 3238 7320 2874  u); 0.233828s (t
│ │ │ │ +000047c0: 6564 2030 2e32 3535 3733 3673 2028 6370  ed 0.255736s (cp
│ │ │ │ +000047d0: 7529 3b20 302e 3139 3738 3233 7320 2874  u); 0.197823s (t
│ │ │ │  000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000047f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00004810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1167,16 +1167,16 @@
│ │ │ │  000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │  00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544  time projectiveD
│ │ │ │  00004910: 6567 7265 6573 2070 7369 2020 2020 2020  egrees psi      
│ │ │ │  00004920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004950: 6564 2034 2e39 3936 3537 7320 2863 7075  ed 4.99657s (cpu
│ │ │ │ -00004960: 293b 2034 2e32 3737 3236 7320 2874 6872  ); 4.27726s (thr
│ │ │ │ +00004950: 6564 2035 2e37 3339 3933 7320 2863 7075  ed 5.73993s (cpu
│ │ │ │ +00004960: 293b 2035 2e33 3537 3834 7320 2874 6872  ); 5.35784s (thr
│ │ │ │  00004970: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00004980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000049a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1213,18 +1213,18 @@
│ │ │ │  00004bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00004be0: 0a7c 6931 3220 3a20 7469 6d65 2070 6869  .|i12 : time phi
│ │ │ │  00004bf0: 203d 2072 6174 696f 6e61 6c4d 6170 206d   = rationalMap m
│ │ │ │  00004c00: 696e 6f72 7328 332c 6d61 7472 6978 7b7b  inors(3,matrix{{
│ │ │ │  00004c10: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e  t_0..t_4},{t_1..
│ │ │ │  00004c20: 745f 357d 2c7b 745f 322e 2e74 5f36 207c  t_5},{t_2..t_6 |
│ │ │ │ -00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3031  .| -- used 0.001
│ │ │ │ -00004c40: 3132 3533 7320 2863 7075 293b 2030 2e30  1253s (cpu); 0.0
│ │ │ │ -00004c50: 3031 3937 3134 3173 2028 7468 7265 6164  0197141s (thread
│ │ │ │ -00004c60: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3032  .| -- used 0.002
│ │ │ │ +00004c40: 3030 3231 3173 2028 6370 7529 3b20 302e  00211s (cpu); 0.
│ │ │ │ +00004c50: 3030 3138 3635 3435 7320 2874 6872 6561  00186545s (threa
│ │ │ │ +00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00004c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004cd0: 0a7c 6f31 3220 3d20 2d2d 2072 6174 696f  .|o12 = -- ratio
│ │ │ │ @@ -1493,17 +1493,17 @@
│ │ │ │  00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869  .|i13 : time phi
│ │ │ │  00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870   = rationalMap(p
│ │ │ │  00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920  hi,Dominant=>2) 
│ │ │ │  00005d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3535  .| -- used 0.155
│ │ │ │ -00005dc0: 3437 3173 2028 6370 7529 3b20 302e 3037  471s (cpu); 0.07
│ │ │ │ -00005dd0: 3535 3436 3773 2028 7468 7265 6164 293b  55467s (thread);
│ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3933  .| -- used 0.193
│ │ │ │ +00005dc0: 3131 3773 2028 6370 7529 3b20 302e 3036  117s (cpu); 0.06
│ │ │ │ +00005dd0: 3633 3736 3873 2028 7468 7265 6164 293b  63768s (thread);
│ │ │ │  00005de0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00005df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00005e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2153,17 +2153,17 @@
│ │ │ │  00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  000086a0: 0a7c 6931 3420 3a20 7469 6d65 2070 6869  .|i14 : time phi
│ │ │ │  000086b0: 5e28 2d31 2920 2020 2020 2020 2020 2020  ^(-1)           
│ │ │ │  000086c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000086d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000086e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000086f0: 0a7c 202d 2d20 7573 6564 2030 2e35 3034  .| -- used 0.504
│ │ │ │ -00008700: 3832 3573 2028 6370 7529 3b20 302e 3432  825s (cpu); 0.42
│ │ │ │ -00008710: 3530 3336 7320 2874 6872 6561 6429 3b20  5036s (thread); 
│ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3335  .| -- used 0.435
│ │ │ │ +00008700: 3333 3273 2028 6370 7529 3b20 302e 3433  332s (cpu); 0.43
│ │ │ │ +00008710: 3538 3039 7320 2874 6872 6561 6429 3b20  5809s (thread); 
│ │ │ │  00008720: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00008730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00008740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00008750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2708,18 +2708,18 @@
│ │ │ │  0000a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000a950: 0a7c 6931 3520 3a20 7469 6d65 2064 6567  .|i15 : time deg
│ │ │ │  0000a960: 7265 6573 2070 6869 5e28 2d31 2920 2020  rees phi^(-1)   
│ │ │ │  0000a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3535  .| -- used 0.355
│ │ │ │ -0000a9b0: 3436 3673 2028 6370 7529 3b20 302e 3237  466s (cpu); 0.27
│ │ │ │ -0000a9c0: 3033 3535 7320 2874 6872 6561 6429 3b20  0355s (thread); 
│ │ │ │ -0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3839  .| -- used 0.489
│ │ │ │ +0000a9b0: 3635 3173 2028 6370 7529 3b20 302e 3336  651s (cpu); 0.36
│ │ │ │ +0000a9c0: 3634 3573 2028 7468 7265 6164 293b 2030  645s (thread); 0
│ │ │ │ +0000a9d0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0000a9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000a9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000aa40: 0a7c 6f31 3520 3d20 7b35 2c20 3135 2c20  .|o15 = {5, 15, 
│ │ │ │ @@ -2743,18 +2743,18 @@
│ │ │ │  0000ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000ab80: 0a7c 6931 3620 3a20 7469 6d65 2064 6567  .|i16 : time deg
│ │ │ │  0000ab90: 7265 6573 2070 6869 2020 2020 2020 2020  rees phi        
│ │ │ │  0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000abc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e31 3230  .| -- used 0.120
│ │ │ │ -0000abe0: 3930 3773 2028 6370 7529 3b20 302e 3034  907s (cpu); 0.04
│ │ │ │ -0000abf0: 3337 3030 3273 2028 7468 7265 6164 293b  37002s (thread);
│ │ │ │ -0000ac00: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3733  .| -- used 0.073
│ │ │ │ +0000abe0: 3435 3532 7320 2863 7075 293b 2030 2e30  4552s (cpu); 0.0
│ │ │ │ +0000abf0: 3231 3830 3037 7320 2874 6872 6561 6429  218007s (thread)
│ │ │ │ +0000ac00: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0000ac10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ac20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ac70: 0a7c 6f31 3620 3d20 7b31 2c20 332c 2039  .|o16 = {1, 3, 9
│ │ │ │ @@ -2778,17 +2778,17 @@
│ │ │ │  0000ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000adb0: 0a7c 6931 3720 3a20 7469 6d65 2064 6573  .|i17 : time des
│ │ │ │  0000adc0: 6372 6962 6520 7068 6920 2020 2020 2020  cribe phi       
│ │ │ │  0000add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000adf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3033  .| -- used 0.003
│ │ │ │ -0000ae10: 3230 3737 3273 2028 6370 7529 3b20 302e  20772s (cpu); 0.
│ │ │ │ -0000ae20: 3030 3331 3633 3131 7320 2874 6872 6561  00316311s (threa
│ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2031 2e35 3239  .| -- used 1.529
│ │ │ │ +0000ae10: 3865 2d30 3573 2028 6370 7529 3b20 302e  8e-05s (cpu); 0.
│ │ │ │ +0000ae20: 3030 3335 3139 3331 7320 2874 6872 6561  00351931s (threa
│ │ │ │  0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000ae40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2843,18 +2843,18 @@
│ │ │ │  0000b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b1c0: 0a7c 6931 3820 3a20 7469 6d65 2064 6573  .|i18 : time des
│ │ │ │  0000b1d0: 6372 6962 6520 7068 695e 282d 3129 2020  cribe phi^(-1)  
│ │ │ │  0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3036  .| -- used 0.006
│ │ │ │ -0000b220: 3433 3239 3673 2028 6370 7529 3b20 302e  43296s (cpu); 0.
│ │ │ │ -0000b230: 3030 3939 3338 3936 7320 2874 6872 6561  00993896s (threa
│ │ │ │ -0000b240: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3038  .| -- used 0.008
│ │ │ │ +0000b220: 3436 3537 3373 2028 6370 7529 3b20 302e  46573s (cpu); 0.
│ │ │ │ +0000b230: 3031 3233 3635 3673 2028 7468 7265 6164  0123656s (thread
│ │ │ │ +0000b240: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  0000b250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b2b0: 0a7c 6f31 3820 3d20 7261 7469 6f6e 616c  .|o18 = rational
│ │ │ │ @@ -2923,18 +2923,18 @@
│ │ │ │  0000b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b6c0: 0a7c 6931 3920 3a20 7469 6d65 2028 662c  .|i19 : time (f,
│ │ │ │  0000b6d0: 6729 203d 2067 7261 7068 2070 6869 5e2d  g) = graph phi^-
│ │ │ │  0000b6e0: 313b 2066 3b20 2020 2020 2020 2020 2020  1; f;           
│ │ │ │  0000b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3038  .| -- used 0.008
│ │ │ │ -0000b720: 3435 3334 7320 2863 7075 293b 2030 2e30  4534s (cpu); 0.0
│ │ │ │ -0000b730: 3038 3831 3833 3873 2028 7468 7265 6164  0881838s (thread
│ │ │ │ -0000b740: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3532  .| -- used 0.052
│ │ │ │ +0000b720: 3039 3235 7320 2863 7075 293b 2030 2e30  0925s (cpu); 0.0
│ │ │ │ +0000b730: 3532 3035 3973 2028 7468 7265 6164 293b  52059s (thread);
│ │ │ │ +0000b740: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  0000b750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b7b0: 0a7c 6f32 3020 3a20 4d75 6c74 6968 6f6d  .|o20 : Multihom
│ │ │ │ @@ -2958,18 +2958,18 @@
│ │ │ │  0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567  .|i21 : time deg
│ │ │ │  0000b900: 7265 6573 2066 2020 2020 2020 2020 2020  rees f          
│ │ │ │  0000b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e31 3931  .| -- used 1.191
│ │ │ │ -0000b950: 3038 7320 2863 7075 293b 2030 2e39 3031  08s (cpu); 0.901
│ │ │ │ -0000b960: 3637 3973 2028 7468 7265 6164 293b 2030  679s (thread); 0
│ │ │ │ -0000b970: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e35 3036  .| -- used 1.506
│ │ │ │ +0000b950: 3532 7320 2863 7075 293b 2031 2e32 3632  52s (cpu); 1.262
│ │ │ │ +0000b960: 3431 7320 2874 6872 6561 6429 3b20 3073  41s (thread); 0s
│ │ │ │ +0000b970: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0000b980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b9e0: 0a7c 6f32 3120 3d20 7b39 3034 2c20 3530  .|o21 = {904, 50
│ │ │ │ @@ -2994,16 +2994,16 @@
│ │ │ │  0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567  .|i22 : time deg
│ │ │ │  0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020  ree f           
│ │ │ │  0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bb70: 0a7c 202d 2d20 7573 6564 2030 2e30 3030  .| -- used 0.000
│ │ │ │ -0000bb80: 3139 3036 3938 7320 2863 7075 293b 2031  190698s (cpu); 1
│ │ │ │ -0000bb90: 2e33 3633 3565 2d30 3573 2028 7468 7265  .3635e-05s (thre
│ │ │ │ +0000bb80: 3134 3336 3139 7320 2863 7075 293b 2031  143619s (cpu); 1
│ │ │ │ +0000bb90: 2e31 3539 3265 2d30 3573 2028 7468 7265  .1592e-05s (thre
│ │ │ │  0000bba0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -3019,16 +3019,16 @@
│ │ │ │  0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573  .|i23 : time des
│ │ │ │  0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020  cribe f         
│ │ │ │  0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3030  .| -- used 0.000
│ │ │ │ -0000bd10: 3932 3635 3738 7320 2863 7075 293b 2030  926578s (cpu); 0
│ │ │ │ -0000bd20: 2e30 3031 3432 3938 3773 2028 7468 7265  .00142987s (thre
│ │ │ │ +0000bd10: 3738 3633 3534 7320 2863 7075 293b 2030  786354s (cpu); 0
│ │ │ │ +0000bd20: 2e30 3031 3139 3332 3773 2028 7468 7265  .00119327s (thre
│ │ │ │  0000bd30: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  0000bd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -4588,17 +4588,17 @@
│ │ │ │  00011eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00011ed0: 3420 3a20 7469 6d65 2070 7369 203d 2061  4 : time psi = a
│ │ │ │  00011ee0: 6273 7472 6163 7452 6174 696f 6e61 6c4d  bstractRationalM
│ │ │ │  00011ef0: 6170 2850 342c 5035 2c66 2920 2020 2020  ap(P4,P5,f)     
│ │ │ │  00011f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00011f20: 2d2d 2075 7365 6420 302e 3030 3131 3838  -- used 0.001188
│ │ │ │ -00011f30: 3436 7320 2863 7075 293b 2030 2e30 3030  46s (cpu); 0.000
│ │ │ │ -00011f40: 3336 3839 3932 7320 2874 6872 6561 6429  368992s (thread)
│ │ │ │ +00011f20: 2d2d 2075 7365 6420 302e 3030 3232 3137  -- used 0.002217
│ │ │ │ +00011f30: 3733 7320 2863 7075 293b 2030 2e30 3030  73s (cpu); 0.000
│ │ │ │ +00011f40: 3239 3433 3132 7320 2874 6872 6561 6429  294312s (thread)
│ │ │ │  00011f50: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  00011f60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00011f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011fb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ @@ -4659,17 +4659,17 @@
│ │ │ │  00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012340: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │  00012350: 2074 696d 6520 7072 6f6a 6563 7469 7665   time projective
│ │ │ │  00012360: 4465 6772 6565 7328 7073 692c 3329 2020  Degrees(psi,3)  
│ │ │ │  00012370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012380: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00012390: 6564 2030 2e32 3939 3535 3273 2028 6370  ed 0.299552s (cp
│ │ │ │ -000123a0: 7529 3b20 302e 3135 3731 3673 2028 7468  u); 0.15716s (th
│ │ │ │ -000123b0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00012390: 6564 2030 2e32 3239 3332 3673 2028 6370  ed 0.229326s (cp
│ │ │ │ +000123a0: 7529 3b20 302e 3136 3930 3936 7320 2874  u); 0.169096s (t
│ │ │ │ +000123b0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000123c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000123d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000123e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000123f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012400: 2020 207c 0a7c 6f35 203d 2032 2020 2020     |.|o5 = 2    
│ │ │ │  00012410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4678,18 +4678,18 @@
│ │ │ │  00012450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00012480: 0a7c 6936 203a 2074 696d 6520 7261 7469  .|i6 : time rati
│ │ │ │  00012490: 6f6e 616c 4d61 7020 7073 6920 2020 2020  onalMap psi     
│ │ │ │  000124a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000124b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000124c0: 202d 2d20 7573 6564 2030 2e34 3032 3738   -- used 0.40278
│ │ │ │ -000124d0: 7320 2863 7075 293b 2030 2e33 3331 3733  s (cpu); 0.33173
│ │ │ │ -000124e0: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │ -000124f0: 2867 6329 2020 2020 2020 207c 0a7c 2020  (gc)       |.|  
│ │ │ │ +000124c0: 202d 2d20 7573 6564 2030 2e34 3330 3331   -- used 0.43031
│ │ │ │ +000124d0: 3573 2028 6370 7529 3b20 302e 3336 3935  5s (cpu); 0.3695
│ │ │ │ +000124e0: 3634 7320 2874 6872 6561 6429 3b20 3073  64s (thread); 0s
│ │ │ │ +000124f0: 2028 6763 2920 2020 2020 207c 0a7c 2020   (gc)      |.|  
│ │ │ │  00012500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012530: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │  00012540: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │  00012550: 202d 2d20 2020 2020 2020 2020 2020 2020   --             
│ │ │ │  00012560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5102,17 +5102,17 @@
│ │ │ │  00013ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013ee0: 2d2d 2d2d 2b0a 7c69 3134 203a 2074 696d  ----+.|i14 : tim
│ │ │ │  00013ef0: 6520 5420 3d20 6162 7374 7261 6374 5261  e T = abstractRa
│ │ │ │  00013f00: 7469 6f6e 616c 4d61 7028 492c 224f 4144  tionalMap(I,"OAD
│ │ │ │  00013f10: 5022 2920 2020 2020 2020 2020 2020 2020  P")             
│ │ │ │  00013f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f30: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -00013f40: 3135 3138 3135 7320 2863 7075 293b 2030  151815s (cpu); 0
│ │ │ │ -00013f50: 2e30 3635 3236 3533 7320 2874 6872 6561  .0652653s (threa
│ │ │ │ -00013f60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00013f40: 3036 3339 3938 3773 2028 6370 7529 3b20  0639987s (cpu); 
│ │ │ │ +00013f50: 302e 3036 3439 3739 3573 2028 7468 7265  0.0649795s (thre
│ │ │ │ +00013f60: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00013f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00013f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00013fd0: 7c6f 3134 203d 202d 2d20 7261 7469 6f6e  |o14 = -- ration
│ │ │ │ @@ -5173,46952 +5173,46953 @@
│ │ │ │  00014340: 6d73 2064 6566 696e 696e 6720 7468 6520  ms defining the 
│ │ │ │  00014350: 6162 7374 7261 6374 206d 6170 2054 2063  abstract map T c
│ │ │ │  00014360: 616e 2062 6520 6f62 7461 696e 6564 2062  an be obtained b
│ │ │ │  00014370: 7920 7468 650a 666f 6c6c 6f77 696e 6720  y the.following 
│ │ │ │  00014380: 636f 6d6d 616e 643a 0a0a 2b2d 2d2d 2d2d  command:..+-----
│ │ │ │  00014390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000143a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000143b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000143c0: 6931 3520 3a20 7469 6d65 2070 726f 6a65  i15 : time proje
│ │ │ │ -000143d0: 6374 6976 6544 6567 7265 6573 2854 2c32  ctiveDegrees(T,2
│ │ │ │ -000143e0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -000143f0: 2020 7c0a 7c20 2d2d 2075 7365 6420 332e    |.| -- used 3.
│ │ │ │ -00014400: 3032 3534 7320 2863 7075 293b 2031 2e36  0254s (cpu); 1.6
│ │ │ │ -00014410: 3932 3838 7320 2874 6872 6561 6429 3b20  9288s (thread); 
│ │ │ │ -00014420: 3073 2028 6763 297c 0a7c 2020 2020 2020  0s (gc)|.|      
│ │ │ │ +000143b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000143c0: 7c69 3135 203a 2074 696d 6520 7072 6f6a  |i15 : time proj
│ │ │ │ +000143d0: 6563 7469 7665 4465 6772 6565 7328 542c  ectiveDegrees(T,
│ │ │ │ +000143e0: 3229 2020 2020 2020 2020 2020 2020 2020  2)              
│ │ │ │ +000143f0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00014400: 332e 3434 3934 3173 2028 6370 7529 3b20  3.44941s (cpu); 
│ │ │ │ +00014410: 312e 3935 3232 3173 2028 7468 7265 6164  1.95221s (thread
│ │ │ │ +00014420: 293b 2030 7320 2867 6329 7c0a 7c20 2020  ); 0s (gc)|.|   
│ │ │ │  00014430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014450: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00014460: 3135 203d 2033 2020 2020 2020 2020 2020  15 = 3          
│ │ │ │ +00014450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014460: 7c0a 7c6f 3135 203d 2033 2020 2020 2020  |.|o15 = 3      
│ │ │ │  00014470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014490: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00014490: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000144a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000144b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000144c0: 2d2d 2d2d 2d2d 2b0a 0a57 6520 7665 7269  ------+..We veri
│ │ │ │ -000144d0: 6679 2074 6861 7420 7468 6520 636f 6d70  fy that the comp
│ │ │ │ -000144e0: 6f73 6974 696f 6e20 6f66 2054 2077 6974  osition of T wit
│ │ │ │ -000144f0: 6820 6974 7365 6c66 2069 7320 6465 6669  h itself is defi
│ │ │ │ -00014500: 6e65 6420 6279 206c 696e 6561 7220 666f  ned by linear fo
│ │ │ │ -00014510: 726d 733a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  rms:..+---------
│ │ │ │ +000144c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ +000144d0: 6520 7665 7269 6679 2074 6861 7420 7468  e verify that th
│ │ │ │ +000144e0: 6520 636f 6d70 6f73 6974 696f 6e20 6f66  e composition of
│ │ │ │ +000144f0: 2054 2077 6974 6820 6974 7365 6c66 2069   T with itself i
│ │ │ │ +00014500: 7320 6465 6669 6e65 6420 6279 206c 696e  s defined by lin
│ │ │ │ +00014510: 6561 7220 666f 726d 733a 0a0a 2b2d 2d2d  ear forms:..+---
│ │ │ │  00014520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014550: 2d2b 0a7c 6931 3620 3a20 7469 6d65 2054  -+.|i16 : time T
│ │ │ │ -00014560: 3220 3d20 5420 2a20 5420 2020 2020 2020  2 = T * T       
│ │ │ │ +00014550: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │ +00014560: 7469 6d65 2054 3220 3d20 5420 2a20 5420  time T2 = T * T 
│ │ │ │  00014570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014590: 7c20 2d2d 2075 7365 6420 302e 3030 3031  | -- used 0.0001
│ │ │ │ -000145a0: 3939 3535 3473 2028 6370 7529 3b20 322e  99554s (cpu); 2.
│ │ │ │ -000145b0: 3737 3132 652d 3035 7320 2874 6872 6561  7712e-05s (threa
│ │ │ │ -000145c0: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │ -000145d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014590: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +000145a0: 302e 3030 3031 3234 3937 3573 2028 6370  0.000124975s (cp
│ │ │ │ +000145b0: 7529 3b20 322e 3731 3265 2d30 3573 2028  u); 2.712e-05s (
│ │ │ │ +000145c0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +000145d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000145e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014600: 2020 2020 2020 2020 7c0a 7c6f 3136 203d          |.|o16 =
│ │ │ │ -00014610: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │ -00014620: 202d 2d20 2020 2020 2020 2020 2020 2020   --             
│ │ │ │ +00014600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014610: 7c6f 3136 203d 202d 2d20 7261 7469 6f6e  |o16 = -- ration
│ │ │ │ +00014620: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │  00014630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014640: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00014650: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -00014660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014640: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00014650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014660: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  00014670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014680: 2020 7c0a 7c20 2020 2020 2073 6f75 7263    |.|      sourc
│ │ │ │ -00014690: 653a 2050 726f 6a28 2d2d 2d2d 2d5b 7820  e: Proj(-----[x 
│ │ │ │ -000146a0: 2c20 7820 2c20 7820 2c20 7820 5d29 2020  , x , x , x ])  
│ │ │ │ -000146b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000146c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000146d0: 2020 2020 2036 3535 3231 2020 3020 2020       65521  0   
│ │ │ │ -000146e0: 3120 2020 3220 2020 3320 2020 2020 2020  1   2   3       
│ │ │ │ -000146f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00014700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014710: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +00014680: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00014690: 2073 6f75 7263 653a 2050 726f 6a28 2d2d   source: Proj(--
│ │ │ │ +000146a0: 2d2d 2d5b 7820 2c20 7820 2c20 7820 2c20  ---[x , x , x , 
│ │ │ │ +000146b0: 7820 5d29 2020 2020 2020 2020 2020 2020  x ])            
│ │ │ │ +000146c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000146d0: 2020 2020 2020 2020 2020 2036 3535 3231             65521
│ │ │ │ +000146e0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ +000146f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00014710: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │  00014720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014730: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014740: 2020 7461 7267 6574 3a20 5072 6f6a 282d    target: Proj(-
│ │ │ │ -00014750: 2d2d 2d2d 5b78 202c 2078 202c 2078 202c  ----[x , x , x ,
│ │ │ │ -00014760: 2078 205d 2920 2020 2020 2020 2020 2020   x ])           
│ │ │ │ -00014770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00014780: 2020 2020 2020 2020 2020 2020 3635 3532              6552
│ │ │ │ -00014790: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ -000147a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147b0: 2020 207c 0a7c 2020 2020 2020 6465 6669     |.|      defi
│ │ │ │ -000147c0: 6e69 6e67 2066 6f72 6d73 3a20 6769 7665  ning forms: give
│ │ │ │ -000147d0: 6e20 6279 2061 2066 756e 6374 696f 6e20  n by a function 
│ │ │ │ -000147e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00014730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014740: 0a7c 2020 2020 2020 7461 7267 6574 3a20  .|      target: 
│ │ │ │ +00014750: 5072 6f6a 282d 2d2d 2d2d 5b78 202c 2078  Proj(-----[x , x
│ │ │ │ +00014760: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │ +00014770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00014780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014790: 2020 3635 3532 3120 2030 2020 2031 2020    65521  0   1  
│ │ │ │ +000147a0: 2032 2020 2033 2020 2020 2020 2020 2020   2   3          
│ │ │ │ +000147b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000147c0: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +000147d0: 3a20 6769 7665 6e20 6279 2061 2066 756e  : given by a fun
│ │ │ │ +000147e0: 6374 696f 6e20 2020 2020 2020 2020 2020  ction           
│ │ │ │ +000147f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00014800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00014830: 6f31 3620 3a20 4162 7374 7261 6374 5261  o16 : AbstractRa
│ │ │ │ -00014840: 7469 6f6e 616c 4d61 7020 2872 6174 696f  tionalMap (ratio
│ │ │ │ -00014850: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -00014860: 3320 746f 2050 505e 3329 7c0a 2b2d 2d2d  3 to PP^3)|.+---
│ │ │ │ -00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014830: 2020 207c 0a7c 6f31 3620 3a20 4162 7374     |.|o16 : Abst
│ │ │ │ +00014840: 7261 6374 5261 7469 6f6e 616c 4d61 7020  ractRationalMap 
│ │ │ │ +00014850: 2872 6174 696f 6e61 6c20 6d61 7020 6672  (rational map fr
│ │ │ │ +00014860: 6f6d 2050 505e 3320 746f 2050 505e 3329  om PP^3 to PP^3)
│ │ │ │ +00014870: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00014880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000148a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20  -------+.|i17 : 
│ │ │ │ -000148b0: 7469 6d65 2070 726f 6a65 6374 6976 6544  time projectiveD
│ │ │ │ -000148c0: 6567 7265 6573 2854 322c 3229 2020 2020  egrees(T2,2)    
│ │ │ │ -000148d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000148e0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -000148f0: 352e 3238 3735 3373 2028 6370 7529 3b20  5.28753s (cpu); 
│ │ │ │ -00014900: 322e 3930 3535 3373 2028 7468 7265 6164  2.90553s (thread
│ │ │ │ -00014910: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00014920: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000148a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000148b0: 6931 3720 3a20 7469 6d65 2070 726f 6a65  i17 : time proje
│ │ │ │ +000148c0: 6374 6976 6544 6567 7265 6573 2854 322c  ctiveDegrees(T2,
│ │ │ │ +000148d0: 3229 2020 2020 2020 2020 2020 2020 2020  2)              
│ │ │ │ +000148e0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +000148f0: 2075 7365 6420 362e 3233 3035 3673 2028   used 6.23056s (
│ │ │ │ +00014900: 6370 7529 3b20 332e 3331 3733 3973 2028  cpu); 3.31739s (
│ │ │ │ +00014910: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +00014920: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00014930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014960: 7c6f 3137 203d 2031 2020 2020 2020 2020  |o17 = 1        
│ │ │ │ +00014950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014960: 2020 2020 7c0a 7c6f 3137 203d 2031 2020      |.|o17 = 1  
│ │ │ │  00014970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014990: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000149a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000149a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000149b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000149c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000149d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520 7665  --------+..We ve
│ │ │ │ -000149e0: 7269 6679 2074 6861 7420 7468 6520 636f  rify that the co
│ │ │ │ -000149f0: 6d70 6f73 6974 696f 6e20 6f66 2054 2077  mposition of T w
│ │ │ │ -00014a00: 6974 6820 6974 7365 6c66 206c 6561 7665  ith itself leave
│ │ │ │ -00014a10: 7320 6120 7261 6e64 6f6d 2070 6f69 6e74  s a random point
│ │ │ │ -00014a20: 2066 6978 6564 3a0a 0a2b 2d2d 2d2d 2d2d   fixed:..+------
│ │ │ │ +000149d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000149e0: 0a57 6520 7665 7269 6679 2074 6861 7420  .We verify that 
│ │ │ │ +000149f0: 7468 6520 636f 6d70 6f73 6974 696f 6e20  the composition 
│ │ │ │ +00014a00: 6f66 2054 2077 6974 6820 6974 7365 6c66  of T with itself
│ │ │ │ +00014a10: 206c 6561 7665 7320 6120 7261 6e64 6f6d   leaves a random
│ │ │ │ +00014a20: 2070 6f69 6e74 2066 6978 6564 3a0a 0a2b   point fixed:..+
│ │ │ │  00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a50: 2d2d 2d2d 2b0a 7c69 3138 203a 2070 203d  ----+.|i18 : p =
│ │ │ │ -00014a60: 2061 7070 6c79 2833 2c69 2d3e 7261 6e64   apply(3,i->rand
│ │ │ │ -00014a70: 6f6d 285a 5a2f 3635 3532 3129 297c 7b31  om(ZZ/65521))|{1
│ │ │ │ -00014a80: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ +00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ +00014a60: 203a 2070 203d 2061 7070 6c79 2833 2c69   : p = apply(3,i
│ │ │ │ +00014a70: 2d3e 7261 6e64 6f6d 285a 5a2f 3635 3532  ->random(ZZ/6552
│ │ │ │ +00014a80: 3129 297c 7b31 7d7c 0a7c 2020 2020 2020  1))|{1}|.|      
│ │ │ │  00014a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014aa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014ab0: 7c6f 3138 203d 207b 3238 3936 332c 2033  |o18 = {28963, 3
│ │ │ │ -00014ac0: 3139 3735 2c20 2d33 3031 3732 2c20 317d  1975, -30172, 1}
│ │ │ │ -00014ad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ab0: 2020 2020 7c0a 7c6f 3138 203d 207b 3238      |.|o18 = {28
│ │ │ │ +00014ac0: 3936 332c 2033 3139 3735 2c20 2d33 3031  963, 31975, -301
│ │ │ │ +00014ad0: 3732 2c20 317d 2020 2020 2020 2020 2020  72, 1}          
│ │ │ │ +00014ae0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b00: 2020 2020 2020 2020 7c0a 7c6f 3138 203a          |.|o18 :
│ │ │ │ -00014b10: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +00014b00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014b10: 7c6f 3138 203a 204c 6973 7420 2020 2020  |o18 : List     
│ │ │ │  00014b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00014b30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00014b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014b60: 2d2d 2b0a 7c69 3139 203a 2071 203d 2054  --+.|i19 : q = T
│ │ │ │ -00014b70: 2070 2020 2020 2020 2020 2020 2020 2020   p              
│ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014b90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014b60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a  --------+.|i19 :
│ │ │ │ +00014b70: 2071 203d 2054 2070 2020 2020 2020 2020   q = T p        
│ │ │ │ +00014b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00014bc0: 3139 203d 207b 3331 3934 332c 2031 3633  19 = {31943, 163
│ │ │ │ -00014bd0: 3436 2c20 2d31 3539 382c 2031 7d20 2020  46, -1598, 1}   
│ │ │ │ -00014be0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014bc0: 2020 7c0a 7c6f 3139 203d 207b 3331 3934    |.|o19 = {3194
│ │ │ │ +00014bd0: 332c 2031 3633 3436 2c20 2d31 3539 382c  3, 16346, -1598,
│ │ │ │ +00014be0: 2031 7d20 2020 2020 2020 2020 2020 207c   1}            |
│ │ │ │ +00014bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00014c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c10: 2020 2020 2020 7c0a 7c6f 3139 203a 204c        |.|o19 : L
│ │ │ │ -00014c20: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00014c10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00014c20: 3139 203a 204c 6973 7420 2020 2020 2020  19 : List       
│ │ │ │  00014c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00014c40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  00014c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014c70: 2b0a 7c69 3230 203a 2054 2071 2020 2020  +.|i20 : T q    
│ │ │ │ -00014c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c70: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 2054  ------+.|i20 : T
│ │ │ │ +00014c80: 2071 2020 2020 2020 2020 2020 2020 2020   q              
│ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ -00014cd0: 203d 207b 3238 3936 332c 2033 3139 3735   = {28963, 31975
│ │ │ │ -00014ce0: 2c20 2d33 3031 3732 2c20 317d 2020 2020  , -30172, 1}    
│ │ │ │ -00014cf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00014cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014cd0: 7c0a 7c6f 3230 203d 207b 3238 3936 332c  |.|o20 = {28963,
│ │ │ │ +00014ce0: 2033 3139 3735 2c20 2d33 3031 3732 2c20   31975, -30172, 
│ │ │ │ +00014cf0: 317d 2020 2020 2020 2020 2020 207c 0a7c  1}           |.|
│ │ │ │  00014d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d20: 2020 2020 7c0a 7c6f 3230 203a 204c 6973      |.|o20 : Lis
│ │ │ │ -00014d30: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00014d20: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ +00014d30: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00014d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00014d50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00014d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00014d80: 0a57 6520 6e6f 7720 636f 6d70 7574 6520  .We now compute 
│ │ │ │ -00014d90: 7468 6520 636f 6e63 7265 7465 2072 6174  the concrete rat
│ │ │ │ -00014da0: 696f 6e61 6c20 6d61 7020 636f 7272 6573  ional map corres
│ │ │ │ -00014db0: 706f 6e64 696e 6720 746f 2054 3a0a 0a2b  ponding to T:..+
│ │ │ │ -00014dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014d80: 2d2d 2d2d 2b0a 0a57 6520 6e6f 7720 636f  ----+..We now co
│ │ │ │ +00014d90: 6d70 7574 6520 7468 6520 636f 6e63 7265  mpute the concre
│ │ │ │ +00014da0: 7465 2072 6174 696f 6e61 6c20 6d61 7020  te rational map 
│ │ │ │ +00014db0: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ +00014dc0: 2054 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d   T:..+----------
│ │ │ │  00014dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014e00: 2d2b 0a7c 6932 3120 3a20 7469 6d65 2066  -+.|i21 : time f
│ │ │ │ -00014e10: 203d 2072 6174 696f 6e61 6c4d 6170 2054   = rationalMap T
│ │ │ │ -00014e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20  -------+.|i21 : 
│ │ │ │ +00014e10: 7469 6d65 2066 203d 2072 6174 696f 6e61  time f = rationa
│ │ │ │ +00014e20: 6c4d 6170 2054 2020 2020 2020 2020 2020  lMap T          
│ │ │ │  00014e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e40: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00014e50: 2034 2e31 3032 3537 7320 2863 7075 293b   4.10257s (cpu);
│ │ │ │ -00014e60: 2032 2e32 3537 3135 7320 2874 6872 6561   2.25715s (threa
│ │ │ │ -00014e70: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ -00014e80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e40: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00014e50: 2d20 7573 6564 2035 2e32 3832 3031 7320  - used 5.28201s 
│ │ │ │ +00014e60: 2863 7075 293b 2033 2e34 3636 3937 7320  (cpu); 3.46697s 
│ │ │ │ +00014e70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00014e80: 2920 2020 2020 2020 2020 2020 2020 207c  )              |
│ │ │ │ +00014e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00014ed0: 6f32 3120 3d20 2d2d 2072 6174 696f 6e61  o21 = -- rationa
│ │ │ │ -00014ee0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ed0: 2020 207c 0a7c 6f32 3120 3d20 2d2d 2072     |.|o21 = -- r
│ │ │ │ +00014ee0: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │  00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00014f20: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ -00014f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00014f20: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +00014f30: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  00014f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f50: 2020 2020 207c 0a7c 2020 2020 2020 736f       |.|      so
│ │ │ │ -00014f60: 7572 6365 3a20 5072 6f6a 282d 2d2d 2d2d  urce: Proj(-----
│ │ │ │ -00014f70: 5b78 202c 2078 202c 2078 202c 2078 205d  [x , x , x , x ]
│ │ │ │ -00014f80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00014f90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014fa0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00014fb0: 3535 3231 2020 3020 2020 3120 2020 3220  5521  0   1   2 
│ │ │ │ -00014fc0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00014fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ff0: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │ +00014f50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00014f60: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ +00014f70: 282d 2d2d 2d2d 5b78 202c 2078 202c 2078  (-----[x , x , x
│ │ │ │ +00014f80: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │ +00014f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014fb0: 2020 2020 2036 3535 3231 2020 3020 2020       65521  0   
│ │ │ │ +00014fc0: 3120 2020 3220 2020 3320 2020 2020 2020  1   2   3       
│ │ │ │ +00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014fe0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00014ff0: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │  00015000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015020: 207c 0a7c 2020 2020 2020 7461 7267 6574   |.|      target
│ │ │ │ -00015030: 3a20 5072 6f6a 282d 2d2d 2d2d 5b78 202c  : Proj(-----[x ,
│ │ │ │ -00015040: 2078 202c 2078 202c 2078 205d 2920 2020   x , x , x ])   
│ │ │ │ -00015050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00015070: 2020 2020 2020 2020 2020 2036 3535 3231             65521
│ │ │ │ -00015080: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ -00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000150b0: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ -000150c0: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │ +00015020: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015030: 7461 7267 6574 3a20 5072 6f6a 282d 2d2d  target: Proj(---
│ │ │ │ +00015040: 2d2d 5b78 202c 2078 202c 2078 202c 2078  --[x , x , x , x
│ │ │ │ +00015050: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │ +00015060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015080: 2036 3535 3231 2020 3020 2020 3120 2020   65521  0   1   
│ │ │ │ +00015090: 3220 2020 3320 2020 2020 2020 2020 2020  2   3           
│ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000150b0: 0a7c 2020 2020 2020 6465 6669 6e69 6e67  .|      defining
│ │ │ │ +000150c0: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │  000150d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000150f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015100: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -00015110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015120: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00015130: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00015140: 2020 2020 2020 2020 2020 202d 2078 2020             - x  
│ │ │ │ -00015150: 2d20 3332 3735 3978 2078 2078 2020 2b20  - 32759x x x  + 
│ │ │ │ -00015160: 3332 3736 3078 2078 202c 2020 2020 2020  32760x x ,      
│ │ │ │ -00015170: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000150e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000150f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015110: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00015120: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00015130: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015150: 202d 2078 2020 2d20 3332 3735 3978 2078   - x  - 32759x x
│ │ │ │ +00015160: 2078 2020 2b20 3332 3736 3078 2078 202c   x  + 32760x x ,
│ │ │ │ +00015170: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00015180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015190: 2020 3120 2020 2020 2020 2020 3020 3120    1         0 1 
│ │ │ │ -000151a0: 3220 2020 2020 2020 2020 3020 3320 2020  2         0 3   
│ │ │ │ -000151b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000151c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015190: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +000151a0: 2020 3020 3120 3220 2020 2020 2020 2020    0 1 2         
│ │ │ │ +000151b0: 3020 3320 2020 2020 2020 2020 2020 207c  0 3            |
│ │ │ │ +000151c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000151d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000151e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000151f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00015220: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015200: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015220: 2020 2032 2020 2020 2020 2020 3220 2020     2        2   
│ │ │ │  00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00015250: 2020 2020 2020 2020 2020 2033 3237 3630             32760
│ │ │ │ -00015260: 7820 7820 202b 2078 2078 2020 2b20 3332  x x  + x x  + 32
│ │ │ │ -00015270: 3736 3078 2078 2078 202c 2020 2020 2020  760x x x ,      
│ │ │ │ -00015280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015240: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015260: 2033 3237 3630 7820 7820 202b 2078 2078   32760x x  + x x
│ │ │ │ +00015270: 2020 2b20 3332 3736 3078 2078 2078 202c    + 32760x x x ,
│ │ │ │ +00015280: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152a0: 2020 2020 2031 2032 2020 2020 3020 3220       1 2    0 2 
│ │ │ │ -000152b0: 2020 2020 2020 2020 3020 3120 3320 2020          0 1 3   
│ │ │ │ -000152c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000152d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000152a0: 2020 2020 2020 2020 2020 2031 2032 2020             1 2  
│ │ │ │ +000152b0: 2020 3020 3220 2020 2020 2020 2020 3020    0 2         0 
│ │ │ │ +000152c0: 3120 3320 2020 2020 2020 2020 2020 207c  1 3            |
│ │ │ │ +000152d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000152e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000152f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015310: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015330: 2032 2020 2020 3220 2020 2020 2020 2020   2    2         
│ │ │ │ +00015330: 2020 2020 2020 2032 2020 2020 3220 2020         2    2   
│ │ │ │  00015340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015350: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00015360: 2020 2020 2020 2020 2020 202d 2033 3237             - 327
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│ │ │ │ +000158e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  00015950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00015960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00015b90: 6163 7452 6174 696f 6e61 6c4d 6170 2c20  actRationalMap, 
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│ │ │ │ +00015a60: 6961 6c52 696e 672c 4675 6e63 7469 6f6e  ialRing,Function
│ │ │ │ +00015a70: 436c 6f73 7572 652c 5a5a 2922 0a20 202a  Closure,ZZ)".  *
│ │ │ │ +00015a80: 2022 6162 7374 7261 6374 5261 7469 6f6e   "abstractRation
│ │ │ │ +00015a90: 616c 4d61 7028 5261 7469 6f6e 616c 4d61  alMap(RationalMa
│ │ │ │ +00015aa0: 7029 220a 0a46 6f72 2074 6865 2070 726f  p)"..For the pro
│ │ │ │ +00015ab0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
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│ │ │ │ +00015ad0: 6f62 6a65 6374 202a 6e6f 7465 2061 6273  object *note abs
│ │ │ │ +00015ae0: 7472 6163 7452 6174 696f 6e61 6c4d 6170  tractRationalMap
│ │ │ │ +00015af0: 3a20 6162 7374 7261 6374 5261 7469 6f6e  : abstractRation
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│ │ │ │ +00015b50: 612e 696e 666f 2c20 4e6f 6465 3a20 6170  a.info, Node: ap
│ │ │ │ +00015b60: 7072 6f78 696d 6174 6549 6e76 6572 7365  proximateInverse
│ │ │ │ +00015b70: 4d61 702c 204e 6578 743a 2042 6c6f 7755  Map, Next: BlowU
│ │ │ │ +00015b80: 7053 7472 6174 6567 792c 2050 7265 763a  pStrategy, Prev:
│ │ │ │ +00015b90: 2061 6273 7472 6163 7452 6174 696f 6e61   abstractRationa
│ │ │ │ +00015ba0: 6c4d 6170 2c20 5570 3a20 546f 700a 0a61  lMap, Up: Top..a
│ │ │ │ +00015bb0: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ +00015bc0: 654d 6170 202d 2d20 7261 6e64 6f6d 206d  eMap -- random m
│ │ │ │ +00015bd0: 6170 2072 656c 6174 6564 2074 6f20 7468  ap related to th
│ │ │ │ +00015be0: 6520 696e 7665 7273 6520 6f66 2061 2062  e inverse of a b
│ │ │ │ +00015bf0: 6972 6174 696f 6e61 6c20 6d61 700a 2a2a  irational map.**
│ │ │ │  00015c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00015c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00015c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00015c30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00015c40: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -00015c50: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ -00015c60: 7361 6765 3a20 0a20 2020 2020 2020 2061  sage: .        a
│ │ │ │ -00015c70: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ -00015c80: 654d 6170 2070 6869 200a 2020 2020 2020  eMap phi .      
│ │ │ │ -00015c90: 2020 6170 7072 6f78 696d 6174 6549 6e76    approximateInv
│ │ │ │ -00015ca0: 6572 7365 4d61 7028 7068 692c 6429 0a20  erseMap(phi,d). 
│ │ │ │ -00015cb0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00015cc0: 202a 2070 6869 2c20 6120 2a6e 6f74 6520   * phi, a *note 
│ │ │ │ -00015cd0: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ -00015ce0: 7469 6f6e 616c 4d61 702c 2c20 6120 6269  tionalMap,, a bi
│ │ │ │ -00015cf0: 7261 7469 6f6e 616c 206d 6170 0a20 2020  rational map.   
│ │ │ │ -00015d00: 2020 202a 2064 2c20 616e 202a 6e6f 7465     * d, an *note
│ │ │ │ -00015d10: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -00015d20: 6c61 7932 446f 6329 5a5a 2c2c 206f 7074  lay2Doc)ZZ,, opt
│ │ │ │ -00015d30: 696f 6e61 6c2c 2062 7574 2069 7420 7368  ional, but it sh
│ │ │ │ -00015d40: 6f75 6c64 2062 6520 7468 650a 2020 2020  ould be the.    
│ │ │ │ -00015d50: 2020 2020 6465 6772 6565 206f 6620 7468      degree of th
│ │ │ │ -00015d60: 6520 666f 726d 7320 6465 6669 6e69 6e67  e forms defining
│ │ │ │ -00015d70: 2074 6865 2069 6e76 6572 7365 206f 6620   the inverse of 
│ │ │ │ -00015d80: 7068 690a 2020 2a20 2a6e 6f74 6520 4f70  phi.  * *note Op
│ │ │ │ -00015d90: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -00015da0: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -00015db0: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -00015dc0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -00015dd0: 732c 3a0a 2020 2020 2020 2a20 2a6e 6f74  s,:.      * *not
│ │ │ │ -00015de0: 6520 4365 7274 6966 793a 2043 6572 7469  e Certify: Certi
│ │ │ │ -00015df0: 6679 2c20 3d3e 202e 2e2e 2c20 6465 6661  fy, => ..., defa
│ │ │ │ -00015e00: 756c 7420 7661 6c75 6520 6661 6c73 652c  ult value false,
│ │ │ │ -00015e10: 2077 6865 7468 6572 2074 6f20 656e 7375   whether to ensu
│ │ │ │ -00015e20: 7265 0a20 2020 2020 2020 2063 6f72 7265  re.        corre
│ │ │ │ -00015e30: 6374 6e65 7373 206f 6620 6f75 7470 7574  ctness of output
│ │ │ │ -00015e40: 0a20 2020 2020 202a 202a 6e6f 7465 2043  .      * *note C
│ │ │ │ -00015e50: 6f64 696d 4273 496e 763a 2043 6f64 696d  odimBsInv: Codim
│ │ │ │ -00015e60: 4273 496e 762c 203d 3e20 2e2e 2e2c 2064  BsInv, => ..., d
│ │ │ │ -00015e70: 6566 6175 6c74 2076 616c 7565 206e 756c  efault value nul
│ │ │ │ -00015e80: 6c2c 200a 2020 2020 2020 2a20 2a6e 6f74  l, .      * *not
│ │ │ │ -00015e90: 6520 5665 7262 6f73 653a 2069 6e76 6572  e Verbose: inver
│ │ │ │ -00015ea0: 7365 4d61 705f 6c70 5f70 645f 7064 5f70  seMap_lp_pd_pd_p
│ │ │ │ -00015eb0: 645f 636d 5665 7262 6f73 653d 3e5f 7064  d_cmVerbose=>_pd
│ │ │ │ -00015ec0: 5f70 645f 7064 5f72 702c 203d 3e20 2e2e  _pd_pd_rp, => ..
│ │ │ │ -00015ed0: 2e2c 0a20 2020 2020 2020 2064 6566 6175  .,.        defau
│ │ │ │ -00015ee0: 6c74 2076 616c 7565 2074 7275 652c 0a20  lt value true,. 
│ │ │ │ -00015ef0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00015f00: 2020 2a20 6120 2a6e 6f74 6520 7261 7469    * a *note rati
│ │ │ │ -00015f10: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -00015f20: 616c 4d61 702c 2c20 6120 7261 6e64 6f6d  alMap,, a random
│ │ │ │ -00015f30: 2072 6174 696f 6e61 6c20 6d61 7020 7768   rational map wh
│ │ │ │ -00015f40: 6963 6820 696e 2073 6f6d 650a 2020 2020  ich in some.    
│ │ │ │ -00015f50: 2020 2020 7365 6e73 6520 6973 2072 656c      sense is rel
│ │ │ │ -00015f60: 6174 6564 2074 6f20 7468 6520 696e 7665  ated to the inve
│ │ │ │ -00015f70: 7273 6520 6f66 2070 6869 2028 652e 672e  rse of phi (e.g.
│ │ │ │ -00015f80: 2c20 7468 6579 2073 686f 756c 6420 6861  , they should ha
│ │ │ │ -00015f90: 7665 2074 6865 2073 616d 650a 2020 2020  ve the same.    
│ │ │ │ -00015fa0: 2020 2020 6261 7365 206c 6f63 7573 290a      base locus).
│ │ │ │ -00015fb0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ -00015fc0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 616c  ========..The al
│ │ │ │ -00015fd0: 676f 7269 7468 6d20 6973 2074 6f20 7472  gorithm is to tr
│ │ │ │ -00015fe0: 7920 746f 2063 6f6e 7374 7275 6374 2074  y to construct t
│ │ │ │ -00015ff0: 6865 2069 6465 616c 206f 6620 7468 6520  he ideal of the 
│ │ │ │ -00016000: 6261 7365 206c 6f63 7573 206f 6620 7468  base locus of th
│ │ │ │ -00016010: 6520 696e 7665 7273 650a 6279 206c 6f6f  e inverse.by loo
│ │ │ │ -00016020: 6b69 6e67 2066 6f72 2074 6865 2069 6d61  king for the ima
│ │ │ │ -00016030: 6765 7320 7669 6120 7068 6920 6f66 2072  ges via phi of r
│ │ │ │ -00016040: 616e 646f 6d20 6c69 6e65 6172 2073 6563  andom linear sec
│ │ │ │ -00016050: 7469 6f6e 7320 6f66 2074 6865 2073 6f75  tions of the sou
│ │ │ │ -00016060: 7263 650a 7661 7269 6574 792e 2047 656e  rce.variety. Gen
│ │ │ │ -00016070: 6572 616c 6c79 2c20 6f6e 6520 6361 6e20  erally, one can 
│ │ │ │ -00016080: 7370 6565 6420 7570 2074 6865 2070 726f  speed up the pro
│ │ │ │ -00016090: 6365 7373 2062 7920 7061 7373 696e 6720  cess by passing 
│ │ │ │ -000160a0: 7468 726f 7567 6820 7468 6520 6f70 7469  through the opti
│ │ │ │ -000160b0: 6f6e 0a2a 6e6f 7465 2043 6f64 696d 4273  on.*note CodimBs
│ │ │ │ -000160c0: 496e 763a 2043 6f64 696d 4273 496e 762c  Inv: CodimBsInv,
│ │ │ │ -000160d0: 2061 2067 6f6f 6420 6c6f 7765 7220 626f   a good lower bo
│ │ │ │ -000160e0: 756e 6420 666f 7220 7468 6520 636f 6469  und for the codi
│ │ │ │ -000160f0: 6d65 6e73 696f 6e20 6f66 2074 6869 730a  mension of this.
│ │ │ │ -00016100: 6261 7365 206c 6f63 7573 2e0a 0a2b 2d2d  base locus...+--
│ │ │ │ -00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ +00015c50: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ +00015c60: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00015c70: 2020 2020 2061 7070 726f 7869 6d61 7465       approximate
│ │ │ │ +00015c80: 496e 7665 7273 654d 6170 2070 6869 200a  InverseMap phi .
│ │ │ │ +00015c90: 2020 2020 2020 2020 6170 7072 6f78 696d          approxim
│ │ │ │ +00015ca0: 6174 6549 6e76 6572 7365 4d61 7028 7068  ateInverseMap(ph
│ │ │ │ +00015cb0: 692c 6429 0a20 202a 2049 6e70 7574 733a  i,d).  * Inputs:
│ │ │ │ +00015cc0: 0a20 2020 2020 202a 2070 6869 2c20 6120  .      * phi, a 
│ │ │ │ +00015cd0: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +00015ce0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +00015cf0: 2c20 6120 6269 7261 7469 6f6e 616c 206d  , a birational m
│ │ │ │ +00015d00: 6170 0a20 2020 2020 202a 2064 2c20 616e  ap.      * d, an
│ │ │ │ +00015d10: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +00015d20: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +00015d30: 2c2c 206f 7074 696f 6e61 6c2c 2062 7574  ,, optional, but
│ │ │ │ +00015d40: 2069 7420 7368 6f75 6c64 2062 6520 7468   it should be th
│ │ │ │ +00015d50: 650a 2020 2020 2020 2020 6465 6772 6565  e.        degree
│ │ │ │ +00015d60: 206f 6620 7468 6520 666f 726d 7320 6465   of the forms de
│ │ │ │ +00015d70: 6669 6e69 6e67 2074 6865 2069 6e76 6572  fining the inver
│ │ │ │ +00015d80: 7365 206f 6620 7068 690a 2020 2a20 2a6e  se of phi.  * *n
│ │ │ │ +00015d90: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ +00015da0: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ +00015db0: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ +00015dc0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +00015dd0: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ +00015de0: 2a20 2a6e 6f74 6520 4365 7274 6966 793a  * *note Certify:
│ │ │ │ +00015df0: 2043 6572 7469 6679 2c20 3d3e 202e 2e2e   Certify, => ...
│ │ │ │ +00015e00: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00015e10: 6661 6c73 652c 2077 6865 7468 6572 2074  false, whether t
│ │ │ │ +00015e20: 6f20 656e 7375 7265 0a20 2020 2020 2020  o ensure.       
│ │ │ │ +00015e30: 2063 6f72 7265 6374 6e65 7373 206f 6620   correctness of 
│ │ │ │ +00015e40: 6f75 7470 7574 0a20 2020 2020 202a 202a  output.      * *
│ │ │ │ +00015e50: 6e6f 7465 2043 6f64 696d 4273 496e 763a  note CodimBsInv:
│ │ │ │ +00015e60: 2043 6f64 696d 4273 496e 762c 203d 3e20   CodimBsInv, => 
│ │ │ │ +00015e70: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00015e80: 7565 206e 756c 6c2c 200a 2020 2020 2020  ue null, .      
│ │ │ │ +00015e90: 2a20 2a6e 6f74 6520 5665 7262 6f73 653a  * *note Verbose:
│ │ │ │ +00015ea0: 2069 6e76 6572 7365 4d61 705f 6c70 5f70   inverseMap_lp_p
│ │ │ │ +00015eb0: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ +00015ec0: 653d 3e5f 7064 5f70 645f 7064 5f72 702c  e=>_pd_pd_pd_rp,
│ │ │ │ +00015ed0: 203d 3e20 2e2e 2e2c 0a20 2020 2020 2020   => ...,.       
│ │ │ │ +00015ee0: 2064 6566 6175 6c74 2076 616c 7565 2074   default value t
│ │ │ │ +00015ef0: 7275 652c 0a20 202a 204f 7574 7075 7473  rue,.  * Outputs
│ │ │ │ +00015f00: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74  :.      * a *not
│ │ │ │ +00015f10: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ +00015f20: 5261 7469 6f6e 616c 4d61 702c 2c20 6120  RationalMap,, a 
│ │ │ │ +00015f30: 7261 6e64 6f6d 2072 6174 696f 6e61 6c20  random rational 
│ │ │ │ +00015f40: 6d61 7020 7768 6963 6820 696e 2073 6f6d  map which in som
│ │ │ │ +00015f50: 650a 2020 2020 2020 2020 7365 6e73 6520  e.        sense 
│ │ │ │ +00015f60: 6973 2072 656c 6174 6564 2074 6f20 7468  is related to th
│ │ │ │ +00015f70: 6520 696e 7665 7273 6520 6f66 2070 6869  e inverse of phi
│ │ │ │ +00015f80: 2028 652e 672e 2c20 7468 6579 2073 686f   (e.g., they sho
│ │ │ │ +00015f90: 756c 6420 6861 7665 2074 6865 2073 616d  uld have the sam
│ │ │ │ +00015fa0: 650a 2020 2020 2020 2020 6261 7365 206c  e.        base l
│ │ │ │ +00015fb0: 6f63 7573 290a 0a44 6573 6372 6970 7469  ocus)..Descripti
│ │ │ │ +00015fc0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00015fd0: 5468 6520 616c 676f 7269 7468 6d20 6973  The algorithm is
│ │ │ │ +00015fe0: 2074 6f20 7472 7920 746f 2063 6f6e 7374   to try to const
│ │ │ │ +00015ff0: 7275 6374 2074 6865 2069 6465 616c 206f  ruct the ideal o
│ │ │ │ +00016000: 6620 7468 6520 6261 7365 206c 6f63 7573  f the base locus
│ │ │ │ +00016010: 206f 6620 7468 6520 696e 7665 7273 650a   of the inverse.
│ │ │ │ +00016020: 6279 206c 6f6f 6b69 6e67 2066 6f72 2074  by looking for t
│ │ │ │ +00016030: 6865 2069 6d61 6765 7320 7669 6120 7068  he images via ph
│ │ │ │ +00016040: 6920 6f66 2072 616e 646f 6d20 6c69 6e65  i of random line
│ │ │ │ +00016050: 6172 2073 6563 7469 6f6e 7320 6f66 2074  ar sections of t
│ │ │ │ +00016060: 6865 2073 6f75 7263 650a 7661 7269 6574  he source.variet
│ │ │ │ +00016070: 792e 2047 656e 6572 616c 6c79 2c20 6f6e  y. Generally, on
│ │ │ │ +00016080: 6520 6361 6e20 7370 6565 6420 7570 2074  e can speed up t
│ │ │ │ +00016090: 6865 2070 726f 6365 7373 2062 7920 7061  he process by pa
│ │ │ │ +000160a0: 7373 696e 6720 7468 726f 7567 6820 7468  ssing through th
│ │ │ │ +000160b0: 6520 6f70 7469 6f6e 0a2a 6e6f 7465 2043  e option.*note C
│ │ │ │ +000160c0: 6f64 696d 4273 496e 763a 2043 6f64 696d  odimBsInv: Codim
│ │ │ │ +000160d0: 4273 496e 762c 2061 2067 6f6f 6420 6c6f  BsInv, a good lo
│ │ │ │ +000160e0: 7765 7220 626f 756e 6420 666f 7220 7468  wer bound for th
│ │ │ │ +000160f0: 6520 636f 6469 6d65 6e73 696f 6e20 6f66  e codimension of
│ │ │ │ +00016100: 2074 6869 730a 6261 7365 206c 6f63 7573   this.base locus
│ │ │ │ +00016110: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  00016120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00016160: 203a 2050 3820 3d20 5a5a 2f39 375b 745f   : P8 = ZZ/97[t_
│ │ │ │ -00016170: 302e 2e74 5f38 5d3b 2020 2020 2020 2020  0..t_8];        
│ │ │ │ +00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016160: 2d2b 0a7c 6931 203a 2050 3820 3d20 5a5a  -+.|i1 : P8 = ZZ
│ │ │ │ +00016170: 2f39 375b 745f 302e 2e74 5f38 5d3b 2020  /97[t_0..t_8];  
│ │ │ │  00016180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000161a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000161b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00016200: 203a 2070 6869 203d 2069 6e76 6572 7365   : phi = inverse
│ │ │ │ -00016210: 4d61 7020 7261 7469 6f6e 616c 4d61 7028  Map rationalMap(
│ │ │ │ -00016220: 7472 696d 286d 696e 6f72 7328 322c 6765  trim(minors(2,ge
│ │ │ │ -00016230: 6e65 7269 634d 6174 7269 7828 5038 2c33  nericMatrix(P8,3
│ │ │ │ -00016240: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000161f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016200: 2d2b 0a7c 6932 203a 2070 6869 203d 2069  -+.|i2 : phi = i
│ │ │ │ +00016210: 6e76 6572 7365 4d61 7020 7261 7469 6f6e  nverseMap ration
│ │ │ │ +00016220: 616c 4d61 7028 7472 696d 286d 696e 6f72  alMap(trim(minor
│ │ │ │ +00016230: 7328 322c 6765 6e65 7269 634d 6174 7269  s(2,genericMatri
│ │ │ │ +00016240: 7828 5038 2c33 2020 2020 2020 2020 2020  x(P8,3          
│ │ │ │ +00016250: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00016260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016290: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -000162a0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ -000162b0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │ +00016290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000162a0: 207c 0a7c 6f32 203d 202d 2d20 7261 7469   |.|o2 = -- rati
│ │ │ │ +000162b0: 6f6e 616c 206d 6170 202d 2d20 2020 2020  onal map --     
│ │ │ │  000162c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000162d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000162e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000162f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016300: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
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│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00016c40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -00016c60: 2020 2020 7820 7820 202d 2078 2078 202c      x x  - x x ,
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│ │ │ │  00016d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00016d50: 2020 2020 7820 7820 202d 2078 2078 202c      x x  - x x ,
│ │ │ │ -00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017be0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017cd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -00017d80: 6f20 5050 5e38 2920 2020 2020 2020 2020  o PP^8)         
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│ │ │ │  00017e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017e90: 202b 2031 3178 2078 2078 2078 2020 2d20   + 11x x x x  - 
│ │ │ │ -00017ea0: 3139 7820 7820 7820 7820 202b 2032 3178  19x x x x  + 21x
│ │ │ │ -00017eb0: 2078 2078 2078 2020 2b20 207c 0a7c 2020   x x x  +  |.|  
│ │ │ │ -00017ec0: 2030 2031 2033 2035 2020 2020 2020 3120   0 1 3 5      1 
│ │ │ │ -00017ed0: 3220 3320 3520 2020 2020 2030 2034 2035  2 3 5      0 4 5
│ │ │ │ -00017ee0: 2020 2020 2020 3020 3220 3420 3520 2020        0 2 4 5   
│ │ │ │ -00017ef0: 2020 2030 2033 2034 2035 2020 2020 2020     0 3 4 5      
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│ │ │ │ +00017e70: 207c 0a7c 3435 7820 7820 7820 7820 202d   |.|45x x x x  -
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│ │ │ │ +00017e90: 7820 7820 7820 202b 2031 3178 2078 2078  x x x  + 11x x x
│ │ │ │ +00017ea0: 2078 2020 2d20 3139 7820 7820 7820 7820   x  - 19x x x x 
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│ │ │ │ +00017ec0: 207c 0a7c 2020 2030 2031 2033 2035 2020   |.|   0 1 3 5  
│ │ │ │ +00017ed0: 2020 2020 3120 3220 3320 3520 2020 2020      1 2 3 5     
│ │ │ │ +00017ee0: 2030 2034 2035 2020 2020 2020 3020 3220   0 4 5      0 2 
│ │ │ │ +00017ef0: 3420 3520 2020 2020 2030 2033 2034 2035  4 5      0 3 4 5
│ │ │ │ +00017f00: 2020 2020 2020 3220 3320 3420 3520 2020        2 3 4 5   
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│ │ │ │  00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00017f60: 2032 2032 2020 2020 2020 2020 2020 3220   2 2          2 
│ │ │ │ -00017f70: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ -00017f80: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00017f90: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00017fa0: 2020 2020 2020 2020 2020 207c 0a7c 3236             |.|26
│ │ │ │ -00017fb0: 7820 7820 202b 2031 3978 2078 2078 2020  x x  + 19x x x  
│ │ │ │ -00017fc0: 2b20 3338 7820 7820 202d 2033 3078 2078  + 38x x  - 30x x
│ │ │ │ -00017fd0: 2078 2020 2d20 3131 7820 7820 7820 7820   x  - 11x x x x 
│ │ │ │ -00017fe0: 202d 2031 3178 2078 2078 2020 2b20 3330   - 11x x x  + 30
│ │ │ │ -00017ff0: 7820 7820 7820 7820 202b 207c 0a7c 2020  x x x x  + |.|  
│ │ │ │ -00018000: 2030 2035 2020 2020 2020 3020 3220 3520   0 5      0 2 5 
│ │ │ │ -00018010: 2020 2020 2032 2035 2020 2020 2020 3120       2 5      1 
│ │ │ │ -00018020: 3220 3620 2020 2020 2031 2032 2033 2036  2 6      1 2 3 6
│ │ │ │ -00018030: 2020 2020 2020 3120 3320 3620 2020 2020        1 3 6     
│ │ │ │ -00018040: 2030 2032 2034 2036 2020 207c 0a7c 2d2d   0 2 4 6   |.|--
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│ │ │ │ +00017f60: 2d7c 0a7c 2020 2032 2032 2020 2020 2020  -|.|   2 2      
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│ │ │ │ +00017f80: 2020 2020 2020 3220 2020 2020 2020 2020        2         
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│ │ │ │ +00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017fb0: 207c 0a7c 3236 7820 7820 202b 2031 3978   |.|26x x  + 19x
│ │ │ │ +00017fc0: 2078 2078 2020 2b20 3338 7820 7820 202d   x x  + 38x x  -
│ │ │ │ +00017fd0: 2033 3078 2078 2078 2020 2d20 3131 7820   30x x x  - 11x 
│ │ │ │ +00017fe0: 7820 7820 7820 202d 2031 3178 2078 2078  x x x  - 11x x x
│ │ │ │ +00017ff0: 2020 2b20 3330 7820 7820 7820 7820 202b    + 30x x x x  +
│ │ │ │ +00018000: 207c 0a7c 2020 2030 2035 2020 2020 2020   |.|   0 5      
│ │ │ │ +00018010: 3020 3220 3520 2020 2020 2032 2035 2020  0 2 5      2 5  
│ │ │ │ +00018020: 2020 2020 3120 3220 3620 2020 2020 2031      1 2 6      1
│ │ │ │ +00018030: 2032 2033 2036 2020 2020 2020 3120 3320   2 3 6      1 3 
│ │ │ │ +00018040: 3620 2020 2020 2030 2032 2034 2036 2020  6      0 2 4 6  
│ │ │ │ +00018050: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
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│ │ │ │  00018080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180b0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000180a0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000180d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180e0: 2020 2020 2020 2020 2020 207c 0a7c 3232             |.|22
│ │ │ │ -000180f0: 7820 7820 7820 7820 202b 2031 3178 2078  x x x x  + 11x x
│ │ │ │ -00018100: 2078 2078 2020 2d20 3232 7820 7820 7820   x x  - 22x x x 
│ │ │ │ -00018110: 202b 2031 3078 2078 2078 2078 2020 2b20   + 10x x x x  + 
│ │ │ │ -00018120: 3131 7820 7820 7820 7820 202d 2034 3278  11x x x x  - 42x
│ │ │ │ -00018130: 2078 2078 2078 2020 2d20 207c 0a7c 2020   x x x  -  |.|  
│ │ │ │ -00018140: 2031 2032 2034 2036 2020 2020 2020 3020   1 2 4 6      0 
│ │ │ │ -00018150: 3320 3420 3620 2020 2020 2030 2034 2036  3 4 6      0 4 6
│ │ │ │ -00018160: 2020 2020 2020 3120 3220 3520 3620 2020        1 2 5 6   
│ │ │ │ -00018170: 2020 2030 2033 2035 2036 2020 2020 2020     0 3 5 6      
│ │ │ │ -00018180: 3120 3320 3520 3620 2020 207c 0a7c 2d2d  1 3 5 6    |.|--
│ │ │ │ -00018190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00018130: 202d 2034 3278 2078 2078 2078 2020 2d20   - 42x x x x  - 
│ │ │ │ +00018140: 207c 0a7c 2020 2031 2032 2034 2036 2020   |.|   1 2 4 6  
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│ │ │ │ +00018160: 2030 2034 2036 2020 2020 2020 3120 3220   0 4 6      1 2 
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│ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181f0: 3220 2020 2020 2020 2032 2032 2020 2020  2        2 2    
│ │ │ │ -00018200: 2020 2020 2020 3220 2020 2020 3220 3220        2     2 2 
│ │ │ │ -00018210: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00018220: 3220 3220 2020 2020 2020 207c 0a7c 3130  2 2        |.|10
│ │ │ │ -00018230: 7820 7820 7820 7820 202b 2034 3278 2078  x x x x  + 42x x
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│ │ │ │ -00018270: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -00018280: 2030 2034 2035 2036 2020 2020 2020 3020   0 4 5 6      0 
│ │ │ │ -00018290: 3520 3620 2020 2020 2032 2036 2020 2020  5 6      2 6    
│ │ │ │ -000182a0: 2020 3220 3320 3620 2020 2020 3320 3620    2 3 6     3 6 
│ │ │ │ -000182b0: 2020 2020 2032 2034 2036 2020 2020 2020       2 4 6      
│ │ │ │ -000182c0: 3420 3620 2020 2020 2020 207c 0a7c 2d2d  4 6        |.|--
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│ │ │ │ +000181e0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +000181f0: 2020 2020 2020 3220 2020 2020 2020 2032        2        2
│ │ │ │ +00018200: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
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│ │ │ │ +00018220: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ +00018230: 207c 0a7c 3130 7820 7820 7820 7820 202b   |.|10x x x x  +
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│ │ │ │ +00018250: 7820 202d 2033 3778 2078 2078 2020 2b20  x  - 37x x x  + 
│ │ │ │ +00018260: 3778 2078 2020 2b20 3238 7820 7820 7820  7x x  + 28x x x 
│ │ │ │ +00018270: 202d 2033 3878 2078 2020 2d20 2020 2020   - 38x x  -     
│ │ │ │ +00018280: 207c 0a7c 2020 2030 2034 2035 2036 2020   |.|   0 4 5 6  
│ │ │ │ +00018290: 2020 2020 3020 3520 3620 2020 2020 2032      0 5 6      2
│ │ │ │ +000182a0: 2036 2020 2020 2020 3220 3320 3620 2020   6      2 3 6   
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│ │ │ │ +000182c0: 2020 2020 2020 3420 3620 2020 2020 2020        4 6       
│ │ │ │ +000182d0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00018330: 2020 2020 2020 2020 2020 3220 2020 2020            2     
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│ │ │ │ -00018360: 2020 2020 2020 2020 2020 207c 0a7c 3378             |.|3x
│ │ │ │ -00018370: 2078 2078 2020 2d20 3239 7820 7820 7820   x x  - 29x x x 
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│ │ │ │ -000183b0: 3078 2078 2078 2078 2020 2d7c 0a7c 2020  0x x x x  -|.|  
│ │ │ │ -000183c0: 3220 3520 3620 2020 2020 2033 2035 2036  2 5 6      3 5 6
│ │ │ │ -000183d0: 2020 2020 2020 3420 3520 3620 2020 2020        4 5 6     
│ │ │ │ -000183e0: 2035 2036 2020 2020 2020 3020 3120 3220   5 6      0 1 2 
│ │ │ │ -000183f0: 3720 2020 2020 2031 2032 2037 2020 2020  7      1 2 7    
│ │ │ │ -00018400: 2020 3120 3220 3320 3720 207c 0a7c 2d2d    1 2 3 7  |.|--
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│ │ │ │ +00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00018390: 2020 2b20 3336 7820 7820 202b 2033 3078    + 36x x  + 30x
│ │ │ │ +000183a0: 2078 2078 2078 2020 2d20 3232 7820 7820   x x x  - 22x x 
│ │ │ │ +000183b0: 7820 202d 2032 3078 2078 2078 2078 2020  x  - 20x x x x  
│ │ │ │ +000183c0: 2d7c 0a7c 2020 3220 3520 3620 2020 2020  -|.|  2 5 6     
│ │ │ │ +000183d0: 2033 2035 2036 2020 2020 2020 3420 3520   3 5 6      4 5 
│ │ │ │ +000183e0: 3620 2020 2020 2035 2036 2020 2020 2020  6      5 6      
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│ │ │ │ +00018410: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
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│ │ │ │  00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00019480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000194a0: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -000194b0: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -000194c0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ -000194d0: 2020 2032 2032 2020 2020 2020 2020 2032     2 2         2
│ │ │ │ -000194e0: 2020 2020 2020 2020 2020 327c 0a7c 7820            2|.|x 
│ │ │ │ -000194f0: 7820 7820 7820 202b 2037 7820 7820 202d  x x x  + 7x x  -
│ │ │ │ -00019500: 2032 3978 2078 2078 2020 2b20 3336 7820   29x x x  + 36x 
│ │ │ │ -00019510: 7820 202d 2031 3178 2078 2078 2020 2b20  x  - 11x x x  + 
│ │ │ │ -00019520: 3438 7820 7820 202b 2032 7820 7820 7820  48x x  + 2x x x 
│ │ │ │ -00019530: 202d 2033 3378 2078 2078 207c 0a7c 2034   - 33x x x |.| 4
│ │ │ │ -00019540: 2035 2037 2038 2020 2020 2030 2038 2020   5 7 8     0 8  
│ │ │ │ -00019550: 2020 2020 3020 3120 3820 2020 2020 2031      0 1 8      1
│ │ │ │ -00019560: 2038 2020 2020 2020 3020 3220 3820 2020   8      0 2 8   
│ │ │ │ -00019570: 2020 2032 2038 2020 2020 2030 2034 2038     2 8     0 4 8
│ │ │ │ -00019580: 2020 2020 2020 3120 3420 387c 0a7c 2d2d        1 4 8|.|--
│ │ │ │ -00019590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000194a0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +000194b0: 2032 2032 2020 2020 2020 2020 2020 3220   2 2          2 
│ │ │ │ +000194c0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +000194d0: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ +000194e0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000194f0: 327c 0a7c 7820 7820 7820 7820 202b 2037  2|.|x x x x  + 7
│ │ │ │ +00019500: 7820 7820 202d 2032 3978 2078 2078 2020  x x  - 29x x x  
│ │ │ │ +00019510: 2b20 3336 7820 7820 202d 2031 3178 2078  + 36x x  - 11x x
│ │ │ │ +00019520: 2078 2020 2b20 3438 7820 7820 202b 2032   x  + 48x x  + 2
│ │ │ │ +00019530: 7820 7820 7820 202d 2033 3378 2078 2078  x x x  - 33x x x
│ │ │ │ +00019540: 207c 0a7c 2034 2035 2037 2038 2020 2020   |.| 4 5 7 8    
│ │ │ │ +00019550: 2030 2038 2020 2020 2020 3020 3120 3820   0 8      0 1 8 
│ │ │ │ +00019560: 2020 2020 2031 2038 2020 2020 2020 3020       1 8      0 
│ │ │ │ +00019570: 3220 3820 2020 2020 2032 2038 2020 2020  2 8      2 8    
│ │ │ │ +00019580: 2030 2034 2038 2020 2020 2020 3120 3420   0 4 8      1 4 
│ │ │ │ +00019590: 387c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  8|.|------------
│ │ │ │  000195a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000195b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000195c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000195d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000195e0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -000195f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000195d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000195e0: 2d7c 0a7c 2020 2020 2020 2020 2032 2020  -|.|         2  
│ │ │ │ +000195f0: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │  00019600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019620: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00019630: 3437 7820 7820 7820 202d 2034 3878 2078  47x x x  - 48x x
│ │ │ │ -00019640: 2020 2b20 3438 7820 7820 7820 7820 202d    + 48x x x x  -
│ │ │ │ -00019650: 2034 3878 2078 2078 2078 2020 2d20 3438   48x x x x  - 48
│ │ │ │ -00019660: 7820 7820 7820 7820 202b 2034 3878 2078  x x x x  + 48x x
│ │ │ │ -00019670: 2078 2078 2020 2b20 2020 207c 0a7c 2020   x x  +    |.|  
│ │ │ │ -00019680: 2020 2032 2034 2038 2020 2020 2020 3420     2 4 8      4 
│ │ │ │ -00019690: 3820 2020 2020 2033 2034 2036 2039 2020  8      3 4 6 9  
│ │ │ │ -000196a0: 2020 2020 3220 3520 3620 3920 2020 2020      2 5 6 9     
│ │ │ │ -000196b0: 2031 2033 2037 2039 2020 2020 2020 3020   1 3 7 9      0 
│ │ │ │ -000196c0: 3520 3720 3920 2020 2020 207c 0a7c 2d2d  5 7 9      |.|--
│ │ │ │ -000196d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019630: 207c 0a7c 2d20 3437 7820 7820 7820 202d   |.|- 47x x x  -
│ │ │ │ +00019640: 2034 3878 2078 2020 2b20 3438 7820 7820   48x x  + 48x x 
│ │ │ │ +00019650: 7820 7820 202d 2034 3878 2078 2078 2078  x x  - 48x x x x
│ │ │ │ +00019660: 2020 2d20 3438 7820 7820 7820 7820 202b    - 48x x x x  +
│ │ │ │ +00019670: 2034 3878 2078 2078 2078 2020 2b20 2020   48x x x x  +   
│ │ │ │ +00019680: 207c 0a7c 2020 2020 2032 2034 2038 2020   |.|     2 4 8  
│ │ │ │ +00019690: 2020 2020 3420 3820 2020 2020 2033 2034      4 8      3 4
│ │ │ │ +000196a0: 2036 2039 2020 2020 2020 3220 3520 3620   6 9      2 5 6 
│ │ │ │ +000196b0: 3920 2020 2020 2031 2033 2037 2039 2020  9      1 3 7 9  
│ │ │ │ +000196c0: 2020 2020 3020 3520 3720 3920 2020 2020      0 5 7 9     
│ │ │ │ +000196d0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000196e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000196f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3438  -----------|.|48
│ │ │ │ -00019720: 7820 7820 7820 7820 202d 2034 3878 2078  x x x x  - 48x x
│ │ │ │ -00019730: 2078 2078 2020 2020 2020 2020 2020 2020   x x            
│ │ │ │ +00019710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019720: 2d7c 0a7c 3438 7820 7820 7820 7820 202d  -|.|48x x x x  -
│ │ │ │ +00019730: 2034 3878 2078 2078 2078 2020 2020 2020   48x x x x      
│ │ │ │  00019740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019770: 2031 2032 2038 2039 2020 2020 2020 3020   1 2 8 9      0 
│ │ │ │ -00019780: 3420 3820 3920 2020 2020 2020 2020 2020  4 8 9           
│ │ │ │ +00019760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019770: 207c 0a7c 2020 2031 2032 2038 2039 2020   |.|   1 2 8 9  
│ │ │ │ +00019780: 2020 2020 3020 3420 3820 3920 2020 2020      0 4 8 9     
│ │ │ │  00019790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000197a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000197b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000197c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000197b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000197c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000197d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000197e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000197f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00019810: 203a 2074 696d 6520 7073 6920 3d20 6170   : time psi = ap
│ │ │ │ -00019820: 7072 6f78 696d 6174 6549 6e76 6572 7365  proximateInverse
│ │ │ │ -00019830: 4d61 7020 7068 6920 2020 2020 2020 2020  Map phi         
│ │ │ │ +00019800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019810: 2d2b 0a7c 6933 203a 2074 696d 6520 7073  -+.|i3 : time ps
│ │ │ │ +00019820: 6920 3d20 6170 7072 6f78 696d 6174 6549  i = approximateI
│ │ │ │ +00019830: 6e76 6572 7365 4d61 7020 7068 6920 2020  nverseMap phi   
│ │ │ │  00019840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019850: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00019860: 2d20 7573 6564 2030 2e32 3138 3636 3773  - used 0.218667s
│ │ │ │ -00019870: 2028 6370 7529 3b20 302e 3136 3930 3535   (cpu); 0.169055
│ │ │ │ -00019880: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00019890: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ -000198a0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -000198b0: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -000198c0: 7273 654d 6170 3a20 7374 6570 2031 206f  rseMap: step 1 o
│ │ │ │ -000198d0: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019860: 207c 0a7c 202d 2d20 7573 6564 2030 2e33   |.| -- used 0.3
│ │ │ │ +00019870: 3732 3331 3273 2028 6370 7529 3b20 302e  72312s (cpu); 0.
│ │ │ │ +00019880: 3235 3439 3238 7320 2874 6872 6561 6429  254928s (thread)
│ │ │ │ +00019890: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +000198a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000198b0: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +000198c0: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +000198d0: 6570 2031 206f 6620 3130 2020 2020 2020  ep 1 of 10      
│ │ │ │  000198e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198f0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019900: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019910: 7273 654d 6170 3a20 7374 6570 2032 206f  rseMap: step 2 o
│ │ │ │ -00019920: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +000198f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019900: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019910: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019920: 6570 2032 206f 6620 3130 2020 2020 2020  ep 2 of 10      
│ │ │ │  00019930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019940: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019950: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019960: 7273 654d 6170 3a20 7374 6570 2033 206f  rseMap: step 3 o
│ │ │ │ -00019970: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019950: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019960: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019970: 6570 2033 206f 6620 3130 2020 2020 2020  ep 3 of 10      
│ │ │ │  00019980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019990: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -000199a0: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -000199b0: 7273 654d 6170 3a20 7374 6570 2034 206f  rseMap: step 4 o
│ │ │ │ -000199c0: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000199a0: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +000199b0: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +000199c0: 6570 2034 206f 6620 3130 2020 2020 2020  ep 4 of 10      
│ │ │ │  000199d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000199e0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -000199f0: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019a00: 7273 654d 6170 3a20 7374 6570 2035 206f  rseMap: step 5 o
│ │ │ │ -00019a10: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +000199e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000199f0: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019a00: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019a10: 6570 2035 206f 6620 3130 2020 2020 2020  ep 5 of 10      
│ │ │ │  00019a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019a30: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019a40: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019a50: 7273 654d 6170 3a20 7374 6570 2036 206f  rseMap: step 6 o
│ │ │ │ -00019a60: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019a40: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019a50: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019a60: 6570 2036 206f 6620 3130 2020 2020 2020  ep 6 of 10      
│ │ │ │  00019a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019a80: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019a90: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019aa0: 7273 654d 6170 3a20 7374 6570 2037 206f  rseMap: step 7 o
│ │ │ │ -00019ab0: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019a90: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019aa0: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019ab0: 6570 2037 206f 6620 3130 2020 2020 2020  ep 7 of 10      
│ │ │ │  00019ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ad0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019ae0: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019af0: 7273 654d 6170 3a20 7374 6570 2038 206f  rseMap: step 8 o
│ │ │ │ -00019b00: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ae0: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019af0: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019b00: 6570 2038 206f 6620 3130 2020 2020 2020  ep 8 of 10      
│ │ │ │  00019b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b20: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019b30: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019b40: 7273 654d 6170 3a20 7374 6570 2039 206f  rseMap: step 9 o
│ │ │ │ -00019b50: 6620 3130 2020 2020 2020 2020 2020 2020  f 10            
│ │ │ │ +00019b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b30: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019b40: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019b50: 6570 2039 206f 6620 3130 2020 2020 2020  ep 9 of 10      
│ │ │ │  00019b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b70: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00019b80: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00019b90: 7273 654d 6170 3a20 7374 6570 2031 3020  rseMap: step 10 
│ │ │ │ -00019ba0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b80: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +00019b90: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +00019ba0: 6570 2031 3020 6f66 2031 3020 2020 2020  ep 10 of 10     
│ │ │ │  00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019bd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00019be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c10: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00019c20: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ -00019c30: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │ +00019c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c20: 207c 0a7c 6f33 203d 202d 2d20 7261 7469   |.|o3 = -- rati
│ │ │ │ +00019c30: 6f6e 616c 206d 6170 202d 2d20 2020 2020  onal map --     
│ │ │ │  00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c80: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00019c80: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │  00019c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019cc0: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ -00019cd0: 2d2d 5b74 202c 2074 202c 2074 202c 2074  --[t , t , t , t
│ │ │ │ -00019ce0: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ -00019cf0: 202c 2074 205d 2920 2020 2020 2020 2020   , t ])         
│ │ │ │ -00019d00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d20: 3937 2020 3020 2020 3120 2020 3220 2020  97  0   1   2   
│ │ │ │ -00019d30: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ -00019d40: 3720 2020 3820 2020 2020 2020 2020 2020  7   8           
│ │ │ │ -00019d50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d70: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ -00019d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019cc0: 207c 0a7c 2020 2020 2073 6f75 7263 653a   |.|     source:
│ │ │ │ +00019cd0: 2050 726f 6a28 2d2d 5b74 202c 2074 202c   Proj(--[t , t ,
│ │ │ │ +00019ce0: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │ +00019cf0: 2074 202c 2074 202c 2074 205d 2920 2020   t , t , t ])   
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│ │ │ │ -00019fe0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
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│ │ │ │  0001a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a020: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a040: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │ -0001a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a030: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a040: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0001a090: 2020 2020 2035 2037 2020 2020 3420 3820       5 7    4 8 
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│ │ │ │  0001a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a130: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │ -0001a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a180: 2020 2020 2032 2037 2020 2020 3120 3820       2 7    1 8 
│ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a190: 2020 3120 3820 2020 2020 2020 2020 2020    1 8           
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│ │ │ │ -0001a1b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a200: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a220: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │ -0001a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a210: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a220: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a250: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a270: 2020 2020 2035 2036 2020 2020 3320 3820       5 6    3 8 
│ │ │ │ -0001a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a260: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a270: 2020 2020 2020 2020 2020 2035 2036 2020             5 6  
│ │ │ │ +0001a280: 2020 3320 3820 2020 2020 2020 2020 2020    3 8           
│ │ │ │  0001a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a310: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │ -0001a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a300: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a310: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
│ │ │ │ +0001a320: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  0001a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a340: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a360: 2020 2020 2034 2036 2020 2020 3320 3720       4 6    3 7 
│ │ │ │ -0001a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a350: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a360: 2020 2020 2020 2020 2020 2034 2036 2020             4 6  
│ │ │ │ +0001a370: 2020 3320 3720 2020 2020 2020 2020 2020    3 7           
│ │ │ │  0001a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a390: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a3a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a400: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │ -0001a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a450: 2020 2020 2032 2036 2020 2020 3020 3820       2 6    0 8 
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│ │ │ │ -0001a480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a490: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001a4f0: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
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│ │ │ │ -0001a570: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a580: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a630: 2020 2020 2032 2034 2020 2020 3120 3520       2 4    1 5 
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│ │ │ │  0001a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6d0: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
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│ │ │ │ +0001a6d0: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │ -0001a700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a720: 2020 2020 2032 2033 2020 2020 3020 3520       2 3    0 5 
│ │ │ │ -0001a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a710: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a720: 2020 2020 2020 2020 2020 2032 2033 2020             2 3  
│ │ │ │ +0001a730: 2020 3020 3520 2020 2020 2020 2020 2020    0 5           
│ │ │ │  0001a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a750: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a760: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7c0: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
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│ │ │ │ +0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a7c0: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a810: 2020 2020 2031 2033 2020 2020 3020 3420       1 3    0 4 
│ │ │ │ -0001a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a800: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a810: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
│ │ │ │ +0001a820: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │  0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a850: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001aa50: 616c 206d 6170 2066 726f 6d20 5050 5e38  al map from PP^8
│ │ │ │ -0001aa60: 2074 6f20 6879 7065 7273 7572 6661 6365   to hypersurface
│ │ │ │ -0001aa70: 2069 6e20 5050 5e39 2920 207c 0a7c 2d2d   in PP^9)  |.|--
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│ │ │ │ +0001aa60: 6d20 5050 5e38 2074 6f20 6879 7065 7273  m PP^8 to hypers
│ │ │ │ +0001aa70: 7572 6661 6365 2069 6e20 5050 5e39 2920  urface in PP^9) 
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│ │ │ │ -0001ac30: 7820 7820 202d 2031 3478 2078 2078 2078  x x  - 14x x x x
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│ │ │ │ +0001ac20: 2b20 7820 7820 202d 2031 3178 2078 2078  + x x  - 11x x x
│ │ │ │ +0001ac30: 2020 2b20 3338 7820 7820 202d 2031 3478    + 38x x  - 14x
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│ │ │ │ +0001ac50: 7820 202b 2034 3578 2078 2078 2078 2020  x  + 45x x x x  
│ │ │ │ +0001ac60: 2d7c 0a7c 2020 2020 2020 3120 3320 3420  -|.|      1 3 4 
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│ │ │ │ -0001b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5d0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
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│ │ │ │ -0001b600: 2020 2020 2020 2020 2020 207c 0a7c 202b             |.| +
│ │ │ │ -0001b610: 2034 7420 7420 202b 2033 3974 2074 2020   4t t  + 39t t  
│ │ │ │ -0001b620: 2d20 3232 7420 7420 202b 2031 3474 2020  - 22t t  + 14t  
│ │ │ │ -0001b630: 2d20 7420 7420 202d 2036 7420 7420 202d  - t t  - 6t t  -
│ │ │ │ -0001b640: 2034 3174 2074 2020 2b20 3432 7420 7420   41t t  + 42t t 
│ │ │ │ -0001b650: 202b 2032 3374 2074 2020 2b7c 0a7c 2020   + 23t t  +|.|  
│ │ │ │ -0001b660: 2020 2030 2034 2020 2020 2020 3120 3420     0 4      1 4 
│ │ │ │ -0001b670: 2020 2020 2033 2034 2020 2020 2020 3420       3 4      4 
│ │ │ │ -0001b680: 2020 2030 2035 2020 2020 2031 2035 2020     0 5     1 5  
│ │ │ │ -0001b690: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ -0001b6a0: 2020 2020 2020 3420 3520 207c 0a7c 2d2d        4 5  |.|--
│ │ │ │ -0001b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001b620: 3974 2074 2020 2d20 3232 7420 7420 202b  9t t  - 22t t  +
│ │ │ │ +0001b630: 2031 3474 2020 2d20 7420 7420 202d 2036   14t  - t t  - 6
│ │ │ │ +0001b640: 7420 7420 202d 2034 3174 2074 2020 2b20  t t  - 41t t  + 
│ │ │ │ +0001b650: 3432 7420 7420 202b 2032 3374 2074 2020  42t t  + 23t t  
│ │ │ │ +0001b660: 2b7c 0a7c 2020 2020 2030 2034 2020 2020  +|.|     0 4    
│ │ │ │ +0001b670: 2020 3120 3420 2020 2020 2033 2034 2020    1 4      3 4  
│ │ │ │ +0001b680: 2020 2020 3420 2020 2030 2035 2020 2020      4    0 5    
│ │ │ │ +0001b690: 2031 2035 2020 2020 2020 3220 3520 2020   1 5      2 5   
│ │ │ │ +0001b6a0: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ +0001b6b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001b700: 2020 2020 2020 2020 2020 2020 2020 3220                2 
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│ │ │ │ -0001b770: 202d 2031 3978 2078 2078 2078 2020 2b20   - 19x x x x  + 
│ │ │ │ -0001b780: 3231 7820 7820 7820 7820 202b 2032 3678  21x x x x  + 26x
│ │ │ │ -0001b790: 2078 2020 2b20 3139 7820 207c 0a7c 2020   x  + 19x  |.|  
│ │ │ │ -0001b7a0: 2031 2032 2033 2035 2020 2020 2020 3020   1 2 3 5      0 
│ │ │ │ -0001b7b0: 3420 3520 2020 2020 2030 2032 2034 2035  4 5      0 2 4 5
│ │ │ │ -0001b7c0: 2020 2020 2020 3020 3320 3420 3520 2020        0 3 4 5   
│ │ │ │ -0001b7d0: 2020 2032 2033 2034 2035 2020 2020 2020     2 3 4 5      
│ │ │ │ -0001b7e0: 3020 3520 2020 2020 2030 207c 0a7c 2020  0 5      0 |.|  
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│ │ │ │ +0001b740: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ +0001b750: 207c 0a7c 3139 7820 7820 7820 7820 202b   |.|19x x x x  +
│ │ │ │ +0001b760: 2031 3478 2078 2078 2020 2b20 3131 7820   14x x x  + 11x 
│ │ │ │ +0001b770: 7820 7820 7820 202d 2031 3978 2078 2078  x x x  - 19x x x
│ │ │ │ +0001b780: 2078 2020 2b20 3231 7820 7820 7820 7820   x  + 21x x x x 
│ │ │ │ +0001b790: 202b 2032 3678 2078 2020 2b20 3139 7820   + 26x x  + 19x 
│ │ │ │ +0001b7a0: 207c 0a7c 2020 2031 2032 2033 2035 2020   |.|   1 2 3 5  
│ │ │ │ +0001b7b0: 2020 2020 3020 3420 3520 2020 2020 2030      0 4 5      0
│ │ │ │ +0001b7c0: 2032 2034 2035 2020 2020 2020 3020 3320   2 4 5      0 3 
│ │ │ │ +0001b7d0: 3420 3520 2020 2020 2032 2033 2034 2035  4 5      2 3 4 5
│ │ │ │ +0001b7e0: 2020 2020 2020 3020 3520 2020 2020 2030        0 5      0
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│ │ │ │  0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b920: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001b930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b9d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001ba60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001bab0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001bac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001bb00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001bbb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001bbf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0001bc40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001bed0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001bfc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c000: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c010: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c050: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c060: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c0b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c100: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0001c110: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c100: 207c 0a7c 2020 2032 2020 2020 2020 2020   |.|   2        
│ │ │ │ +0001c110: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c140: 2020 2020 2020 2020 2020 207c 0a7c 3231             |.|21
│ │ │ │ -0001c150: 7420 202b 2032 3474 2074 2020 2d20 3874  t  + 24t t  - 8t
│ │ │ │ -0001c160: 2074 2020 2d20 3231 7420 202b 2034 3174   t  - 21t  + 41t
│ │ │ │ -0001c170: 2074 2020 2b20 3133 7420 7420 202b 2031   t  + 13t t  + 1
│ │ │ │ -0001c180: 3574 2074 2020 2d20 3232 7420 7420 202b  5t t  - 22t t  +
│ │ │ │ -0001c190: 2033 3874 2074 2020 2d20 207c 0a7c 2020   38t t  -  |.|  
│ │ │ │ -0001c1a0: 2035 2020 2020 2020 3020 3620 2020 2020   5      0 6     
│ │ │ │ -0001c1b0: 3320 3620 2020 2020 2036 2020 2020 2020  3 6      6      
│ │ │ │ -0001c1c0: 3020 3720 2020 2020 2031 2037 2020 2020  0 7      1 7    
│ │ │ │ -0001c1d0: 2020 3320 3720 2020 2020 2034 2037 2020    3 7      4 7  
│ │ │ │ -0001c1e0: 2020 2020 3620 3720 2020 207c 0a7c 2d2d      6 7    |.|--
│ │ │ │ -0001c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c150: 207c 0a7c 3231 7420 202b 2032 3474 2074   |.|21t  + 24t t
│ │ │ │ +0001c160: 2020 2d20 3874 2074 2020 2d20 3231 7420    - 8t t  - 21t 
│ │ │ │ +0001c170: 202b 2034 3174 2074 2020 2b20 3133 7420   + 41t t  + 13t 
│ │ │ │ +0001c180: 7420 202b 2031 3574 2074 2020 2d20 3232  t  + 15t t  - 22
│ │ │ │ +0001c190: 7420 7420 202b 2033 3874 2074 2020 2d20  t t  + 38t t  - 
│ │ │ │ +0001c1a0: 207c 0a7c 2020 2035 2020 2020 2020 3020   |.|   5      0 
│ │ │ │ +0001c1b0: 3620 2020 2020 3320 3620 2020 2020 2036  6     3 6      6
│ │ │ │ +0001c1c0: 2020 2020 2020 3020 3720 2020 2020 2031        0 7      1
│ │ │ │ +0001c1d0: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ +0001c1e0: 2034 2037 2020 2020 2020 3620 3720 2020   4 7      6 7   
│ │ │ │ +0001c1f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
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│ │ │ │ -0001c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c280: 2020 2020 2020 2020 2020 207c 0a7c 7820             |.|x 
│ │ │ │ -0001c290: 7820 202b 2033 3878 2078 2020 2d20 3330  x  + 38x x  - 30
│ │ │ │ -0001c2a0: 7820 7820 7820 202d 2031 3178 2078 2078  x x x  - 11x x x
│ │ │ │ -0001c2b0: 2078 2020 2d20 3131 7820 7820 7820 202b   x  - 11x x x  +
│ │ │ │ -0001c2c0: 2033 3078 2078 2078 2078 2020 2b20 3232   30x x x x  + 22
│ │ │ │ -0001c2d0: 7820 7820 7820 7820 202b 207c 0a7c 2032  x x x x  + |.| 2
│ │ │ │ -0001c2e0: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ -0001c2f0: 2031 2032 2036 2020 2020 2020 3120 3220   1 2 6      1 2 
│ │ │ │ -0001c300: 3320 3620 2020 2020 2031 2033 2036 2020  3 6      1 3 6  
│ │ │ │ -0001c310: 2020 2020 3020 3220 3420 3620 2020 2020      0 2 4 6     
│ │ │ │ -0001c320: 2031 2032 2034 2036 2020 207c 0a7c 2020   1 2 4 6   |.|  
│ │ │ │ -0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001c250: 3220 2020 2020 2020 2032 2020 2020 2020  2        2      
│ │ │ │ +0001c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c270: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c290: 207c 0a7c 7820 7820 202b 2033 3878 2078   |.|x x  + 38x x
│ │ │ │ +0001c2a0: 2020 2d20 3330 7820 7820 7820 202d 2031    - 30x x x  - 1
│ │ │ │ +0001c2b0: 3178 2078 2078 2078 2020 2d20 3131 7820  1x x x x  - 11x 
│ │ │ │ +0001c2c0: 7820 7820 202b 2033 3078 2078 2078 2078  x x  + 30x x x x
│ │ │ │ +0001c2d0: 2020 2b20 3232 7820 7820 7820 7820 202b    + 22x x x x  +
│ │ │ │ +0001c2e0: 207c 0a7c 2032 2035 2020 2020 2020 3220   |.| 2 5      2 
│ │ │ │ +0001c2f0: 3520 2020 2020 2031 2032 2036 2020 2020  5      1 2 6    
│ │ │ │ +0001c300: 2020 3120 3220 3320 3620 2020 2020 2031    1 2 3 6      1
│ │ │ │ +0001c310: 2033 2036 2020 2020 2020 3020 3220 3420   3 6      0 2 4 
│ │ │ │ +0001c320: 3620 2020 2020 2031 2032 2034 2036 2020  6      1 2 4 6  
│ │ │ │ +0001c330: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001c460: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001c500: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001c510: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c550: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c560: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  0001c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c640: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c690: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c740: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c780: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001c830: 2020 2020

TRUNCATED DUE TO SIZE LIMIT: 10485760 bytes